Stochastic costly state verification and dynamic contracts

Journal of Economic Dynamics & Control 64 (2016) 1–22

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Stochastic costly state verification and dynamic contracts$ Latchezar Popov Department of Economics, University of Virginia, PO Box 400182, Charlottesville, VA 22904-4182, USA

a r t i c l e i n f o

abstract

Article history: Received 15 November 2015 Received in revised form 17 December 2015 Accepted 29 December 2015 Available online 6 January 2016

I consider a dynamic costly state verification environment in which a risk-averse agent enters into a contract with a risk-neutral principal. The agent has random income which is unknown to the principal but can be verified at a cost. The principal can commit to executing random verifications. I extend the standard recursive methods to study the problem and show that it is optimal to set verification probabilities strictly less than 1. If the agent's absolute risk aversion declines sufficiently slowly, the principal will use verification regardless of its cost. If the agent's income is verified then he would get consumption and continuation utility strictly higher than if his income were not verified. & 2016 Elsevier B.V. All rights reserved.

JEL classification: C73 D82 Keywords: Stochastic costly state verification

1. Introduction In many economic environments with private information, one party of the relationship can, at a cost, obtain perfect or imperfect knowledge of the other party's private information. Employers try to determine the effort exerted by their employees. Firms and individuals are audited for tax compliance. When employing a new worker, employers may check the new employee's credentials and job experience. Entrepreneurs who employ external financing may be subject to random inspections by their investors. These examples show that in many applied settings the principal can overcome the information asymmetry by a costly state verification of the agent. Therefore, in many settings asymmetric information is not an innate property of the physical environment but an economic one which depends on the cost of verification – one of the different methods of providing incentives. This paper considers the optimal contract between a risk-averse agent and a risk-neutral principal in a private information endowment economy with the possibility of verification. I examine the mix of different tools to provide incentives – both static (verification) and dynamic (continuation utilities) and show how the optimal contract depends on the cost of verification. The principal can commit to stochastic verification of the agent's reports. The focus on stochastic verification schemes is motivated by the fact that, from an optimal contracting standpoint, stochastic verification strictly dominates deterministic verification. With some utility functions the benefit from allowing stochastic verification can be relatively large. ☆ I have benefitted greatly from the discussions with B. Ravikumar and Galina Vereshchagina. I am grateful for their helpful comments and suggestions as well as their support. I also thank Erwan Quintin, Cheng Wang, the associate editor and an anonymous referee for their generous and insightful feedback. I appreciate useful comments from participants in the Iowa Alumni Workshop, the Public Economic Theory conference in Nashville, 2007 and the 2008 Midwest Macro Meetings in Philadelphia. All remaining errors are mine. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jedc.2015.12.006 0165-1889/& 2016 Elsevier B.V. All rights reserved.

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

I introduce the model in Section 2. I consider a repeated and simplified version of the Mookherjee and Png (1989) static model of stochastic costly verification.1 In each period, the agent receives a random endowment of a non-storable consumption good. The stochastic endowment process is an infinite sequence of independently and identically distributed (i.i. d.) random variables. The realization of the agent's endowment is private information but can be verified at a cost. The agent enters into a contractual relationship with a risk-neutral principal. Under full information, efficiency implies that the principal bears all the risk. Private information, however, creates a trade-off between efficiency and incentives. The contract without verification creates incentives for truthful revelation of the endowment realization by changing the future payoffs to the agent. Costly state verification expands the set of incentive-compatible allocations by introducing another instrument for providing incentives. Allowing stochastic verification presents a few technical difficulties. First, if the principal can threaten to impose arbitrarily large punishment if she detects that the agent is not reporting his realized endowment truthfully, then the first-best allocation is incentive compatible with an arbitrarily low verification probability. To keep the problem non-trivial, I assume that the utility function is bounded from below and that the continuation utility is bounded. Second, the principal can offer arbitrarily high consumption (and hence utility) with arbitrary low probability which would make the maximization operation ill-defined. I rule this out by imposing the restriction that the marginal utility of consumption converges to zero as consumption converges to infinity, which is equivalent to imposing the condition that the marginal cost of providing utility converges to positive infinity as utility converges to its upper bound. In the Section 3 I use the usual tools of the dynamic contracts literature, the revelation principle and future expected discounted utility as a state variable, to give a recursive formulation of the problem. Section 4 presents the results of the analysis. I find that an agent whose income has been verified gets strictly higher consumption and continuation utility than an unaudited one. If we interpret this result in terms of taxation, an agent whose income was verified would get a tax rebate and a lower future tax liability. However, even if a verification took place, the agent with low income would still get consumption and continuation utility strictly less than that of an agent with high income realization. The reason is that the costs of spreading consumption and continuation utility are approximately quadratic for small distortions while the verification costs are linear, therefore some distortion is optimal. This shows that verification is never the sole method of providing incentives. I find that even at the lower bound the principal wants to use spreading continuation utility to provide incentives. This implies that an agent with promised utility at the lower bound and high income will have higher continuation utility – the lower bound is not an absorbing state and there is no immiserization. Providing incentives via stochastic verification and continuation utility has important implications for the long-run behavior of promised utility. This is in contrast with Wang (2005) who studies deterministic costly state verification in an environment identical to mine. I find that the agent's continuation utility is always contingent on the income realization. On the other hand, in the case characterized by Wang (2005), continuation utility is not contingent on income, that is, the optimal allocation is repetition of the static allocation. Risk-aversion (and stochastic verification) implies that there is a downward trend in promised utility. This is in contrast with Monnet and Quintin (2005). They consider a relationship between a risk-neutral principal and a risk-neutral agent and stochastic verification. The benefit from considering risk-aversion is that the insurance – incentives trade-off can be studied. Monnet and Quintin find that in the long-run the agent receives the entire stochastic income. The result is due to the fact that when the agent is risk-neutral there is no welfare loss from variable income and it is easier to provide incentives when the agents promised utility is high. In my environment, I show that this result depends crucially on the linearity assumption. As in other environments with private information and risk aversion, there is a downward trend in promised utility. My paper is related to recent work by Salitskiy (2014), which is a dynamic extension of the classic model of Townsend (1979). This work has a sharp characterization of the verification probability as a function of the agent's report: for low reports the project is terminated and for intermediate reports the report is verified with probability one. This result is driven by risk-neutrality and deterministic verification, whereas my focus is on risk-sharing and stochastic verification. My paper is also related to the seminal work by Bernanke et al. (1999) on the financial accelerator. In this paper, information frictions amplify dynamics, but the loan contracts are (scaled) repetitions of the same static costly state verification contract. I find that the allocation and the verification probabilities always depend on past reports. Since it is feasible to verify the agent's income every period, the principal's value function cannot differ from the firstbest solution by more than the present value of permanent verification. This implies that verification will be used in the optimal contract if the cost is low enough. Allowing stochastic verification delivers a stronger result. As promised utility converges to its upper bound, the agent's loss increases if verification reveals misreporting. At the same time the benefit from misreporting converges to zero (in utility terms), implying that the verification probability that ensures the first-best allocation converges to zero. Therefore, the principal's value function converges to the first-best value function for any verification cost. I show that if the agent's risk aversion declines sufficiently slowly, the ratio of the cost imposed by consumption fluctuation to the costs of verification necessary to ensure perfect insurance converges to positive infinity. Therefore, verification takes place for an arbitrarily high cost. Section 5 briefly concludes. 1 Mookherje and Png have hidden action and private information. In this paper I assume that there is no hidden action and that the endowment is exogenous.

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

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2. Model The environment is a dynamic simplified version of Mookherjee and Png (1989).2 Time is discrete and runs forever. A risk-averse agent trades with a risk-neutral principal. Both the agent and the principal can fully commit to a contract. Agent: The agent is endowed with stochastic income of non-storable consumption good. The income process is an infinite sequence of serially independent and identically distributed random variables (i.i.d.) that can take values yA fyL ; yH g (yH 4 yL 4 0) with probability ðπ L ; π H Þ. The income realization is privately observed by the agent but can be verified by the principal at a cost γ. The agent maximizes discounted expected utility with a discount factor β. His utility function satisfies the following assumption: Assumption 1. The period utility function u(c) is defined on ½0; þ 1Þ, with u A C 2 . It is increasing and strictly concave with limc-0 u0 ðcÞ ¼ 1, limc-1 u0 ðcÞ ¼ 0, uð0Þ ¼ 0 and  u″ =u0 is non-increasing. Principal: The principal (also called the planner) is risk-neutral with discount factor q Zβ and maximizes the expected discounted value of the net transfers from the agent to the principal. She can verify the true income realization at a cost γ. I assume that the principal can commit to verifying the agents income with a probability ϕ that is a function of the history. Contract: The timing of the contract is the following. At the beginning of each period the agent observes privately the income realization. Then the agent sends the principal a message m A M, where M is some message space. After that the principal verifies the agents income with a probability conditional on the history of received messages and verifications. Finally, the principal prescribes a transfer b of the consumption good either to or from the agent conditional on the history, message and the result of the verification. In a static setup Townsend (1988) proved that if M is sufficiently rich, then the revelation principle is applicable. The revelation principle applies in this model as well. Before formally defining an allocation, I need to clarify the notion of a state in this setup. At the end of the period the principal has received the message yt A fyL ; yH g and the result of the verification at A f∅; yL ; yH g, where ∅ denotes that no t verification took place. Let ht ¼ fyt ; at g. Then ht is the new information acquired at date t. Define h ¼ ðh0 ; …; ht Þ, that is ht is the history of reports and verifications. Define Ht to be the set of all possible histories up to date t. 1 t 1 Then an allocation is a sequence of functions fϕt g1  fyL ; yH g-½0; 1 and fbt gt ¼ 0 where t ¼ 0 where ϕt : H t t t bt : H -½ yH ; 1Þ. Here bt ðh Þ 40 denotes a transfer of the consumption good to the agent and bt ðh Þ o 0 denotes a transfer of the consumption good from the agent to the principal. The agent can condition her report at date t on the actual history of income realizations yt and the results of the verifications up to period t  1. Therefore an agent's reporting strategy σ is a sequence of functions t t 1 Þ-fyH ; yL g. Let Σ be the set of all possible strategies. Let σ t ðyt ; at  1 Þ denote the vector of reports up to and fσ t g1 t ¼ 0 ; σ t : ðy ; a including date t conditional on a history of realizations yt, verification outcomes at  1 and a reporting strategy σ. Define σ  to be the truthtelling strategy, that is σ t ðyt ; at  1 Þ ¼ yt for all ðyt ; at  1 Þ.3 The allocation must satisfy the following properties: 1. Incentive compatibility σ  ¼ arg maxE0 σ AΣ

1 X

βt u½yt þbt ðσ t ðyt ; at  1 Þ; at Þ

ð1Þ

t¼0

The expectation is taken with respect to the probability measure that the reporting strategy and the verification policy t t fϕt g1 t ¼ 0 implicitly induce on H . Clearly, most histories h will have probability zero. If a history has a probability zero under all possible reporting strategies, then choosing a trivially incentive compatible allocation (constant transfers for example) will not affect the agent's expected utility or the value of the principal. On the other hand, clearly there are histories that have zero probability in the measure induced by σ  but will have nonzero probability under alternative reporting strategies. Then the consumption and verification probabilities for these histories will be relevant since they provide incentives. 2. Transversality condition For all possible yt and sequence of verifications we have: lim βt

t-1

1 X

t þs1

βs u½yt þ s þ bt þ s ðh

; yt þ s ; at þ s Þ ¼ 0

ð2Þ

s¼0

This condition ensures that the set of incentive-compatible allocations has a recursive structure. 2

Or, equivalently, a version of Thomas and Worrall (1990) with costly state verification. It is clear that some pairs ðyt ; at  1 Þ are impossible. For example, for some jr t  1, aj a ∅ and aj a yj can never occur. For simplicity, I assume that σ t ðyt ; at Þ is still defined on such vectors. Obviously this is without loss of generality. 3

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

3. Limited liability For every ht, 1 X

Et

i1

βi  t  1 u½yi þ bi ðh

i1

; σ t ðh

; yi Þ; ai Þ Zw;

