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Risk sharing with private and public information
Journal Pre-proof Risk sharing with private and public information
Piotr Denderski, Christian A. Stoltenberg
PII: DOI: Reference:
S0022-0531(18)30488-5 https://doi.org/10.1016/j.jet.2019.104988 YJETH 104988
To appear in:
Journal of Economic Theory
Received date: Revised date: Accepted date:
20 August 2018 19 November 2019 19 December 2019
Please cite this article as: Denderski, P., Stoltenberg, C.A. Risk sharing with private and public information. J. Econ. Theory (2020), 104988, doi: https://doi.org/10.1016/j.jet.2019.104988.
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Risk sharing with private and public information∗
2
Piotr Denderski† Christian A. Stoltenberg‡
3
December 30, 2019
4
Abstract
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In this paper, we revisit the conventional view on efficient risk sharing that advance information on future shocks is detrimental to welfare. In our model, risk-averse agents receive private and public signals on future income realizations and engage in insurance contracts with limited enforceability. Consistent with the conventional view, better private and public signals are detrimental to welfare when only one type of signal is informative. Our main novel result applies when both signals are informative. In this case, we show that better public information can improve the allocation of risk when private signals are sufficiently precise. More precise public signals spread out the outside option values of highincome agents with high and low public signals and their willingness to transfer resources to low-income agents decreases. With informative private signals, however, more informative public signals increase outside option values of agents with a high signal by less than outside options of agents with a low signal decrease, facilitating more transfers. The latter effect dominates the former when private signals are sufficiently precise.
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JEL classification: D80, F41, E21, D52 Keywords: Risk sharing; Social value of information; Limited commitment
∗ We are especially grateful to Martin Kaae Jensen and Dirk Krueger for their comments. We have also ´ ad Abrah´ ´ benefited greatly from the discussions with Arp´ am, Ufuk Akcigit, Eric Bartelsman, Tobias Broer, Bjoern Bruegemann, Alex Clymo, Pieter Gautier, Marek Kapiˇcka, Per Krusell, Bartosz Ma´ckowiak, Guido Menzio, Thomas Mertens, Ctirad Slav´ık, Thijs van Rens, Gianluca Violante, Chris Wallace, Mirko Wiederholt, Sweder van Wijnbergen, conference and seminar participants at various places. Previous versions circulated under the title On Positive Value of Information in Risk Sharing. † University of Leicester, University Road, LE1 7RH, Leicester, the United Kingdom and Institute of Eco´ nomics, Polish Academy of Sciences, Nowy Swiat 73, 00-330, Warsaw, Poland; email: [email protected]. uk, tel: + 44 (0) 116 252 3703 ‡ MInt, Department of Economics, University of Amsterdam and Tinbergen Institute, Postbus 15867, 1001 NJ Amsterdam, the Netherlands, email: [email protected], tel: +31 20 525 3913.
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1
Introduction
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Should benevolent governments and international institutions strive to provide better public
22
forecasts on real GDP to improve consumption smoothing against country-specific shocks? The
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theoretical literature (Hirshleifer, 1971, Schlee, 2001) offers sharp predictions for welfare effects
24
of issuing such public forecasts. Better information on future realizations of idiosyncratic shocks
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harms risk sharing and welfare by limiting insurance possibilities from an ex-ante perspective.
26
Similar to Hirshleifer, we consider an environment in which better private and public infor-
27
mation on future idiosyncratic shocks are in isolation detrimental to welfare. However, as our
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main theoretical result, we show that if households have private information at their disposal
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to forecast future shock realizations, releasing better publicly available forecasts can improve
30
welfare and risk sharing. In particular, this happens whenever private information is sufficiently
31
precise.
32
As in Krueger and Perri (2011), we consider a model with risk-averse agents that seek
33
insurance against idiosyncratic fluctuations of their income. While there is no restriction on the
34
type of insurance contracts that can be traded, the contracts suffer from limited commitment
35
because every period agents have the option to default to autarky. The option to default is
36
in particular tempting for agents whose current income is high and these agents face a tension
37
between the higher current consumption in autarky and the future benefits of the insurance
38
promised in the contracts. Each period – and this is the new element here – agents receive a
39
private and a public signal on their future income shock realization. Thus, the agents possess
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fore-knowledge of future shock realizations. By changing the expected value of autarky, the
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signals affect the trade-off between higher current consumption in autarky and better insurance
42
proffered in the contracts.
43
As our main novel theoretical contribution, we formally show that better public information
44
can be beneficial for welfare and risk sharing when private information is sufficiently precise
45
and enforcement constraints matter. The positive effect of public information on risk sharing
46
is intriguing but can be understood as a trade-off between costs and benefits from releasing
47
more precise public signals. With enforcement constraints, the degree of risk sharing depends
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on the willingness of high-income agents to transfer resources to low-income agents. First, more
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precise public signals spread out the outside option values of agents with a high and a low public
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signal, that is, high signal agents become less but low-signal agents more willing to share their
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good fortune. With uninformative private signals, high-income agents with a high public signal
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require more resources to be indifferent with autarky than low-signal agents are willing to give
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up, resulting in less transfers to low-income agents, and better public signals are detrimental
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to welfare. With informative private signals, however, more informative public signals increase
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the willingness to transfer resources of low public signal agents by more than the high public
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signal agents’ willingness to transfer resources decreases, facilitating better risk sharing. When
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private information is sufficiently precise, risk sharing improves with better public information
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and agents prefer informative public signals ex-ante.
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We then quantitatively test the robustness of our main theoretical result lifting simplifying
60
assumptions necessary to deliver analytical tractability. We discipline the model with data on
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international risk-sharing. The risk-averse agents here are representative households of small 2
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countries that seek insurance against country-specific shocks to real GDP. The public signals
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capture information on a country’s future GDP that is publicly available to other countries, for
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example, the real GDP growth forecasts provided in the IMF World Economic Outlook (WEO).
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Private signals are exclusively observed by a particular country and can capture how transparent
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a country is. A high precision of private signals indicates secrecy, i.e., useful information on
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future real GDP exists but that information is not publicly shared.
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The model implies that countries with a low degree of insurance are countries in which the
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private information friction is the most severe. In these countries, private signals are sufficiently
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precise and public data quality improvements are beneficial for welfare; countries with either
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low or high quality of public information are countries in which private signals are not precise
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enough and improvements in public information quality have negative but negligible effects on
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welfare. Taking together, these quantitative results confirm the logic of our main theoretical
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result on the positive effect of better public information.
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1.1
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The papers that are most closely related to ours are Hirshleifer (1971), Schlee (2001), Broer,
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Kapicka, and Klein (2017) and Hassan and Mertens (2015).
Related literature
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Hirshleifer (1971) is among the first to point out that perfect information – public or private
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– makes risk-averse agents ex-ante worse off if such information leads to an evaporation of risks
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that otherwise could have been shared in a competitive equilibrium with full insurance. Schlee
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(2001) provides general conditions under which better public information about idiosyncratic
82
risk is undesirable in a competitive equilibrium. There are two main differences to these papers
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in our work. First, we consider optimal insurance contracts with limited enforcement that hinder
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agents from achieving full insurance. Second, we allow for the interplay of both private and
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public information which can overturn Hirshleifer (1971)’s result on the negative effect of public
86
information. Further, our main theoretical result on the positive effect of public information
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only applies in the empirically realistic case of partial insurance and when private information
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is important.
89
Broer et al. (2017) study the effects of unobservable income shocks on consumption risk
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sharing with limited commitment. Agents can be induced to report their income truthfully,
91
and private information reports become contractible but do not increase the state space. The
92
key difference to our paper is that we consider the realistic case with both publicly observable
93
elements of the state vector – public signals and income – as well as unobservable private signals
94
which allows us to analyze how the interaction between private and public information affects
95
welfare and risk sharing.
96
Our information structure with signals about future endowments is similar to Hassan and
97
Mertens (2015) who examine the business cycle implications of private information on future
98
productivity shocks in a closed economy. Our focus is to understand the welfare implications
99
of private and public signals on future income realizations with limited commitment.
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The remainder of the paper is organized as follows. Section 2 presents a general risk-sharing
101
model with private and public signals. In Section 3, we analytically characterize how the
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precision of private and public signals affect social welfare and risk sharing. Section 4 provides 3
103
a quantitative application to international risk sharing. The last section concludes.
104
2
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In this section, we present a tractable and yet general model in which risk-averse households use
106
self-enforceable insurance contracts to hedge their consumption against undesirable fluctuations
107
resulting from idiosyncratic income shocks. The insurance contracts are required to be self-
108
enforceable because agents have the possibility to default on their obligations and live in autarky.
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Agents receive private and public signals on their future income realizations that affect how they
110
value the insurance provided in the contracts relative to their outside option of living in autarky.
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The model is designed to provide the simplest possible framework to analytically show our main
112
result on the positive effect of better public information in Section 3. ]
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Preferences
114
indexed i. The time is discrete and indexed by t from zero onward. Households have identical
115
preferences over consumption streams
A risk-sharing model with private and public information
Consider an economy that is populated by a continuum of risk-averse agents
U0 ({ct }∞ t=0 ) = (1 − β) E0
∞
β t u(ct ),
(1)
t=0
116
where the instantaneous utility function u : R+ → R is strictly increasing, strictly concave and
117
satisfies the Inada conditions.
118
The income stream of agent i, {yti }∞ t=0 , is governed by a stochastic process with two possible
119
time-invariant realizations yti ∈ Y ≡ {yl , yh }, with 0 < yl < yh that are equally likely with
120
realizations that are independent across time and agents such that average income is given
121
by y¯ = (yl + yh )/2. The history of income realizations (y0 , ...yt ) is denoted by y t ∈ Y t+1 .
122
Current-period income is publicly observable.
123
Information
124
and a private signal nit ∈ Y (observed only by agent i) that inform about her income realization
Each period, agent i receives a public signal kti ∈ Y (observed by all agents)
125
y in the next period. As income, both signals have two realizations that are equally likely with
126
realizations that are independent across time and agents. The signals can indicate either a high
127
income (“good” or “high” signals) or a low income (“bad” or “low” signals) in the future. The
128
precision of the public signal is denoted by the conditional probability κ = π(y = yj |k = yj ),
129
while the precision of the private signal is given by ν = π(y = yj |n = yj ), κ, ν ∈ [1/2, 1].
130
Uninformative signals are characterized by precision 1/2 while perfectly informative signals
131
exhibit precision of one. The publicly observable part of the state vector is st = (yt , kt ), st ∈ S,
132
where S = Y × Y . The whole state vector is θt = (st , nt ), with θt ∈ Θ = S × Y . The public
133
history of the state is st = (s0 , ...st ) ∈ S t+1 ; the history of the public and the private state is
134
denoted by θt = (θ0 , ..., θt ) ∈ Θt+1 .
135
Memoryless allocations
136
mation affect social welfare and risk sharing, we follow Coate and Ravallion (1993) and consider
To analytically characterize how better public and private infor-
4
Income
Signals
?
t
Consumption
?
yti
?
kti , nit
cit
-
t+1
Figure 1: Timing of events with public and private information 137
memoryless allocations that depend solely on the current realization of the state.1
138
With memoryless allocations, continuation values depend only on future but not on current
139
realizations of income and signals. It can be shown that this implies that memoryless allocations
140
are only contingent on the publicly observable part of the current state but not on private signal
141
reports.2 Thus, a memoryless allocation is a tuple of consumption levels c = {cji }i,j∈{l,h} , with
142
cji as consumption with current income y = yj and current public signal k = yi .
143
Insurance arrangements
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agents can engage in insurance contracts that suffer from limited enforceability because agents
145
have the option to default to autarky. As illustrated in Figure 1, after the realization of current
146
income and both two signals, agents can decide to participate in insurance contracts implement-
147
ing the allocation c or to deviate into autarky forever consuming only their income. Agents have
148
no incentive to default in any of the eight possible current-period states θ = (y, k, n) when the
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allocation satisfies enforcement or participation constraints. These constraints state that the
150
lifetime utility of an agent in each state θ exceeds the value of his outside option. For example,
151
the enforcement constraints of high-income agents with high public and high private signals are u(chh ) + βzh
To protect their consumption from undesirable fluctuations,
u(chh ) + u(chl ) u(clh ) + u(cll ) β 2 u(chh ) + u(chl ) + u(clh ) + u(cll ) +β(1 − zh ) + ≥ 2 2 4 1−β h Vrs
Vrs
l Vrs
u(yh ) + β [zh u(yh ) + (1 − zh )u(yl )] +
β2
u(yh ) + u(yl ) h ≡ Vout,hh , (2) 1−β 2 Vout
152
with zh as the probability of a high income in the next period conditional on a high public and
153
a high private signal in the current period given by the following expression zh ≡
κν , κν + (1 − κ)(1 − ν)
0.5 ≤ zh ≤ 1.
(3)
154
With the insurance contracts, agents have the possibility to engage in risk sharing with
155
each other. For this reason, we denote the continuation values of high-and low-income agents,
156
h , V l as well as the unconditional value V Vrs rs with the subscript ‘rs’ for risk sharing. We refer rs
157
to the left-hand side of the enforcement constraint as the value of the insurance contract and
158
the right-hand side as the value of the outside option of that agent. Using these definitions, the 1 Our environment is an example of a dynamic contracting problem with private information considered as for example in Fernandes and Phelan (2000). The solution to this type of problems can feature dependence of allocation on the full history of the public state and private signal reports. All our theoretical results are derived for memoryless allocations. In our quantitative application in Section 4, we test and confirm the robustness of our theoretical results. 2 To ease the exposition in the main text, we derive this result in an accompanying Online Appendix.
5
159
enforcement constraints of high-income agents with a low public and a high private signal can
160
be compactly written as β2 h l Vrs ≥ u(chl ) + β zl Vrs + (1 − zl )Vrs + 1−β u(yh ) + β [zl u(yh ) + (1 − zl )u(yl )] +
β2 h Vout ≡ Vout,lh (4) 1−β
161
with zl as the probability of a high income in the next period conditional on a low public signal
162
and a high private signal in the current period zl ≡
(1 − κ)ν , (1 − κ)ν + κ(1 − ν)
0 ≤ zl ≤ 1,
(5)
163
and zl ≤ zh . While the consumption allocation is not contingent on private signal realizations,
164
private information affects consumption allocations indirectly through the enforcement con-
165
straints. In this setting, we analyze optimal memoryless allocation that are defined as follows.
166
Definition 1 (Optimal memoryless allocation) An optimal memoryless allocation {cji }
167
maximizes agents’ expected utility before any risk has been resolved (social welfare) Vrs =
168
u(chh ) + u(chl ) + u(clh ) + u(cll ) 4
(6)
and is constrained feasible, i.e., in each period t, the allocation satisfies resource feasibility chh + chl + clh + cll yh + y l = 4 2
(7)
169
and is for all periods t and all states θ consistent with agents’ rational incentives to participate
170
in the risk-sharing arrangement as summarized by the enforcement constraints.
171
One can think of the optimal memoryless allocation as the outcome of a contracting problem
172
to design an optimal risk-sharing arrangement that respects rational incentives constraints and
173
resource feasibility. For this reason, we use the terms optimal memoryless allocation and optimal
174
memoryless arrangement interchangeably. By construction, autarky (all agents consume their
175
income in all periods and all states) is constrained feasible.
176
The focus on memoryless allocations allows us to analytically derive sharp predictions on
177
the effect of fore-knowledge on risk sharing and welfare. In the quantitative part in Section 4,
178
we document that the simplifying assumption of memoryless allocations is not pivotal for our
179
main result on the positive value of public information. Our main result also applies when we
180
consider allocations that can depend on the history of the public state st or on the whole state
181
θt .
182
3
183
In this section, we analytically characterize optimal memoryless allocations with private and
184
public signals. Before we get to our main result on the positive value of public information,
Analysis
6
185
we provide intermediate results that are essential to understand it. Eventually, we explain the
186
necessary conditions for the positive welfare effect of better public information in risk sharing.
187
3.1
188
To begin with we show that the optimal arrangement exists and is unique and that social
189
welfare is continuous in the precision of both signals. We also show that while the consumption
190
of agents with a low income weakly exceeds their income, high-income agents’ consumption is
191
weakly lower than under autarky. These general properties are summarized in the following
192
Lemma.
193
Lemma 1 (General properties) Consider optimal memoryless allocations.
General properties of optimal memoryless allocations
194
(i) The optimal memoryless allocation exists and is unique. The optimal memoryless allo-
195
cation and social welfare are continuous functions in the precision of public and private
196
signal precision.
197
(ii) The optimal memoryless allocation satisfies: cli ≥ yl and chi ≤ yh for all public signals i.
198
The proof is provided in Appendix A.1. Existence and uniqueness of the optimal memoryless
199
allocation follow from Weierstrass’ theorem. Continuity is a direct implication of the theorem
200
of the maximum. The second part of the Lemma follows from a contradiction argument. Any
201
allocation that does not satisfy cli ≥ yl and chi ≤ yh violates enforcement and resource feasibility
202
constraints and therefore does not constitute an optimal memoryless allocation.
203
The first best allocation is perfect risk sharing (all agents consume y¯ in all states) which
204
can be constrained feasible when this allocation is consistent with agents’ rational incentives
205
to participate in the insurance arrangements. Alternatively, enforcement constraints can be
206
consistent with either only no insurance against income risk (autarky) or partial risk sharing.
207
The effect of signal precision in the case of perfect risk sharing or autarky are analyzed in
208
Appendix A.2. In the main text, we focus on partial risk sharing, that is, perfect risk sharing
209
is not constrained feasible and autarky is not the only constrained feasible allocation.3
210
3.2
211
We continue our analysis with two Lemmas that summarize properties of optimal memoryless
212
allocations that are essential to derive Theorem 1, our main result. In particular, we show in
213
Lemma 2 that only enforcement constraints of high-income agents for a high private signal can
214
be binding and that consumption of low-income agents is equalized across all signal realizations.
215
In Lemma 3, we characterize the optimal consumption allocation in case the enforcement con-
216
straints of high-income agents are binding for both public signals which is the case considered
217
in our main theoretical result.
Properties of optimal memoryless allocations with partial risk sharing
218
When perfect risk sharing cannot be supported, some enforcement constraints are binding at
219
the optimal memoryless allocation and risk sharing is either partial or absent (by “binding” we
220
mean that the multiplier on the constraint is positive). In the following Lemma, we summarize
221
essential properties of optimal allocations with partial risk sharing. 3
The relevant parameter space for partial risk sharing follows from Propositions 4 and 5 in Appendix A.2.
7
222
Lemma 2 (Allocation properties with partial risk sharing) Consider the optimal mem-
223
oryless arrangement. Let some enforcement constraints of high-income agents be binding. As-
224
sume that autarky is not the only memoryless arrangement.
225
226
h < u(y ) and V l > u(y ). (i) Then chi < yh for at least one public signal signal i, Vrs h l rs
(ii) Consider informative private signals, ν ∈ (0.5, 1]. Then only enforcement constraints of agents with a high private signal can be binding.
227
228
(iii) Only enforcement constraints of high-income agents can be binding.
229
(iv) Consumption in the optimal memoryless arrangement satisfies: clh = cll = cl .
230
The proof is provided in Appendix A.3. The main message from Lemma 2 is that only
231
enforcement constraints of high-income agents for a high private signal can be binding. For
232
each income and public signal realization, allocations must be consistent with the participation
233
incentives of households with the highest outside option value among all private signal real-
234
izations, i.e., the outside option value for high private signals. The optimal allocation further
235
prescribes transfers to agents with a low income realization which is why only enforcement
236
constraints of high-income agents can be binding. Thus, enforcement constraints of low-income
237
agents are slack and it is optimal to equalize their consumption.
238
The last part of Lemma 2 implies that the optimal consumption allocation with informative
public signals comprises three elements, chh , chl , cl . Thereby, the optimal consumption alloca-
239 240
tion is the non-autarkic allocation that is pinned down by the binding enforcement constraints
241
and the resource constraint. A natural question is whether one can solve for optimal memoryless
242
allocations – utility or consumption allocations – in closed form making a particular paramet-
243
ric assumption for the utility function. We find that the resulting solutions that we study in
244
an accompanying Online Appendix are fairly complicated and convey little economic intuition
245
in understanding how changes in signal precision affect social welfare. For these reasons, we
246
continue our analysis with the general utility function specified in (1) and resort to the implicit
247
function theorem to derive the welfare effects of increases in public and private signal precision.
248
The results (ii)-(iii) of Lemma 2 point to the particular enforcement constraints that can be
249
binding in the absence of perfect risk sharing; either the enforcement constraints of high-income
250
agents with both a high and a low public signal or only the constraints of agents with a high
251
public signal bind.4 Our main theoretical result on the positive value of public information
252
applies for the first pattern of binding enforcement constraints as summarized in the following
253
assumption.
254
Assumption 1 The optimal memoryless allocation is characterized by partial risk sharing and
255
at the optimal allocation the enforcement constraints of high-income agents with a high private
256
signal are binding for both high and low public signals.
257
In Section 3.4, we show that Assumption 1 is a necessary condition for our main result.5 4
Formal arguments for this claim are provided in Lemma 3 part (iii). In our application to international risk sharing in Section 4, we find this assumption to be satisfied in all the numerical experiments we entertain. 5
8
258
Given Assumption 1 and after using the insights from Lemma 2 (iv) as well as the resource
259
constraint (7) to substitute for consumption of low-income agents, the social planner problem
260
simplifies to choosing {chh , chl } to maximize the following Lagrangian Vrs =
1 h u(ch ) + u(chl ) + 2u 2¯ y − (chl + chh )/2 4 + λhpc,h F (chh , chl , κ, ν) + λhpc,l G(chh , chl , κ, ν),
261
with the binding enforcement constraints of high-income agents with a high public signal h h h l F (chh , chl , κ, ν) ≡ (1−β)u(chh )+β(1−β)zh Vrs (ch , cl )+β(1−β)(1−zh )Vrs (chh , chl )+β 2 Vrs (chh , chl )
− (1 − β)u(yh ) − β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] − β 2 Vout = 0, 262
and with a low public signal h h h l G(chh , chl , κ, ν) ≡ (1 − β)u(chl ) + β(1 − β)zl Vrs (ch , cl ) + β(1 − β)(1 − zl )Vrs (chh , chl ) + β 2 Vrs (chh , chl )
− (1 − β)u(yh ) − β(1 − β) [zl u(yh ) + (1 − zl )u(yl )] − β 2 Vout = 0, 263 264
and λhpc,h , λhpc,l > 0 as the multipliers on the constraints F (chh , chl , κ, ν), G(chh , chl , κ, ν), respec-
tively. The first order conditions for chh and chl are
1 h u (ch ) − u (cl ) + λhpc,h Fch + λhpc,l Gch = 0 h h 4 1 h l h h u (cl ) − u (c ) + λpc,h Fch + λpc,l Gch = 0. l l 4
(8) (9)
265
In the following Lemma, we use the two binding enforcement constraints and the first order
266
conditions (8)-(9) to derive properties of optimal memoryless consumption allocations that we
267
eventually use to study the welfare effects of changes in signal precision.
268
Lemma 3 (Binding enforcement constraints of high-income agents) Let the optimal
269
memoryless allocation satisfy Assumption 1. Then the following holds:
270 271
(i) Consumption in the optimal memoryless arrangement satisfies: chh ≥ chl > cl and ch > y¯ > cl , with average consumption of high-income agents defined as ch =
272
1 h ch + chl . 2
(ii) At the optimal memoryless arrangement the following relationship applies t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2
273
with t2 ≡ λhpc,h Fch + λhpc,l Gch > 0 h
274
(10)
h
t1 ≡ λhpc,h Fch + λhpc,l Gch > 0. l
l
(iii) At the optimal memoryless allocation, the two multipliers are characterized by λhpc,h ≥ λhpc,l . 9
275
The proof is provided in Appendix A.4. The outside option value of agents with a high
276
public signal exceeds the outside option value of agents with a low public signal. With binding
277
enforcement constraints for these agents, this implies that consumption of agents with a high
278
public signal is also higher than consumption of agents with a low public signal. Because of their
279
binding enforcement constraints, consumption of high-income agents with a low public signal
280
exceeds the consumption of low-income agents. From resource feasibility, it follows then that
281
high-income agents consume more while low-income agents consume less than average income.
282
Part (iii) implies that at the optimal allocation either the enforcements constraints are binding
283
for both public signal realizations (as in Assumption 1) or only the enforcement constraints of
284
high-income agents with a high public signal.
285
3.3
286
Our theoretical results on the social value of public and private information operate through
287
the binding enforcement constraints of high-income agents with high and low public signals. To
288
understand the intuition behind the welfare effects of changes in signal precision that we present
289
in Theorem 1 and Proposition 1, it is useful to decompose the total effect of signal precision
290
on the enforcement constraints into a direct and an indirect effect. To fix ideas, consider an
291
increase in ν (an increase in κ follows analogous arguments).
292 293
Social value of public and private information
First, the increase in ν directly affects the enforcement constraints (2) and (4) by changing zh , zl for a given consumption allocation as follows for each public signal i ∈ {l, h} ∂z
i h l β Vrs − Vrs ∂ν
Direct effect of ν =
Effect on value of insurance
∂zi − β [u(yh ) − u(yl )] ∂ν
(11)
Effect on outside option value
294
The first term is the direct change in the value of the insurance contract and the second term
295
is the direct (and total) change in the outside option value. With partial risk sharing, the average
296
h < u(y ), period utility across public signals of high-income agents is lower than in autarky, Vrs h
297
l > u(y ). Thus, in while the average utility of low-income agents is higher than in autarky, Vrs l
298
general, the change in signal precision has a stronger direct effect on the outside option than
299
on the insurance value. Here, the total direct effect is negative for both public signals because
300
∂zi /∂ν ≥ 0.
301
Second, the increase in signal precision affects the value of the insurance contract indirectly
302
by changing the consumption allocation itself. The indirect effect must offset the direct effect
303
because enforcement constraints are binding. For high-income agents with a high public signal,
304
this requires ∂z ∂chh ∂chl 1 h h l Fch + Fc h + β Vrs − Vrs − u(yh ) + u(yl ) = 0. h ∂ν l ∂ν 1−β ∂ν
(12)
Direct effect of ν
Indirect effect of ν
305
In this equation, the indirect effect is dominated by changes in chh as the consumption state
306
that enters the enforcement constraint with the largest weight. At the optimal allocation, the
307
possibilities to transfer resources to low-income agents must have been exhausted. This implies
10
308
that the values of the insurance contracts for both types of high-income agents at the optimal
309
memoryless allocation are increasing in their respective consumption levels. In other words, the
310
left-hand sides of (2) and (4) are increasing in chh and chl such that Fch , Gch > 0.6 This implies h
l
311
that the indirect effect results in changes of chh and chl that mimic the sign of the corresponding
312
changes in the outside option. Here, both outside option values increase and the direct effect
313
in (12) is negative. Consequently, both chh and chl tend to increase, compensating for the direct
314
effect.
