Growth and risk-sharing with private information

Journal of Monetary Economics 47 (2001) 499–521

Growth and risk-sharing with private information$ Aubhik Khana,*, B. Ravikumarb a

Research Department, Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 19106-0066, USA b Department of Economics, Pennsylvania State University, University Park, PA 16802-3306, USA Received 24 September 1997; received in revised form 13 July 2000; accepted 9 August 2000

Abstract We examine the impact of incomplete risk-sharing on growth and welfare. The source of market incompleteness in our economy is private information: a household’s idiosyncratic productivity shock is not observable by others. Risk-sharing between households occurs through long-term contracts with intermediaries. We find that incomplete risk-sharing tends to reduce the rate of growth relative to the complete risksharing benchmark. Numerical examples indicate that the welfare cost and the growth effect of private information are small. r 2001 Elsevier Science B.V. All rights reserved. JEL classification: E20; O16 Keywords: Growth; Long-term contracts; Risk-sharing

$ We would like to thank the participants of the C.V. Starr Center Conference on Skills and Technology, 1997 Society for Economic Dynamics Meetings, 1996 Mid-West Macro Meetings, 1996 NBER Summer Institute, 1996 North American Summer Meetings of the Econometric Society and the seminar participants at The Federal Reserve Banks of Philadelphia and Minneapolis, Pennsylvania State University, University of California-Riverside, University of Iowa, and University of Virginia and, in particular, Fernando Alvarez, Andrew Atkeson, Dean Corbae, Robert King, Young-Sik Kim, Julia Thomas and Cheng Wang for helpful comments. All remaining errors are our sole responsibility. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Philadelphia or of the Federal Reserve System.

*Corresponding author. Tel.: +1-215-574-3905; fax: +1-215-574-4364. E-mail addresses: [email protected] (A. Khan). 0304-3932/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 0 1 ) 0 0 0 5 3 - 8

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1. Introduction Recent research has found evidence that is inconsistent with the full insurance predictions of the complete markets model. For example, Cochrane (1991), Mace (1991) and Hayashi et al. (1996) provide evidence against complete risk-sharing within the US at the individual level; Townsend (1994) and Maitra (1997) reject full insurance across households within Indian villages; and Backus et al. (1992), Baxter and Crucini (1995) and Athanasoulis and Van Wincoop (2000) provide evidence against cross-country consumption risk-sharing. Motivated by this finding, we consider the effect of risk on growth and welfare. We develop an environment where household production is subject to idiosyncratic shocks which are private information, and growth is endogenous. The assumption of private information provides a basis for market incompleteness; the resulting problem of incentive compatibility eliminates the possibility of complete risk-sharing. Households share risk through long-term contracts with competitive intermediaries. The enduring relationship allows intermediaries to exploit intertemporal trade-offs, thereby providing (partial) insurance. Previous work on risk and growth typically has contrasted complete risksharing with autarchy. Since the extent of market incompleteness is endogenous in our environment, we are able to examine an intermediate case. We find that the presence of uninsurable risk reduces the rate of growth relative to the complete risk-sharing benchmark. This, for example, differs from the result in Devereux and Smith (1994). Comparing autarchy with complete risksharing, in a model of capital risk which essentially shares our technology and preferences, they find that the effect of risk on savings, and hence growth, is ambiguous.1 Our results indicate that the impact of risk on growth and welfare is likely to be sensitive to the origin of market incompleteness and the types of insurance arrangements allowed. In related work, Marcet and Marimon (1992) examine a two-agent model with capital accumulation where a risk-neutral investor with unlimited resources invests in the technology of a risk-averse producer whose output is subject to productivity shocks which are private information.2 Our work extends their analysis to a market-clearing economy with endogenous growth. In contrast to Marcet and Marimon, we find that investment, as well as consumption, is affected by shocks to production. As a result, there are growth 1 The ambiguous effect of risk on savings was noted by Levhari and Srinivasan (1969). Specifically, when the elasticity of intertemporal substitution is low (high), risk tends to raise (reduce) savings. (See also Weil, 1990.) Obstfeld (1994) shows that when risk-sharing leads to a portfolio shift into riskier, more productive assets, it may be growth promoting. 2

See also Aiyagari and Williamson (1997) for a model of credit in which only the social planner has access to capital.

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effects of private information. However, numerical examples indicate that, on average, the growth and welfare effects of incomplete risk-sharing are likely to be small. In Section 2 we describe technology, preferences and the contract. Section 3 solves the contract assuming logarithmic utility, while Section B of the appendix examines the contract when utility is iso-elastic. Numerical examples are presented in Section 4; these provide quantitative measures of the size of the growth and welfare effects resulting from private information. Section 5 discusses additional applications of our model. In particular, there are several interesting differences between our long-term contracting economy with production and the more standard model of contracts with risky endowments. These may be of independent interest.

2. The environment In each period, there is a large number of households each of which operates a technology of the form Yt ¼ zt Kt where Yt is output, Kt is capital, and zt is the level of productivity at time t ¼ 0; 1; y . Productivity, which is independently and identically distributed across households at any time, and over time for any household, takes on one of two possible values: it is zi with probability mi > 0, i ¼ 1; 2, where 0oz1 oz P2 and m1 þ m2 ¼ 1. We define the expected value of productivity as x ¼ 2i¼1 mi zi and assume that capital completely depreciates after production. Households are infinitely lived, and possess time separable preferences over sequences of consumption with period utility from current consumption, C, of the form ( 1@s ð1@bÞC1@s for s > 0 and sa1; vðCÞ ¼ ð1@bÞlog C for s ¼ 1: Note that, for convenience, we normalize the utility function by ð1@bÞAð0; 1Þ where future utility is discounted by b. Each household participates in a permanent contract with a risk-neutral, competitive intermediary. As we will characterize the solution to the contract using a functional approach, we find it useful to provide a recursive description of the timing of events within the contractual arrangement here.3 In any period, only the household observes its own productivity, thus there is private information with respect to output. At the beginning of the period the household has a predetermined capital stock, K. Given the household’s capital, the intermediary announces a set of potential transfers, Bi , and investments, 3

However, a formal statement of the contractual arrangement as a sequence problem is contained in Section C of the appendix.