ð3Þ

i ¼ t þ1

where w Z π H uðyH yL Þ=ð1 βÞ. That is, for an arbitrary history, an agent will have continuation utility above some bound w if she tells the truth from this period on. The assumption on the utility function uð:Þ ensures that w 4 0. The expectation is taken with respect to the measure induced by fϕt g and σ t conditional on history ht. It may seem that this restriction is necessary to ensure that the principal cannot use infinite punishment to prevent cheating. Actually assuming that the utility function is bounded from below is sufficient to make the problem non-trivial. The lower bound w ensures that the principal can offer an incentive-compatible allocation (constant transfers of yL ) that would attain this lower bound without the need for verification. That ensures that the value function of the principal is strictly decreasing. This assumption can be interpreted in two ways. First, we may think of it as a constraint on the ability of the current generation to borrow against the income of its descendants. Second, we may allow the agent to break the contract (perhaps incurring some cost) and continue in a state of autarchy. Then a contract will be incentive-compatible only if for each possible history the expected discounted utility is higher than the autarchy utility, or w ¼ ½π H uðyH Þ þ π L uðyL Þ=ð1 βÞ.4 Both of these interpretations are consistent with w Z π H uðyH yL Þ=ð1 βÞ. 4. Promise-keeping E0

1 X

t1

βt u½yt þbt ðh

t1

; σ  ðh

; yt Þ; at Þ Z w0

ð4Þ

t¼0

The agent is guaranteed some expected utility w0 Z w. 5. Feasibility For any ht t

bt ðh Þ Z  yt if at ¼ ∅ t

bt ðh Þ Z  at if at a∅

ð5Þ

That is, the principal cannot require a transfer from the agent that is larger than the report (if no verification took place) or the true income realization (if a verification took place). Mathematically, this simply requires that the expected utility is well-defined. (The utility function is defined on R þ .) The principal's expected value from an allocation which satisfies Eqs. (1)–(5) is given by E0

1 X

t 1

t

qt ½ bt ðh Þ  ϕt ðh

; yt Þγ

t¼0

The expectation is taken with respect to the measure induced by the verification probability. Let Θ ðw0 Þ denote the set of allocations that satisfy conditions Eqs. (1)–(5) for a given w0. Then the supremum function is defined in the usual way: v ðwÞ ¼ sup E0 

θ A Θ ðwÞ

1 X

t

t 1

qt ½  bt ðh Þ  ϕt ðh

; yt Þγ:

ð6Þ

t¼0

I show later that the supremum is actually attained.

3. Recursive formulation The sequential formulation of the problem is very cumbersome to work with. In particular, the structure of the incentive constraint is very complicated, since misreporting in one period affects all future allocations and the probability of future verification. Following Green (1987), I show that a necessary and sufficient condition for incentive compatibility is that the truthtelling strategy σn dominates all the reporting strategies that involve misreporting in a single period and then reporting truthfully in all following periods. (This condition is also known as temporary incentive compatibility.) Thus it is fairly straightforward to put the problem in a more convenient recursive form. As is usual in these kinds of problems, expected discounted utility is a sufficient state statistic. The first step is to restate the timing and the contractual variables in the recursive formulation. 4 Phelan (1995) considers a similar problem in which the agent is free to contract with other insurers. Then w is the utility offered to the agent by other firms at the start of a new contract.

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

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1. At the beginning of the period the agent has a promised utility w. 2. The principal announces the contract, that is, verification probabilities ϕj and transfers and continuation utilities conditional on agent's report and verification outcome. 3. The shock is realized and observed privately by the agent. 4. The agent makes a report to the principal. 5. A stochastic verification takes place. 6. The transfer occurs and the agent is assigned continuation utility. It is easier to work directly with consumption than with transfers. Let ci;j denote consumption allocated to the agent if she reported yi and the verification shows that the true income realization is yj; ciN means that no verification took place. The subscripts of continuation utility work in the same way. Then assuming that the agent misreports only in the current period, the incentive constraints can be rewritten as: ð1 ϕk Þ½uðckN Þ þ βwkN  þϕk ½uðck;k Þ þβwk;k  Z ð1 ϕj Þ½uðyk  yj þ cjN Þ þβwjN  þϕj ½uðcj;k Þ þβwj;k  8 k; j

The right-hand side is the utility of misreporting in the current period. Hence, if not caught, his subsequent utility is given by the continuation utility wjN of an agent who truthfully reported j. Now we can see that the principal's value function is given by the solution of the following Bellman equation: vðwÞ ¼

s:t

sup

2 X

fci;j g;fwi;j g;fϕk g k ¼ 1

2 X

  π k yk þ ð1 ϕk Þ½ ckN þqvðwkN Þþ ϕk ½  ck;k γ þ qvðwk;k Þ

π k fð1  ϕk Þ½uðckN Þ þβwkN  þ ϕk ½uðck;k Þ þ βwk;k g ¼ w

ð7Þ

ð8Þ

k¼1

ð1 ϕk Þ½uðckN Þ þ βwkN  þϕk ½uðck;k Þ þβwk;k  Z ð1 ϕj Þ½uðyk  yj þ cjN Þ þβwjN  þϕj ½uðcj;k Þ þβwj;k  8 k; j wk;j A ½w; wÞ; ckN Z 0; ck;j Z 0; ϕk A ½0; 1;

ð9Þ ð10Þ

where w ¼ supc Z 0 ðuðcÞÞ=1 β, that is, w is the maximum discounted utility that any allocation can give. Eq. (8) is the promise-keeping constraint and Eq. (9) is the incentive constraint. The maximization problem defined by Eqs. (7)–(10) defines an operator T that maps the space of real-valued functions to itself. The Bellman equation can then be written more compactly as v ¼Tv. Clearly, the principal wants to impose the maximum possible punishment if she discovers the agent misrepresenting his income, so wk;j ¼ w; 8 k; j: k a j. Similarly, ck;j ¼ 0; 8 k; j: k a j. The set of feasible and incentive-compatible contracts is not compact. I show that if v is concave, without loss of generality we can restrict the feasible set to a compact subset. Therefore the Weierstrass Theorem is applicable and the supremum Tv is attainable. Moreover, since this subset is a continuous correspondence, the Maximum Theorem implies that Tv is continuous if v is concave (whenever v is concave it is also continuous). Finally, I show that the operator maps concave functions to concave functions and therefore the unique fixed point is concave. Next I show that the supremum function defined in the previous section falls in the function space considered here and solves the Bellman equation. Since the operator T is a contraction on that space, this guarantees that the value function that solves the recursive problem is equal to the supremum function. Moreover, the allocations constructed recursively from the policy functions are incentive-compatible and deliver the promised utility. This completes the recursive characterization of the problem. The first order of business is to show that the operator T is a contraction in an appropriately defined metric space and that the fixed point v ¼ v . Define a “first best” value function to be the expected value for the principal of providing utility w subject to the constraint that the agent's discounted utility is above w for all periods. This constraint will bind if q 4 β. This is a problem with a relaxed constraint set so the first-best value function is an upper bound for the supremum function. Denote the solution of this relaxed problem vFB. P2 1 X s ¼ 1 π s ys  qt ct vFB ðwÞ ¼ max fct g1 1 q t ¼ 0 t¼0 s:t

1 X

βt uðct Þ ¼ w

t¼0 1 X t¼j

βt  j uðct Þ Zw 8 j ¼ 1; 2; …;

ð11Þ

6

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

If the principal always verifies the agent's report, then a full insurance solution is incentive-compatible. Then a lower bound for vn would be: γ : ð12Þ vFV ðwÞ ¼ vFB ðwÞ  1q Here FV means full verification. Clearly, we can restrict our attention to the set of functions that fall between vFB and vFV and are defined on the appropriate interval. I call this function space D and define it by D ¼ ff : ½w; wÞ-R; vFV ðwÞ rf ðwÞ rvFB ðwÞg. Proposition 1. T is a contraction on D. Proof. In Appendix A. It is straightforward to show that the supremum function vn defined above solves the Bellman equation. As I showed that the operator T is a contraction, the fixed point is unique and v ¼ v . Proposition 2. The function vn solves the problem (7) subject to Eqs. (8)–(10). Proof. In Appendix A. 3.1. Concavity of the value function The first step to analyzing the problem is to simplify the constraint set. Since almost any allocation will be incentivecompatible with sufficiently high verification probabilities, the set of incentive-compatible contracts is dramatically larger and more complicated. In this section, I show that if the continuation value of the principal is concave, then the optimal policies satisfy a much simpler set of constraints: the constraint preventing the agent with high income from reporting a low income realization is binding, only low reports are (stochastically) verified and the other incentive constraint can be ignored. With the help of this simplified constraint structure, I show that the Bellman operator maps concave functions to concave functions, which, via the Contraction Mapping Theorem, implies that the principal's value function is concave. Lemma 1. Assume v is concave. Then without loss of generality we can restrict the set of incentive-compatible contracts to those such that (i) ciN r ci;i ; wiN r wi;i and (ii) if the constraint that prevents the agent with income j from reporting i is slack then ϕi ¼ 0. If optimal policies exist, then they satisfy (i) and (ii). Proof. In Appendix A. Starting from the same value, increasing consumption in the verified state has the same effect on the promise-keeping constraint as in the non-verified state; however, its effect on the incentive constraints is (weakly) stronger: as a result ci;i Z ci;N . The argument that wi;i Zwi;N is similar. Clearly, the principal would engage in costly verification only if there is an incentive problem. The next lemma shows that at the optimum an agent with the high income realization is indifferent between reporting yH or yL whereas an agent with low income realization strictly prefers telling the truth. I also find that if the principal uses verification it is always stochastic. Lemma 2. If v is concave then without loss of generality we can restrict the set of incentive-compatible contracts to those that satisfy: (i) if the agent has income yH, then he is indifferent between reporting yH or yL, (ii) if the agent has income yL then he strictly prefers reporting yL, (iii) ϕL o1. If optimal policies exist then they satisfy (i), (ii) and (iii). Proof. I relegate the proof of (i) and (ii) to the Appendix. (iii) Let's relax the constraint set and perform the maximization without the incentive constraint. Then clearly, ϕH ¼ ϕL ¼ 0, cHN ¼ cLN , wHN ¼ wLN . Also uðcHN Þ þβwHN ¼ w. Then cLL and wLL are irrelevant, so without loss of generality, we can set cLL ¼ cHN and wLL ¼ wHN . Define ϕ~ L as: ϕ~ L ¼

uðcHN þ ΔyÞ  uðcHN Þ o1 w  βw þ uðcHN þ ΔyÞ  uðcHN Þ

If the principal verifies with probability ϕ~ L the first best solution is incentive compatible. So the principal would never choose ϕL 4 ϕ~ L .□ The upper bound on the verification probability illustrates the value of stochastic verification: as the promised utility increases, the feasible punishment (a drop to the lowest allowable utility level) increases, so the verification probability required to ensure the first-best is incentive-compatible converges to zero. In Section 4.2 I show that for some utility functions, the value of eliminating consumption risk drops slower than the verification cost, so verification will occur for any cost γ. This result greatly simplifies the analysis of the model. Since the upward constraint is slack, the principal would not verify the high report. Then cHH and wHH are irrelevant. Lemmas 1 and 2 restrict the set of incentive-compatible contracts that need to be considered when taking the supremum if v is concave. I show that I can restrict the set of incentive-