315
Equipped with Lemmas 2 and 3, we present our main theoretical result that improvements
316
in public information can either benefit or worsen welfare depending on the precision of private
317
signals.
318
Theorem 1 (Social Value of Public Information) Let the optimal memoryless allocation
319
satisfy Assumption 1. Then there exists a precision of the private signal ν¯, such that for ν ≥ ν¯
320
and ν ∈ (0.5, 1], social welfare is increasing in the precision of the public signal κ over κ ∈
321
[0.5, 1].
322
The proof is provided in Appendix A.5. The logic of the proof is as follows. We show that
323
for an uninformative private signal welfare decreases in the precision of public signals while
324
for the limiting case of a perfectly informative private signal welfare increases in public signal
325
precision. Continuity then implies that there exists a level of private information precision for
326
which the welfare effect of better public information is positive.
327
The main theorem is illustrated in Figure 2 that depicts social welfare as a function of public
328
information precision for different precisions of private information. When private information is
329
uninformative or not precise enough (see the upper two functions), the negative effect dominates
330
and social welfare is decreasing in public signal precision. For ν = 0.70, the negative and the
331
positive effect neutralize each other. However, when private signals are sufficiently precise the
332
latter positive effect dominates the negative effect, and social welfare increases in public signal
333
precision (see the lower two functions).
334
There is a negative and a positive welfare effect of releasing better public information when
335
private signals are informative. First, more precise public signals spread out the outside option
336
values of agents with a high and a low public signal which causes their consumption to spread
337
out too. The riskier consumption allocation of high-income agents results in lower willingness
338
to transfer resources to low-income agents and thus lower welfare. Second – and this is the new
339
effect here that only occurs for informative private signals – more informative public signals
340
increase outside option values of agents with a high public signal by less than outside options
341
of agents with a low public signal decrease, facilitating more transfers and higher welfare. The
342
higher is the precision of private signals, the stronger is the second positive effect of more precise
343
public signals. 6 To see this, consider an optimal memoryless allocation and suppose alternatively that F (chh , chl , κ, ν) was decreasing in chh , then social welfare could be increased by transferring resources from high-income agents with a high public signal to low-income agents. By construction, these transfers would be consistent with enforcement constraints and with resource feasibility. Thus, the initial allocation could have not been an optimal memoryless allocation which yields a contradiction, and the values of both insurance contracts are increasing in the corresponding consumption.
11
Welfare in certainty equivalent consumption
1.035
1.03
1.025
1.02
1.015
1.01 ν ν ν ν ν
1.005
1 0.5
0.55
0.6
0.65
0.7
0.75
Precision of public information, κ
0.8
= 0.5 = 0.6 = 0.7 = 0.8 = 0.9
0.85
0.9
Figure 2: Welfare effects of public information for different precisions of private information. 344
To gain intuition, consider an increase in the precision of the public signal. Public sig-
345
nal precision affects outside options in (2) and (4) through its effect on the signal-conditional
346
probabilities of a high income in the next period zh and zl h ∂Vout,hh
∂κ 347
= β [u(yh ) − u(yl )]
∂zh ν(1 − ν) = β [u(yh ) − u(yl )] ≥0 ∂κ [κν + (1 − ν)(1 − κ)]2
= β [u(yh ) − u(yl )]
∂zl ν(ν − 1) = β [u(yh ) − u(yl )] ≤ 0. ∂κ [(1 − κ)ν + (1 − ν)κ]2
and h ∂Vout,lh
∂κ 348
Regardless how precise are the private signals, more precise public signals result in an
349
increase in the value of the outside option for high-income agents with a good public signal and
350
a decrease in the outside option for agents with a bad public signal as illustrated in Figure 3.
351
As argued above, the increase in κ directly affects the value of insurance in the same direction
352
as the outside option but to a smaller extent. The direct effect as the difference between the
353
changes in the insurance and outside option value is negative for agents with a high public signal
354
and positive for agents with a low public signal. The offsetting indirect effect therefore tends to
355
increase chh and to decrease chl . Comparing the derivatives of the outside options with respect to
356
h h κ reveals that the marginal effect (in absolute terms) on Vout,lh is larger than on Vout,hh because
∂zl ∂zh ∂κ ≥ ∂κ . 357
The inequality implies that the direct and the compensating indirect effect of the increase in
358
κ on enforcement constraints must be stronger – in absolute terms – for agents with a low public
12
Value of the outside option in utility
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05 0.5
h Vout,hh , ν = 0.50 h Vout,lh , ν = 0.50 h Vout,hh , ν = 0.90 h Vout,lh , ν = 0.90
0.55
0.6
0.65
0.7
0.75
Precision of public information, κ
0.8
0.85
0.9
Figure 3: Outside option values of high-income agents as functions of public information precision for different precisions of private information. 359
signal than for agents with a high public signal. This asymmetric response of the two outside
360
options is where Assumption 1 has its bite. If only the enforcement constraints of high-income
361
agents with high public signals were binding, then the stronger response of low-public signal
362
agents would not impact on the allocation.
363
When private signals are uninformative (ν = 0.50), the changes in the value of the outside
364
h option of high-income agents with a good public signal (Vout,hh ) and with a bad public signal
365
h (Vout,lh ) are symmetric, i.e., |∂zl /∂κ| = ∂zh /∂κ as illustrated in the lower part of Figure 3
366
with ν = 0.50. The consumption of high-income agents with a good public signal exceeds the
367
consumption of agents with a bad public signal. For this reason, those agents have a lower
368
marginal utility of consumption and require more resources to be indifferent to autarky than
369
the high-income agents with a bad public signal are willing to give up. Hence, the increase in
370
chh is larger than the decrease in chl , and average consumption of high-income agents increases
371
which by resource feasibility reduces consumption of low-income agents. As a consequence, the
372
allocation becomes riskier ex-ante, social welfare decreases and only the first (negative) effect
373
of better public information is present.
374
When private signals are informative, better public signals increase the outside option values
375
of agents with a high public signal by less than outside options of agents with a low public signal
376
decrease, i.e., |∂zl /∂κ| > ∂zh /∂κ as illustrated in Figure 3 for ν = 0.90. Thus, more precise
377
public signals affect outside option values in an asymmetric way; the asymmetry is the stronger
378
the more precise are the private signals. When private signals are sufficiently precise, the drop
379
in chl dominates the jump in chh , and average consumption of high-income agents decreases. By
380
resource feasibility, the latter leaves room for more transfers to low-income agents, social welfare
381
improves and the second positive effect of better public signals prevails.
13
382
Increases in private signal precision on the other hand are unambiguously welfare reducing
383
independent from the precision of public signals as summarized in the following proposition.
384
Proposition 1 (Social Value of Private Information) Let the optimal memoryless alloca-
385
tion satisfy Assumption 1. Then social welfare is decreasing in the precision of the private signal
386
over ν ∈ [0.5, 1] for κ ∈ [0.5, 1].
387
The proof is provided in Appendix A.6. The direct effect of an increase in ν on the outside
388
option exceeds the direct effect on the value of the insurance contracts for high-income agents
389
for both public signal realizations. To render enforcement constraint satisfied, the indirect effect
390
of ν must therefore yield an improvement in consumption for both types of high-income agents.
391
By resource feasibility, consumption of low-income agents decreases which is why overall by
392
concavity of utility social welfare worsens.
393 394
We can also characterize how the precision of private and public signals affect risk sharing. Let risk sharing be measured by 2 2 ch − y¯ + cl − y¯
RS = 1 −
(yh − y¯)2 + (yl − y¯)2
.
(13)
395
Under perfect risk sharing, the measure equals unity while in the absence of risk sharing it
396
equals zero. The following proposition summarizes the effects of increases in private or public
397
signal precision on risk sharing.
398
Proposition 2 (Risk Sharing and Information) Let the optimal memoryless allocation
399
satisfy Assumption 1. Let risk sharing be measured by RS.
400
(i) Risk sharing is decreasing in the precision of the private signal over ν ∈ [0.5, 1] for κ ∈ [0.5, 1].
401
402
(ii) Then there exists a precision of the private signal ν¯, such that for ν ≥ ν¯ and ν ∈ (0.5, 1], risk sharing is increasing in the precision of the public signal over κ ∈ [0.5, 1].
403
404
The proof is provided in Appendix A.7. The logic for the results in the Proposition follows
405
the same logic as in Proposition 1 and Theorem 1. As shown in Lemma 3, consumption of
406
high-income agents is on average above mean income while low-income agents consumption
407
is less than mean income. More precise private signals increase consumption of high-income
408
agents on average while consumption of low-income agents decreases by resource feasibility and
409
risk sharing worsens. Improvements in public signal precision decrease average consumption
410
of high-income agent when private signals are sufficiently precise and increase their average
411
consumption otherwise. Thus, risk sharing improves when private signals are sufficiently precise
412
and risk sharing worsens otherwise.
413
3.4
414
The main message from Theorem 1 is that social welfare can increase in public signal precision
415
when private signals are sufficiently informative and if Assumption 1 is satisfied. In particular,
Discussion of Theorem 1
14
416
risk sharing in the optimal memoryless allocation can neither be absent nor perfect and the
417
enforcement constraints of high-income agents with a high and a low public signal must be both
418
binding in optimum. To better understand our main result, we explain first why Assumption 1
419
is a necessary condition for it. Furthermore, we also discuss changes in preliminary assumptions
420
such as exogenous restrictions on the set of feasible contracts that prevent a positive value of
421
public information.
422
Autarky and perfect risk sharing
423
be welfare improving when risk sharing is partial but not when risk sharing is absent (autarky)
424
or full (perfect risk sharing). In the latter cases, the consumption allocation is independent
425
from public (and private) signal precision. Correspondingly, changes in public signal precision
426
do not affect social welfare. For both autarky and perfect risk sharing, changes in public signal
427
precision can only affect welfare when after the increase in signal precision, risk sharing becomes
428
partial. With partial risk sharing, a further increase in public signal precision may or may not
429
improve welfare depending on the precision of private signals and on the pattern of binding
430
enforcement constraints. The relevance of the pattern of binding enforcement constraints is
431
discussed in the following paragraph.
432
Pattern of binding enforcement constraints
433
forcement constraints or only the enforcement constraints of high-income agents with a high
434
public signal can be binding.7 In the first case, perfect risk sharing violates all enforcement
435
constraints of high-income agents, while in the latter case perfect risk sharing violates only the
436
enforcement constraints of high-income agents with a high public signal. The positive effect of
437
better public information only applies when the enforcement constraints for both public sig-
438
nals are binding. Let Assumption 1 be not satisfied but assume alternatively that only the
439
enforcement constraints of high-income agents with a high public signal are binding. Then in-
440
creases in public signal precision are welfare reducing independent from private signal precision
441
as summarized in the following Proposition.
442
Proposition 3 The optimal memoryless allocation is characterized by partial risk sharing and
443
at the optimal allocation only the enforcement constraints of high-income agents with a high
444
public and a high private signal are binding. Then social welfare is decreasing in the precision
445
of private and public signals.
As stated in Theorem 1, better public signals can only
With partial risk sharing, either both en-
446
The proof is provided in Appendix A.8. The rationale behind this result is as follows.
447
Improvements in public signal precision still increase the willingness of agents with a low public
448
signal to engage in risk sharing but this increase does not impact the allocation because their
449
enforcement constraints are not binding. The only binding enforcement constraints are the ones
450
of high-income agents with a high public signal. Thus, only the increase in their outside option
451
and their resulting lower willingness to transfer resources to low-income agents is reflected in
452
the optimal allocation, and welfare decreases in the precision of public signals.8 7 In Lemma 3 part (iii), we show that λhpc,h ≥ λhpc,l such that enforcement constraints are binding for either both public signals or only for high public signals. 8 Following similar arguments, the welfare effects of more precise private signals are also detrimental.
15
As an implication of the discussion in the last two paragraphs, it follows that Assumption
453 454
1 is a necessary condition for the positive effect of better public information.
455
Different insurance arrangements
456
surance contracts that are contingent on public signals and income considered here. Suppose
457
that the environment is the same and risk sharing is partial. However, assume that consumption
458
allocations can only be contingent on income, i.e., cji = cj for income j and public signal i, but
459
have to satisfy the enforcement constraints (2) and (4). Also in this case, the positive effect of
460
better public information does not arise. Consumption offered to agents with different public
461
signals is equalized. Nevertheless consumption offered must be consistent with the tightest of
462
the enforcement constraints of high-income agents. These are the enforcement constraints of
463
agents with a high public signal, and these constraints will be binding in optimum. When
464
public signals become more precise – in direct analogy to Proposition 3 – only the decreased
465
willingness of high-income agents with a high public signal impacts on the allocation and social
466
welfare decreases for the same reasons stated in the previous paragraph.
467
Disconnect between subjective and objective probabilities
468
fects consumption allocations via the signal-conditional probabilities of a high future income,
469
zh , zl , as objective probabilities that depend on both, public and private signal precision. Agents
470
in our model have rational expectations such that their subjective probabilities are equal to these
471
objective probabilities. Whenever there is a strong disconnect between subjective and objective
472
probabilities such that the optimal consumption allocation becomes independent from public
473
signal precision, the positive effect of public information is absent. For example, suppose that
474
agents either assign a subjective probability of one (optimism) or zero (pessimism) to the high-
475
income in the next period for all signal precisions. In both cases, information precision does not
476
affect agents’ subjective probabilities and the optimal consumption allocation does not depend
477
on signal precision. Another example is when agents trust only their private signals and pay no
478
attention to the public signals. In that case, zh = zl = ν, the two enforcement constraints of
479
high-income agents collapse to one and the consumption allocation is independent from public
480
signal precision. Consequently, there is no positive effect of better public signals in these cases.
481
4
482
Our main theoretical result is derived under simplifying assumptions. In particular, we restrict
483
attention to allocations that are contingent only on the current state (and not on the history
484
of the state), and consider i.i.d. income shocks with only two states. In this section, we
485
quantitatively analyze whether these simplifying assumptions are pivotal for the existence of the
486
positive effect of better public information in a context of international risk sharing. To achieve
487
this goal, we start with adjusting the risk-sharing model from Section 2 to an international
488
setting, also allowing for more general insurance contracts that can depend on the history
489
of the state. Afterwards, we confront the model with data on international risk-sharing to
490
investigate whether there is a positive effect of better public information in international risk
491
sharing which is the normative question posed at the beginning of the paper.
Proposition 3 has implications that go beyond the in-
Information precision af-
An application to international risk sharing
16
492
4.1
Environment
493
Preferences, endowments and information
494
by a continuum of representative households or benevolent governments of small countries in-
495
dexed i. Households’ preferences are given by (1). Real GDP in country i is stochastic and takes
496
the form of a country-specific endowment shock yti that evolves independently across countries
Consider a world economy that is populated
497
according to a first-order Markov process with constant transition probabilities π(y |y); the set
498
of possible endowment realizations Y is time-invariant and finite with N elements. The Markov
499
chain induces a unique invariant distribution of income π(y).
500
Each period, household i receives a public signal kti ∈ Y and a private signal nit ∈ Y that
501
inform about her country-specific shock realization in the next period. As before, the precision
502
of the public signal is denoted by the conditional probability κ = π(y = yj |k = yj ), while the
503
precision of the private signal is given by ν = π(y = yj |n = yj ). Both signals have as many
504
possible realizations as income and also follow first order Markov processes.
505
The transition probabilities of the two signals, π(k = yj |k = yi ) and π(n = yj |n = yi )
506
are chosen to yield a joint distribution of income and the two signals that is consistent with
507
household rationality. Household rationality requires that the joint invariant distribution of
508 509
income and the two signals π(y, k, n) exhibits the invariant income distribution as marginal . distribution, i.e., π(y) = (k,n) π(y, k, n). Further, rationality requires that the conditional
510
income distribution π(y |y) follows from integrating the income distribution that is conditional
511
on income and signals over of any pair of signals . π(y |y) = π(y |y, k, n)π(k, n|y). (k,n)
512
Insurance arrangements
Each household can sign a long-term insurance contract with a
513
risk-neutral financial intermediary. As a consequence, the utility allocation is no longer re-
514
stricted to be contingent on the current state as in the case of memoryless allocations analyzed
515
in Section 3. Instead, the utility allocation can depend on the history of the state. The inter-
516
mediaries, unlike the households, can commit to the terms of the contract and seek to minimize
517
the intertemporal resource costs of the insurance contract. While the intermediaries can observe
518
the public state st , they cannot observe the realization of the private signal nt . Households can
519
ex-post repudiate the contract signed with the principal for which they are punished with perma-
520
nent exclusion from the market. In case of repudiation, households are confined to autarky and
521
they consume their endowment in all states of the world in the future. Per construction, long-
522
term contracts will provide higher welfare than the memoryless allocations that we considered
523
in the previous section.
524
Obstfeld and Rogoff (1996) argue in Chapter 6.3 that an efficient revelation mechanism as
525
formalized by truth-telling constraints may not be not realistic in the context of international
526
risk sharing. Insurance contracts with efficient revelation would require that the contract is
527
the only financial commitment countries can make or that countries’ other trades are perfectly
528
observable which is difficult in the context of sovereign risk.
529
As our baseline, we therefore analyze allocations that depend directly only on the publicly
530
observable state, i.e., private information is not contractible. These allocations trivially satisfy 17
531
the truth-telling constraints as there are no incentives to misreport the realization of the private
532
signal. Nevertheless, private information affects the optimal allocation via the enforcement con-
533
straints. As a robustness exercise, we also study optimal allocations when private information
534
is contractible and allocations are contingent on private signal reports (see the accompanying
535
Online Appendix).9
536
There are many financial intermediaries that act under perfect competition and can inter-
537
temporally trade resources with each other at the given shadow price 1/R with R ∈ (1, 1/β].
538
The equilibrium interest rate is the interest rate that guarantees resource feasibility. Given
539
a shadow price R, an utility promise w, today’s publicly observable realization of the state
540
s = (y, k), today’s unobserved private signal realizations n, the intermediary chooses a portfolio
541
of current utility h and state-contingent future promises w (s ). More formally, the financial
542
intermediaries solve the following optimization problem V (w, s) =
543
min
h,{w (s )}
π (n|s)
n
1 1− R
1 C(h) + π(s |s, n)V w (s ), s R
(14)
s
to deliver the promised value w(s) and to satisfy the participation constraints: w=
π (n|s) (1 − β) h + β π s |s, n w s
n
(1 − β) h + β
(15)
s
π s |s, n w s ≥ U Aut (s, n) , ∀n,
(16)
s 544
where π (n|s) is the probability of a given private signal realization conditional on the realization
545
of the public state in the current period.10 The expected continuation value of an agent who
546
defaults after state s, n has occurred is denoted by U Aut (s, n) with U as the solution of the
547
following functional equation:11 U (s, n) = (1 − β)u(y) + β
π(s , n |s, n)U (s , n ).
(17)
s ,n 9
We should also not be silent about the fact considering non-contractible private information as baseline makes the computation of optimal allocation more tractable which allows us to consider realistic income processes with more than two income states in the quantitative exercise below. In case of contractible private information, the interaction of the increase in the state space with two types of occasionally binding incentive constraints – enforcement and truth-telling constraints – makes the computation of optimal allocations a cumbersome task. As other papers in the literature with a less complex setting than ours such as Broer et al. (2017), we restrict ourselves to persistent income processes with two states in this case. 10 In our recursive formulation, the enforcement constraints are imposed for the current period. This is different from Krueger and Perri (2011) who impose the enforcement constraint on continuation values, i.e., w (s ) ≥ U Aut (s , n ), for all s , n . If households can sign contracts that are contractible on the whole state vector, both recursive formulations are equivalent. When private information is not contractible, imposing the enforcement constraint on continuation values is not equivalent to imposing the constraints in the current period. Our recursive formulation implies constraints that are less restrictive than the constraints in case of enforcements constraints on promises, resulting in lower resource costs. 11 In Appendix A.9, we provide details on the derivations of the transition probabilities π (s |s, n), π (θ |θ) = π (s , n |s, n) as functions of the income transition probabilities, the signal transition probabilities and the two precisions κ, ν.
18
548
4.2
Quantitative exercise
549
Empirical estimates
550
provide here empirical counterparts for the quality of public information, the income process
551
and a consumption insurance measure.12
To take the model described in the previous section to the data, we
552
As the first data source, we employ annual data from the Penn World Tables, version 8.1
553
(PWT) to generate a panel of 70 countries for the years 1990-2004. The PWT assign countries
554
to four data-quality grades, ranging from A, the highest quality, to D, the lowest grade. We
555
use the data quality grades to bin the countries in our panel. First, we generate measures
556
of insurance βΔy,i by regressing country-by-country the idiosyncratic component of changes in
557
consumption per capita in country i, on the idiosyncratic component of changes in GDP per
558
capita as follows βΔy,i = 1 −
cov [Δ ln(cit ) − Δ ln (Ct ) , Δ ln(yit ) − Δ ln (Yt )] , var(Δ ln(yit ) − Δ ln (Yt ))
(18)
559
where Δ ln(cit ) (Δ ln(yit )) stand for country i’s per capita growth rate of consumption (real
560
GDP, respectively) between year t and t − 1. Capital letter variables are sample aggregates that
561
capture the uninsurable aggregate risk component of all countries real GDP. When doing so, we
562
control for differences in insurance between countries that are not explained by our model but
563
stem, for example, from the general level of development or the quality of financial and political
564
institutions. The insurance measure attains a value of 0 if there is no insurance against country-
565
specific GDP shocks and the value of 1 when there is perfect insurance, so that consumption
566 567
per capita does not respond to country-specific fluctuations in GDP. Within each bin, we compute average insurance measures βΔy,j , j = A, B, C which are
568
reported in the lower part of Table 1. Data for D countries lacks information on consumption
569
which is why we cannot estimate insurance in these countries. The main message is that
570
improvements in public data quality on country’s idiosyncratic GDP risk do not exhibit a
571
monotone correlation with insurance. Instead, the correlation follows a U-shape; increasing data
572
quality from grade C to B worsens insurance but further improvements ameliorate insurance
573
again. To test the significance of the conditional insurance measure, we regress the averages on
574
A- and C-group dummies. We find the positive differences with regard to the B-group to be
575
significant at p-values that are below 3%.
576
As an empirical counterpart to capture the quality of public information, we employ the
577
one-year ahead GDP growth forecasts as well as the realizations as reported in the IMF World
578
Economic Outlook (WEO). Using this data, we compute mean-squared forecast errors for real
579 580
GDP growth for all countries in our sample and normalize them by the variance of country i’s j GDP growth. In the next step, we average the forecast errors in each bin j to receive M SF E
581
reported in the bottom of Table 1, j = A, B, C, D.
582
We model the logarithm of the idiosyncratic part of income of country i as an AR(1) process
583
with autocorrelation coefficient ρ ∈ [0, 1) and disturbances that are normally distributed with
584
mean zero and standard deviation (SD) σ . We estimate an autocorrelation coefficient of 12 In the accompanying Online Appendix, we provide further details on the data used, estimation strategy and estimation results.
19
Table 1: Baseline parameters and data moments Parameter
Value
σ β κA κB κC ρ σ Θ
Risk aversion Discount factor Precision public information A countries Precision public information B countries Precision public information C countries Auto-correlation SD of i.i.d. term Income-signals states
1 0.96 0.84 0.76 0.73 0.84 0.026 3×8
βΔy,A βΔy,B βΔy,C A M SF E
Insurance coefficient A countries Insurance coefficient B countries Insurance coefficient C countries
0.36 0.09 0.33
Scaled mean-squared forecast error A countries
1.02
M SF E C M SF E
Scaled mean-squared forecast error B countries
1.24
Scaled mean-squared forecast error C countries
1.30
M SF E
Scaled mean-squared forecast error D countries
1.49
B
D
585
ρ = 0.84 and σ = 0.026 as the averages across all countries in the sample.
586
Preferences and endowments
587
erences. The instantaneous utility function features log preferences and we choose an annual
588
discount factor of β = 0.96. We normalize the mean of income to one, and employ the method
589
proposed by Tauchen and Hussey (1991) to approximate the income process by a Markov process
590
with two states and time-invariant transition probabilities to yield the unconditional variance
591
and autocorrelation estimated in our data set.13
592
Calibrating public signal precision using the IMF forecasts The mean-squared forecast
593
errors computed in the publicly available IMF WEO data is a measure of public information
594
quality and can be linked to the corresponding forecast errors in the model as follows. For
We start with standard values for the specification of pref-
595
a given Markov process for income and uninformative private signals (ν = 0.5), we compute
596
the reduction of perceived income risk κ ˜ as the reduction in the mean-squared forecast error
597
resulting from conditioning expectations on public signals with precision κ in the model κ ˜ (κ) =
598
with MSFEy =
y
MSFEy − MSFEs (κ) , MSFEy
π(y)
2 π(y |y) y − E(y |y)
y
13
As a robustness exercise, we also consider a discretization with three income states. The results of this exercise can be found in the accompanying Online Appendix.
20
MSFEs (κ) =
π(s)
s
2 π(y |s) y − E(y |s) ≤ MSFEy ,
y
599
and π(s) as the joint invariant distribution of income and public signals.14 The reduction in the
600
mean-squared forecast error κ ˜ (κ) is a monotone transformation of signal precision. The more
601
precise are the public signals the larger is κ ˜ (κ). If signals are uninformative, κ ˜ is equal to zero
602
and is equal to one if signals are perfectly informative.
603
The income process is identical across country groups and private signals are uninformative
604
per assumption which implies that the model interprets all differences in mean-squared forecast
605
errors between the country groups as differences in public signal precision. We choose D coun-
606
tries as the reference point (implicitly assuming that public signals are uninformative in these
607
countries) to compute κj as follows κ ˜ (κj ) = 1 −
M SF E M SF E
j
D
j = A, B, C.
(19)
608
In the data, A countries’ mean-squared forecast error is 32 percent lower than the corre-
609
sponding forecast error in D countries. We adjust κj such that κ ˜ (κj ) = 0.32 which results in
610
a calibrated precision of the public information of κA = 0.84. For the other country groups we
611
need to match for the B countries κ ˜ (κj ) = 0.17 and κ ˜ (κj ) = 0.13 for the C countries, resulting
612
in κB = 0.76 and κC = 0.73, respectively as reported in Table 1.