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Ki0 , as functions of the impending productivity report. Upon observing its output, zi K, the household determines a report for the intermediary. Subsequently, the intermediary executes the transfer, and implements the investment for the household, which determines its capital stock at the onset of the next period. By definition, if the contract is incentive-compatible, then, at every point in time, the household will truthfully report the level of productivity. Hence, the household’s consumption in state i will be Ci ¼ zi K þ Bi and its investment will be Ki0 . As is evident from above, the intermediary observes the household’s investment but not its consumption. The intermediary can borrow from, or lend to, other intermediaries at a competitive interest rate. This yields a constant discount factor, qAð0; 1Þ, that is determined by equilibrium conditions. In the initial period, a large number of intermediaries compete to establish permanent contracts with households. As a result of perfect competition, intermediaries’ initial expected discounted profits for each contract are driven to zero; consequently, each household accepts a contractual arrangement that yields it the highest feasible level of lifetime utility. Our approach to solving the contract adapts the methods used to characterize the risky endowment model of long-term contracts.4 An important assumption in extending the existing results to our analysis is that the household has no ability to invest in an unobservable manner. The value of misreporting productivity lies in being able to consume hidden output. A parable may be helpful. Let the stock of capital represent trees and let the fruit of these trees be consumed or planted to yield fruit-bearing trees for the next period. Our assumption on the observability of investment is equivalent to assuming that the intermediary is able to observe the quantity of trees, at any point in time, but not the yield per tree. A household may hide output, fruit, from the intermediary. This hidden fruit may be consumed. However, if the household chooses to plant any of it, then it is observable, in the form of trees, in the next period. This allows the intermediary to infer the household’s deception. In order to ensure truth-telling (incentive-compatibility), we constrain the contract so there are no gains from one-period temporary deviations from truth-telling. That is, the contract is temporarily incentive compatible (t.i.c.) in the sense of Green (1987). Provided certain boundary conditions, satisfied by our problem, hold, temporary incentive compatibility is equivalent to incentive compatibility. Next, since the t.i.c. constraints introduce future expected lifetime utility as a state variable, we follow Green in characterizing the contract by solving a dual, expenditure minimization problem for the 4

See for example Green (1987), Taub (1990), Phelan and Townsend (1991), Atkeson and Lucas (1992) and the related analysis of Spear and Srivastava (1987) and Thomas and Worrall (1990).

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intermediary. Standard duality theorems ensure that this solution also solves the utility maximization problem faced by household.5 Finally, we impose an aggregate resource constraint upon our economy: the sum of consumption and investment cannot exceed output. This is related to the approach taken by Atkeson and Lucas (1992) in the context of an endowment economy. However, in contrast to their work, we solve for equilibrium, not efficient, allocations. Let Ui0 represent expected lifetime utility, starting next period, for the household, assuming that it will accurately report productivity from that date onward, given a current productivity report of zi . When the state is z1 , temporary incentive compatibility is ensured by the following constraint: if z1 K þ B2 > 0 then vðz1 K þ B1 Þ þ bU10 Xvðz1 K þ B2 Þ þ bU20 :

ð1Þ

The left-hand side of (1) represents the value to the household with actual output z1 K of truthfully reporting its productivity. Provided that misreporting the level of productivity generates a feasible level of consumption, then the right-hand side of the constraint represents the value of following this strategy. The t.i.c. constraint when productivity is z2 is given below. vðz2 K þ B2 Þ þ bU20 Xvðz2 K þ B1 Þ þ bU10 :

ð2Þ

Note that, as z2 > z1 , non-negativity of C1 ensures that z2 K þ B1 X0, eliminating the need for a conditional constraint. As discussed in Oh and Green (1992), concavity of v implies that if both (1) and (2) are to hold, then B1 XB2 and U10 pU20 . Furthermore, if (2) binds and B1 XB2 ðB1 > B2 Þ then (1) is satisfied (holds with inequality). These results will prove useful below. As indicated earlier, we obtain equilibrium allocations for the contracting economy using a dual approach. Given an initial utility entitlement, U, and capital stock, K, for the household, the intermediary solves an expenditure minimization problem. Hereafter, we will refer to the solution of the expenditure minimization problem as the contract. In this formulation, we must impose a promise-keeping constraint upon the contract which ensures that the household’s expected lifetime utility satisfies its initial entitlement. Up

2 X

mi ðvðzi K þ Bi Þ þ bUi0 Þ:

ð3Þ

i¼1

Let the expected present value of expenditure be EðU; KÞ. Recall that the intermediary discounts future expenditures by q. The contract, which minimizes the intermediary’s net expenditure, by choice of ðBi ; Ki0 ; Ui0 Þ2i¼1 , subject to the incentive compatibility constraints (1)–(2) and the promise 5

Proofs of these results are contained in Sections C:1 and C:2, respectively, of the appendix.

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keeping constraint (3), satisfies the following Bellman equation: 2 X mi ðBi þ Ki0 þ qEðUi0 ; Ki0 ÞÞ: EðU; KÞ ¼ min

ð4Þ

i¼1

The expected present value of expenditure, at the optimum, will equal the sum of expected current expenditure and the discounted expected present value of expenditures incurred from the next period onwards. A competitive intermediary must maximize the household’s expected lifetime utility; in equilibrium, this implies an initial level of expected lifetime utility, U0 , given the household’s starting stock of capital, K0 , such that the zero profit condition EðU0 ; K0 Þ ¼ 0 is satisfied. This zero profit condition is required only at the beginning of each long-term contractual arrangement. As competition between intermediaries is for permanent contracts with households, we cannot impose this condition in subsequent periods. However, we will establish the result that, in equilibrium, the intermediary’s expected profits are indeed zero in every period.

3. Analysis In this section we provide a complete characterization of the private information economy for the case of logarithmic preferences. The case of general iso-elastic utility is similar, and is summarized in Section B of the appendix. 3.1. The contract In order to solve the Bellman equation, we deflate the value function by capital. This allows us to reformulate the contract in an intensive form which, by exploiting a homogeneity property of the constraint correspondence that describes the set of feasible allocations, reduces the dimension of the state vector. Let bi K ¼ Bi and gi K ¼ Ki0 for i ¼ 1; 2. Rewrite the objective as 2 X mi ðbi þ gi þ qgi EðUi0 ; Ki0 Þ=Ki0 Þ: ð5Þ EðU; KÞ=K ¼ min i¼1

Now define a composite state variable, u  U@log K. Consistency requires that the future state, conditional on i, is given by u0i ¼ Ui0 @log Ki0 . Since this implies that Ui0 ¼ u0i þ log gi þ log K, the t.i.c. constraint at z1 may be revised as if z1 þ b2 X0 then ð1@bÞlogðz1 þ b1 Þ þ bðu01 þ log g1 Þ Xð1@bÞlogðz1 þ b2 Þ þ bðu02 þ log g2 Þ ð6Þ