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

7

compatible contracts to lie in a compact set B(w), thus ensuring that the maximum is attained and that optimal policies exist. Moreover, the Maximum Theorem is applicable, so the optimal policies are continuous and Tv is continuous too. Proposition 3. The function v (the unique fixed point of the operator T) is concave. The optimal contract exists and is a continuous function of promised utility w. Proof. In Appendix A. Lemmas 1 and 2 give some properties of the policies that would maximize the right-hand side of the operator Tf if f is concave. With the help of these results, this proposition shows that the operator T maps concave functions to concave functions, therefore the value function is concave. This means that the hypothesis of Lemmas 1 and 2 is satisfied and the optimal policies have the following properties: (i) cLN r cLL ; wLN r wLL , (ii) if the constraint that prevents the agent with income j from reporting i is slack then ϕi ¼ 0, (iii) the downward incentive constraint is binding, (iv) the upward incentive constraint is slack, (v) ϕL o 1. An argument identical to the one in Lemma 1 shows that cHN Z cLL and wHN ZwLL . Finally, I show that the allocation generated recursively by the policy function derived from the recursive problem delivers the promised utility and is incentive-compatible. Theorem 1. The allocation generated recursively by the policy function satisfies the promise-keeping constraint (8) and the incentive constraint (9). Proof. In Appendix A. 3.2. Other properties of the value function After establishing the concavity of the value function, I establish two additional properties of the principal's value function. First, I establish that the value function is strictly decreasing: increasing promised utility to the agent strictly lowers the value to the principal. As a result, the contracts are renegotiation-proof: the principal never has an incentive to increase the value of the agent. Lemma 3. The value function v(w) is strictly decreasing. For any w4 w, cHN 4 0; cLL 4 0; cLN 4 0. Proof. In Appendix A. In the results so far, I have not used the assumption that limc-0 u0 ðcÞ ¼ 1. This assumption implies that if consumption in a state is zero, then delivering extra utility in this state is essentially costless, so we can change the way the principal delivers the promised utility and reduce her costs. This in turn implies that it in a neighborhood of w4 w it is incentivefeasible to reduce cLN ; cLL and cHN, which demonstrates that the value function is strictly decreasing. Another crucial assumption for the result is the lower bound on utility: w Z π L ðuðΔyÞ=1  βÞ. When uð0Þ ¼ 0, π L ðuðΔyÞ=1 βÞ is the minimum utility level that can be achieved without verification.5 Lowering the agent's expected utility below this bound requires costly verification, which may reduce the principal's value. Next, I consider the differentiability of the principal's value function. Since v is a concave function on the real line, its points of non-differentiability are at most countable (see, for example, Rockafellar (1970), Theorem 25.3). However, since spreading continuation utilities is key in providing incentives, kink points in the value function may be absorbing states for the agent's promised utility. I discuss this point further in Section 4.1. Proposition 4. The value function v(w) is continuously differentiable. Proof. In Appendix A.

4. Policy functions After establishing some of the basic properties of the value function, we may consider the optimal policy functions. With the help of the results from Section 3.1, we can rewrite the constraints: uðcHN Þ þ βwHN ¼ ð1  ϕL Þ½uðcLN þyH  yL Þ þ βwLN  þ ϕL βw π H ½uðcHN Þ þ βwHN  þ π L ð1  ϕL Þ½uðcLN Þ þ βwLN  þ ð1  π L ÞϕL ½uðcLL Þ þ βwLL  ¼ w Then the optimal policy will satisfy the following necessary first-order conditions (λ is the multiplier to the incentive constraint and μ to the promise-keeping constraint):  π H þ λu0 ðcHN Þ þ μπ H u0 ðcHN Þ ¼ 0

ð13Þ

5 Aiyagari and Alvarez (1995) are the first to demonstrate that the dynamics of incentive-constrained problems depend crucially on whether the lower bound of the utility function is incentive-compatible.

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

π L ð1  ϕL Þ  λð1  ϕL Þu0 ðcLN þ ΔyÞ þ μπ L ð1 ϕL Þu0 ðcLN Þ ¼ 0

ð14Þ

π L ϕL þμπ L ϕL u0 ðcLL Þ ¼ 0

ð15Þ

π H qv0 ðwHN Þ þλβ þ μπ H β r 0

ð16Þ

π L ð1 ϕL Þqv0 ðwLN Þ λð1 ϕL Þβ þμπ L ð1  ϕL Þβ r 0

ð17Þ

π L ϕL qv0 ðwLL Þ þ μπ L ϕL β r 0

ð18Þ

π L ð  cLL þ qvðwLL Þ þcLN qvðwLN ÞÞ þ λ½uðcLN þ yH  yL Þ þβwLN  þ μπ L ½uðcLL Þ þ βwLL  uðcLN Þ  βwLN   π L γ r0

ð19Þ

The direction of the inequality Eq. (19) follows from the fact that the upper bound on ϕL is never binding. The envelope theorem gives: v0 ðwÞ ¼ μ:

ð20Þ

Then we can determine that: v0 ðwÞ ¼ 

π H u0 ðcLN þ ΔyÞ þ π L u0 ðcHN Þ u0 ðcHN Þ½π H u0 ðcLN þΔyÞ þ π H u0 ðcLN Þ

ð21Þ

Similarly, if ϕL 4 0 then v0 ðwÞ ¼ 

1 : u0 ðcLL Þ

ð22Þ

After taking care of these preliminaries I find some of the properties of the policy functions. 4.1. Consumption and continuation utility The first result states that even an agent whose income is verified would receive less consumption and continuation utility than an agent who got the high shock. Also, the agent receives consumption and continuation utility higher than that of an agent whose income is not verified. Proposition 5. If ϕL ðwÞ ¼ 0, then cHN ðwÞ ZcLN ðwÞ; if ϕL ðwÞ 40 then cHN ðwÞ 4 cLL ðwÞ 4 cLN ðwÞ. Suppose that cHN ¼ cLL . Then the cost spreading consumption by ϵ has costs of the order ϵ2 and benefits in terms of reduced verification costs of order ϵ, so for ϵ sufficiently small the benefits outweigh the costs. Next, I consider how the principal uses variation in continuation utilities in conjunction with verification to provide incentives for truthtelling. Increasing the spread of continuation utility decreases the principal's continuation value (through concavity) but improves insurance in the current period and reduces verification costs. The optimal allocation also affects the limiting distribution of promised utilities. First, I show an analogue of Proposition 5 regarding continuation utilities, with a very similar logic. Proposition 6. If wHN ðwÞ 4 w, then wHN ðwÞ 4 wLL ðwÞ 4wLN ðwÞ; otherwise wHN ðwÞ ¼ wLL ðwÞ ¼ wLN ðwÞ ¼ w. Proof. In Appendix A. Next, we turn to the question of whether the lower bound of promised utility is an absorbing state. Lemma 4. If q is close enough to β then wHN ðwÞ 4w for all w Zw. Proof of Lemma 4. I have proved that for w 4w, cLL 4 0; cLN 40; cH 4 0. By Proposition 4, vðÞ is differentiable. By the Mean Value Theorem,   vðw þδÞ vðwÞ ¼ lim v0 w þ δ lim δ δ↓0 δ↓0

ð23Þ

The limit is well-defined since v is concave. By the Maximum Theorem the policy functions are continuous in w, so (21) and (22) hold for all w Zw. Assume that wHN ¼ w: I will show that we can change the allocation and increase the principal's value. Consider lowering cHN by ϵ and increasing wHN by u0 ðcHN Þϵ=β. This is feasible because by Lemma 3 cHN 4 0. This variation would keep both the incentive and promise-keeping constraints satisfied. Then for small ϵ the change in the principal's value is approximately:             q q q  ð24Þ π H ϵ 1 þ u0 ðcHN Þv0 ðwHN ðwÞÞ ¼ π H ϵ 1 þ u0 ðcHN Þv0 w Z π H ϵ 1 þ u0 cHN w v0 w β β β where cHN ðwÞ denotes consumption cHN if the promised utility is w. The last inequality follows from the fact that v0 o0 and that consumption is (weakly) increasing in promised utility.

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

9

Now assume that q ¼ β. If ϕL 40, then by Proposition 5 cLL o cHN . But v0 ðwÞ ¼  1=u0 ðcLL Þ so u0 ðcHN ðwÞÞv0 ðwÞ 4  1 and then Eq. (24) implies that the variation increases the principal's profit. The other possibility is ϕL ¼ 0. Since w r wLN rwHN ¼ w we know that wLN ¼ w. Therefore the incentive constraint implies that cHN ¼ cLN þ Δy. Then Eq. (21) implies that   v0 w ¼ 

1 1 4 0 π H u0 ðcHN Þ þπ L u0 ðcHN ΔyÞ u ðcHN Þ

Then (24) again implies that the variation increases the principal's value. So in both cases I reach contradictions, therefore it must be that wHN ðwÞ 4w. Now assume q 4β. Since the optimal policies are continuous in q then the statement of the lemma will be true for q sufficiently close to β.□ The logic of the proof is quite straightforward: the principal uses verification and spreading utility to provide incentives. Even if the lower bound on continuation utility for wLN is binding, the principal would still want to set wHN 4w to provide incentives. At the other extreme, an important question is if there is some upper bound on the continuation utility. The next lemma shows that if q ¼ β, wHN ðwÞ 4 w for all w. On the other hand if q 4 β there exists some w such that wHN ðwÞ ow if w4 w. Lemma 5. Continuation utility satisfies wLN ðwÞ r w and the inequality is strict if w 4w. If q ¼ β then wHN ðwÞ 4 w for all wZ w. Proof. In Appendix A. The result shows that verification is never the sole method of providing incentives. If the continuation utilities fall in an interval where the principal's value function is flat, then only the expected continuation utility matters for the principal's value and delivering incentives by spreading continuation utilities is costless, so there is no need for expensive verification. Once the continuation utilities exit the linear spot (which possibly may not exist) providing incentives becomes expensive. However since the slope of the value function changes continuously, the marginal cost of spreading utility increases gradually. On the other hand, the cost of an extra “unit” of verification is linear, so the principal would use spreading utility to some extent. Another way of looking at it is that the cost of spreading utility by ϵ is of the order ð1=2Þv″ ðwÞϵ2 whereas the cost of verification is linear. So for ϵ small enough, the cost of spreading utilities will be lower. After that point verification will be used. The first order conditions show that v0 ðwLL Þ ¼ ðβ=qÞv0 ðwÞ and I showed above that v0 ðwHN ðwÞÞ oðβ=qÞv0 ðwÞ and 0 v ðwLN ðwÞÞ Zðβ=qÞv0 ðwÞ where the inequality is strict for w large enough. Therefore, taken together, Lemmas 4 and 5 give a result about continuation utility similar to what Proposition 5 shows about consumption. The two results depend crucially on the fact that the principal's value function is continuously differentiable everywhere: spreading continuation utility is initially costless. However, if the value function has a kink, this mechanism is not in action. Even further, Koeppl (2006) shows that points of non-differentiability in incentive-constrained problems have measure zero, but can be attractor points for expected utility.6 4.2. Verification probability I turn next to the verification probability ϕL. The first question to answer is if the principal would perform any verification at all. The following proposition shows that if the verification cost is sufficiently low, the principal would set ϕL 40 for some w. The logic is straightforward. The value function can differ from the first-best by at most γπ L =ð1 qÞ. Since the principal's value when verification is not allowed is strictly smaller than the first-best, then if the verification cost is low enough, then the value of the principal is higher than the no-verification value, so the principal must verify. Proposition 7. If γ is low enough, then ϕL ðwÞ 4 0 for some w. Proof. The proof is straightforward. Let vFB ðwÞ be the principal's value associated with the first-best allocation. Recall that if the principal verifies the low shock with probability 1, then the first-best allocation is incentive-compatible. This shows that: vðwÞ Z vFB ðwÞ 

γπ L 1 q

ð25Þ

Denote vNV ðwÞ to mean the value function of the principal if verification is not possible. If ϕL ðwÞ ¼ 0 for all w, then ^ Clearly vNV ðwÞ ^ ovðwÞ. ^ Then if γ o 1  ðq=π L Þ½vðwÞ ^  vNV ðwÞ, ^ (25) implies that vðwÞ ^ 4 vNV ðwÞ. ^ vA ðwÞ ¼ vðwÞ. Fix an arbitrary w. This proves that ϕL ðwÞ 40 for some w.□ 6

I thank the anonymous referee for this insight.