613
Calibrating private signal precision using insurance estimates
614
public signal precision values κj for all country groups using the IMF WEO data, we still
615
need to calibrate private signal precision νj . We consider trade that occurs within each group
616
of countries. For each country group j, we compute optimal stationary allocations and the
617
resulting amount of insurance. Based on the findings of Propositions 1 and 2, we expect that
618
not only welfare and risk sharing but also insurance decreases monotonically in the precision
619
of private information for given public signal precision. Thus, if allocations with uninformative
620
private signals and informative public signals with precision pinned down according to (19)
621
feature insurance that is too high compared to the data, we can choose private signal precision
622
to reduce the gap between the data and the model-implied amount of insurance. Formally,
623
given public information precision κ ¯ j , we seek to identify the precision of private information
624 625
Having backed out the
νj by varying it until insurance in the optimal stationary allocation βΔy,j (¯ κj , νj ) mimics the estimated degrees of insurance in the data, βΔy,j , for j ∈ {A, B, C} βΔy,j (¯ κj , νj ) = βΔy,j .
(20)
626
Eventually, the country groups differ only with respect to the importance of private and public
627
information and the degree of insurance, preferences and endowments are identical.15 14 In Appendix A.9, we provide the formulas for the conditional income probabilities π(y |θ) and π(y |s) as functions of the income transition probabilities and the two precisions κ, ν. 15 As an alternative measure of international risk sharing, we also consider the ratio of consumption to income volatility. The quantitative results for this case are reported in the accompanying Online Appendix.
21
628
4.3
Quantitative results for the baseline calibration
629
As the first step, we confront the theoretical model with the data. Then we demonstrate that
630
when the private information friction is sufficiently severe, better public forecast are indeed
631
beneficial for insurance and social welfare.
632
Insurance: model versus data
633
insurance measure with respect to the private signal precision – larger values of ν are detrimental
634
to insurance as measured by (18). Second, insurance in the model with informative public but
635
uninformative private signals is indeed too high compared to the data for all country groups.
636
This allows us to identify private signal precision according to (20) when insurance reacts
637
sensitively enough to changes in private signal precision. We find that the detrimental effect
638
of more precise private signals in the model is quantitatively strong enough to explain the
639
differences in insurance for groups A, B and C. Thereby, the private information friction plays
640
the most important role in countries of the group B. To capture the low degree of insurance of
First, the allocations exhibit strict monotonicity of the
641
βΔy,B = 0.09, the private information friction is with νB = 0.84 more severe than in country
642
groups A with νA = 0.56, and in C countries, νC = 0.57, respectively. For these values of ν, the
643
model also matches the insurance in A, βΔy,A = 0.36 and in C, βΔy,C = 0.33.
644
The calibrated precisions of private and public information reveal that consumption insur-
645
ance reacts sensitively to changes in private-signal precision but only mildly to changes in public
646
signal precision. When both signals are uninformative, κ = ν = 0.50, the insurance measure
647
equals 0.50. For uninformative private signals, increasing κ to κC = 0.73 reduces insurance to
648
0.45, increasing κ further to κA = 0.84 reduces insurance only mildly to 0.44. The impact of
649
private information on insurance is stronger. Starting at κB = 0.76, increasing private signal
650
precision from 0.5 to νB = 0.84 reduces insurance by 0.36. The same rationale is behind the
651
calibrated values of private signal precision for the country groups A and C. Private infor-
652
mation precision is comparable but the difference in public information precisions is relatively
653
sizeable. Nevertheless the amount of insurance in the two groups is with 0.33 and 0.36 simi-
654
lar because private and not public information is the main driver of the insurance differences
655
between country groups.
656
Improvements in private signal precision reduce welfare by more than changes of public
657
signal precision of the same size. More precise private signals increase the outside option values
658
of agents with a high and a low public signal because only enforcement constraints with a high
659
private signal can be binding. Improvements in public signal precision increase the outside
660
option value of agents with a high public signal but decrease the outside option of agents with a
661
low signals. For this reason, consumption of high-income agents goes up on average by more in
662
case of private than public signals, and insurance and welfare decrease by relatively more when
663
private signal precision increases.
664
Using the Consumer Expenditure Survey 1998–2003 for US households, Broer (2013) finds
665
that the limited commitment model tends to predict too much consumption insurance for stan-
666
dard calibration such as ours. Thus, it may seem surprising that a limited commitment model
667
can capture the relatively low degree of insurance observed across country groups. One differ-
668
ence to Broer (2013) is that we consider private and public signals and we find that consumption
22
Welfare in certainty-equivalent consumption
0.9991
0.9991
0.9990
0.999
0.9989
Welfare, A-countries, νA = 0.56 Welfare, B-countries, νB = 0.84 Welfare, C-countries, νC = 0.57 Welfare ν = 0.60
0.9989
0.9988 0.5
0.55
0.6
0.65
0.7
0.75
Precision of public signals, κ
0.8
0.85
0.9
Figure 4: Non-contractible private information. Welfare of A, B, C countries measured in certainty-equivalent consumption as a function public signal precision. 669
insurance reacts in particular sensitively to changes in private signal precision. Another dif-
670
ference is the size of the income risk in US micro data compared to the international setting
671
we consider. Broer (2013) employs in his calibration a variance of logged income of 0.36 while
672
the country-specific shocks we consider yield a variance of logged income equal to 0.0023 or a
673
standard deviation of 0.048.16
674
Positive value of public information
675
at the beginning of the paper whether governments of countries and international institutions
676
should strive to provide better country-GDP forecasts to the public in order to improve house-
677
holds’ consumption smoothing against country-specific shocks. The main message from The-
678
orem 1 is that increases in public signal precision increase welfare when private signals are
679
sufficiently precise. While this result was derived for memoryless allocations, the allocations we
680
consider here are more general and can be contingent on the history of the public state st . To
681
test whether the positive welfare effect of better public signals applies here as well, we compute
682
welfare in the three country groups as a function of public signal precision.
Here we come back to the normative question posed
683
The results of this exercise are plotted in Figure 4 and are consistent with the findings in
684
Theorem 1. Private signals in A and C-countries are with precisions νA = 0.56 and νC = 0.57
685
not precise enough, and social welfare decreases mildly by less than 0.01 percent in the precision
686
of public signals. The positive and negative effect of better public signals exactly cancel each
687
other out for private signals with precision ν = 0.60. In the B-countries, private signals are 16 This estimate is consistent with other estimates in the related literature. For example, Kose, Prasad, and Terrones (2003) find estimates that range from 0.02 (industrial countries) to 0.05 (less financially integrated countries) for the years 1960 - 1999. Backus, Kehoe, and Kydland (1992) employ in their benchmark case standard deviation of innovations approximately equal to 0.01.
23
688
with νB = 0.84 > 0.60 precise enough and better public information increases welfare by 0.01
689
percent. The main message is that the positive effect of better public signals does not depend on
690
the assumption of memoryless allocations but also emerges for a more general type of insurance
691
contracts in which also the history of the public state is contractible. This conclusion also
692
extends to the case when insurance contracts that can depend on the history of the public state
693
as well as additionally on the history of private signal reports (see the accompanying Online
694
Appendix).
695
5
696
In this paper, we have revisited the value of private and public information in a general con-
697
sumption risk-sharing model under limited commitment. In contrast to conventional wisdom,
698
we show in our main theoretical result that improvements in public information can improve
699
risk sharing and welfare. Better public information has two opposite effects when private sig-
700
nals are informative. First, more precise public signals spread out the outside option values of
701
high-income agents with a high and a low public signal and these agents are less willing to trans-
702
fer resources to low-income agents. Second, more informative public signals affect the outside
703
option values of high-income agents asymmetrically. As a consequence, the willingness of low-
704
signal agents to transfer resources increases by more than the high-signal agents’ willingness to
705
transfer resources decreases, facilitating in total higher transfers from high-to low income agents
706
and benefiting welfare. When private information is sufficiently precise, risk sharing improves
707
with better public information and agents prefer informative public signals ex-ante.
Conclusions
708
We find that this positive effect of better public information is not only present under specific
709
circumstances but is a rather general phenomenon. In particular, we find that the logic behind
710
the positive effect also applies in optimal allocations that are contingent on the history of the
711
public state, the history of truthful reports on private signal realizations and for persistent
712
idiosyncratic shock processes.
713
As a methodological contribution, we formulate a general risk-sharing model in which agents
714
with private and public fore-knowledge of their future idiosyncratic shock realizations can engage
715
in dynamic insurance contracts to smooth their consumption.17 This risk-sharing model can be
716
directly applied to explore ideas related to thought-provoking results in the recent literature on
717
the existence of a private unemployment insurance market and the optimal provision of public
718
insurance.
719
Hendren (2017) analyzes the implications of individuals with private knowledge of future
720
job losses for the existence of an unemployment insurance market. As his main result, he
721
finds that the private information friction gives rise to unemployment insurance contracts with
722
pooled risks that are too adversely selected to be profitable. Hendren (2017) derives this result
723
for static insurance contracts. In this paper, we explicitly model individual fore-knowledge on
724
future shocks as signals and consider a general optimal contracting problem comprising a rich 17 There is ample evidence that individuals possess fore-knowledge on their future risks. Exemplary papers are Dominitz (1998) and Dominitz and Manski (1997) for income risk, Smith, Taylor, and Sloan (2001) for predicting mortality risk, Stephens (2004) for unemployment, or Campbell, Carruth, Dickerson, and Green (2007) for job insecurity.
24
725
variety of static and dynamic insurance contracts. Using our framework, one can ask whether
726
Hendren (2017)’s main result on the non-existence of a private unemployment market due to
727
adverse selection prevails when individuals can also trade dynamic unemployment insurance
728
contracts.
729
Krueger and Perri (2011) find that more public insurance via a progressive tax system
730
can more than one-for-one crowd out private insurance of income shocks provided in contracts
731
with limited enforceability. This crowding out effect arises when private insurance markets
732
are working well and the overall degree of risk sharing is high even without the additional
733
insurance provided by the tax system. When agents receive informative private signals on their
734
future income, private insurance becomes less effective in smoothing consumption. It would
735
be interesting to investigate if better public insurance may now overall improve consumption
736
insurance for parameterizations for which Krueger and Perri (2011) find a more than one-for-one
737
crowding out of the insurance achieved via private insurance markets.
738
References
739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762
Backus, D. K., P. J. Kehoe, and F. E. Kydland (1992): “International Real Business Cycles,” Journal of Political Economy, 100, 745–75. Broer, T. (2013): “The wrong shape of insurance? What cross-sectional distributions tell us about models of consumption smoothing,” American Economic Journal: Macroeconomics, 5, 107–140. Broer, T., M. Kapicka, and P. Klein (2017): “Consumption Risk Sharing with Private Information and Limited Enforcement,” Review of Economic Dynamics, 23, 170–190. Campbell, D., A. Carruth, A. Dickerson, and F. Green (2007): “Job insecurity and wages*,” The Economic Journal, 117, 544–566. Coate, S. and M. Ravallion (1993): “Reciprocity without Commitment: Characterization and Performance of Informal Insurance Arrangements,” Journal of Development Economics, 40, 1–24. Dominitz, J. (1998): “Earnings Expectations, Revisions, And Realizations,” The Review of Economics and Statistics, 80, 374–388. Dominitz, J. and C. F. Manski (1997): “Using Expectations Data To Study Subjective Income Expectations,” Journal of the American Statistical Association, 92, 855–867. Fernandes, A. and C. Phelan (2000): “A Recursive Formulation for Repeated Agency with History Dependence,” Journal of Economic Theory, 91, 223–247. Hassan, T. A. and T. M. Mertens (2015): “Information Aggregation in a Dynamic Stochastic General Equilibrium Model,” NBER Macroeconomics Annual, 29, 159–207. Hendren, N. (2017): “Knowledge of Future Job Loss and Implications for Unemployment Insurance,” American Economic Review, 107, 1778–1823. Hirshleifer, J. (1971): “The Private and Social Value of Information and the Reward to Inventive Activity,” American Economic Review, 61, 561–574. Kose, M. A., E. Prasad, and M. E. Terrones (2003): “Financial Integration and Macroeconomic Volatility,” IMF Working Papers 03/50, International Monetary Fund.
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Krueger, D. and F. Perri (2011): “Public versus Private Risk Sharing,” Journal of Economic Theory, 146, 920–956.
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Lepetyuk, V. and C. A. Stoltenberg (2013): “Policy Announcements and Welfare,” Economic Journal, 123, 962–997.
767
Obstfeld, M. and K. Rogoff (1996): Foundations of International Macroeconomics, MIT Press, 1 ed.
768
Schlee, E. E. (2001): “The Value of Information in Efficient Risk-Sharing Arrangements,” American Economic Review, 91, 509–524.
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Smith, V. K., D. H. Taylor, and F. A. Sloan (2001): “Longevity Expectations and Death: Can People Predict Their Own Demise?” American Economic Review, 91, 1126–1134. Stephens, M. (2004): “Job Loss Expectations, Realizations, and Household Consumption Behavior,” The Review of Economics and Statistics, 86, 253–269. Tauchen, G. and R. Hussey (1991): “Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models,” Econometrica, 59, 371–96.
26
776
A
777
A.1
Appendix: proofs and results in the main text Proof of Lemma 1
778
(i) Let the vector of parameters be γ = (κ, ν). Based on part (i) and given a strictly concave
779
function u (c) : R+ → R satisfying Inada conditions, the problem to compute the optimal
780
memoryless allocation is ∞
1 max E U (c) = (1 − β) β t u cji ≡ Vrs subject to: 4 {cji }i,j∈{l,h} t=0 i,j∈{l,h}
non-negativity: cji ≥ 0 ∀i, j ∈ {l, h} 1 j 1 resource feasibility: ci ≤ yj 4 2
(21) (22) (23)
j
i,j∈{l,h}
enforcement constraints ∀i, j, k ∈ {l, h} :
m (1 − β) u cji + β (1 − β) π y = ym |k = yi , n = yk Vrs + β 2 Vrs ≥ m∈{l,h}
(1 − β) u (yj ) + β (1 − β)
π y = ym |k = yi , n = yk u (ym ) + β 2 Vout ,
(24)
m∈{l,h} 781
with π (y = ym |k = yi , n = yk ) as the probability to receive income ym in the future con-
782
ditional on receiving a public signal indicating yi and a private signal indicating yk . The
783
above constraints define the set of constraints P (γ), any candidate solution consumption
784
allocation will be labelled P (γ) ∈ P (γ). Observe that P (γ) is trivially non-empty, as the
785
autarkic allocation cki,j = yk satisfies the enforcement constraints and is resource-feasible.
786
Observe that the resource feasibility constraint (23) together with non-negativity con-
787
straints (22) define a compact and convex subset of R4 . The target function (social welfare
788
function) is continuous. Hence, by Weierstrass theorem the solution to the maximization
789
problem exists. The participation constraints define a convex subset of the resource fea-
790
sible non-negative set of consumption bundles. Uniqueness follows from strict concavity
791
of the target function and convexity of the set of constraints. In fact, the constraints
792
define a continuous (both upper and lower hemicontinuous) correspondence P (γ). Thus,
793
the continuity and differentiability of the welfare function with respect to γ is a direct
794
application of the theorem of maximum.
795
(ii) Consider the autarky allocation which per construction satisfies enforcement constraints.
796
Without loss of generality, consider the incentives of high-income agents with a low public
797
and a high private signal to transfer an infinitesimal amount of resources to other agents.
798
Such a transfer satisfies the enforcement constraints of these agents if the cost of such a
799
transfer in utility is smaller than the gains. Consider first these agents to transfer resources
800
to high-income agents with a high private and a high public signal. Such a transfer is
801
consistent with their enforcement constraints if
zl β 2 zl β 2 + ≤ u (yh )δ (1 − β)β + , u (yh )δ (1 − β) 1 + β 2 4 2 4 27
802
where the left-hand side captures the utility costs and the right-had side the utility gains of
803
the transfer δ and zl as defined in (5). For β ∈ (0, 1), the costs exceed the gains, rendering
804
such a transfer incompatible with enforcement constraints. Thus, the only possibility
805
is to transfer resources to low-income agents because the consistency with enforcement
806
constraints then requires
zl β 2 1 − zl β 2 u (yh )δ (1 − β) 1 + β + ≤ u (yl )δ (1 − β)β + 2 4 2 4
807
which can be supported when the difference in marginal utilities u (yl )−u (yh ) > 0 is large
808
enough. Starting alternatively from any other enforcement constraints of high-income
809
agents follows the same logic and the same conclusion that transfers from high-income
810
agents to other high-income agents violate enforcement constraints.
811
The only remaining possibility to support chi > yh for any public signal i are transfers
812
stemming from low-income agents. Using the similar logic, we can also exclude transfers
813
from any low-income agent to any high-income agent in an optimal memoryless alloca-
814
tion. Thus, in the optimal memoryless allocation, we get chi ≤ yh , and then by resource feasibility cli ≥ yl for all signals i.
815
816
A.2
817
The first best allocation can be perfect risk sharing (all agents in all states consume y¯). In the
818
following proposition, we summarize how public and private information affect the condition
819
that renders perfect risk sharing constrained feasible and thus optimal.
820
Proposition 4 (Perfect Risk Sharing) Consider the optimal memoryless allocation.
821 822
823
824
825
Information, perfect risk sharing and autarky
¯ ν) < 1, such that for any discount factor (i) There exists a unique cutoff point, 0 < β(κ, ¯ ν) the optimal allocation for any signal precision is perfect risk sharing. 1 > β ≥ β(κ, ¯ ν) is increasing in the precision of the public and private signal. (ii) The cutoff point β(κ, Proof. (i) Let V¯rs = u(¯ y ) be social welfare under perfect risk sharing. First, perfect risk sharing
827
provides the highest ex-ante utility among the consumption-feasible allocations. The exis¯ ν) follows from monotonicity of participation constraints in β and V¯rs > Vout . tence of β(κ,
828
A higher β increases the future value of perfect risk sharing relative to the allocation in the
826
830
equilibrium without transfers, leaving the current incentives to deviate unaffected. There¯ ν), they are not binding for fore, if the participation constraints are not binding for β(κ,
831
¯ ν). The cutoff point is characterized by the tightest participation constraint, any β ≥ β(κ,
832
i.e., by the participation constraint with the highest value of the outside option
829
h l + β 2 Vrs ≥ (1 − β)u(chh ) + β(1 − β) zh Vrs + (1 − zh )Vrs (1 − β)u(yh ) + β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] + β 2 Vout ,
28
833 834
835
with zh defined in the main text (3). Solving this constraint yields a unique solution for ¯ ν) in (0, 1) because u(¯ β(κ, y ) < u(yh ). (ii) The tightest constraint at the first best allocation is u(¯ y ) ≥ (1 − β)u(yh ) + β(1 − β)zh u(yh ) + β(1 − β)(1 − zh )u(yl ) + β 2 Vout .
836
Differentiating the constraint fulfilled with equality with respect to κ using the implicit
837
function theorem gives h ¯ ν) β(1 − β)[u(yh ) − u(yl )] ∂z ∂ β(κ, ∂κ = > 0, ∂κ D
838
results in a positive sign for yh > yl , ∂zh ν(1 − ν) = >0 ∂κ [κν + (1 − κ)(1 − ν)]2
839
and D ≡ (1 − β)[u(yh ) − zh u(yh ) − (1 − zh )u(yl )] + β[zh u(yh ) + (1 − zh )u(yl ) − u(yl )] > 0.
840
For ν ∈ [0.5, 1), the cutoff point increases strictly in precision of the public signal. Simi-
841
larly, taking the derivative with respect to the ν results in a positive sign as well, a strictly
842
positive sign for κ ∈ [0.5, 1).
843 844
On the other extreme, the absence of risk sharing in autarky may be the only constrained-
845
feasible memoryless allocation. In the next proposition, we provide conditions for this case.
846
Proposition 5 (Autarky) Consider the case when the participation constraints of high-
847
income agents with a high and low public signal are binding. If and only if u (yh ) + β
z h + zl β2 1 [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] ≥ 0 2 1−β2
(25)
848
autarky is the optimal memoryless allocation with zh and zl defined in (3), (5).
849
Proof. Optimal memoryless allocations can be analyzed as a fixed-point problem expressed in
850
terms of the period value of the arrangement.
851
The fixed-point problem is constructed as follows. Let W = E U (c) be the unconditional
853
expected value of an allocation c before any shock has realized. We restrict attention to W ∈ [Vout , V¯rs ) because per assumption participation constraints for high-income households are
854
binding. The binding participation constraints are given by the following
852
u(chh ) + β zh [u(chh ) + u(chl )]/2 + (1 − zh )u(2¯ y − (chh + chl )/2) = u(yh ) + β [zh u(yh ) + (1 − zh )u(yl )] +
29
β2 (Vout − W ), 1−β
(26)
855
u(chl ) + β zl [u(chh ) + u(chl )]/2 + (1 − zl )u(2¯ y − (chh + chl )/2) = u(yh ) + β [zl u(yh ) + (1 − zl )u(yl )] +
β2 (Vout − W ), 1−β
(27)
856
and resource feasibility is used. The objective function of the problem to compute the optimal
857
memoryless arrangement is given by the following expression Vrs (W ) ≡
1 h u(ch (W )) + u(chl (W )) + 2u(2¯ y − (chh (W ) + chl (W ))/2) . 4
858
The optimal memoryless arrangement should necessary solve the fixed-point problem W =
859
Vrs (W ). We will show that Vrs (W ) is strictly increasing. V (W ) is also strictly concave, therefore
860
there exist at most two solutions to the fixed-point problem.
861
From the participation constraints (26) and (27), the derivative of V (W ) is given by Vrs (W )
∂chh ∂chl 1 h h (u (ch ) − u (cl )) + (u (cl ) − u (cl )) = 4 ∂W ∂W
862
which is strictly increasing in W because perfect risk sharing is not constrained feasible which
863
implies that ∂chh /∂W and ∂chl /∂W are negative and chh , chl = y¯.
864
By construction, one solution to the fixed-point problem is Vout . The concavity of Vrs (W )
865
implies that the derivative of Vrs (W ) at Vout is higher than at any partial risk-sharing allocation.
866
(w) at V Therefore, autarky is the optimal memoryless arrangement if the derivative of Vrs out
867
must be smaller than or equal to 1 which implies Vrs (W )
868
1 (u (yh ) − u (yl ) = 4
∂ch ∂chh + l ∂W ∂W
{cji }={yj }
≤1
The two derivatives can be computed using the implicit function theorem and are given by β2 P h cl 2 β Q ch ∂chh l , = − Ph Ph ∂W ch cl Q ch Q ch h
P h β2 ch 2 h Q ch β ∂cl h , = − Ph Ph ∂W ch cl Q ch Q ch
l
h
l
869
with the auxiliary derivatives P, Q as the partial derivatives of the binding participation con-
870
straints (26) and (27) evaluated at the autarky allocation given by β β zh u (yh ) − (1 − zh )u (yl )] 2 2 − (1 − β)u (yh )
Pch = (1 − β)[u (yh ) + h
Pch = Pch l
h
β β zl u (yh ) − (1 − zl )u (yl )] 2 2 − (1 − β)u (yh ).
Qch = (1 − β)[u (yh ) + l
Q c h = Qc h h
l
30
871 872
Using these expressions, the sum of the partial derivatives with respect to W evaluated at {cji } = {yj } is given by
∂ch ∂chh + l ∂W ∂W
=−
h
=
873
2β 2 (1 − β)u (yh ) (1 − β)u (yh )[Pch + Qch − (1 − β)u (yh )]
(1 − β) u (yh ) zh +
l
−4β 2 β
. + zl − (2 − zh − zl )u (yl )
(W ) and collecting terms eventually results in Using this expression in Vrs
u (yh ) + β
z h + zl β2 1 [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] ≥ 0. 2 1−β2
874
Under this condition, the optimal memoryless arrangement is the outside option. If the condi-
875
tion is strictly negative then there exists an alternative constrained feasible allocation that is
876
preferable to the outside option. This result is used in the main theorem.
877 878
This condition can be related to the corresponding condition in case of uninformative signals (κ = ν = 0.5). In this case, zh = zl = 0.5, and the condition reads 1 − β β2 1−β β + − u (yl )β 1 − β − + ≥0 u (yh ) (1 − β) + β 2 2 2 2 β β ⇔ u (yh ) 1 − − u (yl ) ≥ 0 2 2 β u (yh ) ≥ u (yl ), 2−β
879
which corresponds to the condition derived in Krueger and Perri (2011) and in Lepetyuk and
880
Stoltenberg (2013). From the other end, suppose that autarky is the optimal memoryless
881
arrangement and that both participation constraints of high-income agents are binding. Then,
882
the value of this arrangement Wout = Vout must be a solution to the fixed-point problem. This
883
requires that the slope of Vrs (W ) at {cji } = {yj } must be necessarily smaller than or equal to
884
unity. Otherwise, due to the concavity of Vrs (W ), there exists another solution to the fixed-point
885
problem with an allocation that Pareto dominates the autarky allocation.
886
A.3
Proof of Lemma 2
887
(i) Autarky is not the only constrained-feasible memoryless allocation, thus, Vrs > Vout ;
888
h < u(y ) chi ≤ yh for all i (see of part (iii) of Lemma 1) and strict concavity implies, Vrs h
889 890 891
such that chi < yh for at least one i; the resource constraint is binding because utility is l > u(y ) because strictly increasing in consumption, resource feasibility then results in Vrs l
cli > yl for at least one i.
892
(ii) Suppose that the enforcement constraints of high-income agents with a high public signal
893
are binding (the case of a low public signal follows analogous arguments and is omitted
31
894
here).18 To establish a contradiction, suppose that constraints with high and low private
895
signals are binding. Let zhl denote the conditional probability of a high future income
896
conditional on a high public and low private signal and be defined as follows: zhl ≡
(1 − ν)κ , (1 − ν)κ + ν(1 − κ)
0 ≤ zhl ≤ zh ≤ 1
897
and zh as defined in (3).
898
The enforcement constraints of high-income agents with high public signal and with low
899
private signals are h l (1 − β)u(chh ) + β(1 − β) zhl Vrs + (1 − zhl )Vrs + β 2 Vrs = (1 − β)u(yh ) + β(1 − β) zhl u(yh ) + (1 − zhl )u(yl ) + β 2 Vout . Subtracting these constraints from the ones of agents with a high private signal gives
900
h l = (zh − zhl ) [u(yh ) − u(yl )] . (zh − zhl ) Vrs − Vrs l > y . Strict concavity With ν ∈ (0.5, 1], zh = zhl . From part (i), we have chi < yh and Vrs l
901 902
h − V l < u(y ) − u(y ) which contradicts that the enforcement constraints of implies Vrs h l rs
903
agents with a high and a low private signals are binding. What remains to be shown is
904
whether enforcement constraints with a low or a high private signal are binding. To see
905
this, suppose first that only the enforcement constraints of high-income agents with a low
906
private signal (and a high public signal) were binding. In that case, we have that β2 h h − u(yh )] + β(1 − zhl )[Vrs − u(yh )]. (Vout − Vrs ) = u(chh ) − u(yh ) + βzhl [Vrs 1−β
907
Combining this expression with slack enforcement constraints of high-income agents with
908
a high private signal (and a high public signal) it must be that
h l > (zh − zhl ) [u(yh ) − u(yl )] , (zh − zhl ) Vrs − Vrs
909
h −Vl < which results in a contradiction because with partial risk sharing, we have Vrs rs
910
u(yh ) − u(yl ). Assume alternatively that only enforcement constraints of high-income
911
agents with a high private signal (and a high public signal) are binding. Following similar
912
steps as before, now it must be that
h l (zhl − zh ) Vrs − Vrs > (zhl − zh ) [u(yh ) − u(yl )]
h l ⇔ Vrs − Vrs < [u(yh ) − u(yl )] which is satisfied.