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while the t.i.c. constraint at z2 is equivalent to ð1@bÞlogðz2 þ b2 Þ þ bðu02 þ log g2 ÞXð1@bÞlogðz2 þ b1 Þ þ bðu01 þ log g1 Þ: ð7Þ Subtracting log K from both sides of (3), the promise-keeping constraint becomes up

2 X

mi ðð1@bÞlogðzi þ bi Þ þ bðu0i þ log gi ÞÞ:

ð8Þ

i¼1

Since the constraints above depend only upon the composite state variable, we are able to define WðuÞ ¼ EðU; KÞ=K. The intensive form problem, which describes expenditures per unit capital, satisfies the following Bellman equation: WðuÞ ¼ min

2 X

mi ðbi þ gi þ qgi Wðu0i ÞÞ

ð9Þ

i¼1

where the minimization is with respect to ðbi ; gi ; u0i Þ2i¼1 . We now analyze the intensive form contract. Let l and y be the multipliers for the constraints (7) and (8). We suppress (6) which never binds, as is shown below in Proposition 1. The first order conditions, with respect to ðbi ; gi ; u0i Þ2i¼1 , are listed below.6 lð1@bÞ ym1 ð1@bÞ m1 þ @ ¼ 0; ð10Þ z 2 þ b1 z 1 þ b1 lð1@bÞ ym2 ð1@bÞ m2 @ @ ¼ 0; z 2 þ b2 z 2 þ b2

ð11Þ

b m b m1 ð1 þ qWðu01 ÞÞ þ l @y 1 ¼ 0; g1 g1

ð12Þ

b m b m2 ð1 þ qWðu02 ÞÞ@l @y 2 ¼ 0; g2 g2

ð13Þ

m1 qg1 W 0 ðu01 Þ þ lb@ym1 b ¼ 0;

ð14Þ

m2 qg2 W 0 ðu02 Þ@lb@ym2 b ¼ 0:

ð15Þ

The Benveniste–Scheinkman theorem implies W 0 ðuÞ ¼ y. Conditions (10)–(15) allow a strong characterization of the risk-sharing contract. Firstly, the introduction of productive capital offers a channel for adjusting utility entitlements absent in the endowment model. The linear 6

Standard methods (Stokey et al., 1989) may be used to show that the value function is unique, strictly increasing, convex and differentiable.

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production structure implies that utility entitlements, U, are linear functions of the logarithm of the capital stock, K. As a result, the continuation value of the state variable, u0i , is independent of productivity and the initial state, u. Changes in expected lifetime utility which occur in response to productivity reports are implemented through changes in the household’s stock of capital. Secondly, risk-aversion on part of the household implies that the contract insures current consumption: when the household reports low productivity the net transfer is higher than when it reports high productivity (b1 > b2 ). Alternatively, repayment is lower. However, the presence of private information limits the extent of risk-sharing. Households must be prevented from under-reporting income during periods when income is relatively high. As a result, reports of low productivity reduce lifetime consumption. Given diminishing marginal utility, the cost minimizing intermediary will spread this fall in lifetime consumption over time. Consequently, our third result is that low productivity results in both lower current consumption and reduced investment (c1 oc2 and g1 og2 ). These qualitative characteristics of the contract are summarized in the following proposition. (All proofs are in Appendix A.) Proposition 1. In the log case; u01 ¼ u02 ; g1 og2 ; b1 > b2 and c1 oc2 . The higher transfer when z ¼ z1 , given the binding incentive constraint at z2 , implies that the t.i.c. constraint at z1 does not bind, as assumed above. 3.2. Equilibrium As noted earlier, since E is strictly increasing in U given K, the zero profit condition EðU; KÞ ¼ 0 will determine the highest level of (initial) expected lifetime utility feasible for the household given its (initial) stock of capital, K. Since EðU; KÞ ¼ KWðU@log KÞ, this zero profit condition implies WðuÞ ¼ 0. Given strict monotonicity of W, this pins down u and implies that U is proportional to log K. Hence u and the contract ðbi ; gi ; u0i Þ2i¼1 are the same for all households. As a result, any household with capital stock K and productivity zi will be allocated current consumption ðzi þ bi ÞK and investment gi K. Average output for all households with K units of capital will be xK, assuming a positive for this P measure of such households; average consumption P group will be 2i¼1 mi ðzi þ bi ÞK and average investment will be 2i¼1 mi gi K. Economy-wide market clearing requires that aggregate output equal the sum of aggregate consumption and investment. Let cðKÞ represent the distribution of capital across households over the space of current capital holdings, K. Equilibrium requires that ! Z Z 2 X xKcðKÞ ¼ mi ðzi þ bi þ gi Þ KcðKÞ: K

K

i¼1

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P2 This market-clearing condition implies i ðbi þ gi Þ ¼ 0. Next, using P2 that i¼1 m (9), we have the result, WðuÞ ¼ i¼1 mi qgi Wðu0i Þ. Recalling u01 ¼ u02 and WðuÞ ¼ 0, this implies Wðu0i Þ ¼ WðuÞ ¼ 0; i ¼ 1; 2. Thus, u0i ¼ u; i ¼ 1; 2, and EðU; KÞ ¼ 0 in all states along the equilibrium path. This establishes the important result that zero expected profits are a period by period feature of our risk-sharing agreement. Hence the initial zero-profit condition is sustained over time in the equilibrium contract. Finally, (12) and (14) yield the equilibrium condition qy ¼ 1. Note that the recursive equilibrium is stationary in the sense that ðbi ; gi Þ2i¼1 ; q and u are timeinvariant. This verifies our earlier conjecture that q is constant. We now contrast growth between our incomplete risk-sharing economy and the complete risk-sharing benchmark. The latter, a well-known problem, may be retrieved by suppressing (7) (setting l ¼ 0 everywhere) and repeating the above analysis. The solution, denoted by superscript f , is characterized by cfi ¼ ð1@bÞx and gfi ¼ bx for i ¼ 1; 2. Furthermore, under complete risksharing qf x ¼ 1 and uf ¼ logð1@bÞðbb xÞ1=ð1@bÞ . The introduction of private information reduces the mean rate of P2 growth, m g , relative to the complete risk-sharing value of bx. i i i¼1 As a result, the intermediary’s discount rate, q@1 @1 falls. We suggest the following explanation. If, upon observing z2 , the household truthfully reports the productivity then it consumes C2 , while misrepresentation yields consumption equal to ðz2 @z1 ÞK þ C1 . All else being equal, higher levels of capital tend to increase the current gains to deviations from truth-telling. The contract then requires larger variations in both Ci and Ui0 in order to ensure incentive-compatibility. Given convexity of preferences, this tends to reduce welfare for any given level of resources. This welfare reducing aspect of additional capital makes investment less attractive in the private information economy relative to the complete risk-sharing economy. Hence the overall rate of capital accumulation is lower under private information. Proposition 2. In equilibrium; q > x@1 and

P2

i¼1

mi gi obx.