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

Remarkably, the converse is not true. I will give a class of utility functions such that the principal would use verification for arbitrarily high costs. I have shown that if the principal verifies with probability ϕ~ L ðwÞ ¼

uðcðwÞ þ ΔyÞ  uðcðwÞÞ ; w βw þuðcðwÞ þ ΔyÞ  uðcðwÞÞ

where c(w) is the consumption in the corresponding full information problem, then the first-best allocation is incentivecompatible. Then clearly, vðwÞ Z vFB ðwÞ γπ L ϕ~ L ðwÞ=ð1 qÞ. Define w ¼ sup uðcÞ=ð1  βÞ. By Assumption 2 limc-1 u0 ðcÞ ¼ 0, therefore limw-w ϕ~ L ðwÞ ¼ 0. This shows that the cost of ensuring complete insurance is becoming negligible. On the other hand, as a richer agent can bear consumption risk more easily, the benefits of complete insurance may also fall. In the proof of the following proposition, I show that if absolute risk aversion does not decrease too rapidly, the benefits dominate the costs: limw-w ½vFB ðwÞ  vNV ðwÞ=ϕL ðwÞ ¼ 1, which imply that for arbitrarily high costs, verification will occur.7 Proposition 8. Suppose that the agent's absolute risk aversion satisfies u″ ðcÞ=u0 ðcÞ Zbcα  1 for all c, where α A ð0; 1 and b 4 0 are some constants. Also suppose that q ¼ β.8 Then ϕL ðwÞ 4 0 for some w for arbitrarily high verification cost γ. Proof. In Appendix A. Utility functions in the Constant Absolute Risk Aversion (CARA) class satisfy the assumption with α ¼ 1.9 For Constant Relative Risk Aversion preferences (CRRA)  u″ ðcÞ=u0 ðcÞ ¼ bc  1 ; therefore the proposition above is applicable if absolute riskaversion decreases slower than for CRRA preferences. To summarize the argument, the cost of making full insurance possible, π l ϕL ðwÞγ=ð1 qÞ converges to zero. I have shown that if verification is not possible, for some utility functions the loss arising from consumption variability is proportional to ½u  1 ðð1 βÞwÞα  1 . Since the verification probability declines exponentially, the benefits will exceed the costs. Sharper characterization of the verification probability is difficult in general since verification is not the only way to provide incentives. The costs of using continuation utility to provide incentives depend on the change of the curvature of the value function. Since we do not have results about the second derivative of the value function at this point, analytical results about the verification probability appear intractable.

5. Conclusions In this work I study optimal stochastic verification in a dynamic model of random endowment shocks with private information. Two important features distinguish the current model from closely related work: the agent is risk-averse and verification is allowed to be stochastic. These two features allow the study of the interaction between insurance and incentives (including verification) in complete generality. I show that if verification is used, then it will be with probability less than one, therefore stochastic verification dominates strictly. The relationship between principal and agent has a standard recursive representation. For each history of endowments and reports, a sufficient statistic that summarizes the principal's constraints from that point onward is the agent's promised utility. For a given promised utility, the recursive contract specifies two verification probabilities – one for each possible agent's report – and consumption and continuation utility conditional on the agent's report and the outcome of the verification. The contract must satisfy incentives and provide the promised level of utility if the agent reports truthfully. Allowing verification expands the set of incentive-compatible contracts. Setting consumption and continuation utility at their lower bound gives the strongest incentives for verification. The qualitative features of the optimal contract are the following: the agent is verified only if a low endowment is reported; consumption and continuation utility are strictly ranked – highest if a high endowment is reported, middle if verification took place and lowest if low endowment was reported and no verification took place. The reason for this is that consumption and continuation utility in the high endowment state provide incentives for truthtelling and deliver promised utility, consumption and continuation utility in the verified state are irrelevant for incentive provision, so help only with delivering promised utility and, finally, in the low endowment unverified state higher consumption and continuation utility distort incentives so the planner optimally reduces them. The two innovations in the model have the biggest influence on the process of the agent's continuation utility. If verification is deterministic as in Wang (2005) and if the principal verifies an agent with some utility promise, there is no incentive to spread continuation utilities. Moreover, if the discount factors of the principal and the agent are the same the contract would be completely static – a repetition of one-shot verification scheme. Stochastic verification implies that continuation utilities are always used to provide incentives. That is explained by the fact that the cost of distorting the contract satisfies the Inada conditions (zero marginal cost at 0), whereas the cost of verification is linear. This implies that continuation utility has a downward trend if the utility is high enough. Since the lower bound on promised utility is not 7

Recall that vNV ðwÞ denotes the principal's value function if verification is not possible. The result can be generalized to q Z β. 9 The CARA utility function does not satisfy the assumption that limc-0 u0 ðcÞ ¼ 1. However, it is easy to show that there exist utility functions that satisfy this condition and have constant absolute risk aversion when consumption is above a certain threshold. 8

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

11

absorbing, the result is that at each date there is a non-trivial distribution of promised utilities with mobility. This in contrast with the linear utility case, where promised utility has an upward trend until the agent gets the entire endowment stream (payments to the principal are front-loaded) in order to economize on verification. Numerical calculations show that the shape of the optimal verification probability is complex, and depends on the curvature of the utility function in complicated ways.

Appendix A. Additional proofs Proof of Proposition 1. Equip D with a metric in the usual way, ρðf ; gÞ ¼ supx A ½w;wÞ jf ðxÞ  gðxÞj. Then for any f ; g A Dρðf ; gÞ rγ=ð1 qÞ. The Blackwell theorem is applicable in this case. The monotonicity and discounting conditions of Blackwell's theorem are clearly satisfied. All I need to show is that T: D-D. P t1 uðct Þ, bi;i ¼ c0 yi , where fct g1 It is always feasible to set ϕi ¼ 1, wi;i ¼ 1 t¼1β t ¼ 0 are derived from the first-best problem (without the incentive constraints). Then ! 2 1 X X  FV FV t 1 Tv ðwÞ Z π s ys  c0 γ þ qv β uðct Þ ¼ s¼1

t¼1

! 1 X  q γ FB t 1 ¼ vFB ðwÞ  ¼ vFV ðwÞ π s ys  c0  γ þ qv β uðct Þ γ 1  q 1 q s¼1 t¼1 2 X

Let Tn be the maximization operator Eq. (7) subject to constraints Eqs. (8) and (10), that is Tn is the operator when we ignore the incentive constraints. Obviously T  vFB ¼ vFB . TvFB r T  vFB ¼ vFB . As T is monotone and vFV rf r vFB we have vFV r TvFV r Tf r TvFB rvFB Therefore T: D-D. Then by Blackwell's theorem T is a contraction on a complete metric space and has a unique fixed point on D.□ Proof of Proposition 2. The proof follows a standard argument. Set some arbitrary ϵ 40. Let ϕi ; bi;j ; wi;j be such that they satisfy constraints Eqs. (8)–(10) and 2 X

π i ½yi þ ϕi ð bi;i  γ þqv ðwi;i ÞÞ þ ð1 ϕi Þð  bi;N þ qv ðwi;N ÞÞ 4ðTv ÞðwÞ ϵ=2

i¼1

i;j

For each wi;j there exists an allocation fϕi;j t gfbt g such that E0

1 X

i;j

t 1

t

qt ½ bt ðh Þ  ϕi;j t ðh

; yt Þγ 4 v ðwi;j Þ  ϵ=ð2qÞ

t¼0

Then we can construct an allocation fϕt g; fbt g such that the transfers and verification probabilities in period 0 are given by i;j ϕi ; bi;j and the transfers and verification probability in the subsequent periods are given by the relevant ϕi;j ; b . This allocation is incentive-compatible and delivers the promised utility. So: v ðwÞ Z E0

1 X

t

t 1

qt ½ bt ðh Þ  ϕt ðh

; yt ÞγZ

t¼0

2 X

π i ½yi þ ϕi ð  bi;i  γ þ qv ðwi;i ÞÞ

i¼1

þ ð1 ϕi Þð  bi;N þ qv ðwi;N ÞÞ ϵ=2 ZðTv ÞðwÞ ϵ But ϵ is arbitrary, so v ðwÞ Z ðTv ÞðwÞ. Now, assume that v ðwÞ 4 ðTv ÞðwÞ. Then there exists an allocation that satisfies Eqs. (1)–(5) and E0

1 X

t

qt ½ bt ðh Þ ϕt ðh

t 1

; yt Þγ 4 ðTv ÞðwÞ

t¼0

Then define wi;j be the expected utility in state (i,j) implied by this allocation. Then bi;j , wi;j and ϕi satisfy Eqs. (8)–(10) and 2 X

π i ½yi þ ϕi ð bi;i  γ þqv ðwi;i ÞÞ þ ð1 ϕi Þð  bi;N þ qv ðwi;N ÞÞ ZE0

i¼1

which gives a contradiction.□

1 X t¼0

t

t 1

qt ½  bt ðh Þ  ϕt ðh

; yt Þγ 4 ðTv ÞðwÞ;

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

Proof of Lemma 1. In both cases I will show that if a contract does not satisfy the condition, I can construct a contract that does (satisfy the condition) and also strictly increases the principal's value. Thus without loss of generality we can restrict the set of contracts in such a manner. (i) Assume ciN 4 ci;i Then replace ciN and ci;i with ci defined by uðci Þ ¼ ϕi uðci;i Þ þð1  ϕi ÞuðciN Þ. This variation keeps all the constraints satisfied and increases the principal's profit. The proof of the other statement is analogous. (ii) Assume ϕi 40 and the constraint is slack. ϕj ½uðcj;j Þ þ βwj;j  þ ð1 ϕj Þ½uðcjN Þ þ βwjN  4 ð1 ϕi Þ½uðciN þyj yi Þ þ βwiN  þ ϕi βw If ϕi ¼ 1, set, without loss of generality ci;N ¼ ci;i and wi;N ¼ wi;i . Define c0iN ðzÞ by uðc0iN ðzÞÞ ¼ uðciN Þ þz and c0i;i ðzÞ by uðc0i;i ðzÞÞ ¼ uðci;i Þ  ϕi z=ð1 ϕi Þ. Pick the largest z such that the incentive constraint is still satisfied and ciN ðzÞ rci;i ðzÞ and replace ci;N with c0i;N ðzÞ and ci;i with c0i;i ðzÞ. If the constraint is binding, then we are done. If the constraint is still not binding, then repeat the same procedure on wi;N and wi;i . Both these variations keep the promise-keep constraint intact and increase the principal's value. If the constraint is still not binding, ci;N ¼ ci;i and wi;N ¼ wi;i . Then we can lower ϕ to the level where the incentive constraint will be binding (or to 0 if this level is negative). This strictly increases the principal's value by reducing verification costs. If optimal policies exist and they do not satisfy (i) or (ii) then we can change the contract in a way described above and increase the principal's value which is a contradiction.□ Proof of Lemma 2. The strategy of the proof is exactly the same as in Lemma 1. (i) Assume the relevant incentive constraint is slack. Then by Lemma 1, I know that ϕL ¼ 0. There are two cases to consider. Assume the other incentive constraint is slack too. Then ϕH ¼ 0 and I have: uðcHN Þ þβwHN 4uðcLN þ ΔyÞ þ βwLN uðcLN Þ þ βwLN 4uðcHN  ΔyÞ þ βwHN Summing the constraints and rearranging I get: uðcHN Þ uðcHN  ΔyÞ 4 uðcLN þ ΔyÞ  uðcLN Þ Since u is strictly concave this is possible if and only if cHN ocLN þΔy. This and the downward incentive constraint implies that wHN 4 wLN . In this case I can increase wLN and decrease wHN in such a way to keep the expected utility constant and make the downward constraint bind. This variation satisfies all the constraints and increases the principal's value. In the second case the downward constraint is binding. ϕH ðuðcHH Þ þ βwHH Þ þ ð1  ϕH ÞðuðcHN Þ þβwHN Þ 4 uðcLN þ ΔyÞ þ βwLN uðcLN Þ þ βwLN ¼ ð1 ϕH ÞðuðcHN  ΔyÞ þ βwHN Þ þ ϕH βw