913 18
In principle, the argument also applies to low-income agents but given that we show in part (iii) that their enforcement constraints cannot be binding, we do not consider this case here.
32
914 915
(iii) From part (ii), we have that only participation constraints of agents with a high private signal can be binding. The optimal memoryless allocation thus maximizes 1 h u(ch ) + u(chl ) + u(clh ) + u(cll ) 4 + λhpc,h
u(chh ) − u(yh ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) + λlpc,h u(clh ) − u(yl ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) + λlpc,l u(cll ) − u(yl ) (1 − β) h l + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) + β 2 (Vrs − Vout ) h l + λhpc,l u(chl ) − u(yh ) (1 − β) + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) 1
1 h + β 2 (Vrs − Vout ) − λres ch + chl + clh + cll − (yh + yl ) , 4 2
916
with λjpc,i ≥ 0 as the multiplier on the enforcement constraints of agents with signal i and
917
income j, and λres > 0 is the multiplier on the resource constraint.
918
In general, chh ≤ yh , and the enforcement constraints of high-income agents with a high
919
public signal (and a high private signal) imply h l (1 − β)β zh Vrs − u(yh ) + (1 − zh ) Vrs − u(yl ) + β 2 (Vrs − Vout ) ≡ xh ≥ 0.
920
Suppose first per contradiction that enforcement constraints of low-income agents with
921
a high public signal were binding such that λlpc,h > 0. From the first order conditions
922 923 924
of these agents, we immediately get clh > cll . Autarky is not the optimal memoryless allocation such that either chh < yh or chl < yh or both strict inequalities apply. In any case, (chh + chl )/2 < yh , resource feasibility then implies that (cll + clh )/2 > yl , and clh > yl
925
because clh > cll . The enforcement constraints of low-income agents with a high public
926
signal are binding if
927
which requires xh < 0 because clh > yl . This results in a contradiction because the
u(clh ) − u(yl ) (1 − β) + xh = 0
928
enforcement constraints of high-income agents with a high public signal require xh ≥ 0.
929
Suppose alternatively per contradiction that enforcement constraints of low-income agents
930
with a low public signal were binding such that λlpc,l > 0. From the first order conditions
931 932 933
of these agents, we immediately get cll > clh . Autarky is not the memoryless allocation such that either chh < yh or chl < yh or both strict inequalities apply. In any case, (chh + chl )/2 < yh , resource feasibility then implies that (cll + clh )/2 > yl , and cll > yl
934
because cll > clh . The enforcement constraints of low-income agents with a low public
935
signal are binding if
u(cll ) − u(yl ) (1 − β) + xl = 0, 33
with
936
h l xl ≡ (1 − β)β zl Vrs − u(yh ) + (1 − zl ) Vrs − u(yl ) + β 2 (Vrs − Vout ), 937
and zl as defined in (5). It is cll > yl , thus unless xl < 0, there is a contradiction. The
938
participation constraints of high-income agents with a low public signal read
u(chl ) − u(yh ) (1 − β) + xl ≥ 0.
939
Since chl ≤ yh , it must be that xl ≥ 0 which contradicts that the enforcement constraints
940
of low-income agents with a low signals can be binding.
941
(iv) From parts (ii) and (iii), we have that only enforcement constraints of high-income agents
942
with high private signals can be binding. Thus, the optimal memoryless allocation maxi-
943
mizes 1 h u(ch ) + u(chl ) + u(clh ) + u(cll ) 4 + λhpc,h
u(chh ) − u(yh ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) h l + λhpc,l u(chl ) − u(yh ) (1 − β) + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) 1
1 h 2 h l l + β (Vrs − Vout ) − λres c + cl + ch + cl − (yh + yl ) , 4 h 2
with λhpc,h ≥ 0 as the multiplier on the enforcement constraints of high-income agents
944 945
with a high public signal (and a high private signal), λhpc,l ≥ 0 as the multiplier on the
946
enforcement constraints of high-income agents with a low public signal (and a high private
947
signal), and λres > 0 is the multiplier on the resource constraint. The first order conditions
948
for low-income agents are u (cll ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 = λres u (clh ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 = λres . Combining these equations gives immediately cll = clh = cl .
949
950
951
A.4
Proof of Lemma 3
(i) Subtract the binding enforcement constraint G(chh , chl , κ, ν) from F (chh , chl , κ, ν) to receive
h l u(chh ) − u(chl ) = β(zh − zl ) u(yh ) − Vrs + Vrs − u(yl ) .
952 953 954
h < u(y ) and V l > u(y ), and the expression in From Lemma 2 part (i), we have that Vrs h l rs
the square brackets on the right-hand side is strictly positive; zh ≥ zl , and thus, chh ≥ chl ; if zh > zl for κ ∈ (0.5, 1] and ν ∈ [0.5, 1), the strict inequality holds, chh > chl .
34
955
From the proof of Lemma 2 part (iv), the first order conditions for chl and cl can be
956
combined to receive u (chl ) 1 + 2λhpc,h β(1 − β)zh + 4(1 − β)λhpc,l + 2λhpc,l β(1 − β)zl + (λhpc,h + λhpc,l )β 2 = u (cl ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + (λhpc,h + λhpc,l )β 2 .
957
From strict concavity, it follows that chl > cl if 1 + 2λhpc,h β(1 − β)zh + 4(1 − β)λhpc,l + 2λhpc,l β(1 − β)zl + β 2 (λhpc,h + λhpc,l ) > 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 (λhpc,h + λhpc,l ) ⇔ 2βλhpc,h (2zh − 1) + 2λhpc,l [2 + β (2zl − 1)] > 0
958
which is satisfied as long as one type of the high-income agents’ enforcement constraints
959
is binding, since zh ≥ 0.5, and β ∈ (0, 1); here, it is λhpc,h , λhpc,l > 0. At the beginning, we
960
961 962
established chh ≥ chl such that ch > cl and by resource feasibility, we get ch > y¯ > cl .
(ii) Per assumption, autarky is not the only memoryless allocation. From Proposition 5, we have that the determinant of the Jacobian evaluated at autarky is J F (ch , ch , κ, ν), G(ch , ch , κ, ν) h l h l
{cji }={yj }
= u (yh ) + β 963
− Fch Gch l h
= Fch Gch h
l
β2
{cji }={yj }
1 z h + zl [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] < 0, 2 1−β2
(28)
where the partial derivatives are given by
F ch h
F ch l
Gch l
Gch h
zh h 1 − zh l β2 h u (c ) + u (ch ) − u (cl ) = (1 − β) u + β u (ch ) − β 2 2 4 2 zh 1 − zh l β u (c ) + u (chl ) − u (cl ) = (1 − β) β u (chl ) − β 2 2 4 zl h 1 − zl l β2 h h = (1 − β) u (cl ) + β u (cl ) − β u (c ) + u (cl ) − u (cl ) 2 2 4 2 zl 1 − zl l β = (1 − β) β u (chh ) − β u (c ) + u (chh ) − u (cl ) , 2 2 4
(chh )
(29) (30) (31) (32)
964
When evaluated at the optimal memoryless allocation, concavity of utility necessarily
965
implies that the determinant of the Jacobian is positive J F (ch , ch , κ, ν), G(ch , ch , κ, ν) h l h l
{cji }={yj }
966
= Fch Gch h
l
− Fch Gch l h
{cji }={yj }
> 0.
(33)
Thus, the implicit function theorem applies, chh , chl are differentiable functions of (κ, ν),
967
and partial derivatives of chh , chl with respect to (κ, ν) can be found by differentiating
968
implicitly.
969
From part (i), we have chh , chl > cl which together with the first order conditions (8)-(9)
35
970
yield λhpc,h Fch + λhpc,l Gch ≡ t2 > 0 h
971
λhpc,h Fch + λhpc,l Gch ≡ t1 > 0.
h
l
l
Combining the first order conditions (8)-(9) gives t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) . t2
972
(iii) Recall the first-order conditions of high-income agents imply t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) ⇒ t2 ≡ λhpc,h Fch + λhpc,l Gch ≥ t1 ≡ λhpc,h Fch + λhpc,l Gch , h h l l t2
973
where the inequality uses chh ≥ chl (see Lemma 3 part (i)). The inequality can we written
974
as λhpc,h (Fch − Fch ) ≥ λhpc,l (Gch − Gch ) h
l
l
h
⇔ λhpc,h ≥ λhpc,l , 975
where the last step follows because Gch − Gch ≥ Fch − Fch ≥ 0; the second inequality,
976
Gch − Gch , Fch − Fch ≥ 0 follows from t2 ≥ t1 and
l
l
h
h
h
l
l
Gch − Gch = (1 − β)u (chl ) + β(1 − β) l
h
h
zl h u (cl ) − u (chh ) > 0 for 2
chl ≤ chh
977
such that Fch − Fch ≥ 0. For the first inequality, Gch − Gch ≥ Fch − Fch , it immediately
978
follows Gch + Fch ≥ Fch + Gch . The left-hand side of the inequality is
h
l
l
l
l
h
G c h + Fc h l
979
l
l
h
(chl )
The right-hand side of the inequality is Fc h + G c h = u h
h
(chh )
z h + zl β 2 2 − zh − z l β 2 l + − u (c ) (1 − β)β + . (1 − β) + (1 − β)β 2 2 2 2
Comparing the two expressions results in
G c h + Fc h − Fc h − G c h = u l
l
h
h
(chl )
−u
(chh )
z h + zl β 2 + (1 − β) + (1 − β)β 2 2
because chl ≤ chh
≥0 981
h
zl h 1 − zl l β2 h u (c ) + u (cl ) − u (cl ) = (1 − β) u + β u (cl ) − β 2 2 4 1 − zh l β2 h zh h u (c ) + u (cl ) − u (cl ) + (1 − β) β u (cl ) − β 2 2 4 2 z h + zl β 2 − zh − z l β 2 + − u (cl ) (1 − β)β + . = u (chl ) (1 − β) + (1 − β)β 2 2 2 2
980
h
such that Gch − Gch ≥ Fch − Fch , and λhpc,h ≥ λhpc,l . l
h
h
l
36
(34) (35)
982
A.5
983
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
984
Proof of Theorem 1
derivative of social welfare with respect to κ is
h ∂ch ∂ch ∂chl ∂Vrs 1 h ∂chh = u (ch ) + u (chl ) l − u (cl ) + , ∂κ 4 ∂κ ∂κ ∂κ ∂κ 985
The derivatives of consumption with respect to the signal precision follow from the implicit
986
function theorem which is applicable because according to Lemma 3 part (ii), Fch Gch −Fch Gch >
987
0. Further, in optimum, the following condition holds (see also Lemma 3 part(ii))
h
l
l
h
t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2 988 989
with t1 , t2 > 0 due to chh ≥ chl > cl (see Lemma 3, part (i)). Using this expression to substitute
for u (chl ) − u (cl ) in the derivative of social welfare with respect to κ ∂Vrs 1 h = u (ch ) − u (cl ) ∂κ 4
990
∂chh t1 ∂chl + ∂κ t2 ∂κ
≡ I(ν, κ).
Given that chh > cl , I(ν, κ) ≥ 0 for
∂chh t1 ∂chl + ∂κ t2 ∂κ
≤ 0,
991
and I(ν, κ) < 0 otherwise. Use the implicit function theorem and the binding enforcement
992
constraints F (chh , chl ), G(chh , chl ) as stated in Lemma 3 to receive x2 Fch + x1 Gch ∂chh t1 ∂chl t1 x2 Fchh + x1 Gchh l l + = − ∂κ t2 ∂κ Fc h G c h − F c h G c h t 2 Fc h G c h − F c h G c h h l l h h l l h x F + x G x F + x G h h h h 2 1 2 1 1 cl cl ch ch ⇔= − t1 t2 t2 Fc h G c h − Fc h G c h F c h G c h − Fc h G c h h l l h h l l h
x 2 Fc h + x 1 G c h
x 2 Fc h + x 1 G c h 1 h h h h l l h h ⇔= λpc,h Fch + λpc,l Gch − λpc,h Fch + λpc,l Gch h h l l t2 Fc h G c h − Fc h G c h Fc h G c h − F c h G c h h l l h h l l h
1 h h ⇔= x1 λpc,h − x2 λpc,l t2
993
where λhpc,h , λhpc,l > 0 per assumption, the partial derivatives Fch , Fch and Gch , Gch are given by
994
(29)-(32), and
h
x1 ≡ − 995
x2 ≡
l
h
l
ν(1 − ν) ∂F h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥0 ∂κ [κν + (1 − κ)(1 − ν)]2
ν(1 − ν) ∂G h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥ 0, ∂κ [(1 − κ)ν + κ(1 − ν)]2
996
with x1 , x2 > 0 for ν ∈ [0.5, 1). The first line uses the implicit function theorem to compute the
997
derivatives ∂chh /∂κ, ∂chl /∂κ, the second line collects terms, the third line employs the definitions 37
998
of t1 , t2 and the last line simplifies the third line by collecting terms.
999
In the following, we evaluate I(κ, ν) for ν = 0.5 and ν → 1 and show that I(κ, 0.5) ≤ 0
1000
while limν→1 I(κ, ν) ≥ 0. Existence of ν¯ then follows from continuity of I(ν, κ) for ν ∈ [0.5, 1)
1001
(see Lemma 1).
1002 1003
For ν = 0.5 , x1 = x2 = x > 0. As before, t2 > 0, chh > cl , chh ≥ chl , and the derivative of
social welfare with respect to κ is
1 x h I(0.5, κ) = [u (chh ) − u (cl )] λpc,h − λhpc,l . 4 t2 1004 1005 1006 1007 1008
Thus, whenever λhpc,h ≥ λlpc,l , I(0.5, κ) ≤ 0 which is shown in Lemma 3 part (iii). The strict
inequality λhpc,h > λhpc,l applies when chh > chl for κ ∈ (0.5, 1] and ν ∈ [0.5, 1).
For ν → 1, outside option values of high and low public signal agents converge such that
λhpc,l
= λhpc,h = λhpc , t1 = t2 = t > 0 and chh → chl = ch > cl . The derivative of social welfare with
respect to κ is
λhpc x2 1 h l x1 1 − lim I(ν, κ) = [u (c ) − u (c )] ν→1 4 t x1 h λ 1 − 2κ 1 pc x1 ≥0 ⇔ = [u (ch ) − u (cl )] 4 t (1 − κ)2 1009
where the second line results after using the definitions of x1 , x2 and the inequality in the last
1010
line follows from κ ∈ [0.5, 1] and ch > cl ; social welfare is strictly increasing in κ for κ ∈ (0.5, 1].
1011
A.6
1012
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
1013
Proof of Proposition 1
derivative of social welfare with respect to ν is
h h ∂ch ∂chl ∂Vrs 1 h ∂chh h ∂cl l = u (ch ) + u (cl ) − u (c ) + , ∂ν 4 ∂ν ∂ν ∂ν ∂ν 1014
The derivatives of consumption with respect to the signal precision follow from the implicit
1015
function theorem which is applicable because according to Lemma 3 part (ii), Fch Gch −Fch Gch >
1016
0 holds at the optimal memoryless allocation. Further, in optimum, the following condition holds
1017
(see also Lemma 3 part(ii))
h
l
l
h
t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2 1018 1019
with t1 , t2 > 0 due to chh ≥ chl > cl (see Lemma 3, part (i)). Using this expression to substitute
for u (chl ) − u (cl ) in the derivative of social welfare with respect to ν yields ∂ch t ∂ch ∂Vrs 1 h 1 l h l = u (ch ) − u (c ) + . ∂ν 4 ∂ν t2 ∂ν
38
1020
Given that u (chh ) − u (cl ) < 0, social welfare is decreasing in ν if
α 1 G c h − α 2 Fc h
l l + λhpc,h Fch ⇔ λhpc,h Fch + λhpc,l Gch h h l F c h G c h − Fc h G c h h
l
l
h
∂ch ∂ch 1 t 2 h + t1 l ≥ 0 t2 ∂ν ∂ν α 2 Fc h − α 1 G c h h h + λhpc,l Gch ≥0 l Fc h G c h − F c h G c h
⇔ (λhpc,h α1 + λhpc,l α2 )
h
l
l
h
h
l
l
h
h
l
l
h
Fch Gch − Fch Gch Fc h G c h − F c h G c h
≥0
⇔ λhpc,h α1 + λhpc,l α2 ≥ 0, 1021
with the partial derivatives Fch , Fch and Gch , Gch given by (29)-(32), t1 , t2 as defined in Lemma
1022
3, and α1 , α2 defined as α1 ≡ −
1023
α2 ≡ − 1024 1025
h
l
h
l
κ(1 − κ) ∂F h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥0 ∂ν [κν + (1 − κ)(1 − ν)]2
κ(1 − κ) ∂G h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥ 0. ∂ν [(1 − κ)ν + κ(1 − ν)]2
The second line uses the implicit function theorem to compute the derivatives ∂chh /∂ν, ∂chl /∂ν,
the third collects terms. The inequality λhpc,h α1 + λhpc,l α2 ≥ 0 is satisfied because λhpc,h , λhpc, > 0
1026
per assumption and the coefficients α1 , α2 are non-negative such that social welfare is decreasing
1027
in ν; social welfare is strictly decreasing in ν for κ ∈ [0.5, 1) because then α1 , α2 > 0.
1028
A.7
1029 1030
Proof of Proposition 2
(i) From the definition of the risk-sharing measure, it follows that risk-sharing decreases in ν whenever ch is increasing in ν. The derivative of ch with respect to ν is α 1 G c h − α 2 Fc h α 2 Fc h − α 1 G c h ∂chh ∂chl 1 l l h h + = + ∂ν ∂ν 2 Fc h G c h − F c h G c h Fc h G c h − F c h G c h l h l l h
h l ⎤h ⎡ 1 α1 Gchl − Gchh + α2 Fchh − Fchl ⎦ = ⎣ ≥0 2 F c h G c h − Fc h G c h
∂ch 1 = ∂ν 2
h
l
l
h
1031
where the inequality follows from Fch Gch − Fch Gch > 0 (see Lemma 3 part (ii)) and
1032
Gch − Gch ≥ Fch − Fch ≥ 0 (see Lemma 3 part (iii)), and risk sharing RS decreases in ν.
1033
(ii) Risk-sharing decreases (increases) in κ whenever ch is increasing (decreasing) in κ. The
1034
h
l
h
h
l
l
h
l
derivative of ch with respect to κ is ∂ch 1 = ∂κ 2
∂chh ∂chl + ∂κ ∂κ
1 = 2
x 2 Fc h + x 1 G c h l
l
Fc h G c h − Fc h G c h h
l
l
h
−
x 2 F ch + x 1 G c h h
h
Fc h G c h − Fc h G c h h
l
l
h
≡
D(κ, ν) . 2
1035
Similar to the proof of the Theorem 1, we evaluate D(κ, ν) for ν = 0.5 and ν → 1 and show
1036
that D(κ, 0.5) ≤ 0 while limν→1 D(κ, ν) ≥ 0. Existence of ν¯ then follows from continuity
1037
of D(ν, κ) for ν ∈ [0.5, 1) (see Lemma 1). For ν = 0.5, it is x1 = x2 = x > 0 and D(κ, 0.5) 39
reads
1038
D(κ, 0.5) = x
Fch + Gch − Fch − Gch l
h
=x
l
h
l
l
h
Fc h G c h − F c h G c h h
l (1 − β)(1 + β zh +z + 2
β2 1 h (1−β) 2 )(u (cl )
− u (chh ))
Fc h G c h − Fc h G c h h
l
l
≥ 0.
h
1039
The denominator is strictly positive (see the proof of Lemma 3 part (ii)) as are the first
1040
two expressions in the numerator. The inequality then follows from concavity of utility
1041
and chh ≥ chl (see Lemma 3 part (i)); for κ > 0.5, chh > chl and D(κ, 0.5) > 0.
1042
For ν → 1, we get chh → chl = ch and thus lim D(κ, ν) =
x 1 G c h − G c h − x 2 F c h − Fc h l
h
h
l
Fc h G c h − F c h G c h
ν→1
h
l
l
= (1 − β)
h
(x1 − x2 )u (ch ) ≤ 0, Fc h G c h − F c h G c h h
l
l
h
1043
The inequality follows because for ν → 1, outside option values of high and low public
1044
signals converge such that chh → chl = ch , Fch → Gch , Fch → Gch and x2 ≥ x1 for l
h
h
l
1045
κ ∈ [0.5, 1]. Thus, risk sharing increase for ν → 1; for κ > 0.5, the strict inequality
1046
applies. Existence of ν¯ follows from continuity of D(ν, κ) for ν ∈ [0.5, 1].
1047
A.8
1048
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
1049
Proof of Proposition 3
derivative of social welfare with respect to κ is
h ∂ch ∂ch ∂chl ∂Vrs 1 h ∂chh = u (ch ) + u (chl ) l − u (cl ) + . ∂κ 4 ∂κ ∂κ ∂κ ∂κ 1050
(36)
With λhpc,h > 0 and λhpc,l = 0, optimal allocations are characterized by the first order condition for
1051
optimal substitution between chh and chl and the binding enforcement constraint F (chh , chl , κ, ν).
1052
The first order condition is
1 h u (ch ) − u (chl ) + λhpc,h Fch − Fch = 0, h l 4
1053
with chh ≥ chl > cl such that Fch , Fch > 0 and Fch − Fch ≥ 0. The latter difference can be
1054
re-written such that we get
h
T ≡ 1055
l
h
l
1 h u (ch ) − u (chl ) 1 + λhpc,h 2β(1 − β)zh + β 2 + λhpc,h (1 − β)u (chh ) = 0. 4
(37)
The binding enforcement constraint is h l F (chh , chl , κ, ν) ≡ (1 − β)u(chh ) + β(1 − β)zh Vrs + β(1 − β)(1 − zh )Vrs + β 2 Vrs
− (1 − β)u(yh ) − β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] − β 2 Vout = 0. (38)
40
1056
The first order condition (37) can be also expressed as (see also Lemma 3 with λhpc,l = 0) Fch l u (chl ) − u (cl ) = u (chh ) − u (cl ) , Fc h h
1057
such that the derivative of social welfare (36), can be written as 1 1 h u (ch ) − u (cl ) 4 Fc h
∂chh ∂ch + F ch l h ∂κ l ∂κ
.
Fc h
h
(39)
1058
The derivatives of chh , chl follow again from applying the implicit function theorem in (37) and
1059
(38).
1060
T c h F κ − T κ Fc h ∂chl h h = ∂κ T c h F c h − T c h Fc h
(40)
T κ F c h − T c h Fκ ∂chh l l = . ∂κ T c h F c h − T c h Fc h
(41)
l
h
h
l
l
h
h
l
The partial derivatives are zh 1 − zh l β2 h u (c ) + u (ch ) − u (cl ) > 0 Fch = (1 − β) u (chh ) + β u (chh ) − β h 2 2 4 2 zh 1 − zh l β u (c ) + u (chl ) − u (cl ) > 0 Fch = (1 − β) β u (chl ) − β l 2 2 4 ν(1 − ν) h l − u(yl ) + Vrs ≤0 Fκ = −x1 = −β(1 − β) u(yh ) − Vrs [κν + (1 − κ)(1 − ν)]2 1 h ν(1 − ν) Tκ = u (ch ) − u (chl ) λhpc,h 2(1 − β)β ≤0 4 [κν + (1 − κ)(1 − ν)]2 1 Tch = u (chh ) 1 + λhpc,h 2β(1 − β)zh + β 2 + λhpc,h (1 − β)u (chh ) < 0 h 4 1 Tch = − u (chl ) 1 + λhpc,h 2β(1 − β)zh + β 2 > 0. l 4
1061 1062
The implicit function theorem is applicable because Tch Fch − Tch Fch = 0. Given that u (chh ) − l
h
h
l
u (cl ) < 0 and Fch > 0, the derivative (39) is negative if the following expression is positive h
T κ F ch − T c h Fκ T c h Fκ − T κ Fc h ∂chh ∂ch l l h + Fch l = Fch + Fc h h l ∂κ h T hF h − T hF h h ∂κ l T hF h − T hF h c c c c c c c c
Fc h
= −Fκ 1063
T c h F c h − T c h Fc h l
h
h
l
l
h
h
l
T c h F c h − T c h Fc h
l
h
h
l
l
h
h
l
= −Fκ ≥ 0.
The first line uses (40)-(41) to substitute for ∂chh /∂κ, ∂chl /∂κ, the second line collects terms
1064
and the inequality in the second line follows because Fκ ≤ 0, and social welfare is decreasing in
1065
κ; social welfare is strictly decreasing in κ for ν ∈ [0.5, 1) because then Fκ < 0. The proof that
1066
social welfare is decreasing in ν follows the same steps and is omitted here.
41
1067
A.9
Transition probabilities
1068
There are five transition and conditional probabilities of interest, namely:
1069
• π (s |s) = π (y , k |y, k),
1070
• π (θ |θ) = π (y , k , n |y, k, n).
1071
• π (s |s, n) = π (y , k |y, k, n),
1072
• π (y |s, n) = π (y |y, k, n)
1073
• π (y |s) = π (y |y, k) The conditional probability for income is π (y = yj , k = ym , n = yl , y = yi ) π y = yj |k = ym , n = yl , y = yi = . π (k = ym , n = yl , y = yi ) The denominator can be derived by using the following identity for N income states N π y = yz |k = ym , n = yl , y = yi = 1 z=1
which implies π (k = ym , n = yl , y = yi ) =
N π y = yz , k = ym , n = yl , y = yi . z=1
1074
The elements of the sum on the right hand side are products that depend on z, m, l, i and
1075
the precisions of signals (we treat the current income simply as a yet another signal). It follows
π y = yz , k = ym , n = yl , y = yi = piz κ
1m=z
1−κ N −1
1−1m=z
ν
1l=z
1−ν N −1
1−1l=z
,
1076
where piz is the Markov transition probability for moving from income i to income z. When
1077
the signal is wrong, m = z, we assume that all income states are equally likely. The general
1078
formula for the conditional probability of income with both signals therefore reads pij κ1m=j
π y = yj |k = ym , n = yl , y = yi = N
1−κ N −1
1m=z z=1 piz κ
1079 1080
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν N −1
1−1l=z .