P2 We calculate the expected increase in lifetime utility as i¼1 P mi ðUi0 @UÞ ¼ 2i¼1 mi log gi . Proposition 2 and Jensen’s Inequality jointly imply that the expected increase in welfare is lower under private information. However, this does not imply that welfare has a negative trend leading to the immiserization of almost all households. This result, due to the possibility of economic growth, is in sharp contrast to the endowment model. Since we cannot assume that the equilibrium allocation is efficient, it is important to emphasize that the lack of a negative trend may not characterize the efficient allocation.

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Table 1 Baseline parameters z1

z2

l1

b

r

0.9585

1.1715

0.5

0.9577

1.0

4. Numerical examples We examine several numerical examples. These allow us to describe the risk sharing arrangement in more detail and obtain preliminary evidence on the magnitude of the growth and welfare effects of the incomplete risk-sharing environment. The baseline parameter values we use are in Table 1. The average level of productivity is set equal to the long run return on equity in the US, x ¼ 1:065, as indicated in Mehra and Prescott (1985). We allow productivity to vary symmetrically around its mean. Thus we assume that m1 ¼ 0:5 and x ¼ x@z1 ¼ z2 @x. The parameter x is difficult to calibrate. In our baseline case we set its value to imply that the coefficient of variation of z is 0:1. This value implies a standard deviation of consumption growth of 0:0468, which is close to 0:044 predicted by the base case of Heaton and Lucas (1996, Table 4, p. 458).7 Finally we choose b so that bx ¼ 1:02. The aggregate rate of growth for the complete risk-sharing economy, when household preferences are logarithmic, matches the long run growth data, as documented in Parente and Prescott (1993). This is also the average rate of growth under autarchy, and, as we shall see, not significantly different from the rate of growth under incomplete risk-sharing. We first examine the case of logarithmic preferences. Across the three different allocations, autarchy (A), incomplete risk-sharing (I) and complete risk-sharing (C), u þ log K represents the level of expected lifetime utility for a household with capital K. Thus u is the expected lifetime utility for a household with one unit of capital. Each entry in the rows of Tables 2–5 marked loss represents the percentage decrease necessary in the level of consumption under complete risk-sharing, at every point in time, to match the level of welfare associated with the other economies. We deflate all quantity variables by the level of capital. Thus, given a shock zi , the household’s savings is @bi and ci is consumption, per unit capital. Investment per unit capital is denoted gi , which is also the gross rate of growth of capital. The average rate of growth is denoted EðgÞ, while r is the percentage discount rate ðq ¼ 1=ð1 þ rÞ). Finally, DU represents the expected increase in lifetime utility. In Table 2 we see that, in the complete risk-sharing allocation, consumption and investment are unresponsive to the productivity shock. The household’s 7

Below, we will examine examples involving different values of x.

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u Loss (%) b1 b2 c1 c2 g1 g2 EðgÞ DU

A

I

C

@2.7712 11.2000 @0.9180 @1.1220 0.0405 0.0495 0.9180 1.1220 1.0200 0.0148

@2.6687 1.6000 @0.9138 @1.1248 0.0447 0.0467 0.9809 1.0577 1.0193 0.0184

@2.6522 0.0000 @0.9135 @1.1265 0.0450 0.0450 1.0200 1.0200 1.0200 0.0198

Table 3 Varying the coefficient of risk aversion r: Growth (%) Loss (%)

C I A I A

0.5

0.75

1

2

3

4.03 3.93 3.77 2.57 9.77

2.67 2.58 2.54 1.96 9.79

2.00 1.93 2.00 1.63 11.21

0.99 0.96 1.50 0.83 18.45

0.66 0.64 1.67 0.48 25.99

4 0.50 0.48 2.01 0.31 33.40

Table 4 Varying the coefficient of variation

Growth (%) Loss (%)

Coeff. var.:

0.05

0.10

0.20

C I A I A

2.00 1.96 2.00 0.79 2.92

2.00 1.93 2.00 1.63 11.21

2.00 1.90 2.00 2.93 38.31

savings varies with productivity so as to completely smooth the consumption profile. The incomplete risk-sharing economy induces fluctuations in current consumption, but this variability in consumption is low relative to that under autarchy. There is a net transfer of resources from households with high current productivity to those with low current productivity: c1 þ g1 > z1 while c2 þ g2 oz2 . For those experiencing below average productivity, this reduces savings, while boosting both consumption and investment, relative to autarchy.

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Table 5 Varying the discount factor b: Growth (%) Loss (%)

C I A I A

0.9390

0.9577

0.9765

0.000 @0.001 0.000 1.65 7.90

2.000 1.930 2.000 1.63 11.21

4.000 3.970 4.000 1.50 19.25

The residual variability in consumption, and the reduced average growth rate, causes expected welfare to increase more slowly than under complete risksharing, DU ¼ 0:0184o0:0198. The inability to smooth consumption under autarchy implies high variability in both consumption and investment rates. Consequently, welfare increases yet more slowly, DU ¼ 0:0148. In Fig. 1, we illustrate initial expected lifetime utility for the complete risksharing, incomplete risk-sharing and autarchy economies. As indicated by the loss measures in Table 2, the move from complete to incomplete insurance is equivalent to a 1.6% decrease in the level of consumption, while autarchy implies an 11.2% decrease. In this example, we see that incentive compatible arrangements are relatively successful in smoothing consumption. The switch from such an economy to autarchy results in a significant loss in expected utility, measured in units of full insurance consumption, for the typical household. In Figs. 2 and 3 we graph the evolution of the distribution of capital ðKÞ, deflated by the compounded growth factor, and expected lifetime utility ðUÞ for the incomplete risk-sharing economy. All households are initially identical. Recall that the intermediary enforces truth-telling by offering relatively higher lifetime utility entitlements for high productivity reports than for low productivity reports at each point in time. As a result, both distributions of wealth and utility, are characterized by increasing dispersion over time. For this example, the distribution of utility entitlements within each period is symmetric. Convexity of preferences then implies a skewed distribution of capital. In Proposition 1 we showed that, in the private information economy, changes in welfare are implemented through changes in capital. This log-linear mapping is also, of course, present in the autarchic model. The greater variability in investment present in the autarchy model implies that the private information economy dampens dispersion over time relative to autarchy. Next, in Table 3, maintaining our other baseline parameters, we allow the coefficient of relative risk aversion, s, to vary between 12 and 4. We find that the rate of growth under private information is consistently below the complete risk-sharing equivalent. This result, which we have found to be robust,