If ϕH ¼ 1, then the binding upward constraint implies that cLN ¼ 0; wLN ¼ w. Then it will be possible to decrease the spread between cLN and cHH and between wLN and wHH in such a way to keep the promise-keeping constraint binding, make the downward constraint bind and the upward slack, and increase the principal's value. Next, assume that wHN 4wLN and ϕH o1. Then I can modify wHN and wLN by: w0HN ðzÞ ¼ wHN  z, 0 wLN ðzÞ ¼ wL N þπ H ð1 ϕH Þz=π L , where z 4 0. If z is small enough, all the constraints are still satisfied, wHN0 ðzÞ Z wLN0 ðzÞ and the principal's value is strictly increased since v is strictly concave. Let z1 denote the value of z such that the downward incentive constraint is binding. If w0HN ðz1 Þ Z w0LN ðzÞ, then we are done. If not, we set w0HN ¼ w0LN ¼

π H ð1  ϕH ÞwHN þπ L wLN π H ð1  ϕH Þ þπ L

This would keep all the constraints satisfied and reduce the spread between continuation utilities thus increasing the principal's profit. Then the last possibility is ϕH o1 and wLN Z wHN . Then the upward incentive constraint implies that cLN r cHN Δy. In this case I can increase cLN slightly and decrease cHN to keep the expected utility constant and ensure that the upward constraint binds. This variation satisfies all the constraints and increases the principal's value. (ii) Let us take an arbitrary allocation that satisfies the promise-keeping and the downward incentive constraint. I will show that there exists another allocation that satisfies all the constraints and gives the principal higher value. This will prove that we can ignore the downward incentive constraint when performing the maximization. Using the construction from Lemma 1, we can change the contract to ensure that ϕH ¼ 0 and increase the principal's value. Similarly, we can assume that the downward constraint will be binding: uðcHN Þ þβwHN ¼ ð1 ϕL Þ½uðcLN þ ΔyÞ þ βwLN  þ ϕL βw

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

13

Then the upward constraint is equivalent to: uðcHN Þ  uðcHN  ΔyÞ Z ð1 ϕL ÞðuðcLN þ ΔyÞ  uðcLN ÞÞ þ ϕL ½βw  uðcLL Þ  βwLL  A sufficient condition for this inequality to be satisfied is cHN  Δyr cLN . Assume this is not true, i.e. cHN 4 cLN þ Δy. Then the downward constraint implies that wHN owLN : Then increase cLN slightly and decrease cHN to keep expected utility constant. Then increase wHN and decrease wLN in such a way to keep the downward constraint satisfied and keep the expected continuation utility constant. This strictly increases the principal's value. Thus the new contract satisfies all the constraints of the original problem and increases the principal's value.□ Next we introduce some auxiliary results for the proof of Proposition 3. Let A ¼ ðfci;j g; fwi;j g; fϕk gÞ denote a contract. To simplify exposition I define the following notation: ξf ðAÞ ¼

2 X

  π k yk þð1  ϕk Þ½  ckN þ βf ðwkN Þ þ ϕk ½  ck;k γ þ βf ðwk;k Þ

ðA:1Þ

k¼1

where it is understood that ci;j … are the ones specified in A. Then we can write succinctly: Tf ðwÞ ¼ supξf ðAÞ A

s:t ð8Þ–ð10Þ

Lemma A1. The set of incentive-compatible contracts is closed. Proof. All the constraints can be written in the form ρi ðAÞ Z0, where ρi are continuous functions. Then if An -A, ρi ðAÞ Z0 therefore A is incentive-compatible, so the set is closed.□ Let f be a concave function with f A D. I will use Lemmas 1 and 2 to construct a compact set B(w) with the property that for any incentive-compatible contract A, there exists A0 A BðwÞ such that A0 is incentive-compatible and the value for the principal ξf ðA0 Þ Z ξf ðAÞ. Lemma A2. There exists a compact-valued correspondence B(w) such that if T 0 f ðwÞ ¼ supξf ðAÞ A

s:t ð8–ð10Þ and A A BðwÞ and f A D is concave, then T 0 f ¼ Tf . Proof. The proof is constructive. Let 2 0 2 πH y þ πL y 34 π H yH þπ L yL 134 H L  vFV ðwÞ vFV ðwÞ  1β 1 β 6 C7 6 7 2 FB  1 B BðwÞ ¼ 40; @ A5  fwg2  ½0; 12 5  f0g  4w; v πH πH β

ðA:2Þ

For any incentive-compatible contract, setting cLH ¼ cHL ¼ 0 and wLH ¼ wHL ¼ w keeps the contract incentive-compatible and does not change the principal's value, so this can be assumed without loss of generality. Lemmas 1 and 2 show that if f is concave for any incentive-compatible contract A we can find A0 such that the downwards constraint is slack, ϕH ¼ 0, cHH ZcLL Z cLN , wHH ZwLL Z wLN and that ξf ðA0 Þ Z ξf ðAÞ. Since cHH and wHH are irrelevant, we can assume that cHH ¼ 0 and wHH ¼ w. Then proving the lemma is equivalent to proving that for any contract A such that cHN Z cLL ZcLN and wHN Z wLL Z wLN and A2 = BðwÞ I can find an incentive-compatible contract A0 A BðwÞ such that ξf ðA0 Þ 4 ξf ðAÞ. It is easy to see that for any w, the contract A0 that specifies consumption cHN ¼ cLL ¼ cLN ¼ u  1 ðw=ð1 βÞÞ, cHL ¼ cLH ¼ cHH ¼ 0, wHN ¼ wLL ¼ wLN ¼ w, wHL ¼ wLH ¼ wHH ¼ w, ϕL ¼ 1, ϕH ¼ 0 is incentive-compatible and delivers utility w. ξf ðA0 Þ ¼ π H yH þπ L yL  u  1 ðw=ð1 βÞÞ  π L γ þ βf ðwÞ 4 π H yH þ π L yL u  1 ðw=ð1  βÞÞ γ þ βf

FV

ðwÞ ¼ vFV ðwÞ 0

ðA:3Þ

= BðwÞ then either Then it is sufficient to show that ξf ðAÞ o vFV ðwÞ. Because for A cHH Z cLL Z cLN and wHH Z wLL Z wLN and A 2 π H yH þ π L yL  vFV ðwÞ 1β cH 4 πH or wH 4v

0 π H yH þ π L yL 1 vFV ðwÞ  1β C @ A πH β

FB  1 B

or both. I will take these two possibilities in turn.

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L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

From the definitions we have f ðwÞ r vFB ðwÞ rðπ H yH þ π L yL Þ=ð1  βÞ Consider the first possibility.

  ξf ðAÞ ¼ π H yH þ π L yL þ π H ð cHH þ βf ðwHH ÞÞ þ π L ϕL ð  cLL þ βf ðwLL ÞÞ þπ L 1 ϕL ð  cLN þ βf ðwLN ÞÞ π H yH þ π L yL π H yH þ π L yL π H yH þ π L yL o þvFV ðwÞ  ¼ vFV ðwÞ  π L ϕL γ rπ H yH þπ L yL π H cHH þ β 1 β 1β 1β

Now the second case. Using the same chain of inequalities, we get:   π H yH þ π L yL þβπ H f ðwHH Þ ξf ðAÞ r π H yH þ π L yL þ βπ H f ðwHH Þ þ βπ L ϕL f ðwLL Þ þ βπ L 1  ϕL f ðwLN Þ r 1β π H yH þ π L yL þβπ H vFB ðwHH Þ o vFV ðwHH Þ r 1β In the last inequality the fact that vFB is strictly decreasing is used. So in both cases, the contract specified in advance, A0 , dominates strictly.□ Proof of Proposition 3. First, I show that given a concave function f A D, the supremum is attained. f is a concave on R1 , so f is continuous. As shown in Lemma A2, without loss of generality Tf ðwÞ ¼ supξf ðwÞ A

s:t ð8Þ–ð10Þ; A A BðwÞ But B(w) is compact and ξf ðAÞ is continuous, therefore the max is attained. Since {A: A satisfies Eqs. (8)–(10), A A BðwÞ} is a continuous compact-valued correspondence the Maximum theorem implies the required result. Given that, it is sufficient to show that if f is concave then Tf is concave too. Let w rw0 ow″ and λ A ð0; 1Þ. Define: ϕ‴L ¼ λϕ0L þ ð1 λÞϕ″L

w‴HN ¼ λw0HN þ ð1 λÞw″HN w‴LN ¼

λð1  ϕ0L Þw0LN þ ð1  λÞð1 ϕ″L Þw″LN 1 ϕ‴L

w‴LL ¼

λϕ0L w0LL þ ð1 λÞϕ″L w″LL ϕ‴L

Define c‴HN ; c‴LN ; c‴LL by uðc‴HN Þ ¼ λuðc0HN Þ þð1  λÞuðc″HN Þ 

uðc‴LN Þ ¼ λð1 ϕ0L Þuðc0LN Þ þ ð1  λÞð1 ϕ″L Þuðc″LN Þ ð1  ϕ‴L Þ 

uðc‴LL Þ ¼ λϕ0L uðc0LL Þ þ ð1  λÞϕ″L uðc″LL Þ ϕ‴L

If ϕ0 ¼ ϕ″ ¼ 0, then without loss of generality, we can set c‴LL ¼ c‴LN and w‴LL ¼ w‴LN . It is easy to verify that the expected utility the agent gets from this allocation is: π½uðc‴HN Þ þ βw‴HN  þ ð1 πÞð1  ϕ‴L Þ½u0 ðc‴LN Þ þ βw‴LN  þ ð1 πÞϕ‴L ½u0 ðc‴LL Þ þ βw‴LL  ¼ λw0 þ ð1 λÞw″ Consider the profit of the principal if this solution is incentive-compatible:      c‴HN ¼ u  1 λu c0HN þ ð1  λÞu c″HN r λc0HN þ ð1 λÞc″HN " # λð1  ϕ0L Þuðc0LN Þ þð1  λÞð1  ϕ″L Þuðc″LN Þ λð1  ϕ0L Þc0LN ð1  λÞð1 ϕ″L Þc″LN c‴LN ¼ u  1 þ r ‴ 1 ϕ‴L 1  ϕ‴L 1  ϕL " # λϕ0L uðc0LN Þ þ ð1  λÞϕ″L uðc″LN Þ λϕ0 c0 ð1  λÞϕ″L c″LL c‴LL ¼ u  1 r L‴ LL þ ϕL ϕ‴L ϕ‴L Then using these inequalities and the assumption that f is concave: Ey  πc‴HN ð1  πÞð1 ϕ‴L Þc‴LN ð1  πÞϕ‴L c‴LL þqπf ðw‴HN Þ þ qð1  πÞð1 ϕ‴L Þf ðw‴LN Þ þ qð1  πÞϕ‴L f ðw‴LL Þ  γð1  πÞϕ‴L ZλðTf Þðw0 Þ þð1  λÞðTf Þðw″ Þ:

ðA:4Þ

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

15

I will show that I can further modify this allocation to ensure that the downward incentive constraint holds with equality, cHN ocLN þ Δy, and the new allocation does not reduce the principal's profit. From Lemma 2 I know that: uðc0HN Þ þ βw0HN ¼ ð1  ϕ0L Þðuðc0LN þΔyÞ þβw0LN Þ þ ϕ0L βw uðc″HN Þ þ βw″HN ¼ ð1  ϕ″L Þðuðc″LN þΔyÞ þβw″LN Þ þ ϕ″L βw

So using the definitions we get: " #      λð1  ϕ0L Þ ð1  λÞð1 ϕ″L Þ  ″ ‴ ‴ 0 u ð c þ Δy Þ þ u c þ Δy þβw u c‴HN þ βw‴HN ¼ 1 ϕ‴L LN LN LN þ ϕL βw: 1 ϕ‴L 1  ϕ‴L

ðA:5Þ

~ by: Define cLN ~ Þ¼ uðcLN

 ð1 λÞð1  ϕ″L Þ  ″  λð1  ϕ0L Þ  0 u cLN þ Δy þ u cLN þ Δy 1  ϕ‴L 1  ϕ‴L

Recall that   λð1 ϕ0L Þ  0  ð1  λÞð1  ϕ″L Þ  ″  u c‴LN ¼ u cLN þ u cLN 1  ϕ‴L 1  ϕ‴L Then using proposition 6.C.3. of Mas-Collel et al. (1995) and the assumption of non-increasing absolute risk aversion ~ c‴LN þ Δyr cLN