(42)
The conditional probability π(y |y, k) follows from setting ν = 1/N in (42). With both signals following independent exogenous first-order Markov processes, the condi-
42
1081
tional probability for the full state is for all k , n π y = yj , k , n |k = ym , n = yl , y = yi = π(k |k = ym )π(n |n = yl )
1082
pij κ1m=j
1−κ N −1
N 1m=z z=1 piz κ
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν K−1
1−1l=z . (43)
Finally, the formula for π (y , k |k, y) naturally follows and is given for all k by π y = yj , k |k = ym , n = yl , y = yi = π(k |k = ym )
pij κ1m=j
1−κ N −1
N 1m=z z=1 piz κ
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν N −1
1−1l=z . (44)
1083
For example, with two equally likely i.i.d. income states, the conditional probability of
1084
receiving a low income in the future conditional on a low private high, a high public signal and
1085
a low income today is given according to (42) π y = yl |k = yh , n = yl , y = yl = π y = yl |k = yh , n = yl =
43
(1 − κ)ν (1 − κ)ν + (1 − ν)κ
.
Piotr Denderski, Christian A. Stoltenberg
PII: DOI: Reference:
S0022-0531(18)30488-5 https://doi.org/10.1016/j.jet.2019.104988 YJETH 104988
To appear in:
Journal of Economic Theory
Received date: Revised date: Accepted date:
20 August 2018 19 November 2019 19 December 2019
Please cite this article as: Denderski, P., Stoltenberg, C.A. Risk sharing with private and public information. J. Econ. Theory (2020), 104988, doi: https://doi.org/10.1016/j.jet.2019.104988.
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2020 Published by Elsevier.
1
Risk sharing with private and public information∗
2
Piotr Denderski† Christian A. Stoltenberg‡
3
December 30, 2019
4
Abstract
5
In this paper, we revisit the conventional view on efficient risk sharing that advance information on future shocks is detrimental to welfare. In our model, risk-averse agents receive private and public signals on future income realizations and engage in insurance contracts with limited enforceability. Consistent with the conventional view, better private and public signals are detrimental to welfare when only one type of signal is informative. Our main novel result applies when both signals are informative. In this case, we show that better public information can improve the allocation of risk when private signals are sufficiently precise. More precise public signals spread out the outside option values of highincome agents with high and low public signals and their willingness to transfer resources to low-income agents decreases. With informative private signals, however, more informative public signals increase outside option values of agents with a high signal by less than outside options of agents with a low signal decrease, facilitating more transfers. The latter effect dominates the former when private signals are sufficiently precise.
6 7 8 9 10 11 12 13 14 15 16 17 18 19
JEL classification: D80, F41, E21, D52 Keywords: Risk sharing; Social value of information; Limited commitment
∗ We are especially grateful to Martin Kaae Jensen and Dirk Krueger for their comments. We have also ´ ad Abrah´ ´ benefited greatly from the discussions with Arp´ am, Ufuk Akcigit, Eric Bartelsman, Tobias Broer, Bjoern Bruegemann, Alex Clymo, Pieter Gautier, Marek Kapiˇcka, Per Krusell, Bartosz Ma´ckowiak, Guido Menzio, Thomas Mertens, Ctirad Slav´ık, Thijs van Rens, Gianluca Violante, Chris Wallace, Mirko Wiederholt, Sweder van Wijnbergen, conference and seminar participants at various places. Previous versions circulated under the title On Positive Value of Information in Risk Sharing. † University of Leicester, University Road, LE1 7RH, Leicester, the United Kingdom and Institute of Eco´ nomics, Polish Academy of Sciences, Nowy Swiat 73, 00-330, Warsaw, Poland; email: [email protected]. uk, tel: + 44 (0) 116 252 3703 ‡ MInt, Department of Economics, University of Amsterdam and Tinbergen Institute, Postbus 15867, 1001 NJ Amsterdam, the Netherlands, email: [email protected], tel: +31 20 525 3913.
20
1
Introduction
21
Should benevolent governments and international institutions strive to provide better public
22
forecasts on real GDP to improve consumption smoothing against country-specific shocks? The
23
theoretical literature (Hirshleifer, 1971, Schlee, 2001) offers sharp predictions for welfare effects
24
of issuing such public forecasts. Better information on future realizations of idiosyncratic shocks
25
harms risk sharing and welfare by limiting insurance possibilities from an ex-ante perspective.
26
Similar to Hirshleifer, we consider an environment in which better private and public infor-
27
mation on future idiosyncratic shocks are in isolation detrimental to welfare. However, as our
28
main theoretical result, we show that if households have private information at their disposal
29
to forecast future shock realizations, releasing better publicly available forecasts can improve
30
welfare and risk sharing. In particular, this happens whenever private information is sufficiently
31
precise.
32
As in Krueger and Perri (2011), we consider a model with risk-averse agents that seek
33
insurance against idiosyncratic fluctuations of their income. While there is no restriction on the
34
type of insurance contracts that can be traded, the contracts suffer from limited commitment
35
because every period agents have the option to default to autarky. The option to default is
36
in particular tempting for agents whose current income is high and these agents face a tension
37
between the higher current consumption in autarky and the future benefits of the insurance
38
promised in the contracts. Each period – and this is the new element here – agents receive a
39
private and a public signal on their future income shock realization. Thus, the agents possess
40
fore-knowledge of future shock realizations. By changing the expected value of autarky, the
41
signals affect the trade-off between higher current consumption in autarky and better insurance
42
proffered in the contracts.
43
As our main novel theoretical contribution, we formally show that better public information
44
can be beneficial for welfare and risk sharing when private information is sufficiently precise
45
and enforcement constraints matter. The positive effect of public information on risk sharing
46
is intriguing but can be understood as a trade-off between costs and benefits from releasing
47
more precise public signals. With enforcement constraints, the degree of risk sharing depends
48
on the willingness of high-income agents to transfer resources to low-income agents. First, more
49
precise public signals spread out the outside option values of agents with a high and a low public
50
signal, that is, high signal agents become less but low-signal agents more willing to share their
51
good fortune. With uninformative private signals, high-income agents with a high public signal
52
require more resources to be indifferent with autarky than low-signal agents are willing to give
53
up, resulting in less transfers to low-income agents, and better public signals are detrimental
54
to welfare. With informative private signals, however, more informative public signals increase
55
the willingness to transfer resources of low public signal agents by more than the high public
56
signal agents’ willingness to transfer resources decreases, facilitating better risk sharing. When
57
private information is sufficiently precise, risk sharing improves with better public information
58
and agents prefer informative public signals ex-ante.
59
We then quantitatively test the robustness of our main theoretical result lifting simplifying
60
assumptions necessary to deliver analytical tractability. We discipline the model with data on
61
international risk-sharing. The risk-averse agents here are representative households of small 2
62
countries that seek insurance against country-specific shocks to real GDP. The public signals
63
capture information on a country’s future GDP that is publicly available to other countries, for
64
example, the real GDP growth forecasts provided in the IMF World Economic Outlook (WEO).
65
Private signals are exclusively observed by a particular country and can capture how transparent
66
a country is. A high precision of private signals indicates secrecy, i.e., useful information on
67
future real GDP exists but that information is not publicly shared.
68
The model implies that countries with a low degree of insurance are countries in which the
69
private information friction is the most severe. In these countries, private signals are sufficiently
70
precise and public data quality improvements are beneficial for welfare; countries with either
71
low or high quality of public information are countries in which private signals are not precise
72
enough and improvements in public information quality have negative but negligible effects on
73
welfare. Taking together, these quantitative results confirm the logic of our main theoretical
74
result on the positive effect of better public information.
75
1.1
76
The papers that are most closely related to ours are Hirshleifer (1971), Schlee (2001), Broer,
77
Kapicka, and Klein (2017) and Hassan and Mertens (2015).
Related literature
78
Hirshleifer (1971) is among the first to point out that perfect information – public or private
79
– makes risk-averse agents ex-ante worse off if such information leads to an evaporation of risks
80
that otherwise could have been shared in a competitive equilibrium with full insurance. Schlee
81
(2001) provides general conditions under which better public information about idiosyncratic
82
risk is undesirable in a competitive equilibrium. There are two main differences to these papers
83
in our work. First, we consider optimal insurance contracts with limited enforcement that hinder
84
agents from achieving full insurance. Second, we allow for the interplay of both private and
85
public information which can overturn Hirshleifer (1971)’s result on the negative effect of public
86
information. Further, our main theoretical result on the positive effect of public information
87
only applies in the empirically realistic case of partial insurance and when private information
88
is important.
89
Broer et al. (2017) study the effects of unobservable income shocks on consumption risk
90
sharing with limited commitment. Agents can be induced to report their income truthfully,
91
and private information reports become contractible but do not increase the state space. The
92
key difference to our paper is that we consider the realistic case with both publicly observable
93
elements of the state vector – public signals and income – as well as unobservable private signals
94
which allows us to analyze how the interaction between private and public information affects
95
welfare and risk sharing.
96
Our information structure with signals about future endowments is similar to Hassan and
97
Mertens (2015) who examine the business cycle implications of private information on future
98
productivity shocks in a closed economy. Our focus is to understand the welfare implications
99
of private and public signals on future income realizations with limited commitment.
100
The remainder of the paper is organized as follows. Section 2 presents a general risk-sharing
101
model with private and public signals. In Section 3, we analytically characterize how the
102
precision of private and public signals affect social welfare and risk sharing. Section 4 provides 3
103
a quantitative application to international risk sharing. The last section concludes.
104
2
105
In this section, we present a tractable and yet general model in which risk-averse households use
106
self-enforceable insurance contracts to hedge their consumption against undesirable fluctuations
107
resulting from idiosyncratic income shocks. The insurance contracts are required to be self-
108
enforceable because agents have the possibility to default on their obligations and live in autarky.
109
Agents receive private and public signals on their future income realizations that affect how they
110
value the insurance provided in the contracts relative to their outside option of living in autarky.
111
The model is designed to provide the simplest possible framework to analytically show our main
112
result on the positive effect of better public information in Section 3. ]
113
Preferences
114
indexed i. The time is discrete and indexed by t from zero onward. Households have identical
115
preferences over consumption streams
A risk-sharing model with private and public information
Consider an economy that is populated by a continuum of risk-averse agents
U0 ({ct }∞ t=0 ) = (1 − β) E0
∞
β t u(ct ),
(1)
t=0
116
where the instantaneous utility function u : R+ → R is strictly increasing, strictly concave and
117
satisfies the Inada conditions.
118
The income stream of agent i, {yti }∞ t=0 , is governed by a stochastic process with two possible
119
time-invariant realizations yti ∈ Y ≡ {yl , yh }, with 0 < yl < yh that are equally likely with
120
realizations that are independent across time and agents such that average income is given
121
by y¯ = (yl + yh )/2. The history of income realizations (y0 , ...yt ) is denoted by y t ∈ Y t+1 .
122
Current-period income is publicly observable.
123
Information
124
and a private signal nit ∈ Y (observed only by agent i) that inform about her income realization
Each period, agent i receives a public signal kti ∈ Y (observed by all agents)
125
y in the next period. As income, both signals have two realizations that are equally likely with
126
realizations that are independent across time and agents. The signals can indicate either a high
127
income (“good” or “high” signals) or a low income (“bad” or “low” signals) in the future. The
128
precision of the public signal is denoted by the conditional probability κ = π(y = yj |k = yj ),
129
while the precision of the private signal is given by ν = π(y = yj |n = yj ), κ, ν ∈ [1/2, 1].
130
Uninformative signals are characterized by precision 1/2 while perfectly informative signals
131
exhibit precision of one. The publicly observable part of the state vector is st = (yt , kt ), st ∈ S,
132
where S = Y × Y . The whole state vector is θt = (st , nt ), with θt ∈ Θ = S × Y . The public
133
history of the state is st = (s0 , ...st ) ∈ S t+1 ; the history of the public and the private state is
134
denoted by θt = (θ0 , ..., θt ) ∈ Θt+1 .
135
Memoryless allocations
136
mation affect social welfare and risk sharing, we follow Coate and Ravallion (1993) and consider
To analytically characterize how better public and private infor-
4
Income
Signals
?
t
Consumption
?
yti
?
kti , nit
cit
-
t+1
Figure 1: Timing of events with public and private information 137
memoryless allocations that depend solely on the current realization of the state.1
138
With memoryless allocations, continuation values depend only on future but not on current
139
realizations of income and signals. It can be shown that this implies that memoryless allocations
140
are only contingent on the publicly observable part of the current state but not on private signal
141
reports.2 Thus, a memoryless allocation is a tuple of consumption levels c = {cji }i,j∈{l,h} , with
142
cji as consumption with current income y = yj and current public signal k = yi .
143
Insurance arrangements
144
agents can engage in insurance contracts that suffer from limited enforceability because agents
145
have the option to default to autarky. As illustrated in Figure 1, after the realization of current
146
income and both two signals, agents can decide to participate in insurance contracts implement-
147
ing the allocation c or to deviate into autarky forever consuming only their income. Agents have
148
no incentive to default in any of the eight possible current-period states θ = (y, k, n) when the
149
allocation satisfies enforcement or participation constraints. These constraints state that the
150
lifetime utility of an agent in each state θ exceeds the value of his outside option. For example,
151
the enforcement constraints of high-income agents with high public and high private signals are u(chh ) + βzh
To protect their consumption from undesirable fluctuations,
u(chh ) + u(chl ) u(clh ) + u(cll ) β 2 u(chh ) + u(chl ) + u(clh ) + u(cll ) +β(1 − zh ) + ≥ 2 2 4 1−β h Vrs
Vrs
l Vrs
u(yh ) + β [zh u(yh ) + (1 − zh )u(yl )] +
β2
u(yh ) + u(yl ) h ≡ Vout,hh , (2) 1−β 2 Vout
152
with zh as the probability of a high income in the next period conditional on a high public and
153
a high private signal in the current period given by the following expression zh ≡
κν , κν + (1 − κ)(1 − ν)
0.5 ≤ zh ≤ 1.
(3)
154
With the insurance contracts, agents have the possibility to engage in risk sharing with
155
each other. For this reason, we denote the continuation values of high-and low-income agents,
156
h , V l as well as the unconditional value V Vrs rs with the subscript ‘rs’ for risk sharing. We refer rs
157
to the left-hand side of the enforcement constraint as the value of the insurance contract and
158
the right-hand side as the value of the outside option of that agent. Using these definitions, the 1 Our environment is an example of a dynamic contracting problem with private information considered as for example in Fernandes and Phelan (2000). The solution to this type of problems can feature dependence of allocation on the full history of the public state and private signal reports. All our theoretical results are derived for memoryless allocations. In our quantitative application in Section 4, we test and confirm the robustness of our theoretical results. 2 To ease the exposition in the main text, we derive this result in an accompanying Online Appendix.
5
159
enforcement constraints of high-income agents with a low public and a high private signal can
160
be compactly written as β2 h l Vrs ≥ u(chl ) + β zl Vrs + (1 − zl )Vrs + 1−β u(yh ) + β [zl u(yh ) + (1 − zl )u(yl )] +
β2 h Vout ≡ Vout,lh (4) 1−β
161
with zl as the probability of a high income in the next period conditional on a low public signal
162
and a high private signal in the current period zl ≡
(1 − κ)ν , (1 − κ)ν + κ(1 − ν)
0 ≤ zl ≤ 1,
(5)
163
and zl ≤ zh . While the consumption allocation is not contingent on private signal realizations,
164
private information affects consumption allocations indirectly through the enforcement con-
165
straints. In this setting, we analyze optimal memoryless allocation that are defined as follows.
166
Definition 1 (Optimal memoryless allocation) An optimal memoryless allocation {cji }
167
maximizes agents’ expected utility before any risk has been resolved (social welfare) Vrs =
168
u(chh ) + u(chl ) + u(clh ) + u(cll ) 4
(6)
and is constrained feasible, i.e., in each period t, the allocation satisfies resource feasibility chh + chl + clh + cll yh + y l = 4 2
(7)
169
and is for all periods t and all states θ consistent with agents’ rational incentives to participate
170
in the risk-sharing arrangement as summarized by the enforcement constraints.
171
One can think of the optimal memoryless allocation as the outcome of a contracting problem
172
to design an optimal risk-sharing arrangement that respects rational incentives constraints and
173
resource feasibility. For this reason, we use the terms optimal memoryless allocation and optimal
174
memoryless arrangement interchangeably. By construction, autarky (all agents consume their
175
income in all periods and all states) is constrained feasible.
176
The focus on memoryless allocations allows us to analytically derive sharp predictions on
177
the effect of fore-knowledge on risk sharing and welfare. In the quantitative part in Section 4,
178
we document that the simplifying assumption of memoryless allocations is not pivotal for our
179
main result on the positive value of public information. Our main result also applies when we
180
consider allocations that can depend on the history of the public state st or on the whole state
181
θt .
182
3
183
In this section, we analytically characterize optimal memoryless allocations with private and
184
public signals. Before we get to our main result on the positive value of public information,
Analysis
6
185
we provide intermediate results that are essential to understand it. Eventually, we explain the
186
necessary conditions for the positive welfare effect of better public information in risk sharing.
187
3.1
188
To begin with we show that the optimal arrangement exists and is unique and that social
189
welfare is continuous in the precision of both signals. We also show that while the consumption
190
of agents with a low income weakly exceeds their income, high-income agents’ consumption is
191
weakly lower than under autarky. These general properties are summarized in the following
192
Lemma.
193
Lemma 1 (General properties) Consider optimal memoryless allocations.
General properties of optimal memoryless allocations
194
(i) The optimal memoryless allocation exists and is unique. The optimal memoryless allo-
195
cation and social welfare are continuous functions in the precision of public and private
196
signal precision.
197
(ii) The optimal memoryless allocation satisfies: cli ≥ yl and chi ≤ yh for all public signals i.
198
The proof is provided in Appendix A.1. Existence and uniqueness of the optimal memoryless
199
allocation follow from Weierstrass’ theorem. Continuity is a direct implication of the theorem
200
of the maximum. The second part of the Lemma follows from a contradiction argument. Any
201
allocation that does not satisfy cli ≥ yl and chi ≤ yh violates enforcement and resource feasibility
202
constraints and therefore does not constitute an optimal memoryless allocation.
203
The first best allocation is perfect risk sharing (all agents consume y¯ in all states) which
204
can be constrained feasible when this allocation is consistent with agents’ rational incentives
205
to participate in the insurance arrangements. Alternatively, enforcement constraints can be
206
consistent with either only no insurance against income risk (autarky) or partial risk sharing.
207
The effect of signal precision in the case of perfect risk sharing or autarky are analyzed in
208
Appendix A.2. In the main text, we focus on partial risk sharing, that is, perfect risk sharing
209
is not constrained feasible and autarky is not the only constrained feasible allocation.3
210
3.2
211
We continue our analysis with two Lemmas that summarize properties of optimal memoryless
212
allocations that are essential to derive Theorem 1, our main result. In particular, we show in
213
Lemma 2 that only enforcement constraints of high-income agents for a high private signal can
214
be binding and that consumption of low-income agents is equalized across all signal realizations.
215
In Lemma 3, we characterize the optimal consumption allocation in case the enforcement con-
216
straints of high-income agents are binding for both public signals which is the case considered
217
in our main theoretical result.
Properties of optimal memoryless allocations with partial risk sharing
218
When perfect risk sharing cannot be supported, some enforcement constraints are binding at
219
the optimal memoryless allocation and risk sharing is either partial or absent (by “binding” we
220
mean that the multiplier on the constraint is positive). In the following Lemma, we summarize
221
essential properties of optimal allocations with partial risk sharing. 3
The relevant parameter space for partial risk sharing follows from Propositions 4 and 5 in Appendix A.2.
7
222
Lemma 2 (Allocation properties with partial risk sharing) Consider the optimal mem-
223
oryless arrangement. Let some enforcement constraints of high-income agents be binding. As-
224
sume that autarky is not the only memoryless arrangement.
225
226
h < u(y ) and V l > u(y ). (i) Then chi < yh for at least one public signal signal i, Vrs h l rs
(ii) Consider informative private signals, ν ∈ (0.5, 1]. Then only enforcement constraints of agents with a high private signal can be binding.
227
228
(iii) Only enforcement constraints of high-income agents can be binding.
229
(iv) Consumption in the optimal memoryless arrangement satisfies: clh = cll = cl .
230
The proof is provided in Appendix A.3. The main message from Lemma 2 is that only
231
enforcement constraints of high-income agents for a high private signal can be binding. For
232
each income and public signal realization, allocations must be consistent with the participation
233
incentives of households with the highest outside option value among all private signal real-
234
izations, i.e., the outside option value for high private signals. The optimal allocation further
235
prescribes transfers to agents with a low income realization which is why only enforcement
236
constraints of high-income agents can be binding. Thus, enforcement constraints of low-income
237
agents are slack and it is optimal to equalize their consumption.
238
The last part of Lemma 2 implies that the optimal consumption allocation with informative
public signals comprises three elements, chh , chl , cl . Thereby, the optimal consumption alloca-
239 240
tion is the non-autarkic allocation that is pinned down by the binding enforcement constraints
241
and the resource constraint. A natural question is whether one can solve for optimal memoryless
242
allocations – utility or consumption allocations – in closed form making a particular paramet-
243
ric assumption for the utility function. We find that the resulting solutions that we study in
244
an accompanying Online Appendix are fairly complicated and convey little economic intuition
245
in understanding how changes in signal precision affect social welfare. For these reasons, we
246
continue our analysis with the general utility function specified in (1) and resort to the implicit
247
function theorem to derive the welfare effects of increases in public and private signal precision.
248
The results (ii)-(iii) of Lemma 2 point to the particular enforcement constraints that can be
249
binding in the absence of perfect risk sharing; either the enforcement constraints of high-income
250
agents with both a high and a low public signal or only the constraints of agents with a high
251
public signal bind.4 Our main theoretical result on the positive value of public information
252
applies for the first pattern of binding enforcement constraints as summarized in the following
253
assumption.
254
Assumption 1 The optimal memoryless allocation is characterized by partial risk sharing and
255
at the optimal allocation the enforcement constraints of high-income agents with a high private
256
signal are binding for both high and low public signals.
257
In Section 3.4, we show that Assumption 1 is a necessary condition for our main result.5 4
Formal arguments for this claim are provided in Lemma 3 part (iii). In our application to international risk sharing in Section 4, we find this assumption to be satisfied in all the numerical experiments we entertain. 5
8
258
Given Assumption 1 and after using the insights from Lemma 2 (iv) as well as the resource
259
constraint (7) to substitute for consumption of low-income agents, the social planner problem
260
simplifies to choosing {chh , chl } to maximize the following Lagrangian Vrs =
1 h u(ch ) + u(chl ) + 2u 2¯ y − (chl + chh )/2 4 + λhpc,h F (chh , chl , κ, ν) + λhpc,l G(chh , chl , κ, ν),
261
with the binding enforcement constraints of high-income agents with a high public signal h h h l F (chh , chl , κ, ν) ≡ (1−β)u(chh )+β(1−β)zh Vrs (ch , cl )+β(1−β)(1−zh )Vrs (chh , chl )+β 2 Vrs (chh , chl )
− (1 − β)u(yh ) − β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] − β 2 Vout = 0, 262
and with a low public signal h h h l G(chh , chl , κ, ν) ≡ (1 − β)u(chl ) + β(1 − β)zl Vrs (ch , cl ) + β(1 − β)(1 − zl )Vrs (chh , chl ) + β 2 Vrs (chh , chl )
− (1 − β)u(yh ) − β(1 − β) [zl u(yh ) + (1 − zl )u(yl )] − β 2 Vout = 0, 263 264
and λhpc,h , λhpc,l > 0 as the multipliers on the constraints F (chh , chl , κ, ν), G(chh , chl , κ, ν), respec-
tively. The first order conditions for chh and chl are
1 h u (ch ) − u (cl ) + λhpc,h Fch + λhpc,l Gch = 0 h h 4 1 h l h h u (cl ) − u (c ) + λpc,h Fch + λpc,l Gch = 0. l l 4
(8) (9)
265
In the following Lemma, we use the two binding enforcement constraints and the first order
266
conditions (8)-(9) to derive properties of optimal memoryless consumption allocations that we
267
eventually use to study the welfare effects of changes in signal precision.
268
Lemma 3 (Binding enforcement constraints of high-income agents) Let the optimal
269
memoryless allocation satisfy Assumption 1. Then the following holds:
270 271
(i) Consumption in the optimal memoryless arrangement satisfies: chh ≥ chl > cl and ch > y¯ > cl , with average consumption of high-income agents defined as ch =
272
1 h ch + chl . 2
(ii) At the optimal memoryless arrangement the following relationship applies t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2
273
with t2 ≡ λhpc,h Fch + λhpc,l Gch > 0 h
274
(10)
h
t1 ≡ λhpc,h Fch + λhpc,l Gch > 0. l
l
(iii) At the optimal memoryless allocation, the two multipliers are characterized by λhpc,h ≥ λhpc,l . 9
275
The proof is provided in Appendix A.4. The outside option value of agents with a high
276
public signal exceeds the outside option value of agents with a low public signal. With binding
277
enforcement constraints for these agents, this implies that consumption of agents with a high
278
public signal is also higher than consumption of agents with a low public signal. Because of their
279
binding enforcement constraints, consumption of high-income agents with a low public signal
280
exceeds the consumption of low-income agents. From resource feasibility, it follows then that
281
high-income agents consume more while low-income agents consume less than average income.
282
Part (iii) implies that at the optimal allocation either the enforcements constraints are binding
283
for both public signal realizations (as in Assumption 1) or only the enforcement constraints of
284
high-income agents with a high public signal.
285
3.3
286
Our theoretical results on the social value of public and private information operate through
287
the binding enforcement constraints of high-income agents with high and low public signals. To
288
understand the intuition behind the welfare effects of changes in signal precision that we present
289
in Theorem 1 and Proposition 1, it is useful to decompose the total effect of signal precision
290
on the enforcement constraints into a direct and an indirect effect. To fix ideas, consider an
291
increase in ν (an increase in κ follows analogous arguments).
292 293
Social value of public and private information
First, the increase in ν directly affects the enforcement constraints (2) and (4) by changing zh , zl for a given consumption allocation as follows for each public signal i ∈ {l, h} ∂z
i h l β Vrs − Vrs ∂ν
Direct effect of ν =
Effect on value of insurance
∂zi − β [u(yh ) − u(yl )] ∂ν
(11)
Effect on outside option value
294
The first term is the direct change in the value of the insurance contract and the second term
295
is the direct (and total) change in the outside option value. With partial risk sharing, the average
296
h < u(y ), period utility across public signals of high-income agents is lower than in autarky, Vrs h
297
l > u(y ). Thus, in while the average utility of low-income agents is higher than in autarky, Vrs l
298
general, the change in signal precision has a stronger direct effect on the outside option than
299
on the insurance value. Here, the total direct effect is negative for both public signals because
300
∂zi /∂ν ≥ 0.