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Fig. 1. The primal value functions.

indicates that the growth reducing effect of incomplete risk-sharing, found for logarithmic preferences, extends to the case of iso-elastic utility. Interestingly, both the growth and welfare effects of private information fall as s rises. Recall that higher values of s are associated with increased reluctance to substitute consumption across time. As shown in Proposition 1, potential deviations from truth-telling raise current consumption at the expense of future consumption. As s increases, the attractiveness of such behaviour is reduced. This reduces the costs of private information and shifts the incomplete risk-sharing allocation closer to full insurance. For all s, the contract is relatively efficient. Even when the intertemporal elasticity of substitution is high (s ¼ 0:5), the loss is only 2.57%. Note that, for the same s, the loss under autarchy is more than three times as large, 9.77%. For higher values of s, autarchy yields larger welfare losses relative to incomplete risk-sharing. Furthermore, under autarchy, income uncertainty generates a strong motive to self-insure through savings when risk aversion is large. This drives the high rates of growth relative to complete risk-sharing. As is well known, the sign of the risk effect on savings changes when s crosses one. In contrast, the growth rate under incomplete risksharing is always below that under complete risk-sharing, and the growth effects are small.

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Fig. 2. The distribution of capital.

Fig. 3. The distribution of utility entitlements.

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In Table 4, we vary the coefficient of variation of z. This implies changes in z1 and z2 . All other parameters are maintained at the values listed in Table 1. Higher variability of productivity implies higher risk and tends to reduce both growth and welfare under private information. However, the differences in rates of growth never rise above one-tenth of 1% and the associated welfare loss is small relative to autarchy. Table 5 considers changes in the discount factor, b, while maintaining all other parameters at the Table 1 values. The three discount factors we consider, b ¼ 0:9390, 0:9577 and 0:9765 imply 0%, 2% and 4% average growth, respectively. Note that higher values of b imply an increased emphasis on future consumption. As indicated by the negative trend in loss, the welfare cost of private information decreases. These numerical examples indicate that, across a range of parameter values, (1) the growth effects of incomplete risk-sharing are small and (2) the incomplete markets economy achieves levels of welfare close to the levels attained under complete risk-sharing. The relative efficiency of the private information economy arises from the ability to adjust capital, and hence output, in response to the changes necessary in lifetime utility entitlement over time. This implies that changes in a household’s utility entitlement are matched by proportionate movements in the gain from understating productivity, ðz2 @z1 ÞK.

5. Concluding remarks We have examined the impact of incomplete risk-sharing on growth and welfare, in an environment with private information. In our economy, households share risk by entering into enduring relationships with competitive intermediaries. We have found that the aggregate growth rate is lower under private information than under full insurance. Furthermore, the risk-sharing arrangement, while incomplete, is relatively efficient and the growth effects of private information are generally small. Our work adapts the methods used to study long-term contracting with risky, unobservable endowments to an economy with production and capital accumulation. The contract with capital exhibits several properties which contrast with the standard model.8 First, expected lifetime utility, while growing more slowly than under complete risk-sharing, does not necessarily contain a negative trend. Second, the contract exhibits the property that all changes in welfare are implemented through changes in the household’s stock of capital. Consequently, welfare always exceeds the autarchy value of capital. 8 Examples, already mentioned above, include Green (1987) and the equilibrium model of Atkeson and Lucas (1992). As noted earlier, the properties that distinguish our model of risksharing with production and capital accumulation may not hold in efficient allocations.

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Finally, while both the endowment and production economies share the property that the distribution of wealth or utility entitlements is characterized by increasing dispersion, in the production economy this rising inequality is larger under autarchy.9 The contract implements risk-sharing by conditioning the household’s future lifetime utility, or wealth, on the current report of productivity. Thus we emphasize the problem of unobservable returns to investment, the common emphasis of the literature on private information in development economics. If investment were itself unobservable, then our risk-sharing arrangement would be infeasible.10 In particular, the intermediary cannot exploit differences in the rates of intertemporal substitution across households. It is, however, unclear what types of risk-sharing arrangements are then feasible. We view this as an area for future research. An implication of our findings is that if resources may be devoted towards either (1) reducing the effects of informational asymmetries and thereby implementing improved insurance services (allowing for observable returns to investment) or (2) developing the legal basis for implementing state-contingent enforceable contracts (allowing for observable investment), such as those we have assumed, then expenditures on the latter may be far more important for welfare gains.

Appendix A A.1. Proofs Proof of Proposition 1. We divide the proof into 5 parts. (1) u01 ¼ u02 ¼ u0 : Eqs. (12) and (14) jointly imply that 1 þ qWðu01 Þ ¼ qW 0 ðu01 Þ while (13) and (15) together yield 1 þ qWðu02 Þ ¼ qW 0 ðu02 Þ. Thus we see that u01 ¼ u02 and this common value, labeled u0 is independent of u and thus common to all contracts. (2) l > 0: (By contradiction) Given part (1), assume that l ¼ 0. Next, from (12) and (13) we have g1 ¼ g2 while (10) and (11) yield z1 þ b1 ¼ z2 þ b2 . Since this implies that z2 þ b1 > z2 þ b2 we have violated (7). (3) g1 og2 : Given parts (1) and (2), we may solve (14) and (15) to obtain     y@ðl=m1 Þ y þ ðl=m2Þ g1 ¼ b ob ¼ g2 : qW 0 ðu0 Þ qW 0 ðu0 Þ

9 See Aiyagari and Alvarez (1995) for an interesting example of an endowment economy where lower bounds on the consumption possibilities set ensure that the economy is characterized by an invariant distribution of wealth. 10

See Cole and Kocherlakota (1999) for an economy with risky endowments and unobservable storage.