ðA:6Þ

  λð1 ϕ0L Þ  0  ð1  λÞð1  ϕ″L Þ  ″  u cLN þ Δy þ u cLN þΔy u c‴LN þ Δy r 1  ϕ‴L 1  ϕ‴L

ðA:7Þ

So from (A.5) and (A.7) we get:  uðc‴HN Þ þ βw‴HN Z ð1  ϕ‴L Þ uðc‴LN þΔyÞ þβw‴LN þϕ‴L βw:

ðA:8Þ

If the constraint is slack, I can perform the modifications described in Lemmas 1 and 2 to ensure that the constraint binds and cHN o cLN þ Δy. Therefore ðTf Þðλw0 þ ð1  λÞw″ Þ ZλðTf Þðw0 Þ þ ð1  λÞðTf Þðw″ Þ so T maps concave functions to concave functions. But the space of concave function is complete, therefore, v is concave.□ Next I turn to proving Theorem 1. I prove some technical results first. t1

Þg is generated recursively by the policy Lemma A3. If the sequence of continuation utilities for the truthtelling agent fwt ðw0 ; h functions then limt-1 βt E  1 wt ¼ 0 where the expectation is taken with respect to the measure induced by the verification function ϕðwÞ. Proof. Since wt Zw we know that lim inf t-1 βt E0 wt Z0. Then it is sufficient to show that lim supt-1 βt E0 wt r0. Assume 0 not. Then there exists A 40 such that for every t there exists t 0 Z t and E0 wt0 Z Aβ  t . By definition: vðw0 Þ ¼

TX 1

  n o t t 1 T 1 qt E0 π H yH þ π L yL  ct ðh Þ γϕt ðh ; yt Þ þqT E0 v wT h

t¼0

r

  1  qT      1  qT  T 1 T 1 r π H yH þπ L yL þ qT E0 vFB wT h π H yH þπ L yL þ qT vFB E0 wT h 1q 1q

I define ηðwÞ by: ηðwÞ ¼  vFB ðwÞ þ

π H yH þ π L yL 1q

That is ηðwÞ is the cost of delivering utility w when there are no information frictions. With this notation we can write:   π H yH þπ L yL T 1 :  qT η E0 wT h vðw0 Þ r 1q ~ such that if w 4 w, ~ ηðwÞ 4 Bw. By It is clear that η is strictly concave and limw-w η0 ðwÞ ¼ 1. Then for any B 4 0, there exists w

16

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22 0

~ Then: assumption there exists t 0 such that E0 wt 0 ZAβ  t 4 w.

t0   q T 1 Z BE0 wt0 Z AB qT η E0 wT h β Then we get: vðw0 Þ r

t 0 π H yH þ π L yL q π H yH þ π L yL  AB AB r β 1q 1q

Since A is fixed and B is an arbitrary positive number vðw0 Þ ¼  1. But vðw0 Þ ZvFB ðw0 Þ γ=ð1  qÞ 4  1. Thus we get the contradiction required.□ We break the proof of Theorem 1 in 2 stages. t1

t

Proposition A1. If for given w0 an allocation fbt ðh Þg, fϕt ðh allocation delivers the promised utility w0.

; yÞg is generated recursively by the policy functions then the

Proof. By repeated application of the recursive promise-keeping constraint Eqs. (8)–(10) and the law of iterated expectations we get: w0 ¼ w0 ¼

T X t¼0 1 X

t

T

βt E0 uðct ðh ÞÞ þβT E0 wT ðh Þ; 8 T t

T

βt E0 uðct ðh ÞÞ þ lim βT E0 wT ðh Þ T-1

t¼0

So, finally using Lemma A3 we get: w0 ¼

1 X

t

βt E0 uðct ðh ÞÞ:□

t¼0

Now I turn to showing that the allocation generated recursively by the policy functions satisfies the incentivecompatibility constraint. I will show that a strategy that involves deviations in finite number of periods is inferior to the truthtelling strategy. Then I show that if a strategy dominates truthtelling then there exists a strategy with finite deviations that dominates truthtelling, thus getting a contradiction. Let Σ T ¼ fσ A Σ: σ t ðyt ; at  1 Þ ¼ yt ; t 4Tg, i.e. the set of strategies that involve misreporting in at most T þ 1 periods. t 1

t

Lemma A4. Suppose that an allocation fbt ðh Þg, fϕt ðh σ  ¼ arg maxE0 σ A ΣT

1 X

t1

βt u½yt þbt ðh

; yÞg is generated recursively by the optimal policy functions. For any T

; σðyt ; at  1 Þ; at Þ

t¼0

Proof. The proof is by induction. Assume that the hypothesis is true for some T. I will prove that it is true for T þ 1. Assume not. Then for some σ A Σ T þ 1 TX þ1

t1

βt E0 u½yt þbt ðh

; σðyt ; at  1 Þ; at Þ þ

t¼0

1 X

t 1

βt E0 u½yt þ bt ðh

; yt ; at Þ 4

t ¼ T þ2

t¼0

By the previous theorem we know that 1 X

t1

βt E0 u½yt þ bt ðh

t þ1

; yt Þ ¼ βT þ 2 wT þ 2 ðh

Þ

t ¼ T þ2 tþ1

where wT þ 2 ðh Þ is the continuation utility from the recursive problem. Then from incentive compatibility we know that for any state hT and yT þ 1 T

T

Efu½yT þ 1 þ bT þ 1 ðh ; yT þ 1 ; aT þ 1 Þ þβwT þ 2 ðh ; yT þ 1 ; aT þ 1 Þg Z Efu½yT þ 1 þ bT þ 1 ðh ; σðyT þ 1 ; aT Þ; aT þ 1 Þ þ βwT þ 2 ðh ; σðyT þ 1 ; aT Þ; aT þ 1 Þg T

T

Thus we get: T X t¼0

t1

βt E0 u½yt þbt ðh

; σðyt ; at  1 Þ; at Þ þ

1 X t ¼ T þ1

1 X

t 1

βt E0 u½yt þ bt ðh

; yt ; at Þ Z

t1

βt E0 u½yt þbt ðh

; yt ; at Þ

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22 TX þ1

t 1

βt E0 u½yt þ bt ðh

1 X

; σðyt ; at  1 Þ; at Þ þ

t¼0

t1

βt E0 u½yt þbt ðh

17

; yt ; at Þ

t ¼ T þ2

Then define a new strategy σ 0 equivalent to σ in which truthtelling starts one period earlier. ( σðyt ; at  1 Þ; t r T σ 0 ðyt ; at  1 Þ ¼ t 4T yt ; I have shown that σ 0 dominates σ  but σ 0 A Σ T thus reaching a contradiction. To complete the inductive proof, note that the same argument as above shows that the agent will not deviate in period 0.□ Proof of Theorem 1. I have already proved that the allocation satisfies promise-keeping. I prove that it also satisfies incentive compatibility by contradiction. Assume that there exists a reporting strategy σ that gives strictly higher utility. E0

1 X

t1

βt u½yt þbt ðh

; σðyt ; at  1 Þ; at Þ 4E0

t¼0

1 X

t1

βt u½yt þ bt ðh

; yt ; at Þ

t¼0

Note that since uðcÞ Z0 all the integrals and sums are well-defined. Also the Lebesgue dominated convergence theorem applies and we get: 1 X

t1

βt E0 u½yt þ bt ðh

; σðyt ; at  1 Þ; at Þ4

t¼0

1 X

t 1

βt E0 u½yt þ bt ðh

; yt ; at Þ:

t¼0

By Proposition A1 the expected utility from truthtelling is w0 o 1 and the expected utility from the alternative strategy is well-defined even though it can be infinity. Then there exists T such that T X

t1

βt E0 u½yt þ bt ðh

; σðyt ; at  1 Þ; at Þ4

t¼0

1 X

t 1

βt E0 u½yt þ bt ðh

; yt ; at Þ:

t¼0

Since uðcÞ Z0 we get: T X

t1

βt E0 u½yt þ bt ðh

; σðyt ; at  1 Þ; at Þþ

t¼0

1 X t ¼ T þ1

βt E0 u½yt þ bt ðh

t 1

; yt ; at Þ 4

1 X

t1

βt E0 u½yt þ bt ðh

; yt ; at Þ:

t¼0

Then a strategy with finite periods of deviation dominates truthtelling. In particular, let: ( σðyt ; at  1 Þ; t r T σ 0 ðyt ; at  1 Þ ¼ t 4T yt ; σ 0 A Σ T but it dominates truthtelling which contradicts Lemma A4. Therefore the allocation satisfies incentive compatibility.□ Proof of Lemma 3. Let v(w) be weakly decreasing. I will show that TvðwÞ is weakly decreasing. Let w rw1 o w2 . Let c2HN ; c2LL , etc. denote the optimal choices for w2. I will modify that allocation to deliver utility w1, satisfy the incentive-constraint and increase the principal's profit. Decrease cLN until the expected utility is w2 or cLN ¼ 0. If necessary, repeat the same procedure with cLL ; wLN ; wLL . Each of these modifications will increase the principal's value and keep the incentive constraint satisfied. If necessary, decrease wHN and cHN until the agent's expected utility is w1 or the incentive constraint is binding. Assume that the incentive-constraint is binding but the agent's expected utility is still higher than w1. Then we have: uðc1HN Þ þ βw1HN ¼ ð1  ϕL ÞuðΔyÞ þ βw: Then the expected utility is: πðuðc1HN Þ þ βw1HN Þ þ ð1  πÞβw ¼ πð1  ϕL ÞuðΔyÞ þ βw r πuðΔyÞ þβw rð1  βÞw þ βw ¼ w:

So we get a contradiction. Therefore, the value function is weakly increasing. Now, I need to prove two more things: (a) v is strictly decreasing, and (b) cHN 4 0; cLL 4 0; cLN 4 0. Recall that: wZ

πuðΔyÞ 1β

18

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

and uð0Þ ¼ 0. w is the lowest promised utility that is incentive-compatible without verification. To force the agent to have lower utility, it is necessary to spend resources on verification. To simplify the exposition, the proof is broken up in 4 steps. 1. If cLL 4 0 or cLN 4 0 then v is strictly decreasing in a neighborhood of w. (Clearly if w1 ow2 and it is feasible to decrease cLL or cLN, then vðw1 Þ 4 vðw2 Þ). 2. cLL 4 0 and ϕL 4 0 implies that cHN 4 0; cLN 4 0. Clearly cHN Z cLL . Assume cLN ¼ 0. Then increase cLN by ϵ and cHN by ϵ0 to keep the incentive-constraint satisfied. Decrease cLL to keep PK. For ϵ small enough this variation will increase the principal's value. 3. Assume v is strictly decreasing. I want to show cLN 4 0 if w 4w. The first step is to show that if wij 4 w then the corresponding cij 40. Let w 4 w Pick some δ4 0 small enough to ensure w δ 4 w. Define K¼