301
Second, the increase in signal precision affects the value of the insurance contract indirectly
302
by changing the consumption allocation itself. The indirect effect must offset the direct effect
303
because enforcement constraints are binding. For high-income agents with a high public signal,
304
this requires ∂z ∂chh ∂chl 1 h h l Fch + Fc h + β Vrs − Vrs − u(yh ) + u(yl ) = 0. h ∂ν l ∂ν 1−β ∂ν
(12)
Direct effect of ν
Indirect effect of ν
305
In this equation, the indirect effect is dominated by changes in chh as the consumption state
306
that enters the enforcement constraint with the largest weight. At the optimal allocation, the
307
possibilities to transfer resources to low-income agents must have been exhausted. This implies
10
308
that the values of the insurance contracts for both types of high-income agents at the optimal
309
memoryless allocation are increasing in their respective consumption levels. In other words, the
310
left-hand sides of (2) and (4) are increasing in chh and chl such that Fch , Gch > 0.6 This implies h
l
311
that the indirect effect results in changes of chh and chl that mimic the sign of the corresponding
312
changes in the outside option. Here, both outside option values increase and the direct effect
313
in (12) is negative. Consequently, both chh and chl tend to increase, compensating for the direct
314
effect.
315
Equipped with Lemmas 2 and 3, we present our main theoretical result that improvements
316
in public information can either benefit or worsen welfare depending on the precision of private
317
signals.
318
Theorem 1 (Social Value of Public Information) Let the optimal memoryless allocation
319
satisfy Assumption 1. Then there exists a precision of the private signal ν¯, such that for ν ≥ ν¯
320
and ν ∈ (0.5, 1], social welfare is increasing in the precision of the public signal κ over κ ∈
321
[0.5, 1].
322
The proof is provided in Appendix A.5. The logic of the proof is as follows. We show that
323
for an uninformative private signal welfare decreases in the precision of public signals while
324
for the limiting case of a perfectly informative private signal welfare increases in public signal
325
precision. Continuity then implies that there exists a level of private information precision for
326
which the welfare effect of better public information is positive.
327
The main theorem is illustrated in Figure 2 that depicts social welfare as a function of public
328
information precision for different precisions of private information. When private information is
329
uninformative or not precise enough (see the upper two functions), the negative effect dominates
330
and social welfare is decreasing in public signal precision. For ν = 0.70, the negative and the
331
positive effect neutralize each other. However, when private signals are sufficiently precise the
332
latter positive effect dominates the negative effect, and social welfare increases in public signal
333
precision (see the lower two functions).
334
There is a negative and a positive welfare effect of releasing better public information when
335
private signals are informative. First, more precise public signals spread out the outside option
336
values of agents with a high and a low public signal which causes their consumption to spread
337
out too. The riskier consumption allocation of high-income agents results in lower willingness
338
to transfer resources to low-income agents and thus lower welfare. Second – and this is the new
339
effect here that only occurs for informative private signals – more informative public signals
340
increase outside option values of agents with a high public signal by less than outside options
341
of agents with a low public signal decrease, facilitating more transfers and higher welfare. The
342
higher is the precision of private signals, the stronger is the second positive effect of more precise
343
public signals. 6 To see this, consider an optimal memoryless allocation and suppose alternatively that F (chh , chl , κ, ν) was decreasing in chh , then social welfare could be increased by transferring resources from high-income agents with a high public signal to low-income agents. By construction, these transfers would be consistent with enforcement constraints and with resource feasibility. Thus, the initial allocation could have not been an optimal memoryless allocation which yields a contradiction, and the values of both insurance contracts are increasing in the corresponding consumption.
11
Welfare in certainty equivalent consumption
1.035
1.03
1.025
1.02
1.015
1.01 ν ν ν ν ν
1.005
1 0.5
0.55
0.6
0.65
0.7
0.75
Precision of public information, κ
0.8
= 0.5 = 0.6 = 0.7 = 0.8 = 0.9
0.85
0.9
Figure 2: Welfare effects of public information for different precisions of private information. 344
To gain intuition, consider an increase in the precision of the public signal. Public sig-
345
nal precision affects outside options in (2) and (4) through its effect on the signal-conditional
346
probabilities of a high income in the next period zh and zl h ∂Vout,hh
∂κ 347
= β [u(yh ) − u(yl )]
∂zh ν(1 − ν) = β [u(yh ) − u(yl )] ≥0 ∂κ [κν + (1 − ν)(1 − κ)]2
= β [u(yh ) − u(yl )]
∂zl ν(ν − 1) = β [u(yh ) − u(yl )] ≤ 0. ∂κ [(1 − κ)ν + (1 − ν)κ]2
and h ∂Vout,lh
∂κ 348
Regardless how precise are the private signals, more precise public signals result in an
349
increase in the value of the outside option for high-income agents with a good public signal and
350
a decrease in the outside option for agents with a bad public signal as illustrated in Figure 3.
351
As argued above, the increase in κ directly affects the value of insurance in the same direction
352
as the outside option but to a smaller extent. The direct effect as the difference between the
353
changes in the insurance and outside option value is negative for agents with a high public signal
354
and positive for agents with a low public signal. The offsetting indirect effect therefore tends to
355
increase chh and to decrease chl . Comparing the derivatives of the outside options with respect to
356
h h κ reveals that the marginal effect (in absolute terms) on Vout,lh is larger than on Vout,hh because
∂zl ∂zh ∂κ ≥ ∂κ . 357
The inequality implies that the direct and the compensating indirect effect of the increase in
358
κ on enforcement constraints must be stronger – in absolute terms – for agents with a low public
12
Value of the outside option in utility
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05 0.5
h Vout,hh , ν = 0.50 h Vout,lh , ν = 0.50 h Vout,hh , ν = 0.90 h Vout,lh , ν = 0.90
0.55
0.6
0.65
0.7
0.75
Precision of public information, κ
0.8
0.85
0.9
Figure 3: Outside option values of high-income agents as functions of public information precision for different precisions of private information. 359
signal than for agents with a high public signal. This asymmetric response of the two outside
360
options is where Assumption 1 has its bite. If only the enforcement constraints of high-income
361
agents with high public signals were binding, then the stronger response of low-public signal
362
agents would not impact on the allocation.
363
When private signals are uninformative (ν = 0.50), the changes in the value of the outside
364
h option of high-income agents with a good public signal (Vout,hh ) and with a bad public signal
365
h (Vout,lh ) are symmetric, i.e., |∂zl /∂κ| = ∂zh /∂κ as illustrated in the lower part of Figure 3
366
with ν = 0.50. The consumption of high-income agents with a good public signal exceeds the
367
consumption of agents with a bad public signal. For this reason, those agents have a lower
368
marginal utility of consumption and require more resources to be indifferent to autarky than
369
the high-income agents with a bad public signal are willing to give up. Hence, the increase in
370
chh is larger than the decrease in chl , and average consumption of high-income agents increases
371
which by resource feasibility reduces consumption of low-income agents. As a consequence, the
372
allocation becomes riskier ex-ante, social welfare decreases and only the first (negative) effect
373
of better public information is present.
374
When private signals are informative, better public signals increase the outside option values
375
of agents with a high public signal by less than outside options of agents with a low public signal
376
decrease, i.e., |∂zl /∂κ| > ∂zh /∂κ as illustrated in Figure 3 for ν = 0.90. Thus, more precise
377
public signals affect outside option values in an asymmetric way; the asymmetry is the stronger
378
the more precise are the private signals. When private signals are sufficiently precise, the drop
379
in chl dominates the jump in chh , and average consumption of high-income agents decreases. By
380
resource feasibility, the latter leaves room for more transfers to low-income agents, social welfare
381
improves and the second positive effect of better public signals prevails.
13
382
Increases in private signal precision on the other hand are unambiguously welfare reducing
383
independent from the precision of public signals as summarized in the following proposition.
384
Proposition 1 (Social Value of Private Information) Let the optimal memoryless alloca-
385
tion satisfy Assumption 1. Then social welfare is decreasing in the precision of the private signal
386
over ν ∈ [0.5, 1] for κ ∈ [0.5, 1].
387
The proof is provided in Appendix A.6. The direct effect of an increase in ν on the outside
388
option exceeds the direct effect on the value of the insurance contracts for high-income agents
389
for both public signal realizations. To render enforcement constraint satisfied, the indirect effect
390
of ν must therefore yield an improvement in consumption for both types of high-income agents.
391
By resource feasibility, consumption of low-income agents decreases which is why overall by
392
concavity of utility social welfare worsens.
393 394
We can also characterize how the precision of private and public signals affect risk sharing. Let risk sharing be measured by 2 2 ch − y¯ + cl − y¯
RS = 1 −
(yh − y¯)2 + (yl − y¯)2
.
(13)
395
Under perfect risk sharing, the measure equals unity while in the absence of risk sharing it
396
equals zero. The following proposition summarizes the effects of increases in private or public
397
signal precision on risk sharing.
398
Proposition 2 (Risk Sharing and Information) Let the optimal memoryless allocation
399
satisfy Assumption 1. Let risk sharing be measured by RS.
400
(i) Risk sharing is decreasing in the precision of the private signal over ν ∈ [0.5, 1] for κ ∈ [0.5, 1].
401
402
(ii) Then there exists a precision of the private signal ν¯, such that for ν ≥ ν¯ and ν ∈ (0.5, 1], risk sharing is increasing in the precision of the public signal over κ ∈ [0.5, 1].
403
404
The proof is provided in Appendix A.7. The logic for the results in the Proposition follows
405
the same logic as in Proposition 1 and Theorem 1. As shown in Lemma 3, consumption of
406
high-income agents is on average above mean income while low-income agents consumption
407
is less than mean income. More precise private signals increase consumption of high-income
408
agents on average while consumption of low-income agents decreases by resource feasibility and
409
risk sharing worsens. Improvements in public signal precision decrease average consumption
410
of high-income agent when private signals are sufficiently precise and increase their average
411
consumption otherwise. Thus, risk sharing improves when private signals are sufficiently precise
412
and risk sharing worsens otherwise.
413
3.4
414
The main message from Theorem 1 is that social welfare can increase in public signal precision
415
when private signals are sufficiently informative and if Assumption 1 is satisfied. In particular,
Discussion of Theorem 1
14
416
risk sharing in the optimal memoryless allocation can neither be absent nor perfect and the
417
enforcement constraints of high-income agents with a high and a low public signal must be both
418
binding in optimum. To better understand our main result, we explain first why Assumption 1
419
is a necessary condition for it. Furthermore, we also discuss changes in preliminary assumptions
420
such as exogenous restrictions on the set of feasible contracts that prevent a positive value of
421
public information.
422
Autarky and perfect risk sharing
423
be welfare improving when risk sharing is partial but not when risk sharing is absent (autarky)
424
or full (perfect risk sharing). In the latter cases, the consumption allocation is independent
425
from public (and private) signal precision. Correspondingly, changes in public signal precision
426
do not affect social welfare. For both autarky and perfect risk sharing, changes in public signal
427
precision can only affect welfare when after the increase in signal precision, risk sharing becomes
428
partial. With partial risk sharing, a further increase in public signal precision may or may not
429
improve welfare depending on the precision of private signals and on the pattern of binding
430
enforcement constraints. The relevance of the pattern of binding enforcement constraints is
431
discussed in the following paragraph.
432
Pattern of binding enforcement constraints
433
forcement constraints or only the enforcement constraints of high-income agents with a high
434
public signal can be binding.7 In the first case, perfect risk sharing violates all enforcement
435
constraints of high-income agents, while in the latter case perfect risk sharing violates only the
436
enforcement constraints of high-income agents with a high public signal. The positive effect of
437
better public information only applies when the enforcement constraints for both public sig-
438
nals are binding. Let Assumption 1 be not satisfied but assume alternatively that only the
439
enforcement constraints of high-income agents with a high public signal are binding. Then in-
440
creases in public signal precision are welfare reducing independent from private signal precision
441
as summarized in the following Proposition.
442
Proposition 3 The optimal memoryless allocation is characterized by partial risk sharing and
443
at the optimal allocation only the enforcement constraints of high-income agents with a high
444
public and a high private signal are binding. Then social welfare is decreasing in the precision
445
of private and public signals.
As stated in Theorem 1, better public signals can only
With partial risk sharing, either both en-
446
The proof is provided in Appendix A.8. The rationale behind this result is as follows.
447
Improvements in public signal precision still increase the willingness of agents with a low public
448
signal to engage in risk sharing but this increase does not impact the allocation because their
449
enforcement constraints are not binding. The only binding enforcement constraints are the ones
450
of high-income agents with a high public signal. Thus, only the increase in their outside option
451
and their resulting lower willingness to transfer resources to low-income agents is reflected in
452
the optimal allocation, and welfare decreases in the precision of public signals.8 7 In Lemma 3 part (iii), we show that λhpc,h ≥ λhpc,l such that enforcement constraints are binding for either both public signals or only for high public signals. 8 Following similar arguments, the welfare effects of more precise private signals are also detrimental.
15
As an implication of the discussion in the last two paragraphs, it follows that Assumption
453 454
1 is a necessary condition for the positive effect of better public information.
455
Different insurance arrangements
456
surance contracts that are contingent on public signals and income considered here. Suppose
457
that the environment is the same and risk sharing is partial. However, assume that consumption
458
allocations can only be contingent on income, i.e., cji = cj for income j and public signal i, but
459
have to satisfy the enforcement constraints (2) and (4). Also in this case, the positive effect of
460
better public information does not arise. Consumption offered to agents with different public
461
signals is equalized. Nevertheless consumption offered must be consistent with the tightest of
462
the enforcement constraints of high-income agents. These are the enforcement constraints of
463
agents with a high public signal, and these constraints will be binding in optimum. When
464
public signals become more precise – in direct analogy to Proposition 3 – only the decreased
465
willingness of high-income agents with a high public signal impacts on the allocation and social
466
welfare decreases for the same reasons stated in the previous paragraph.
467
Disconnect between subjective and objective probabilities
468
fects consumption allocations via the signal-conditional probabilities of a high future income,
469
zh , zl , as objective probabilities that depend on both, public and private signal precision. Agents
470
in our model have rational expectations such that their subjective probabilities are equal to these
471
objective probabilities. Whenever there is a strong disconnect between subjective and objective
472
probabilities such that the optimal consumption allocation becomes independent from public
473
signal precision, the positive effect of public information is absent. For example, suppose that
474
agents either assign a subjective probability of one (optimism) or zero (pessimism) to the high-
475
income in the next period for all signal precisions. In both cases, information precision does not
476
affect agents’ subjective probabilities and the optimal consumption allocation does not depend
477
on signal precision. Another example is when agents trust only their private signals and pay no
478
attention to the public signals. In that case, zh = zl = ν, the two enforcement constraints of
479
high-income agents collapse to one and the consumption allocation is independent from public
480
signal precision. Consequently, there is no positive effect of better public signals in these cases.
481
4
482
Our main theoretical result is derived under simplifying assumptions. In particular, we restrict
483
attention to allocations that are contingent only on the current state (and not on the history
484
of the state), and consider i.i.d. income shocks with only two states. In this section, we
485
quantitatively analyze whether these simplifying assumptions are pivotal for the existence of the
486
positive effect of better public information in a context of international risk sharing. To achieve
487
this goal, we start with adjusting the risk-sharing model from Section 2 to an international
488
setting, also allowing for more general insurance contracts that can depend on the history
489
of the state. Afterwards, we confront the model with data on international risk-sharing to
490
investigate whether there is a positive effect of better public information in international risk
491
sharing which is the normative question posed at the beginning of the paper.
Proposition 3 has implications that go beyond the in-
Information precision af-
An application to international risk sharing
16
492
4.1
Environment
493
Preferences, endowments and information
494
by a continuum of representative households or benevolent governments of small countries in-
495
dexed i. Households’ preferences are given by (1). Real GDP in country i is stochastic and takes
496
the form of a country-specific endowment shock yti that evolves independently across countries
Consider a world economy that is populated
497
according to a first-order Markov process with constant transition probabilities π(y |y); the set
498
of possible endowment realizations Y is time-invariant and finite with N elements. The Markov
499
chain induces a unique invariant distribution of income π(y).
500
Each period, household i receives a public signal kti ∈ Y and a private signal nit ∈ Y that
501
inform about her country-specific shock realization in the next period. As before, the precision
502
of the public signal is denoted by the conditional probability κ = π(y = yj |k = yj ), while the
503
precision of the private signal is given by ν = π(y = yj |n = yj ). Both signals have as many
504
possible realizations as income and also follow first order Markov processes.
505
The transition probabilities of the two signals, π(k = yj |k = yi ) and π(n = yj |n = yi )
506
are chosen to yield a joint distribution of income and the two signals that is consistent with
507
household rationality. Household rationality requires that the joint invariant distribution of
508 509
income and the two signals π(y, k, n) exhibits the invariant income distribution as marginal . distribution, i.e., π(y) = (k,n) π(y, k, n). Further, rationality requires that the conditional
510
income distribution π(y |y) follows from integrating the income distribution that is conditional
511
on income and signals over of any pair of signals . π(y |y) = π(y |y, k, n)π(k, n|y). (k,n)
512
Insurance arrangements
Each household can sign a long-term insurance contract with a
513
risk-neutral financial intermediary. As a consequence, the utility allocation is no longer re-
514
stricted to be contingent on the current state as in the case of memoryless allocations analyzed
515
in Section 3. Instead, the utility allocation can depend on the history of the state. The inter-
516
mediaries, unlike the households, can commit to the terms of the contract and seek to minimize
517
the intertemporal resource costs of the insurance contract. While the intermediaries can observe
518
the public state st , they cannot observe the realization of the private signal nt . Households can
519
ex-post repudiate the contract signed with the principal for which they are punished with perma-
520
nent exclusion from the market. In case of repudiation, households are confined to autarky and
521
they consume their endowment in all states of the world in the future. Per construction, long-
522
term contracts will provide higher welfare than the memoryless allocations that we considered
523
in the previous section.
524
Obstfeld and Rogoff (1996) argue in Chapter 6.3 that an efficient revelation mechanism as
525
formalized by truth-telling constraints may not be not realistic in the context of international
526
risk sharing. Insurance contracts with efficient revelation would require that the contract is
527
the only financial commitment countries can make or that countries’ other trades are perfectly
528
observable which is difficult in the context of sovereign risk.
529
As our baseline, we therefore analyze allocations that depend directly only on the publicly
530
observable state, i.e., private information is not contractible. These allocations trivially satisfy 17
531
the truth-telling constraints as there are no incentives to misreport the realization of the private
532
signal. Nevertheless, private information affects the optimal allocation via the enforcement con-
533
straints. As a robustness exercise, we also study optimal allocations when private information
534
is contractible and allocations are contingent on private signal reports (see the accompanying
535
Online Appendix).9
536
There are many financial intermediaries that act under perfect competition and can inter-
537
temporally trade resources with each other at the given shadow price 1/R with R ∈ (1, 1/β].
538
The equilibrium interest rate is the interest rate that guarantees resource feasibility. Given
539
a shadow price R, an utility promise w, today’s publicly observable realization of the state
540
s = (y, k), today’s unobserved private signal realizations n, the intermediary chooses a portfolio
541
of current utility h and state-contingent future promises w (s ). More formally, the financial
542
intermediaries solve the following optimization problem V (w, s) =
543
min
h,{w (s )}
π (n|s)
n
1 1− R
1 C(h) + π(s |s, n)V w (s ), s R
(14)
s
to deliver the promised value w(s) and to satisfy the participation constraints: w=
π (n|s) (1 − β) h + β π s |s, n w s
n
(1 − β) h + β
(15)
s
π s |s, n w s ≥ U Aut (s, n) , ∀n,
(16)
s 544
where π (n|s) is the probability of a given private signal realization conditional on the realization
545
of the public state in the current period.10 The expected continuation value of an agent who
546
defaults after state s, n has occurred is denoted by U Aut (s, n) with U as the solution of the
547
following functional equation:11 U (s, n) = (1 − β)u(y) + β
π(s , n |s, n)U (s , n ).
(17)
s ,n 9
We should also not be silent about the fact considering non-contractible private information as baseline makes the computation of optimal allocation more tractable which allows us to consider realistic income processes with more than two income states in the quantitative exercise below. In case of contractible private information, the interaction of the increase in the state space with two types of occasionally binding incentive constraints – enforcement and truth-telling constraints – makes the computation of optimal allocations a cumbersome task. As other papers in the literature with a less complex setting than ours such as Broer et al. (2017), we restrict ourselves to persistent income processes with two states in this case. 10 In our recursive formulation, the enforcement constraints are imposed for the current period. This is different from Krueger and Perri (2011) who impose the enforcement constraint on continuation values, i.e., w (s ) ≥ U Aut (s , n ), for all s , n . If households can sign contracts that are contractible on the whole state vector, both recursive formulations are equivalent. When private information is not contractible, imposing the enforcement constraint on continuation values is not equivalent to imposing the constraints in the current period. Our recursive formulation implies constraints that are less restrictive than the constraints in case of enforcements constraints on promises, resulting in lower resource costs. 11 In Appendix A.9, we provide details on the derivations of the transition probabilities π (s |s, n), π (θ |θ) = π (s , n |s, n) as functions of the income transition probabilities, the signal transition probabilities and the two precisions κ, ν.
18
548
4.2
Quantitative exercise
549
Empirical estimates
550
provide here empirical counterparts for the quality of public information, the income process
551
and a consumption insurance measure.12
To take the model described in the previous section to the data, we
552
As the first data source, we employ annual data from the Penn World Tables, version 8.1
553
(PWT) to generate a panel of 70 countries for the years 1990-2004. The PWT assign countries
554
to four data-quality grades, ranging from A, the highest quality, to D, the lowest grade. We
555
use the data quality grades to bin the countries in our panel. First, we generate measures
556
of insurance βΔy,i by regressing country-by-country the idiosyncratic component of changes in
557
consumption per capita in country i, on the idiosyncratic component of changes in GDP per
558
capita as follows βΔy,i = 1 −
cov [Δ ln(cit ) − Δ ln (Ct ) , Δ ln(yit ) − Δ ln (Yt )] , var(Δ ln(yit ) − Δ ln (Yt ))
(18)
559
where Δ ln(cit ) (Δ ln(yit )) stand for country i’s per capita growth rate of consumption (real
560
GDP, respectively) between year t and t − 1. Capital letter variables are sample aggregates that
561
capture the uninsurable aggregate risk component of all countries real GDP. When doing so, we
562
control for differences in insurance between countries that are not explained by our model but
563
stem, for example, from the general level of development or the quality of financial and political
564
institutions. The insurance measure attains a value of 0 if there is no insurance against country-
565
specific GDP shocks and the value of 1 when there is perfect insurance, so that consumption
566 567
per capita does not respond to country-specific fluctuations in GDP. Within each bin, we compute average insurance measures βΔy,j , j = A, B, C which are
568
reported in the lower part of Table 1. Data for D countries lacks information on consumption
569
which is why we cannot estimate insurance in these countries. The main message is that
570
improvements in public data quality on country’s idiosyncratic GDP risk do not exhibit a
571
monotone correlation with insurance. Instead, the correlation follows a U-shape; increasing data
572
quality from grade C to B worsens insurance but further improvements ameliorate insurance
573
again. To test the significance of the conditional insurance measure, we regress the averages on
574
A- and C-group dummies. We find the positive differences with regard to the B-group to be
575
significant at p-values that are below 3%.
576
As an empirical counterpart to capture the quality of public information, we employ the
577
one-year ahead GDP growth forecasts as well as the realizations as reported in the IMF World
578
Economic Outlook (WEO). Using this data, we compute mean-squared forecast errors for real
579 580
GDP growth for all countries in our sample and normalize them by the variance of country i’s j GDP growth. In the next step, we average the forecast errors in each bin j to receive M SF E
581
reported in the bottom of Table 1, j = A, B, C, D.
582
We model the logarithm of the idiosyncratic part of income of country i as an AR(1) process
583
with autocorrelation coefficient ρ ∈ [0, 1) and disturbances that are normally distributed with
584
mean zero and standard deviation (SD) σ . We estimate an autocorrelation coefficient of 12 In the accompanying Online Appendix, we provide further details on the data used, estimation strategy and estimation results.
19
Table 1: Baseline parameters and data moments Parameter
Value
σ β κA κB κC ρ σ Θ
Risk aversion Discount factor Precision public information A countries Precision public information B countries Precision public information C countries Auto-correlation SD of i.i.d. term Income-signals states
1 0.96 0.84 0.76 0.73 0.84 0.026 3×8
βΔy,A βΔy,B βΔy,C A M SF E
Insurance coefficient A countries Insurance coefficient B countries Insurance coefficient C countries
0.36 0.09 0.33
Scaled mean-squared forecast error A countries
1.02
M SF E C M SF E
Scaled mean-squared forecast error B countries
1.24
Scaled mean-squared forecast error C countries
1.30
M SF E
Scaled mean-squared forecast error D countries
1.49
B
D
585
ρ = 0.84 and σ = 0.026 as the averages across all countries in the sample.
586
Preferences and endowments
587
erences. The instantaneous utility function features log preferences and we choose an annual
588
discount factor of β = 0.96. We normalize the mean of income to one, and employ the method
589
proposed by Tauchen and Hussey (1991) to approximate the income process by a Markov process
590
with two states and time-invariant transition probabilities to yield the unconditional variance
591
and autocorrelation estimated in our data set.13
592
Calibrating public signal precision using the IMF forecasts The mean-squared forecast
593
errors computed in the publicly available IMF WEO data is a measure of public information
594
quality and can be linked to the corresponding forecast errors in the model as follows. For
We start with standard values for the specification of pref-
595
a given Markov process for income and uninformative private signals (ν = 0.5), we compute
596
the reduction of perceived income risk κ ˜ as the reduction in the mean-squared forecast error
597
resulting from conditioning expectations on public signals with precision κ in the model κ ˜ (κ) =
598
with MSFEy =
y
MSFEy − MSFEs (κ) , MSFEy
π(y)
2 π(y |y) y − E(y |y)
y
13
As a robustness exercise, we also consider a discretization with three income states. The results of this exercise can be found in the accompanying Online Appendix.
20
MSFEs (κ) =
π(s)
s
2 π(y |s) y − E(y |s) ≤ MSFEy ,
y
599
and π(s) as the joint invariant distribution of income and public signals.14 The reduction in the
600
mean-squared forecast error κ ˜ (κ) is a monotone transformation of signal precision. The more
601
precise are the public signals the larger is κ ˜ (κ). If signals are uninformative, κ ˜ is equal to zero
602
and is equal to one if signals are perfectly informative.
603
The income process is identical across country groups and private signals are uninformative
604
per assumption which implies that the model interprets all differences in mean-squared forecast
605
errors between the country groups as differences in public signal precision. We choose D coun-
606
tries as the reference point (implicitly assuming that public signals are uninformative in these
607
countries) to compute κj as follows κ ˜ (κj ) = 1 −
M SF E M SF E
j
D
j = A, B, C.