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(4) b1 > b2 : Given parts (1)–(3), we know that b log g2 @b log g1 > 0 which requires that ð1@bÞðlogðz2 þ b2 Þ@logðz2 þ b1 ÞÞo0 for (7) to hold with equality. (5) c1 ¼ z1 þ b1 oz2 þ b2 ¼ c2 : Given part (2), rearranging (10) and (11) we have m1 ð1@yð1@bÞ=ðz1 þ b1 ÞÞ ¼ @lð1@bÞ=ðz2 þ b1 Þo0 while m2 ð1@yð1@bÞ=ðz2 þ b2 ÞÞ ¼ lð1@bÞ=ðz2 þ b2 Þ > 0. This requires that z1 þ b1 oz2 þ b2 . & Proof of Proposition 2. As W 0 ðu0i Þ ¼ y and qy ¼ 1, (14) and (15) may be solved as g1 ¼ yb@lb=m1 and g2 ¼ yb þ lb=m2 . Next (10) and (11) may be rearranged as z 1 þ b1 ¼ m1 yð1@bÞ; m1 ðz1 þ b1 Þ þ lð1@bÞ z 2 þ b1 m2 ðz2 þ b2 Þ@lð1@bÞ ¼ m2 yð1@bÞ: It then follows that

X2 z 1 þ b1 m ðb þ gi Þ@lð1@bÞ 1@ ¼ y@x: i¼1 i i z 2 þ b1 Given Proposition 1, we know that l > 0, so that, as z1 oz2 , we know z 1 þ b1 that lð1@bÞ 1@ > 0. Recalling the equilibrium condition z 2 þ b1 P2 mi ðbi þ gi Þ ¼ 0, we have proven yox. Therefore qx > 1 and Pi¼1 2 i¼1 mi gi ¼ byobx. & Appendix B B.1. Iso-elastic preferences We solve the iso-elastic case, drawing heavily on the analysis of Section 3. The intensive form composite state variable, for this case, is given by u ¼ U=K 1@s . The contract is determined by solving (9) subject to ð16Þ–ð18Þ. ðz1 þ b1 Þ1@s þ bg1@s if z1 þ b2 X0 then ð1@bÞ u01 1 1@s X ð1@bÞ ð1@bÞ

ðz1 þ b2 Þ1@s þ bg1@s u02 ; 2 1@s

ðz2 þ b2 Þ1@s ðz2 þ b1 Þ1@s 0 þ bg1@s þ bg1@s u Xð1@bÞ u01 ; 2 2 1 1@s 1@s

ðB:1Þ ðB:2Þ

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up

ðzi þ bi Þ1@s 1@s 0 þ bgi m ð1@bÞ ui : i¼1 i 1@s

X2



ðB:3Þ

Suppressing (B.1) which, as before, does not bind, defining l to be the multiplier for (B.2) and y the multiplier for (B.3), we derive the following efficiency conditions with respect to ðbi ; gi ; u0i Þ2i¼1 . m1 þ l1 ð1@bÞðz2 þ b1 Þ@s @m1 yð1@bÞðz1 þ b1 Þ@s ¼ 0;

ðB:4Þ

m2 @l1 ð1@bÞðz2 þ b2 Þ@s @m2 yð1@bÞðz2 þ b2 Þ@s ¼ 0;

ðB:5Þ

0 @s 0 m1 ð1 þ qWðu01 ÞÞ þ l1 bg@s 1 ð1@sÞu1 @m1 ybg1 ð1@sÞu1 ¼ 0;

ðB:6Þ

0 @s 0 m2 ð1 þ qWðu02 ÞÞ@l1 bg@s 2 ð1@sÞu2 @m2 ybg2 ð1@sÞu2 ¼ 0;

ðB:7Þ

@m1 ybg11@s ¼ 0; m1 qg1 W 0 ðu01 Þ þ l1 bg1@s 1

ðB:8Þ

¼ 0: m2 qg2 W 0 ðu02 Þ@l1 bg21@s @m2 ybg1@s 2

ðB:9Þ

It is straightforward to show that Proposition 1 holds for the general isoelastic case. Furthermore, equilibrium in the economy with iso-elastic preferences may be calculated using the method described in Section 3.2. An examination of the growth effects of private information given iso-elastic utility, which requires numerical methods, is contained in Table 3 and discussed in Section 4.

Appendix C. The contractual arrangement In this section, we adapt two standard results on incentive compatibility and duality which allow us to solve the contract using a recursive dual approach. Recall that an individual’s productivity in period t is zt AZ ¼ fz1 ; z2 g, let Z  2Z be the complete s-algebra and define m : Z-½0; 1 as a probability measure on Z, summarized by mðzi Þ ¼ mi > 0; i ¼ 1; 2. The contract requires that the agent report his productivity in each period. A reporting strategy is Z ¼ ðZ0 ; Z1 ; yÞ such that Zt : Ztþ1 -Z is the time t report of productivity by the agent. Let S represent the set of all measurable reporting strategies. Let Bt ðZt Þ : Ztþ1 -R be the net transfer to the agent as a function of his reported history of productivity shocks and Ktþ1 ðZt Þ : Ztþ1 -Rþ be investment in the agent’s technology for the next period. Output at time t is zt Kt ðZt@1 Þ and consumption at time t is Ct ðzt ; Zt Þ ¼ zt Kt ðZt@1 Þ þ Bt ðZt Þ. We adopt the convention that Z@1 ¼ | and K0 ðZ@1 Þ ¼ K0 ARþ is given. A plan is a sequence N of vector-valued functions ðK; BÞ, where K ¼ fKt gN t¼0 and B ¼ fBt gt¼0 , which allows for a feasible level of consumption at every point in time, in every state of nature: zt Kt ðzt@1 Þ þ Bt ðzt ÞX0; 8zt AZtþ1 ; t ¼ 0; 1; y. Let

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PðK0 Þ be the set of all plans and let SðK; BÞDS be the set of all measurable reporting strategies which allow for feasible consumption given a plan ðK; BÞAPðK0 Þ.11 The expected lifetime utility from a plan ðK; BÞAPðK0 Þ and a reporting strategy gASðK; BÞ is denoted VðK; B; ZÞ. Z XN s VðK; B; gÞ ¼ b vðzs Ks ½Zs@1 ðzs@1 Þ þ Bs ½Zs ðzs Þ Þmsþ1 ðdzs Þ: s¼0 Zsþ1

A plan ðK; BÞAPðK0 Þ is incentive compatible if VðK; B; zÞXVðK; B; gÞ;

8gASðK; BÞ:

ðC:1Þ

Under the assumption that the intermediary’s expenditures will be discounted at a constant rate q@1 @1 Z XN s E0 ðK; BÞ  q ½Ksþ1 ðzs Þ þ Bs ðzs Þ msþ1 ðdzs Þ: ðC:2Þ s¼0 Zsþ1

is the discounted cost implied by the plan ðK; BÞAPðK0 Þ: Definition 3. The contract between an intermediary and an agent is the solution to the following problem: given K0 , maximize VðK; B; zÞ over the set of all plans ðK; BÞAPðK0 Þ satisfying the incentive compatibility constraint ð25Þ and the financial constraint E0 ðK; BÞp0. Let the value of the optimum, be denoted V * ðK0 Þ. C.1. Temporary incentive compatibility We adapt an important result from Green (1987) which establishes conditions under which incentive compatibility is equivalent to the simpler temporary incentive compatibility used in the text. Given a plan ðK; BÞAPðK0 Þ, and the reporting of a partial history zt@1 AZt , we define the expected value of truthful future reporting as Z XN s@t VðK; B; z; zt@1 Þ ¼ b vðzs Ks ½zt@1 ;t zs@1

s¼t þ Bs ½z

t@1

Zsþ1@t s sþ1@t

;t z Þm

ðdt zs Þ;

ðC:3Þ

11 The finite dimensional product spaces ðZt ; Zt ; mt Þ; t ¼ 0; 1; y ; and the infinite dimensional space of sequences of productivity shocks ðZN ; ZN ; mN Þ are defined in the standard manner. See Stokey et al. (1989, pp. 195–199).