vðwij Þ  vðwij  δÞ o 0: δ

Then since v is concave, for any 0 oδ0 o δ vðwij Þ  vðwij  δ0 Þ rK: δ0 Assume that cij ¼ 0. Then we increase the utility from consumption by ϵ oδβ and decrease wij by ϵ=β. So the change of the principal's value from this variation is:          u  1 ðϵÞ q vðwij ϵ=βÞ  vðwij Þ u  1 ðϵÞ q þ  K : Zϵ   u  1 ðϵÞ þq v wij  ϵ=β v wij ¼ ϵ  ϵ β ϵ β ϵ=β For ϵ small enough the term in the brackets is positive. This variation keeps the promise keeping constraint (and incentive constraint for cLN) and increases the principal's value. Now assume cLN ¼ 0. If cLL 4 0 then by step 2 cLN 40. So cLL ¼ 0. By the result shown just above, wLL ¼ w and wLN ¼ w. Then by IC: uðcHN Þ þ βwHN ¼ ð1  ϕL ÞuðΔyÞ þ βw: So the expected utility of the agent is: π½uðcHN Þ þβwHN  þ ð1  πÞð1  ϕL Þ½uðcLN Þ þ βwLN  þ ð1  πÞϕL ½uðcLL Þ þ βwLL  ¼ πð1  ϕL ÞuðΔyÞ þ βw r ð1 βÞw þβw ¼ w: This gives us a contradiction. Therefore cLN 40. 4. Now assume v is not strictly decreasing. Since it is concave and weakly decreasing, v must be constant in some interval ½w; w1  and strictly decreasing elsewhere. As shown in (1), it must be that cLN ¼ 0 and cLL ¼ 0 for all w on the flat spot. I will concentrate on w1. From the incentive-constraint: uðcHN Þ þ βwHN ¼ ð1  ϕL ÞuðΔyÞ þ βð1 ϕL ÞwLN þβϕL w: By the same argument as in part (3) wLN rw1 , wLL rw1 (otherwise consumption would be positive). So the expected utility for this agent would be: π½uðcHN Þ þβwHN  þ ð1  πÞð1  ϕL Þ½uðcLN Þ þ βwLN  þ ð1  πÞϕL ½uðcLL Þ þ βwLL  ¼ π½ð1  ϕL ÞuðΔyÞ þβð1  ϕL ÞwLN þ βϕL w þ ð1  πÞð1 ϕL ÞβwLN þ ð1 πÞϕL βwLL r πð1 ϕL ÞuðΔyÞ þ βð1 πϕL Þw1 þ βπϕL w r ð1 βÞw þ βw1 o w1 : So we get a contradiction. Therefore, v is strictly decreasing if w 4w and from (3) cLN 40; cLL 4 0. □ Proof of Proposition 4. I will use the Benveniste–Scheinkman theorem (Stokey and Lucas with Prescott, 1989 thm 4.10, ^ I will construct a function ψðwÞ, defined on some open neighborhood of w ^ such that ψðwÞ r vðwÞ, page 84.) For each w ^ ¼ vðwÞ ^ and ψðwÞ is concave and differentiable at w. ^ Define ψðwÞ ψðwÞ ¼ Ey  πcHN ðwÞ  ð1 πÞð1  ϕL ÞcLN ðwÞ ð1  πÞϕL cLL þ q½πvðwHN Þ þ ð1  πÞð1  ϕL ÞvðwLN Þ þ ð1  πÞϕL vðwLL Þ  γð1  πÞϕL ;

where cHN(w) and cLN(w) are defined implicitly by: uðcHN ðwÞÞ þ βwHN ¼ ð1  ϕL Þ½uðcLN ðwÞ þΔyÞ þβwLN  þ ϕL βw πðuðcHN ðwÞÞ þ βwHN Þ þð1  πÞð1 ϕL ÞðuðcLN ðwÞÞ þ βwLN Þ þ ð1  πÞϕL ðuðcLL Þ þ βwLL Þ ¼ w ^ 4 0 and cLN ðwÞ ^ 4 0. Then clearly, cHN(w) and cLN(w) are well-defined over some open neighborhood of From Lemma 3 cHN ðwÞ ^ and satisfy all the constraints. Then by construction ψðwÞ r vðwÞ and ψðwÞ ^ ¼ vðwÞ. ^ I need to show that ψðwÞ is concave and w

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

19

continuously differentiable. By the implicit function theorem, cHN(w) and cLN(w) are differentiable and dcLN 1 ¼ ð1  ϕL Þ½πu0 ðcLN ðwÞ þΔyÞ þð1  πÞu0 ðcLN ðwÞÞ dw dcHN ð1  ϕL Þu0 ðcLN ðwÞ þΔyÞ dcLN ¼ u0 ðcHN ðwÞÞ dw dw It is sufficient to show that dcLN =dw and dcHN =dw are increasing. Since cLN(w) is increasing in w and u is strictly concave, then clearly dcLN =dw is increasing. Similarly 1=u0 ðcHN ðwÞÞ is also increasing. So I need to show that ð1 ϕL Þu0 ðcLN ðwÞ þ ΔyÞdcLN =dw is increasing.    d  dcLN d u0 ðcLN þ ΔyÞ 1  ϕL u0 ðcLN ðwÞ þΔyÞ ¼ 0 dw dw πu ðcLN þ ΔyÞ þ ð1  πÞu0 ðcLN Þ dw 

 1π u″ ðcLN Þ u″ ðcLN þ ΔyÞ   40;  ¼ u0 ðcLN Þ u0 ðcLN þΔyÞ ½πu0 ðcLN þΔyÞ þð1  πÞu0 ðcLN Þ3 u0 ðcLN þ ΔyÞu0 ðcLN Þ where the last inequality follows from decreasing absolute risk aversion. Therefore πcHN ðwÞ and πð1  ϕL ÞcLN ðwÞ are concave functions and ψðwÞ is concave and differentiable as a sum of concave and differentiable functions. Then the result follows from the Benveniste–Scheinkman theorem.□ Proof of Proposition 5. The proof is by contradiction. Assume cLL 4 cHN . Then define c0HN and c0LL by     π H uðcH Þ þ π L ϕL uðcLL Þ u c0HN ¼ u c0LL ¼ π H þπ L ϕL This new allocation keeps all the constraints satisfied and increases the principal's profit since it decreases the spread between utilities. By the same argument we get cLL Z cLN . So assume that at the optimum cHN ¼ cLL and ϕL ðwÞ 4 0. Let's fix wHN ; wLL ; wLN . I will concentrate on Eqs. (13)–(15). If cHN ¼ cLL then Eqs. (13) and (15) imply that λ ¼ 0. It follows that cHN ¼ cLN and v0 ðwH Þ ¼ v0 ðwLN Þ ¼ v0 ðwLL Þ. Consider increasing cHN to c0HN . Define c0LN ¼ c0LL implicitly by keeping the expected utility from consumption in the current period constant. This variation will make the downward-incentive constraint slack. Then we get: uðcHN Þ þ βwHN ¼ ð1  ϕL Þ½uðcLN þΔyÞ þβwLN  þϕL βw

ðA:9Þ

Consider increasing cHN to c0HN . Now let us change c0LN and c0LL to keep the promise-keeping constraint. Define c0LN ¼ c0LL implicitly by keeping the expected utility from consumption in the current period constant. This variation would make the incentive-constraint slack. This implies that we can reduce the verification probability ϕL. ϕ0L ¼

uðc0LN þΔyÞ uðc0HN Þ þ βwLN βwHN uðc0LN þ ΔyÞ þ βðwLN  wÞ

Then by the implicit function theorem: 0 π H u0 ðc0HN Þ dcLL ¼ 0 π L u0 ðc0LL Þ dcHN 0 uðc0HN Þ þ βðwHN  wÞ u0 ðc0HN Þ dϕ0L dcLL o0 ¼  0 0 dcHN ½uðc0LN þ ΔyÞ þ βðwLN  wÞ2 dcHN uðc0LN þ ΔyÞ þβðwLN  wÞ

Changing ϕL may change expected continuation utility since it is possible that wLN o wLL . Then increase wLL to keep the expected utility constant. Since v0 ðwHN Þ ¼ v0 ðwLL Þ ¼ v0 ðwLN Þ this does not affect the principal's value since in this interval only the mean of continuation utilities is important. Then the marginal effect of increasing cHN and decreasing cLL from equality is: 0

 πH  πL

dcL dϕ0L πH dϕ0  γπ L 0 L 40 ¼  π H þπ L 0  πL c 0 ðπ L Þ dcH dcHN dcHN

So we get a contradiction. In a similar way it can be shown that cLL 4 cLN .□ Proof of Proposition 6. First, suppose that ϕL ðwÞ ¼ 0. Then wLL is clearly irrelevant. If wHN o wLN , it follows that cHN 4cLN þ Δy. Then I can increase cLN and decrease cHN keeping expected utility the same. This strictly increases the principal's profit. Lower wLN and raise wHN (keeping their expected value constant) to make the incentive-constraint bind. This increases the principal's profit weakly. So we reach a contradiction, so we know that wHN Z wLN .

20

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

If wLN ¼ w, then we are done. Suppose that wLN 4 w. Then wHN 4 w, so (Eqs. (16) and 17) hold with equality. Then if wHN ¼ wLN , the multiplier of the incentive constraint λ ¼ 0, so cHN ¼ cLN , which contradicts the downward-incentive constraint. Finally, suppose that ϕL ðwÞ 40. Again, if wLN ðwÞ ¼ w, we are done. By Proposition 5, it follows that cH 4 cLL 4cLN , which implies that λ 4 0. Since Eqs. (16)–(18) are binding, v0 ðwHN Þ o v0 ðwLL Þ o v0 ðwLN Þ which, via concavity of the value function implies the result.□ Proof of Lemma 5. The proof follows Thomas and Worrall Lemma 5 very closely. Summing Eqs. (16) and (17) gives: β π H v0 ðwHN Þ þ π L v0 ðwLN Þ r v0 ðwÞ: q

ðA:10Þ

Since wHN ZwLN , β ¼ q and v is concave (A.10) shows that v0 ðwHN Þ 4 v0 ðwÞ is impossible. Assume v0 ðwHN Þ ¼ v0 ðwÞ. Then (A.10) implies that v0 ðwLN Þ r v0 ðwÞ therefore v0 ðwLN Þ ¼ v0 ðwÞ Then Eqs. (16)–(17) imply that λ ¼ 0. Then by evaluating the First-Order Condition on verification probability Eq. (19) we see that ϕL ¼ 0. Since λ ¼ 0, then u0 ðcLN Þ ¼ u0 ðcHN Þ ¼  μ so cLN ¼ cHN . Then the incentive-constraint implies that wHN 4 wLN . Therefore there exist an interval ½w1 ; w2 , wA ½w1 ; w2 , w1 o w2 such that on it v0 ðÞ is constant. Then take any w0 A ½w1 ; w2  such that w0 aw. I will show that wHN ðw0 Þ A ½w1 ; w2  and wLN ðw0 Þ A ½w1 ; w2  which automatically implies that wLL ðw0 Þ A ½w1 ; w2 . By the argument above v0 ðwHN ðw0 ÞÞ 4 v0 ðw0 Þ is not possible. Assume that v0 ðwHN ðw0 ÞÞ ov0 ðw0 Þ. This implies that λ 4 0. Then cHN ðw0 Þ 4 cHN ðwÞ and cLN ðw0 Þ o cLN ðwÞ. λ 40 implies that either v0 ðwLN ðw0 ÞÞ 4 v0 ðwLN ðwÞÞ or wLN ¼ w. In both cases wLN ðw0 Þ r wLN ðwÞ and wHN ðw0 Þ 4wHN ðwÞ. Therefore: uðcHN ðwÞÞ þ βwHN ðwÞ ouðcHN ðw0 ÞÞ þβwHN ðw0 Þ

ðA:11Þ

uðcHN ðw0 ÞÞ þβwHN ðw0 Þ ¼ ð1  ϕL ðw0 ÞÞðuðcLN ðw0 Þ þΔyÞ þβwLN ðw0 ÞÞ þβϕL ðw0 Þw

ðA:12Þ

ð1  ϕL ðw0 ÞÞðuðcLN ðw0 Þ þ ΔyÞ þ βwLN ðw0 ÞÞ þ βϕL ðw0 Þw r uðcLN ðw0 Þ þ ΔyÞ þ βwLN ðw0 Þ

ðA:13Þ

uðcLN ðw0 Þ þ ΔyÞ þ βwLN ðw0 Þ ouðcLN ðwÞ þ ΔyÞ þ βwLN ðwÞ:

ðA:14Þ

Equality (A.12) follows from the fact that at the optimum the incentive-constraint holds with equality (Lemma 2). Then (A.11)–(A.14) implies: uðcHN ðwÞÞ þ βwHN ðwÞ ouðcLN ðwÞ þ ΔyÞ þ βwLN ðwÞ This gives a contradiction. Therefore v0 ðwHN ðw0 ÞÞ ¼ v0 ðwÞ. Clearly, v0 ðwLN ðw0 ÞÞ 4v0 ðwÞ is impossible since it contradicts (A.10), therefore v0 ðwLN ðw0 ÞÞ rv0 ðwÞ. But wLN ðw0 Þ rwHN ðw0 Þ, hence v0 ðwLN ðw0 ÞÞ Z v0 ðwHN ðw0 ÞÞ ¼ v0 ðwÞ, so v0 ðwLN ðw0 ÞÞ ¼ v0 ðwÞ. Therefore w1 r wLN ðw0 Þ r wHN ðw0 Þ r w2 . Then for any w A ½w1 ; w2  consumption at any date is determined by u0 ðcÞ ¼ v0 ðwÞ. Then expected utility for any w is uðcÞ=ð1 βÞ which gives us a contradiction. Then v0 ðwHN ðwÞÞ ov0 ðwÞ, so we have wHN ðwÞ 4 w. Next, I show that wLN ðwÞ o w if w4 w. If wLN ðwÞ ¼ w, then we are done. Suppose not. Then (A.10) holds with equality. If β ¼ q, we showed that v0 ðwHN Þ o v0 ðwÞ, so (A.10) implies v0 ðwLN Þ 4v0 ðwÞ, so wLN ðwÞ o w. Lastly, suppose that β=q o1. Since v0 ðwLN Þ Z v0 ðwHN Þ, v0 ðwHN Þ 4ðβ=qÞv0 ðwÞ is impossible. Then v0 ðwHN Þ rðβ=qÞv0 ðwÞ and (A.10) at equality implies that v0 ðwLN Þ Zðβ=qÞv0 ðwÞ 4 v0 ðwÞ (we used the fact that v0 ðwÞ o 0) and therefore wLN o w.□ Lemma A5. Let aðcÞ  u″ ðcÞ=u0 ðcÞ denote the absolute risk aversion of some utility function u, which is strictly concave and twice continuously differentiable. Suppose that aðÞ is non-increasing and that aðcÞ Z bcα  1 4 0 for all c, where α A ð0; 1 is some constant. Let X be some Borel-measurable lottery with zero mean, which is bounded: x1 r X r x2 . For all c 4  x1 , define rp(c) as: Euðc þ x þrpðcÞÞ ¼ uðcÞ: Then rpðcÞ Zð1=2Þbðc þ x2 þ x1 Þα  1 e  ðx2  x1 Það  x1 Þ var x for all c 4  x1 . Proof. First we show that u″ is weakly increasing. Suppose not. Then for some c1 o c2 , u″ ðc1 Þ 4 u″ ðc2 Þ. Then 0 o u″ ðc1 Þ o  u″ ðc2 Þ. Then we have aðc1 Þ ¼ 

u″ ðc1 Þ u″ ðc1 Þ u″ ðc2 Þ o 0 o 0 ¼ aðc2 Þ; 0 u ðc1 Þ u ðc2 Þ u ðc2 Þ

which contradicts the assumption that absolute risk aversion is non-increasing. By second-order Taylor expansion,   1 E uðcÞ þ u0 ðcÞðx þ rpðcÞÞ þ u″ ½c þ ξðxÞðrpðcÞ þxÞðx þrpðcÞÞ2 ¼ uðcÞ; 2

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

21

where ξðxÞ A ½0; 1. Then taking expectations, 1 u0 ðcÞrpðcÞ þ Efu″ ½c þξðxÞðrpðcÞ þ xÞðx þ rpðcÞÞ2 g ¼ 0 2 We showed that u″ is increasing, so Efu″ ½c þξðxÞðrpðcÞ þ xÞðx þ rpðcÞÞ2 g r Efu″ ðc þ x2 þ rpðcÞÞðx þ rpðcÞÞ2 g ¼ u″ ðc þ x2 þ rpðcÞÞðvar x þrpðcÞ2 Þ Therefore rpðcÞ Z 

1 1 u″ ðc þ rpðcÞ þ x2 Þ u0 ðc þrpðcÞ þ x2 Þ var x þ rpðcÞ2 Z bðc þx2  x1 Þα  1 var x; 2 u0 ðcÞ 2 u0 ðcÞ

where we used the bound on absolute risk-aversion and the fact that rpðcÞ r  x1 . R cþz  aðsÞ ds Since u satisfies the following differential equation: u″ ðcÞ þaðcÞu0 ðcÞ ¼ 0, we have that u0 ðc þ zÞ ¼ u0 ðcÞe c . Clearly, rpðcÞ r  x1 , so R c þ x2  x1 u0 ðc þrpðcÞ þ x2 Þ u0 ðc þ x2  x1 Þ  aðsÞds Z ¼e c Ze  ðx2  x1 Það  x1 Þ 0 0 u ðcÞ u ðcÞ So, we get that rpðcÞ Z 12bðc þ x2  x1 Þα  1 e  ðx2  x1 Það  x1 Þ var x.□ Lemma A6. Suppose that the utility function satisfies the hypothesis of Lemma A5 and q ¼ β. Then there exists some k 4 0 such that vNV ðwÞ o vFB ðwÞ k½u  1 ðð1  βÞwÞα  1 for all w sufficiently large. Proof. Let H denote the Bellman operator on D when no verification is allowed. Also vNV is the value function of a principal who cannot verify. By the same arguments as for the rest of the model, HvNV ¼ vNV ; similarly we can show that the operator H is monotone. Since vNV rvFB and H is monotone, vNV ¼ HvNV r HvFB . Next, we show that HvFB ðwÞ rvFB ðwÞ  k for all w Zuð2ΔÞ=ð1 βÞ, where k is some constant. Since q ¼ β, it is immediate that vðwÞ ¼ Ey=ð1  βÞ  u  1 ðwÞ=ð1  βÞ. This implies that we can write



 Ey β 0 β 0  π H cH þ cH  π L cL þ cL HvFB ðwÞ ¼ max 1 β 1β 1β uðcL þΔÞ þ

β  0 β  0  u cL Z uðcH Þ þ u cH 1 β 1β



 β 0 β 0 cH þ π L cL þ cL ¼ w π H cH þ 1β 1β Let an allocation in this problem be denoted ðcL ; c0L ; cH ; c0H Þ. By standard arguments, the incentive constraint binds and cH ¼ c0H . Also, at the optimum cH(w) is strictly increasing and unbounded from above, and uðcH ðwÞÞ 4 ð1  βÞw, which in turn implies that if wZ uð2ΔÞ=ð1 βÞ, cH Z2Δ. 0 Let ðcL ; cL ; cH ; cH Þ be the optimal allocation in the problem above. There are two cases:   1. cH  cL ZΔ=2. Let c~ be the certainty equivalent of the lottery cL ; cH . Then by Lemma A5, 1 1     π H cH þ π L cL  c~ Z be  ðcH  cL Þaðπ H ðcH  cL ÞÞ π H π L ðcH cL Þ2 Z bðc þΔÞα  1 e  ΔaðπH Δ=2Þ π H π L Δ2 2 8

 1 3 α  1  Δaðπ H Δ=2Þ 2 α1 Z b e π H π L Δ  k1 c ; 8 2 where we used the fact that ½ðc þΔÞ=cα  1 is increasing and c Z2Δ. 0 Then the allocation ðc~ ; cL ; c~ ; cH Þ satisfy the promise-keeping constraint and



 Ey β 0 β  Ey β  β  c~ π L cL π H cH Z  π H cH þ cH  π L cL þ cL 0 þk1 ½u  1 ðð1  βwÞÞα  1 vFB ðwÞ Z 1β 1β 1β 1β 1β 1β ZvNV ðwÞ þ k1 ½u  1 ðð1  βwÞÞα  1 2. Suppose that cH cL o Δ=2. Define z  cL þΔ  cH . The fact that cL r cH and the assumption we made implies that z A ½Δ=2; Δ. 0 Also, let t  cH cL . Since z ZΔ=2, the incentive constraint implies that t 40. 0 In addition, guess that cH cL r Δ. By the strict concavity of u, we get that         u0 ðcH  ΔÞ 0    u cH t: u cH  u cH  t o u0 cH  t t ru0 cH  Δ t ¼ u0 ðcH Þ Non-increasing

absolute

risk

aversion

implies

that

u0 ðc  ΔÞ=u0 ðcÞ

is

decreasing,

so

if

cH Z2Δ,

22

L. Popov / Journal of Economic Dynamics & Control 64 (2016) 1–22

u0 ðcH  ΔÞ=u0 ðcH Þ r u0 ðΔÞ=u0 ð2ΔÞ. Then      0   u0 ðΔÞ 0    u cH t: u cH  u cL ¼ u cH  u cH  t o 0 u ð2ΔÞ

ðA:15Þ

On the other hand, by the same reasoning           u0 ðcH þ zÞ 0 u0 ð3ΔÞ 0    u ðcH Þz Z 0 u cH z u cL þΔ  u cH ¼ u cH þ z  u cH Zu0 cH þ z z ¼ u0 ðcH Þ u ð2ΔÞ

ðA:16Þ

0

Since the incentive-constraint is binding, uðcL þ ΔÞ uðcH Þ ¼ β=1 β½uðcH Þ  uðcL Þ. Then combining this with (A.15) and (A.16), we get that 0

cH  cL Z

 1 β u0 ð3ΔÞ Δ 1  β u0 ð3ΔÞ cL  þ Δ cH Z 0 β u ðΔÞ β u0 ðΔÞ 2

Then define Δ0 ¼ minfΔ; ð1 β=βÞu0 ð3ΔÞ=u0 ðΔÞΔ=2g. We have shown that cH  cL oΔ=2 implies that cH  cL0 ZΔ0 . 0 Again, let c~ denote the certainty equivalent of the lottery ðcL ; cH Þ. Since the risk premium increases with a meanpreserving spread, applying Lemma A5 in a similar fashion implies that

 1 3 α  1  Δ0 aðπ H ΔÞ 0 02 π L cL þ π H cH  c~ Z be π H π L Δ ½u  1 ðð1 βÞwÞα  1 : 2 2  α  1  Δ0 aðπ ΔÞ H Defining k2  ðβ=1 βÞð1=2Þ ð3=2Þ be π H π L Δ02 , yields vFB ðwÞ Z vNV ðwÞ þ k2 ½u  1 ðð1 βÞwÞα  1 :

Then the result follows by setting k  minfk1 ; k2 g.□ Proof of Proposition 8. It is sufficient to show that limw-w ½vFV ðwÞ vNV ðwÞ=ϕ~ L ðwÞ ¼ 1. For brevity, we will denote c ðwÞ ¼ u  1 ðð1  βÞwÞ. From the definition of ϕ~ L ðwÞ and the assumption that q ¼ β, uðc ðwÞ þΔÞ uðc ðwÞÞ u0 ðc ðwÞÞΔ w u0 ðc ðwÞÞΔ u0 ðc ðwÞÞΔ ¼ r r : ϕ~ L ðwÞ ¼ w βw w w βw ð1  βÞw uðc ðwÞÞ From Lemma A6, it follows that vFV ðwÞ  vNV ðwÞ Zkðc ðwÞÞα  1 . Since c ðwÞ is unbounded from above and strictly increasing in 0 1α w, it follows that it is sufficient to show that limc-1 cα  1 =u0 ðcÞ=uðcÞ ¼ 0.  ¼ 1, which is implied by limc-1 u ðcÞc Rc 0 0 As shown in the proof of Lemma A5, u ðcÞ ¼ u ð1Þexp  1 aðsÞ ds , where a(s) is absolute risk aversion. Since aðsÞ rbsα  1 ,   u0 ðcÞ ru0 ð1Þ exp  b=αðcα 1Þ . 0 1α Then logðu ðcÞc Þ rlogðu0 ð1ÞÞ þ b=α ðb=αÞcα þð1  αÞlogðcÞ. Then a standard Calculus exercise shows that limc-1 0 1α logðu ðcÞc Þ ¼ 1, which implies the result.□

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