(19)
608
In the data, A countries’ mean-squared forecast error is 32 percent lower than the corre-
609
sponding forecast error in D countries. We adjust κj such that κ ˜ (κj ) = 0.32 which results in
610
a calibrated precision of the public information of κA = 0.84. For the other country groups we
611
need to match for the B countries κ ˜ (κj ) = 0.17 and κ ˜ (κj ) = 0.13 for the C countries, resulting
612
in κB = 0.76 and κC = 0.73, respectively as reported in Table 1.
613
Calibrating private signal precision using insurance estimates
614
public signal precision values κj for all country groups using the IMF WEO data, we still
615
need to calibrate private signal precision νj . We consider trade that occurs within each group
616
of countries. For each country group j, we compute optimal stationary allocations and the
617
resulting amount of insurance. Based on the findings of Propositions 1 and 2, we expect that
618
not only welfare and risk sharing but also insurance decreases monotonically in the precision
619
of private information for given public signal precision. Thus, if allocations with uninformative
620
private signals and informative public signals with precision pinned down according to (19)
621
feature insurance that is too high compared to the data, we can choose private signal precision
622
to reduce the gap between the data and the model-implied amount of insurance. Formally,
623
given public information precision κ ¯ j , we seek to identify the precision of private information
624 625
Having backed out the
νj by varying it until insurance in the optimal stationary allocation βΔy,j (¯ κj , νj ) mimics the estimated degrees of insurance in the data, βΔy,j , for j ∈ {A, B, C} βΔy,j (¯ κj , νj ) = βΔy,j .
(20)
626
Eventually, the country groups differ only with respect to the importance of private and public
627
information and the degree of insurance, preferences and endowments are identical.15 14 In Appendix A.9, we provide the formulas for the conditional income probabilities π(y |θ) and π(y |s) as functions of the income transition probabilities and the two precisions κ, ν. 15 As an alternative measure of international risk sharing, we also consider the ratio of consumption to income volatility. The quantitative results for this case are reported in the accompanying Online Appendix.
21
628
4.3
Quantitative results for the baseline calibration
629
As the first step, we confront the theoretical model with the data. Then we demonstrate that
630
when the private information friction is sufficiently severe, better public forecast are indeed
631
beneficial for insurance and social welfare.
632
Insurance: model versus data
633
insurance measure with respect to the private signal precision – larger values of ν are detrimental
634
to insurance as measured by (18). Second, insurance in the model with informative public but
635
uninformative private signals is indeed too high compared to the data for all country groups.
636
This allows us to identify private signal precision according to (20) when insurance reacts
637
sensitively enough to changes in private signal precision. We find that the detrimental effect
638
of more precise private signals in the model is quantitatively strong enough to explain the
639
differences in insurance for groups A, B and C. Thereby, the private information friction plays
640
the most important role in countries of the group B. To capture the low degree of insurance of
First, the allocations exhibit strict monotonicity of the
641
βΔy,B = 0.09, the private information friction is with νB = 0.84 more severe than in country
642
groups A with νA = 0.56, and in C countries, νC = 0.57, respectively. For these values of ν, the
643
model also matches the insurance in A, βΔy,A = 0.36 and in C, βΔy,C = 0.33.
644
The calibrated precisions of private and public information reveal that consumption insur-
645
ance reacts sensitively to changes in private-signal precision but only mildly to changes in public
646
signal precision. When both signals are uninformative, κ = ν = 0.50, the insurance measure
647
equals 0.50. For uninformative private signals, increasing κ to κC = 0.73 reduces insurance to
648
0.45, increasing κ further to κA = 0.84 reduces insurance only mildly to 0.44. The impact of
649
private information on insurance is stronger. Starting at κB = 0.76, increasing private signal
650
precision from 0.5 to νB = 0.84 reduces insurance by 0.36. The same rationale is behind the
651
calibrated values of private signal precision for the country groups A and C. Private infor-
652
mation precision is comparable but the difference in public information precisions is relatively
653
sizeable. Nevertheless the amount of insurance in the two groups is with 0.33 and 0.36 simi-
654
lar because private and not public information is the main driver of the insurance differences
655
between country groups.
656
Improvements in private signal precision reduce welfare by more than changes of public
657
signal precision of the same size. More precise private signals increase the outside option values
658
of agents with a high and a low public signal because only enforcement constraints with a high
659
private signal can be binding. Improvements in public signal precision increase the outside
660
option value of agents with a high public signal but decrease the outside option of agents with a
661
low signals. For this reason, consumption of high-income agents goes up on average by more in
662
case of private than public signals, and insurance and welfare decrease by relatively more when
663
private signal precision increases.
664
Using the Consumer Expenditure Survey 1998–2003 for US households, Broer (2013) finds
665
that the limited commitment model tends to predict too much consumption insurance for stan-
666
dard calibration such as ours. Thus, it may seem surprising that a limited commitment model
667
can capture the relatively low degree of insurance observed across country groups. One differ-
668
ence to Broer (2013) is that we consider private and public signals and we find that consumption
22
Welfare in certainty-equivalent consumption
0.9991
0.9991
0.9990
0.999
0.9989
Welfare, A-countries, νA = 0.56 Welfare, B-countries, νB = 0.84 Welfare, C-countries, νC = 0.57 Welfare ν = 0.60
0.9989
0.9988 0.5
0.55
0.6
0.65
0.7
0.75
Precision of public signals, κ
0.8
0.85
0.9
Figure 4: Non-contractible private information. Welfare of A, B, C countries measured in certainty-equivalent consumption as a function public signal precision. 669
insurance reacts in particular sensitively to changes in private signal precision. Another dif-
670
ference is the size of the income risk in US micro data compared to the international setting
671
we consider. Broer (2013) employs in his calibration a variance of logged income of 0.36 while
672
the country-specific shocks we consider yield a variance of logged income equal to 0.0023 or a
673
standard deviation of 0.048.16
674
Positive value of public information
675
at the beginning of the paper whether governments of countries and international institutions
676
should strive to provide better country-GDP forecasts to the public in order to improve house-
677
holds’ consumption smoothing against country-specific shocks. The main message from The-
678
orem 1 is that increases in public signal precision increase welfare when private signals are
679
sufficiently precise. While this result was derived for memoryless allocations, the allocations we
680
consider here are more general and can be contingent on the history of the public state st . To
681
test whether the positive welfare effect of better public signals applies here as well, we compute
682
welfare in the three country groups as a function of public signal precision.
Here we come back to the normative question posed
683
The results of this exercise are plotted in Figure 4 and are consistent with the findings in
684
Theorem 1. Private signals in A and C-countries are with precisions νA = 0.56 and νC = 0.57
685
not precise enough, and social welfare decreases mildly by less than 0.01 percent in the precision
686
of public signals. The positive and negative effect of better public signals exactly cancel each
687
other out for private signals with precision ν = 0.60. In the B-countries, private signals are 16 This estimate is consistent with other estimates in the related literature. For example, Kose, Prasad, and Terrones (2003) find estimates that range from 0.02 (industrial countries) to 0.05 (less financially integrated countries) for the years 1960 - 1999. Backus, Kehoe, and Kydland (1992) employ in their benchmark case standard deviation of innovations approximately equal to 0.01.
23
688
with νB = 0.84 > 0.60 precise enough and better public information increases welfare by 0.01
689
percent. The main message is that the positive effect of better public signals does not depend on
690
the assumption of memoryless allocations but also emerges for a more general type of insurance
691
contracts in which also the history of the public state is contractible. This conclusion also
692
extends to the case when insurance contracts that can depend on the history of the public state
693
as well as additionally on the history of private signal reports (see the accompanying Online
694
Appendix).
695
5
696
In this paper, we have revisited the value of private and public information in a general con-
697
sumption risk-sharing model under limited commitment. In contrast to conventional wisdom,
698
we show in our main theoretical result that improvements in public information can improve
699
risk sharing and welfare. Better public information has two opposite effects when private sig-
700
nals are informative. First, more precise public signals spread out the outside option values of
701
high-income agents with a high and a low public signal and these agents are less willing to trans-
702
fer resources to low-income agents. Second, more informative public signals affect the outside
703
option values of high-income agents asymmetrically. As a consequence, the willingness of low-
704
signal agents to transfer resources increases by more than the high-signal agents’ willingness to
705
transfer resources decreases, facilitating in total higher transfers from high-to low income agents
706
and benefiting welfare. When private information is sufficiently precise, risk sharing improves
707
with better public information and agents prefer informative public signals ex-ante.
Conclusions
708
We find that this positive effect of better public information is not only present under specific
709
circumstances but is a rather general phenomenon. In particular, we find that the logic behind
710
the positive effect also applies in optimal allocations that are contingent on the history of the
711
public state, the history of truthful reports on private signal realizations and for persistent
712
idiosyncratic shock processes.
713
As a methodological contribution, we formulate a general risk-sharing model in which agents
714
with private and public fore-knowledge of their future idiosyncratic shock realizations can engage
715
in dynamic insurance contracts to smooth their consumption.17 This risk-sharing model can be
716
directly applied to explore ideas related to thought-provoking results in the recent literature on
717
the existence of a private unemployment insurance market and the optimal provision of public
718
insurance.
719
Hendren (2017) analyzes the implications of individuals with private knowledge of future
720
job losses for the existence of an unemployment insurance market. As his main result, he
721
finds that the private information friction gives rise to unemployment insurance contracts with
722
pooled risks that are too adversely selected to be profitable. Hendren (2017) derives this result
723
for static insurance contracts. In this paper, we explicitly model individual fore-knowledge on
724
future shocks as signals and consider a general optimal contracting problem comprising a rich 17 There is ample evidence that individuals possess fore-knowledge on their future risks. Exemplary papers are Dominitz (1998) and Dominitz and Manski (1997) for income risk, Smith, Taylor, and Sloan (2001) for predicting mortality risk, Stephens (2004) for unemployment, or Campbell, Carruth, Dickerson, and Green (2007) for job insecurity.
24
725
variety of static and dynamic insurance contracts. Using our framework, one can ask whether
726
Hendren (2017)’s main result on the non-existence of a private unemployment market due to
727
adverse selection prevails when individuals can also trade dynamic unemployment insurance
728
contracts.
729
Krueger and Perri (2011) find that more public insurance via a progressive tax system
730
can more than one-for-one crowd out private insurance of income shocks provided in contracts
731
with limited enforceability. This crowding out effect arises when private insurance markets
732
are working well and the overall degree of risk sharing is high even without the additional
733
insurance provided by the tax system. When agents receive informative private signals on their
734
future income, private insurance becomes less effective in smoothing consumption. It would
735
be interesting to investigate if better public insurance may now overall improve consumption
736
insurance for parameterizations for which Krueger and Perri (2011) find a more than one-for-one
737
crowding out of the insurance achieved via private insurance markets.
738
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26
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A
777
A.1
Appendix: proofs and results in the main text Proof of Lemma 1
778
(i) Let the vector of parameters be γ = (κ, ν). Based on part (i) and given a strictly concave
779
function u (c) : R+ → R satisfying Inada conditions, the problem to compute the optimal
780
memoryless allocation is ∞
1 max E U (c) = (1 − β) β t u cji ≡ Vrs subject to: 4 {cji }i,j∈{l,h} t=0 i,j∈{l,h}
non-negativity: cji ≥ 0 ∀i, j ∈ {l, h} 1 j 1 resource feasibility: ci ≤ yj 4 2
(21) (22) (23)
j
i,j∈{l,h}
enforcement constraints ∀i, j, k ∈ {l, h} :
m (1 − β) u cji + β (1 − β) π y = ym |k = yi , n = yk Vrs + β 2 Vrs ≥ m∈{l,h}
(1 − β) u (yj ) + β (1 − β)
π y = ym |k = yi , n = yk u (ym ) + β 2 Vout ,
(24)
m∈{l,h} 781
with π (y = ym |k = yi , n = yk ) as the probability to receive income ym in the future con-
782
ditional on receiving a public signal indicating yi and a private signal indicating yk . The
783
above constraints define the set of constraints P (γ), any candidate solution consumption
784
allocation will be labelled P (γ) ∈ P (γ). Observe that P (γ) is trivially non-empty, as the
785
autarkic allocation cki,j = yk satisfies the enforcement constraints and is resource-feasible.
786
Observe that the resource feasibility constraint (23) together with non-negativity con-
787
straints (22) define a compact and convex subset of R4 . The target function (social welfare
788
function) is continuous. Hence, by Weierstrass theorem the solution to the maximization
789
problem exists. The participation constraints define a convex subset of the resource fea-
790
sible non-negative set of consumption bundles. Uniqueness follows from strict concavity
791
of the target function and convexity of the set of constraints. In fact, the constraints
792
define a continuous (both upper and lower hemicontinuous) correspondence P (γ). Thus,
793
the continuity and differentiability of the welfare function with respect to γ is a direct
794
application of the theorem of maximum.
795
(ii) Consider the autarky allocation which per construction satisfies enforcement constraints.
796
Without loss of generality, consider the incentives of high-income agents with a low public
797
and a high private signal to transfer an infinitesimal amount of resources to other agents.
798
Such a transfer satisfies the enforcement constraints of these agents if the cost of such a
799
transfer in utility is smaller than the gains. Consider first these agents to transfer resources
800
to high-income agents with a high private and a high public signal. Such a transfer is
801
consistent with their enforcement constraints if
zl β 2 zl β 2 + ≤ u (yh )δ (1 − β)β + , u (yh )δ (1 − β) 1 + β 2 4 2 4 27
802
where the left-hand side captures the utility costs and the right-had side the utility gains of
803
the transfer δ and zl as defined in (5). For β ∈ (0, 1), the costs exceed the gains, rendering
804
such a transfer incompatible with enforcement constraints. Thus, the only possibility
805
is to transfer resources to low-income agents because the consistency with enforcement
806
constraints then requires
zl β 2 1 − zl β 2 u (yh )δ (1 − β) 1 + β + ≤ u (yl )δ (1 − β)β + 2 4 2 4
807
which can be supported when the difference in marginal utilities u (yl )−u (yh ) > 0 is large
808
enough. Starting alternatively from any other enforcement constraints of high-income
809
agents follows the same logic and the same conclusion that transfers from high-income
810
agents to other high-income agents violate enforcement constraints.
811
The only remaining possibility to support chi > yh for any public signal i are transfers
812
stemming from low-income agents. Using the similar logic, we can also exclude transfers
813
from any low-income agent to any high-income agent in an optimal memoryless alloca-
814
tion. Thus, in the optimal memoryless allocation, we get chi ≤ yh , and then by resource feasibility cli ≥ yl for all signals i.
815
816
A.2
817
The first best allocation can be perfect risk sharing (all agents in all states consume y¯). In the
818
following proposition, we summarize how public and private information affect the condition
819
that renders perfect risk sharing constrained feasible and thus optimal.
820
Proposition 4 (Perfect Risk Sharing) Consider the optimal memoryless allocation.
821 822
823
824
825
Information, perfect risk sharing and autarky
¯ ν) < 1, such that for any discount factor (i) There exists a unique cutoff point, 0 < β(κ, ¯ ν) the optimal allocation for any signal precision is perfect risk sharing. 1 > β ≥ β(κ, ¯ ν) is increasing in the precision of the public and private signal. (ii) The cutoff point β(κ, Proof. (i) Let V¯rs = u(¯ y ) be social welfare under perfect risk sharing. First, perfect risk sharing
827
provides the highest ex-ante utility among the consumption-feasible allocations. The exis¯ ν) follows from monotonicity of participation constraints in β and V¯rs > Vout . tence of β(κ,
828
A higher β increases the future value of perfect risk sharing relative to the allocation in the
826
830
equilibrium without transfers, leaving the current incentives to deviate unaffected. There¯ ν), they are not binding for fore, if the participation constraints are not binding for β(κ,
831
¯ ν). The cutoff point is characterized by the tightest participation constraint, any β ≥ β(κ,
832
i.e., by the participation constraint with the highest value of the outside option
829
h l + β 2 Vrs ≥ (1 − β)u(chh ) + β(1 − β) zh Vrs + (1 − zh )Vrs (1 − β)u(yh ) + β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] + β 2 Vout ,
28
833 834
835
with zh defined in the main text (3). Solving this constraint yields a unique solution for ¯ ν) in (0, 1) because u(¯ β(κ, y ) < u(yh ). (ii) The tightest constraint at the first best allocation is u(¯ y ) ≥ (1 − β)u(yh ) + β(1 − β)zh u(yh ) + β(1 − β)(1 − zh )u(yl ) + β 2 Vout .
836
Differentiating the constraint fulfilled with equality with respect to κ using the implicit
837
function theorem gives h ¯ ν) β(1 − β)[u(yh ) − u(yl )] ∂z ∂ β(κ, ∂κ = > 0, ∂κ D
838
results in a positive sign for yh > yl , ∂zh ν(1 − ν) = >0 ∂κ [κν + (1 − κ)(1 − ν)]2
839
and D ≡ (1 − β)[u(yh ) − zh u(yh ) − (1 − zh )u(yl )] + β[zh u(yh ) + (1 − zh )u(yl ) − u(yl )] > 0.
840
For ν ∈ [0.5, 1), the cutoff point increases strictly in precision of the public signal. Simi-
841
larly, taking the derivative with respect to the ν results in a positive sign as well, a strictly
842
positive sign for κ ∈ [0.5, 1).
843 844
On the other extreme, the absence of risk sharing in autarky may be the only constrained-
845
feasible memoryless allocation. In the next proposition, we provide conditions for this case.
846
Proposition 5 (Autarky) Consider the case when the participation constraints of high-
847
income agents with a high and low public signal are binding. If and only if u (yh ) + β
z h + zl β2 1 [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] ≥ 0 2 1−β2
(25)
848
autarky is the optimal memoryless allocation with zh and zl defined in (3), (5).
849
Proof. Optimal memoryless allocations can be analyzed as a fixed-point problem expressed in
850
terms of the period value of the arrangement.
851
The fixed-point problem is constructed as follows. Let W = E U (c) be the unconditional
853
expected value of an allocation c before any shock has realized. We restrict attention to W ∈ [Vout , V¯rs ) because per assumption participation constraints for high-income households are
854
binding. The binding participation constraints are given by the following
852
u(chh ) + β zh [u(chh ) + u(chl )]/2 + (1 − zh )u(2¯ y − (chh + chl )/2) = u(yh ) + β [zh u(yh ) + (1 − zh )u(yl )] +
29
β2 (Vout − W ), 1−β
(26)
855
u(chl ) + β zl [u(chh ) + u(chl )]/2 + (1 − zl )u(2¯ y − (chh + chl )/2) = u(yh ) + β [zl u(yh ) + (1 − zl )u(yl )] +
β2 (Vout − W ), 1−β
(27)
856
and resource feasibility is used. The objective function of the problem to compute the optimal
857
memoryless arrangement is given by the following expression Vrs (W ) ≡
1 h u(ch (W )) + u(chl (W )) + 2u(2¯ y − (chh (W ) + chl (W ))/2) . 4
858
The optimal memoryless arrangement should necessary solve the fixed-point problem W =
859
Vrs (W ). We will show that Vrs (W ) is strictly increasing. V (W ) is also strictly concave, therefore
860
there exist at most two solutions to the fixed-point problem.
861
From the participation constraints (26) and (27), the derivative of V (W ) is given by Vrs (W )
∂chh ∂chl 1 h h (u (ch ) − u (cl )) + (u (cl ) − u (cl )) = 4 ∂W ∂W
862
which is strictly increasing in W because perfect risk sharing is not constrained feasible which
863
implies that ∂chh /∂W and ∂chl /∂W are negative and chh , chl = y¯.
864
By construction, one solution to the fixed-point problem is Vout . The concavity of Vrs (W )
865
implies that the derivative of Vrs (W ) at Vout is higher than at any partial risk-sharing allocation.
866
(w) at V Therefore, autarky is the optimal memoryless arrangement if the derivative of Vrs out
867
must be smaller than or equal to 1 which implies Vrs (W )
868
1 (u (yh ) − u (yl ) = 4
∂ch ∂chh + l ∂W ∂W
{cji }={yj }
≤1
The two derivatives can be computed using the implicit function theorem and are given by β2 P h cl 2 β Q ch ∂chh l , = − Ph Ph ∂W ch cl Q ch Q ch h
P h β2 ch 2 h Q ch β ∂cl h , = − Ph Ph ∂W ch cl Q ch Q ch
l
h
l
869
with the auxiliary derivatives P, Q as the partial derivatives of the binding participation con-
870
straints (26) and (27) evaluated at the autarky allocation given by β β zh u (yh ) − (1 − zh )u (yl )] 2 2 − (1 − β)u (yh )
Pch = (1 − β)[u (yh ) + h
Pch = Pch l
h
β β zl u (yh ) − (1 − zl )u (yl )] 2 2 − (1 − β)u (yh ).
Qch = (1 − β)[u (yh ) + l
Q c h = Qc h h
l
30
871 872
Using these expressions, the sum of the partial derivatives with respect to W evaluated at {cji } = {yj } is given by
∂ch ∂chh + l ∂W ∂W
=−
h
=
873
2β 2 (1 − β)u (yh ) (1 − β)u (yh )[Pch + Qch − (1 − β)u (yh )]
(1 − β) u (yh ) zh +
l
−4β 2 β
. + zl − (2 − zh − zl )u (yl )
(W ) and collecting terms eventually results in Using this expression in Vrs
u (yh ) + β
z h + zl β2 1 [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] ≥ 0. 2 1−β2
874
Under this condition, the optimal memoryless arrangement is the outside option. If the condi-
875
tion is strictly negative then there exists an alternative constrained feasible allocation that is
876
preferable to the outside option. This result is used in the main theorem.
877 878
This condition can be related to the corresponding condition in case of uninformative signals (κ = ν = 0.5). In this case, zh = zl = 0.5, and the condition reads 1 − β β2 1−β β + − u (yl )β 1 − β − + ≥0 u (yh ) (1 − β) + β 2 2 2 2 β β ⇔ u (yh ) 1 − − u (yl ) ≥ 0 2 2 β u (yh ) ≥ u (yl ), 2−β
879
which corresponds to the condition derived in Krueger and Perri (2011) and in Lepetyuk and
880
Stoltenberg (2013). From the other end, suppose that autarky is the optimal memoryless
881
arrangement and that both participation constraints of high-income agents are binding. Then,
882
the value of this arrangement Wout = Vout must be a solution to the fixed-point problem. This
883
requires that the slope of Vrs (W ) at {cji } = {yj } must be necessarily smaller than or equal to
884
unity. Otherwise, due to the concavity of Vrs (W ), there exists another solution to the fixed-point
885
problem with an allocation that Pareto dominates the autarky allocation.
886
A.3
Proof of Lemma 2
887
(i) Autarky is not the only constrained-feasible memoryless allocation, thus, Vrs > Vout ;
888
h < u(y ) chi ≤ yh for all i (see of part (iii) of Lemma 1) and strict concavity implies, Vrs h
889 890 891
such that chi < yh for at least one i; the resource constraint is binding because utility is l > u(y ) because strictly increasing in consumption, resource feasibility then results in Vrs l
cli > yl for at least one i.
892
(ii) Suppose that the enforcement constraints of high-income agents with a high public signal
893
are binding (the case of a low public signal follows analogous arguments and is omitted
31
894
here).18 To establish a contradiction, suppose that constraints with high and low private
895
signals are binding. Let zhl denote the conditional probability of a high future income
896
conditional on a high public and low private signal and be defined as follows: zhl ≡
(1 − ν)κ , (1 − ν)κ + ν(1 − κ)
0 ≤ zhl ≤ zh ≤ 1
897
and zh as defined in (3).
898
The enforcement constraints of high-income agents with high public signal and with low
899
private signals are h l (1 − β)u(chh ) + β(1 − β) zhl Vrs + (1 − zhl )Vrs + β 2 Vrs = (1 − β)u(yh ) + β(1 − β) zhl u(yh ) + (1 − zhl )u(yl ) + β 2 Vout . Subtracting these constraints from the ones of agents with a high private signal gives
900
h l = (zh − zhl ) [u(yh ) − u(yl )] . (zh − zhl ) Vrs − Vrs l > y . Strict concavity With ν ∈ (0.5, 1], zh = zhl . From part (i), we have chi < yh and Vrs l
901 902
h − V l < u(y ) − u(y ) which contradicts that the enforcement constraints of implies Vrs h l rs
903
agents with a high and a low private signals are binding. What remains to be shown is
904
whether enforcement constraints with a low or a high private signal are binding. To see
905
this, suppose first that only the enforcement constraints of high-income agents with a low
906
private signal (and a high public signal) were binding. In that case, we have that β2 h h − u(yh )] + β(1 − zhl )[Vrs − u(yh )]. (Vout − Vrs ) = u(chh ) − u(yh ) + βzhl [Vrs 1−β
907
Combining this expression with slack enforcement constraints of high-income agents with
908
a high private signal (and a high public signal) it must be that
h l > (zh − zhl ) [u(yh ) − u(yl )] , (zh − zhl ) Vrs − Vrs
909
h −Vl < which results in a contradiction because with partial risk sharing, we have Vrs rs
910
u(yh ) − u(yl ). Assume alternatively that only enforcement constraints of high-income
911
agents with a high private signal (and a high public signal) are binding. Following similar
912
steps as before, now it must be that
h l (zhl − zh ) Vrs − Vrs > (zhl − zh ) [u(yh ) − u(yl )]
h l ⇔ Vrs − Vrs < [u(yh ) − u(yl )] which is satisfied.
913 18
In principle, the argument also applies to low-income agents but given that we show in part (iii) that their enforcement constraints cannot be binding, we do not consider this case here.
32
914 915
(iii) From part (ii), we have that only participation constraints of agents with a high private signal can be binding. The optimal memoryless allocation thus maximizes 1 h u(ch ) + u(chl ) + u(clh ) + u(cll ) 4 + λhpc,h
u(chh ) − u(yh ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) + λlpc,h u(clh ) − u(yl ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) + λlpc,l u(cll ) − u(yl ) (1 − β) h l + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) + β 2 (Vrs − Vout ) h l + λhpc,l u(chl ) − u(yh ) (1 − β) + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) 1
1 h + β 2 (Vrs − Vout ) − λres ch + chl + clh + cll − (yh + yl ) , 4 2
916
with λjpc,i ≥ 0 as the multiplier on the enforcement constraints of agents with signal i and
917
income j, and λres > 0 is the multiplier on the resource constraint.
918
In general, chh ≤ yh , and the enforcement constraints of high-income agents with a high
919
public signal (and a high private signal) imply h l (1 − β)β zh Vrs − u(yh ) + (1 − zh ) Vrs − u(yl ) + β 2 (Vrs − Vout ) ≡ xh ≥ 0.