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where note that mk ¼ mk@1 m.12 Given (C.3), preferences will satisfy Z ½vðzt Kt ½zt@1 þ Bt ½zt@1 ; zt Þ VðK; B; z; zt@1 Þ ¼ Z

þ bVðK; B; z; ½zt@1 ; zt Þ mðdzt Þ:

ðC:4Þ

A plan ðK; BÞ is ðz; tÞ-incentive compatible ði:c:Þ if VðK; B; z; zt@1 Þ ¼ maxfZASðK;BÞjZt@1 ðzt@1 Þ¼zt@1 g VðK; B; g; zt@1 Þ:

ðC:5Þ

Note that incentive compatibility is equivalent to ðz; 0Þ-incentive compatibility. A plan ðK; BÞ is ðz; tÞ-temporarily incentive compatible (t.i.c.) if given zi ; zj AZ with jai, vðzi Kt ðzt@1 Þ þ Bt ðzt@1 ; zi ÞÞ þ bVðK; B; z; ½zt@1 ; zi Þ Xvðzi Kt ðzt@1 Þ þ Bt ðzt@1 ; zj ÞÞ þ bVðK; B; z; ½zt@1 ; zj Þ:

ðC:6Þ

A temporarily incentive compatible plan will ensure that, conditional on having truthfully reported productivity in the past and being constrained to do so in the future, the agent has no incentive to misrepresent zt . Lemma 4. The plan ðK; BÞAPðK0 Þ is ðz; tÞ-i:c: if and only if it is ðz; tÞ-t:i:c: and ðz; t þ 1Þ-i:c: Proof. ð)Þ Let ðK; BÞ be a plan that is ðz; tÞ-i:c: and assume that it is not ðz; t þ 1Þ-i:c: Then there exists Za ASðK; BÞ such that Zta ðzt Þ ¼ zt ; 8zt AZtþ1 and VðK; B; Za ; z#t Þ > VðK; B; z; z#t Þ for some z#t AZtþ1 . Consider the plan Zb where Zsb ðzs Þ ¼ zs for s ¼ 0; 1; y; t and Zsb ðzs Þ ¼ zs 8sXt þ 1 unless zs ¼ ½ðz#0 ; y; z#t Þ ;tþ1 zs for any tþ1 zs AZs@t . In this case, let Zsb ðzs Þ ¼ Zsa ðzs Þ. Eq. (C.4) implies that VðK; B; Zb ; zt@1 Þ > VðK; B; z; zt@1 Þ contradicting ðz; tÞ-i:c: of ðK; BÞ. Alternatively, assume that ðK; BÞ is not ðz; tÞ-t:i:c:, then for some zi AZ, (C.6) fails. Given (C.4), this contradicts (C.5). ð(Þ If ðK; BÞ is a plan that is both ðz; tÞ-t:i:c: and ðz; t þ 1Þ-i:c:, then (C.4) immediately establishes (C.5). & Using this result, and assuming that a plan ðK; BÞ satisfies the boundary condition limt-N bt VðK; B; z; zt Þ ¼ 0;

8zAZN

ðC:7Þ

we are able to prove that incentive compatibility is equivalent to temporary incentive compatibility. Towards proving this stronger result, it is useful to 12

If sXt then t zs ¼ ðzt ; ztþ1 ; y; zs ÞAZs@tþ1 , otherwise t zs ¼ |.

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rewrite (C.3) as VðK; B; z; zt@1 Þ Z Xn@1 s@t ¼ b vðzs Ks ½zt@1 ;t zs þ Bs ½zt@1 ;t zs Þmsþ1@t ðdt zs Þ s¼t sþ1@t Z Z n@t b VðK; B; z; ½zt@1 ;t zn Þmnþ1@t ðdt zn Þ: ðC:8Þ þ Znþ1@t

Lemma 5. Given (C.7) holds; the plan ðK; BÞ is ðz; 0Þ-i:c: if and only if it is ðz; tÞt.i.c. for all t ¼ 0; 1; y : Proof. ð)Þ Repeated applications of Lemma 4 proves that if ðK; BÞ is ðz; tÞ-i:c: then it is ðz;tÞ-t:i:c:, for t ¼ 0; 1; y : ð(Þ Let ðK; BÞ be ðz; tÞ-t:i:c: for all t ¼ 0; 1; y ; but assume that it is not ðz; 0Þi:c: Then there exists g# ASðK; BÞ such that e ¼ VðK; B; g# Þ@VðK; B; zÞ > 0. Given (C.7) and (C.8), (nAN such that VðK; B; g* Þ > VðK; B; zÞ where Z* t ðzt Þ ¼ Z# t ðzt Þ for ton and Z* t ðzt Þ ¼ zt for tXn. This strategy involves misrepresentation at only a finite number of nodes, a maximum of n2 . Let zl AZlþ1 describe the last node at which there is misrepresentation. As ðK; BÞ is t:i:c: at zl , truth-telling dominates any other strategy. We see that g* allows for misrepresentation at a maximum of ðn@1Þ2 nodes. Backwards induction proves that g* is ðz; 0Þ-i:c: &

C.2. Duality We now adapt the duality results developed in Green (1987) and Oh and Green (1992). In the dual problem, allocations are constrained so that the agent receives some pre-determined level of expected lifetime utility entitlement, which we denote V * . VðK; B; zÞXV * :

ðC:9Þ

Definition 6. The dual contract minimizes E0 ðK; BÞ subject to ðK; BÞAPðK0 Þ satisfying (C.3) and (C.9). Lemma 7. Let ðK * ; B * ÞAPðK0 Þ solve the contract and define V * ¼ VðK * ; B * ; zÞ, then ðK * ; B * Þ solves the dual contract. Furthermore; if # BÞAPðK # # # there exists ðK; 0 Þ that also solves the dual contract; then ðK; BÞ solves the contract.