920
Suppose first per contradiction that enforcement constraints of low-income agents with
921
a high public signal were binding such that λlpc,h > 0. From the first order conditions
922 923 924
of these agents, we immediately get clh > cll . Autarky is not the optimal memoryless allocation such that either chh < yh or chl < yh or both strict inequalities apply. In any case, (chh + chl )/2 < yh , resource feasibility then implies that (cll + clh )/2 > yl , and clh > yl
925
because clh > cll . The enforcement constraints of low-income agents with a high public
926
signal are binding if
927
which requires xh < 0 because clh > yl . This results in a contradiction because the
u(clh ) − u(yl ) (1 − β) + xh = 0
928
enforcement constraints of high-income agents with a high public signal require xh ≥ 0.
929
Suppose alternatively per contradiction that enforcement constraints of low-income agents
930
with a low public signal were binding such that λlpc,l > 0. From the first order conditions
931 932 933
of these agents, we immediately get cll > clh . Autarky is not the memoryless allocation such that either chh < yh or chl < yh or both strict inequalities apply. In any case, (chh + chl )/2 < yh , resource feasibility then implies that (cll + clh )/2 > yl , and cll > yl
934
because cll > clh . The enforcement constraints of low-income agents with a low public
935
signal are binding if
u(cll ) − u(yl ) (1 − β) + xl = 0, 33
with
936
h l xl ≡ (1 − β)β zl Vrs − u(yh ) + (1 − zl ) Vrs − u(yl ) + β 2 (Vrs − Vout ), 937
and zl as defined in (5). It is cll > yl , thus unless xl < 0, there is a contradiction. The
938
participation constraints of high-income agents with a low public signal read
u(chl ) − u(yh ) (1 − β) + xl ≥ 0.
939
Since chl ≤ yh , it must be that xl ≥ 0 which contradicts that the enforcement constraints
940
of low-income agents with a low signals can be binding.
941
(iv) From parts (ii) and (iii), we have that only enforcement constraints of high-income agents
942
with high private signals can be binding. Thus, the optimal memoryless allocation maxi-
943
mizes 1 h u(ch ) + u(chl ) + u(clh ) + u(cll ) 4 + λhpc,h
u(chh ) − u(yh ) (1 − β) h l + (1 − β)β zh (Vrs − u(yh )) + (1 − zh )(Vrs − u(yl )) + β 2 (Vrs − Vout ) h l + λhpc,l u(chl ) − u(yh ) (1 − β) + (1 − β)β zl (Vrs − u(yh )) + (1 − zl )(Vrs − u(yl )) 1
1 h 2 h l l + β (Vrs − Vout ) − λres c + cl + ch + cl − (yh + yl ) , 4 h 2
with λhpc,h ≥ 0 as the multiplier on the enforcement constraints of high-income agents
944 945
with a high public signal (and a high private signal), λhpc,l ≥ 0 as the multiplier on the
946
enforcement constraints of high-income agents with a low public signal (and a high private
947
signal), and λres > 0 is the multiplier on the resource constraint. The first order conditions
948
for low-income agents are u (cll ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 = λres u (clh ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 = λres . Combining these equations gives immediately cll = clh = cl .
949
950
951
A.4
Proof of Lemma 3
(i) Subtract the binding enforcement constraint G(chh , chl , κ, ν) from F (chh , chl , κ, ν) to receive
h l u(chh ) − u(chl ) = β(zh − zl ) u(yh ) − Vrs + Vrs − u(yl ) .
952 953 954
h < u(y ) and V l > u(y ), and the expression in From Lemma 2 part (i), we have that Vrs h l rs
the square brackets on the right-hand side is strictly positive; zh ≥ zl , and thus, chh ≥ chl ; if zh > zl for κ ∈ (0.5, 1] and ν ∈ [0.5, 1), the strict inequality holds, chh > chl .
34
955
From the proof of Lemma 2 part (iv), the first order conditions for chl and cl can be
956
combined to receive u (chl ) 1 + 2λhpc,h β(1 − β)zh + 4(1 − β)λhpc,l + 2λhpc,l β(1 − β)zl + (λhpc,h + λhpc,l )β 2 = u (cl ) 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + (λhpc,h + λhpc,l )β 2 .
957
From strict concavity, it follows that chl > cl if 1 + 2λhpc,h β(1 − β)zh + 4(1 − β)λhpc,l + 2λhpc,l β(1 − β)zl + β 2 (λhpc,h + λhpc,l ) > 1 + 2λhpc,h β(1 − β)(1 − zh ) + 2λhpc,l β(1 − β)(1 − zl ) + β 2 (λhpc,h + λhpc,l ) ⇔ 2βλhpc,h (2zh − 1) + 2λhpc,l [2 + β (2zl − 1)] > 0
958
which is satisfied as long as one type of the high-income agents’ enforcement constraints
959
is binding, since zh ≥ 0.5, and β ∈ (0, 1); here, it is λhpc,h , λhpc,l > 0. At the beginning, we
960
961 962
established chh ≥ chl such that ch > cl and by resource feasibility, we get ch > y¯ > cl .
(ii) Per assumption, autarky is not the only memoryless allocation. From Proposition 5, we have that the determinant of the Jacobian evaluated at autarky is J F (ch , ch , κ, ν), G(ch , ch , κ, ν) h l h l
{cji }={yj }
= u (yh ) + β 963
− Fch Gch l h
= Fch Gch h
l
β2
{cji }={yj }
1 z h + zl [u (yh ) + u (yl )] − βu (yl ) − [u (yl ) − u (yh )] < 0, 2 1−β2
(28)
where the partial derivatives are given by
F ch h
F ch l
Gch l
Gch h
zh h 1 − zh l β2 h u (c ) + u (ch ) − u (cl ) = (1 − β) u + β u (ch ) − β 2 2 4 2 zh 1 − zh l β u (c ) + u (chl ) − u (cl ) = (1 − β) β u (chl ) − β 2 2 4 zl h 1 − zl l β2 h h = (1 − β) u (cl ) + β u (cl ) − β u (c ) + u (cl ) − u (cl ) 2 2 4 2 zl 1 − zl l β = (1 − β) β u (chh ) − β u (c ) + u (chh ) − u (cl ) , 2 2 4
(chh )
(29) (30) (31) (32)
964
When evaluated at the optimal memoryless allocation, concavity of utility necessarily
965
implies that the determinant of the Jacobian is positive J F (ch , ch , κ, ν), G(ch , ch , κ, ν) h l h l
{cji }={yj }
966
= Fch Gch h
l
− Fch Gch l h
{cji }={yj }
> 0.
(33)
Thus, the implicit function theorem applies, chh , chl are differentiable functions of (κ, ν),
967
and partial derivatives of chh , chl with respect to (κ, ν) can be found by differentiating
968
implicitly.
969
From part (i), we have chh , chl > cl which together with the first order conditions (8)-(9)
35
970
yield λhpc,h Fch + λhpc,l Gch ≡ t2 > 0 h
971
λhpc,h Fch + λhpc,l Gch ≡ t1 > 0.
h
l
l
Combining the first order conditions (8)-(9) gives t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) . t2
972
(iii) Recall the first-order conditions of high-income agents imply t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) ⇒ t2 ≡ λhpc,h Fch + λhpc,l Gch ≥ t1 ≡ λhpc,h Fch + λhpc,l Gch , h h l l t2
973
where the inequality uses chh ≥ chl (see Lemma 3 part (i)). The inequality can we written
974
as λhpc,h (Fch − Fch ) ≥ λhpc,l (Gch − Gch ) h
l
l
h
⇔ λhpc,h ≥ λhpc,l , 975
where the last step follows because Gch − Gch ≥ Fch − Fch ≥ 0; the second inequality,
976
Gch − Gch , Fch − Fch ≥ 0 follows from t2 ≥ t1 and
l
l
h
h
h
l
l
Gch − Gch = (1 − β)u (chl ) + β(1 − β) l
h
h
zl h u (cl ) − u (chh ) > 0 for 2
chl ≤ chh
977
such that Fch − Fch ≥ 0. For the first inequality, Gch − Gch ≥ Fch − Fch , it immediately
978
follows Gch + Fch ≥ Fch + Gch . The left-hand side of the inequality is
h
l
l
l
l
h
G c h + Fc h l
979
l
l
h
(chl )
The right-hand side of the inequality is Fc h + G c h = u h
h
(chh )
z h + zl β 2 2 − zh − z l β 2 l + − u (c ) (1 − β)β + . (1 − β) + (1 − β)β 2 2 2 2
Comparing the two expressions results in
G c h + Fc h − Fc h − G c h = u l
l
h
h
(chl )
−u
(chh )
z h + zl β 2 + (1 − β) + (1 − β)β 2 2
because chl ≤ chh
≥0 981
h
zl h 1 − zl l β2 h u (c ) + u (cl ) − u (cl ) = (1 − β) u + β u (cl ) − β 2 2 4 1 − zh l β2 h zh h u (c ) + u (cl ) − u (cl ) + (1 − β) β u (cl ) − β 2 2 4 2 z h + zl β 2 − zh − z l β 2 + − u (cl ) (1 − β)β + . = u (chl ) (1 − β) + (1 − β)β 2 2 2 2
980
h
such that Gch − Gch ≥ Fch − Fch , and λhpc,h ≥ λhpc,l . l
h
h
l
36
(34) (35)
982
A.5
983
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
984
Proof of Theorem 1
derivative of social welfare with respect to κ is
h ∂ch ∂ch ∂chl ∂Vrs 1 h ∂chh = u (ch ) + u (chl ) l − u (cl ) + , ∂κ 4 ∂κ ∂κ ∂κ ∂κ 985
The derivatives of consumption with respect to the signal precision follow from the implicit
986
function theorem which is applicable because according to Lemma 3 part (ii), Fch Gch −Fch Gch >
987
0. Further, in optimum, the following condition holds (see also Lemma 3 part(ii))
h
l
l
h
t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2 988 989
with t1 , t2 > 0 due to chh ≥ chl > cl (see Lemma 3, part (i)). Using this expression to substitute
for u (chl ) − u (cl ) in the derivative of social welfare with respect to κ ∂Vrs 1 h = u (ch ) − u (cl ) ∂κ 4
990
∂chh t1 ∂chl + ∂κ t2 ∂κ
≡ I(ν, κ).
Given that chh > cl , I(ν, κ) ≥ 0 for
∂chh t1 ∂chl + ∂κ t2 ∂κ
≤ 0,
991
and I(ν, κ) < 0 otherwise. Use the implicit function theorem and the binding enforcement
992
constraints F (chh , chl ), G(chh , chl ) as stated in Lemma 3 to receive x2 Fch + x1 Gch ∂chh t1 ∂chl t1 x2 Fchh + x1 Gchh l l + = − ∂κ t2 ∂κ Fc h G c h − F c h G c h t 2 Fc h G c h − F c h G c h h l l h h l l h x F + x G x F + x G h h h h 2 1 2 1 1 cl cl ch ch ⇔= − t1 t2 t2 Fc h G c h − Fc h G c h F c h G c h − Fc h G c h h l l h h l l h
x 2 Fc h + x 1 G c h
x 2 Fc h + x 1 G c h 1 h h h h l l h h ⇔= λpc,h Fch + λpc,l Gch − λpc,h Fch + λpc,l Gch h h l l t2 Fc h G c h − Fc h G c h Fc h G c h − F c h G c h h l l h h l l h
1 h h ⇔= x1 λpc,h − x2 λpc,l t2
993
where λhpc,h , λhpc,l > 0 per assumption, the partial derivatives Fch , Fch and Gch , Gch are given by
994
(29)-(32), and
h
x1 ≡ − 995
x2 ≡
l
h
l
ν(1 − ν) ∂F h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥0 ∂κ [κν + (1 − κ)(1 − ν)]2
ν(1 − ν) ∂G h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥ 0, ∂κ [(1 − κ)ν + κ(1 − ν)]2
996
with x1 , x2 > 0 for ν ∈ [0.5, 1). The first line uses the implicit function theorem to compute the
997
derivatives ∂chh /∂κ, ∂chl /∂κ, the second line collects terms, the third line employs the definitions 37
998
of t1 , t2 and the last line simplifies the third line by collecting terms.
999
In the following, we evaluate I(κ, ν) for ν = 0.5 and ν → 1 and show that I(κ, 0.5) ≤ 0
1000
while limν→1 I(κ, ν) ≥ 0. Existence of ν¯ then follows from continuity of I(ν, κ) for ν ∈ [0.5, 1)
1001
(see Lemma 1).
1002 1003
For ν = 0.5 , x1 = x2 = x > 0. As before, t2 > 0, chh > cl , chh ≥ chl , and the derivative of
social welfare with respect to κ is
1 x h I(0.5, κ) = [u (chh ) − u (cl )] λpc,h − λhpc,l . 4 t2 1004 1005 1006 1007 1008
Thus, whenever λhpc,h ≥ λlpc,l , I(0.5, κ) ≤ 0 which is shown in Lemma 3 part (iii). The strict
inequality λhpc,h > λhpc,l applies when chh > chl for κ ∈ (0.5, 1] and ν ∈ [0.5, 1).
For ν → 1, outside option values of high and low public signal agents converge such that
λhpc,l
= λhpc,h = λhpc , t1 = t2 = t > 0 and chh → chl = ch > cl . The derivative of social welfare with
respect to κ is
λhpc x2 1 h l x1 1 − lim I(ν, κ) = [u (c ) − u (c )] ν→1 4 t x1 h λ 1 − 2κ 1 pc x1 ≥0 ⇔ = [u (ch ) − u (cl )] 4 t (1 − κ)2 1009
where the second line results after using the definitions of x1 , x2 and the inequality in the last
1010
line follows from κ ∈ [0.5, 1] and ch > cl ; social welfare is strictly increasing in κ for κ ∈ (0.5, 1].
1011
A.6
1012
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
1013
Proof of Proposition 1
derivative of social welfare with respect to ν is
h h ∂ch ∂chl ∂Vrs 1 h ∂chh h ∂cl l = u (ch ) + u (cl ) − u (c ) + , ∂ν 4 ∂ν ∂ν ∂ν ∂ν 1014
The derivatives of consumption with respect to the signal precision follow from the implicit
1015
function theorem which is applicable because according to Lemma 3 part (ii), Fch Gch −Fch Gch >
1016
0 holds at the optimal memoryless allocation. Further, in optimum, the following condition holds
1017
(see also Lemma 3 part(ii))
h
l
l
h
t 1 u (chl ) − u (cl ) = u (chh ) − u (cl ) , t2 1018 1019
with t1 , t2 > 0 due to chh ≥ chl > cl (see Lemma 3, part (i)). Using this expression to substitute
for u (chl ) − u (cl ) in the derivative of social welfare with respect to ν yields ∂ch t ∂ch ∂Vrs 1 h 1 l h l = u (ch ) − u (c ) + . ∂ν 4 ∂ν t2 ∂ν
38
1020
Given that u (chh ) − u (cl ) < 0, social welfare is decreasing in ν if
α 1 G c h − α 2 Fc h
l l + λhpc,h Fch ⇔ λhpc,h Fch + λhpc,l Gch h h l F c h G c h − Fc h G c h h
l
l
h
∂ch ∂ch 1 t 2 h + t1 l ≥ 0 t2 ∂ν ∂ν α 2 Fc h − α 1 G c h h h + λhpc,l Gch ≥0 l Fc h G c h − F c h G c h
⇔ (λhpc,h α1 + λhpc,l α2 )
h
l
l
h
h
l
l
h
h
l
l
h
Fch Gch − Fch Gch Fc h G c h − F c h G c h
≥0
⇔ λhpc,h α1 + λhpc,l α2 ≥ 0, 1021
with the partial derivatives Fch , Fch and Gch , Gch given by (29)-(32), t1 , t2 as defined in Lemma
1022
3, and α1 , α2 defined as α1 ≡ −
1023
α2 ≡ − 1024 1025
h
l
h
l
κ(1 − κ) ∂F h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥0 ∂ν [κν + (1 − κ)(1 − ν)]2
κ(1 − κ) ∂G h l = β(1 − β) u(yh ) − Vrs − u(yl ) + Vrs ≥ 0. ∂ν [(1 − κ)ν + κ(1 − ν)]2
The second line uses the implicit function theorem to compute the derivatives ∂chh /∂ν, ∂chl /∂ν,
the third collects terms. The inequality λhpc,h α1 + λhpc,l α2 ≥ 0 is satisfied because λhpc,h , λhpc, > 0
1026
per assumption and the coefficients α1 , α2 are non-negative such that social welfare is decreasing
1027
in ν; social welfare is strictly decreasing in ν for κ ∈ [0.5, 1) because then α1 , α2 > 0.
1028
A.7
1029 1030
Proof of Proposition 2
(i) From the definition of the risk-sharing measure, it follows that risk-sharing decreases in ν whenever ch is increasing in ν. The derivative of ch with respect to ν is α 1 G c h − α 2 Fc h α 2 Fc h − α 1 G c h ∂chh ∂chl 1 l l h h + = + ∂ν ∂ν 2 Fc h G c h − F c h G c h Fc h G c h − F c h G c h l h l l h
h l ⎤h ⎡ 1 α1 Gchl − Gchh + α2 Fchh − Fchl ⎦ = ⎣ ≥0 2 F c h G c h − Fc h G c h
∂ch 1 = ∂ν 2
h
l
l
h
1031
where the inequality follows from Fch Gch − Fch Gch > 0 (see Lemma 3 part (ii)) and
1032
Gch − Gch ≥ Fch − Fch ≥ 0 (see Lemma 3 part (iii)), and risk sharing RS decreases in ν.
1033
(ii) Risk-sharing decreases (increases) in κ whenever ch is increasing (decreasing) in κ. The
1034
h
l
h
h
l
l
h
l
derivative of ch with respect to κ is ∂ch 1 = ∂κ 2
∂chh ∂chl + ∂κ ∂κ
1 = 2
x 2 Fc h + x 1 G c h l
l
Fc h G c h − Fc h G c h h
l
l
h
−
x 2 F ch + x 1 G c h h
h
Fc h G c h − Fc h G c h h
l
l
h
≡
D(κ, ν) . 2
1035
Similar to the proof of the Theorem 1, we evaluate D(κ, ν) for ν = 0.5 and ν → 1 and show
1036
that D(κ, 0.5) ≤ 0 while limν→1 D(κ, ν) ≥ 0. Existence of ν¯ then follows from continuity
1037
of D(ν, κ) for ν ∈ [0.5, 1) (see Lemma 1). For ν = 0.5, it is x1 = x2 = x > 0 and D(κ, 0.5) 39
reads
1038
D(κ, 0.5) = x
Fch + Gch − Fch − Gch l
h
=x
l
h
l
l
h
Fc h G c h − F c h G c h h
l (1 − β)(1 + β zh +z + 2
β2 1 h (1−β) 2 )(u (cl )
− u (chh ))
Fc h G c h − Fc h G c h h
l
l
≥ 0.
h
1039
The denominator is strictly positive (see the proof of Lemma 3 part (ii)) as are the first
1040
two expressions in the numerator. The inequality then follows from concavity of utility
1041
and chh ≥ chl (see Lemma 3 part (i)); for κ > 0.5, chh > chl and D(κ, 0.5) > 0.
1042
For ν → 1, we get chh → chl = ch and thus lim D(κ, ν) =
x 1 G c h − G c h − x 2 F c h − Fc h l
h
h
l
Fc h G c h − F c h G c h
ν→1
h
l
l
= (1 − β)
h
(x1 − x2 )u (ch ) ≤ 0, Fc h G c h − F c h G c h h
l
l
h
1043
The inequality follows because for ν → 1, outside option values of high and low public
1044
signals converge such that chh → chl = ch , Fch → Gch , Fch → Gch and x2 ≥ x1 for l
h
h
l
1045
κ ∈ [0.5, 1]. Thus, risk sharing increase for ν → 1; for κ > 0.5, the strict inequality
1046
applies. Existence of ν¯ follows from continuity of D(ν, κ) for ν ∈ [0.5, 1].
1047
A.8
1048
From Lemma 2 part (iv), clh = cll = cl . Using resource feasibility and that u is Inada, the
1049
Proof of Proposition 3
derivative of social welfare with respect to κ is
h ∂ch ∂ch ∂chl ∂Vrs 1 h ∂chh = u (ch ) + u (chl ) l − u (cl ) + . ∂κ 4 ∂κ ∂κ ∂κ ∂κ 1050
(36)
With λhpc,h > 0 and λhpc,l = 0, optimal allocations are characterized by the first order condition for
1051
optimal substitution between chh and chl and the binding enforcement constraint F (chh , chl , κ, ν).
1052
The first order condition is
1 h u (ch ) − u (chl ) + λhpc,h Fch − Fch = 0, h l 4
1053
with chh ≥ chl > cl such that Fch , Fch > 0 and Fch − Fch ≥ 0. The latter difference can be
1054
re-written such that we get
h
T ≡ 1055
l
h
l
1 h u (ch ) − u (chl ) 1 + λhpc,h 2β(1 − β)zh + β 2 + λhpc,h (1 − β)u (chh ) = 0. 4
(37)
The binding enforcement constraint is h l F (chh , chl , κ, ν) ≡ (1 − β)u(chh ) + β(1 − β)zh Vrs + β(1 − β)(1 − zh )Vrs + β 2 Vrs
− (1 − β)u(yh ) − β(1 − β) [zh u(yh ) + (1 − zh )u(yl )] − β 2 Vout = 0. (38)
40
1056
The first order condition (37) can be also expressed as (see also Lemma 3 with λhpc,l = 0) Fch l u (chl ) − u (cl ) = u (chh ) − u (cl ) , Fc h h
1057
such that the derivative of social welfare (36), can be written as 1 1 h u (ch ) − u (cl ) 4 Fc h
∂chh ∂ch + F ch l h ∂κ l ∂κ
.
Fc h
h
(39)
1058
The derivatives of chh , chl follow again from applying the implicit function theorem in (37) and
1059
(38).
1060
T c h F κ − T κ Fc h ∂chl h h = ∂κ T c h F c h − T c h Fc h
(40)
T κ F c h − T c h Fκ ∂chh l l = . ∂κ T c h F c h − T c h Fc h
(41)
l
h
h
l
l
h
h
l
The partial derivatives are zh 1 − zh l β2 h u (c ) + u (ch ) − u (cl ) > 0 Fch = (1 − β) u (chh ) + β u (chh ) − β h 2 2 4 2 zh 1 − zh l β u (c ) + u (chl ) − u (cl ) > 0 Fch = (1 − β) β u (chl ) − β l 2 2 4 ν(1 − ν) h l − u(yl ) + Vrs ≤0 Fκ = −x1 = −β(1 − β) u(yh ) − Vrs [κν + (1 − κ)(1 − ν)]2 1 h ν(1 − ν) Tκ = u (ch ) − u (chl ) λhpc,h 2(1 − β)β ≤0 4 [κν + (1 − κ)(1 − ν)]2 1 Tch = u (chh ) 1 + λhpc,h 2β(1 − β)zh + β 2 + λhpc,h (1 − β)u (chh ) < 0 h 4 1 Tch = − u (chl ) 1 + λhpc,h 2β(1 − β)zh + β 2 > 0. l 4
1061 1062
The implicit function theorem is applicable because Tch Fch − Tch Fch = 0. Given that u (chh ) − l
h
h
l
u (cl ) < 0 and Fch > 0, the derivative (39) is negative if the following expression is positive h
T κ F ch − T c h Fκ T c h Fκ − T κ Fc h ∂chh ∂ch l l h + Fch l = Fch + Fc h h l ∂κ h T hF h − T hF h h ∂κ l T hF h − T hF h c c c c c c c c
Fc h
= −Fκ 1063
T c h F c h − T c h Fc h l
h
h
l
l
h
h
l
T c h F c h − T c h Fc h
l
h
h
l
l
h
h
l
= −Fκ ≥ 0.
The first line uses (40)-(41) to substitute for ∂chh /∂κ, ∂chl /∂κ, the second line collects terms
1064
and the inequality in the second line follows because Fκ ≤ 0, and social welfare is decreasing in
1065
κ; social welfare is strictly decreasing in κ for ν ∈ [0.5, 1) because then Fκ < 0. The proof that
1066
social welfare is decreasing in ν follows the same steps and is omitted here.
41
1067
A.9
Transition probabilities
1068
There are five transition and conditional probabilities of interest, namely:
1069
• π (s |s) = π (y , k |y, k),
1070
• π (θ |θ) = π (y , k , n |y, k, n).
1071
• π (s |s, n) = π (y , k |y, k, n),
1072
• π (y |s, n) = π (y |y, k, n)
1073
• π (y |s) = π (y |y, k) The conditional probability for income is π (y = yj , k = ym , n = yl , y = yi ) π y = yj |k = ym , n = yl , y = yi = . π (k = ym , n = yl , y = yi ) The denominator can be derived by using the following identity for N income states N π y = yz |k = ym , n = yl , y = yi = 1 z=1
which implies π (k = ym , n = yl , y = yi ) =
N π y = yz , k = ym , n = yl , y = yi . z=1
1074
The elements of the sum on the right hand side are products that depend on z, m, l, i and
1075
the precisions of signals (we treat the current income simply as a yet another signal). It follows
π y = yz , k = ym , n = yl , y = yi = piz κ
1m=z
1−κ N −1
1−1m=z
ν
1l=z
1−ν N −1
1−1l=z
,
1076
where piz is the Markov transition probability for moving from income i to income z. When
1077
the signal is wrong, m = z, we assume that all income states are equally likely. The general
1078
formula for the conditional probability of income with both signals therefore reads pij κ1m=j
π y = yj |k = ym , n = yl , y = yi = N
1−κ N −1
1m=z z=1 piz κ
1079 1080
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν N −1
1−1l=z .
(42)
The conditional probability π(y |y, k) follows from setting ν = 1/N in (42). With both signals following independent exogenous first-order Markov processes, the condi-
42
1081
tional probability for the full state is for all k , n π y = yj , k , n |k = ym , n = yl , y = yi = π(k |k = ym )π(n |n = yl )
1082
pij κ1m=j
1−κ N −1
N 1m=z z=1 piz κ
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν K−1
1−1l=z . (43)
Finally, the formula for π (y , k |k, y) naturally follows and is given for all k by π y = yj , k |k = ym , n = yl , y = yi = π(k |k = ym )
pij κ1m=j
1−κ N −1
N 1m=z z=1 piz κ
1−1m=j
1−κ N −1
ν 1l=j
1−1m=z
1−ν N −1
ν 1l=z
1−1l=j
1−ν N −1
1−1l=z . (44)
1083
For example, with two equally likely i.i.d. income states, the conditional probability of
1084
receiving a low income in the future conditional on a low private high, a high public signal and
1085
a low income today is given according to (42) π y = yl |k = yh , n = yl , y = yl = π y = yl |k = yh , n = yl =
43
(1 − κ)ν (1 − κ)ν + (1 − ν)κ
.