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Proof. ð)Þ Since ðK * ; B * Þ solves the contract, it satisfies (C.3), (C.9) and # BÞAPðK # E0 ðK * ; B * Þp0 by definition. Assume there exists ðK; 0 Þ which also # BÞ # > 0. As ðK; # BÞ # satisfies (C.1) and (C.9) where e  E0 ðK * ; B * Þ@E0 ðK; satisfies (C.1), it is ðz; 0Þ-i:c: which by Lemma 4 implies that it is ðz; 0Þ-t:i:c: and ðz; 1Þ-i:c:. For (C.6) to hold for both i ¼ 1; 2, given concavity of v, it must be true that B0 ðz1 ÞXB0 ðz2 Þ. Next, temporary incentive compatibility for i ¼ 2, requires VðK; B; z; z1 ÞpVðK; B; z; z2 Þ. There are three possibilities: (a) Eq. (C.6) does not bind for either i ¼ 1 or 2, (b) Eq. (C.6) binds only for i ¼ 1 or 2, or (c) # BÞ % Eq. (C.6) binds for both i ¼ 1 and i ¼ 2. We design an alternative plan ðK; t t t t % # % # for each case. Let Bt ðz Þ ¼ Bt ðz Þ for all t > 0 and K t ðz Þ ¼ K t ðz Þ for all t. If # BÞ # is characterized by (a), then let temporary incentive compatibility for ðK; % # B0 ðzi Þ > B0 ðzi Þ for either i ¼ 1 or 2 until (C.6) binds for either i or B% 0 ðzi Þ ¼ B# 0 ðzi Þ þ e. Alternatively if case (b) applies, let B% 0 ðzi Þ > B# 0 ðzi Þ until the t:i:c: for jai binds or B% 0 ðzi Þ ¼ B# 0 ðzi Þ þ e. If case (c) applies then it must be true that B# 0 ðz1 Þ ¼ B# 0 ðz2 Þ; let B% 0 ðzi Þ ¼ B# 0 ðzi Þ þ e=2. In each case, the alternate # BÞ % is temporary incentive compatible at t ¼ 0 and, by construction, plan ðK; # BÞp0 % ðz; 1Þ-i.c., thus it satisfies (C.1). Furthermore, E0 ðK; and # % VðK; B; zÞ > V * ðK0 Þ. This contradicts the assumption that ðK * ; B * Þ solves the contract. # BÞ # solves the dual problem, but not the primal. ð(Þ Next assume that ðK; # # Since ðK; BÞAPðK0 Þ by definition, and satisfies both (C.1) and (C.9), then it # BÞ # > 0XE0 ðK * ; B * Þ. This contradicts the assumption that must be that E0 ðK; # # ðK; BÞ solves the dual. & The equivalence of solving the contracting problem described in Definition 3 and the expenditure minimization problem described in Section 2 follows from Lemmas 5 and 7.

References Aiyagari, S.R., Alvarez, F., 1995. Stationary efficient distributions with private information and monitoring: A tale of kings and slaves. University of Chicago working paper. Aiyagari, S.R., Williamson, S.D., 1997. Credit in a random matching model with private information. University of Iowa working paper #97-03. Atkeson, A., Lucas, R.E., 1992. On efficient distribution with private information. Review of Economic Studies 59, 427–453. Athanasoulis, S., Van Wincoop, E., 2000. Growth uncertainty and risk-sharing. Journal of Monetary Economics 45, 477–505. Backus, D.K., Kehoe, P.J., Kydland, F.E., 1992. International real business cycles. Journal of Political Economy 100, 745–775. Baxter, M.B., Crucini, M.J., 1995. Business cycles and the asset structure of foreign trade. International Economic Review 36, 821–854. Cochrane, J.H., 1991. A simple test of consumption insurance. Journal of Political Economy 99, 957–976.

A. Khan, B. Ravikumar / Journal of Monetary Economics 47 (2001) 499–521

521

Cole, H.L., Kocherlakota, N.R., 1999. Efficient allocations with hidden income and hidden storage. Federal Reserve Bank of Minneapolis Research Department Staff Report 238. Devereux, M.B., Smith, G.W., 1994. International risk sharing and economic growth. International Economic Review 35, 535–550. Green, E.J., 1987. Lending and the smoothing of uninsurable income. In: Prescott, E.C., Wallace, N. (Eds.), Minnesota Studies in Macroeconomics, Vol. I: Contractual Arrangements for Intertemporal Trade, University of Minnesota. Hayashi, F., Altonji, J., Kotlikoff, L., 1996. Risk-sharing between and within families. Econometrica 64, 261–294. Heaton, J., Lucas, D., 1996. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104, 443–487. Levhari, D., Srinivasan, T.N., 1969. Optimal savings under uncertainty. Review of Economic Studies 36, 153–163. Mace, B.J., 1991. Full insurance in the presence of aggregate uncertainty. Journal of Political Economy 99, 928–956. Maitra, P., 1997. Is consumption smooth at the cost of volatile leisure? An Investigation of Rural India. University of Southern California working paper. Marcet, A., Marimon, R., 1992. Communication, commitment, and growth. Journal of Economic Theory 58, 219–249. Mehra, R., Prescott, E.C., 1985. The equity premium puzzle. Journal of Monetary Economics 15, 145–161. Obstfeld, M., 1994. Risk-taking, global diversification, and growth. American Economic Review 84, 1310–1329. Oh, S., Green, E.J., 1992. A re-examination of optimal contracts. Federal Reserve Bank of Minneapolis, Research department working paper, No. 490. Parente, S.L., Prescott, E.C., 1993. Changes in the wealth of nations. Federal Reserve Bank of Minneapolis Quarterly Review 17, 3–16. Phelan, C., Townsend, R.M., 1991. Computing multi-period, information-constrained optima. Review of Economic Studies 58, 853–882. Spear, S.E., Srivastava, S., 1987. On moral hazard with repeated discounting. Review of Economic Studies 54, 599–617. Stokey, N.L., Lucas Jr. R.E., Prescott, E.C., 1989. Recursive Methods in Economic Dynamics. Harvard University Press, Washington, DC. Taub, B., 1990. The equivalence of lending equilibria and signalling based insurance under asymmetric information. The Rand Journal of Economics 21, 388–408. Thomas, J., Worrall, T., 1990. Income fluctuations and asymmetric information: an example of a repeated principal-agent problem. Journal of Economic Theory 51, 367–390. Townsend, R.M., 1994. Risk and insurance in village India. Econometrica 62, 539–592. Weil, P., 1990. Nonexpected utility in macroeconomics. Quarterly Journal of Economics 105, 29–42.