Optimal Asset Management Contracts with Hidden

Savings

Sebastian Di Tella and Yuliy Sannikov∗

Stanford University

November 2016

Abstract

We characterize optimal asset management contracts in a classic portfolio-investmentsetting. When the agent has access to hidden savings, his incentives to misbehave de-pend on his precautionary saving motive. The contract distorts his access to capitalto manipulate his precautionary saving motive and reduce incentives for misbehav-ior. As a result, implementing the optimal contract requires history-dependent equityand leverage constraints. We extend our results to incorporate market risk, hiddeninvestment, and renegotiation. We provide a sufficient condition for the validity ofthe first-order approach: if the agent’s precautionary saving motive weakens after badoutcomes, the contract is globally incentive compatible.

1 Introduction

Delegated asset management is ubiquitous in modern economies, from fund managers in-vesting in financial assets to CEOs or entrepreneurs managing real capital assets. Theseintermediaries facilitate the flow of funds to the most productive uses, but their activity ishindered by financial frictions. To align incentives, asset managers must retain a stake intheir investment activity. However, hidden savings pose a significant challenge to incentive∗sditella@stanford.edu and sannikov@gmail.com. We are grateful to Andy Skrzypacz, V.V. Chari, Brett

Green, Mike Fishman, Zighuo He, Hugo Hopenhayn, Costas Skiadas, and seminar participants at Stanford,Berkeley Haas, Minnesota, Northwestern, Chicago, and UTDT for valuable comments and suggestions.Alex Bloedel and Erik Madsen provided outstanding research assistance.

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schemes. The asset manager can undo incentives by saving to self insure against bad out-comes. We characterize the optimal dynamic asset management contract when the agenthas access to hidden savings.

The problem of hidden savings in a dynamic principal-agent framework is an old, hardproblem because of the possibility of double deviations. As a result, a large part of theliterature on agency problems assumes that the agent doesn’t have access to hidden savings.How much hidden savings matter for the attainable surplus and for the shape of the optimalcontract is still an open question. Our results show that hidden savings are far frominnocuous and have important implications for the kind of financial frictions generated bythe agency problem and the dynamic behavior of the optimal contract.

This paper has two main contributions. First, we build a model of delegated asset man-agement in a classic portfolio-investment setting that can be embedded in macroeconomicor financial frameworks: an agent with CRRA preferences continuously invests in riskyassets, but can secretly divert returns and has access to hidden savings. Hidden savingslead to dynamic distortions in access to capital. As a result, the optimal contract is im-plemented with history-dependent retained equity and leverage constraints. In contrast,the optimal contract without hidden savings requires only a constant retained equity con-straint. Second, on the methodological side, we provide a general verification theorem forglobal incentive compatibility that is valid for a wide class of contracts. Global incentivecompatibility is ensured as long as the agent’s precautionary saving motive weakens afterbad outcomes.

The agent’s precautionary saving motive plays a prominent role in the analysis. Whenthe agent expects a risky consumption stream in the future he places a high value on hid-den savings that he can use to self insure, which makes fund diversion more attractive.The optimal contract must therefore manage the agent’s precautionary saving motive bycommitting to limit his future risk exposure, especially after bad outcomes. This is accom-plished by controlling the agent’s access to capital. Giving the agent capital to managerequires exposing him to risk in order to align incentives. By promising an inefficiently lowamount of capital (and therefore risk) in the future, especially after bad outcomes, the prin-cipal makes fund diversion less attractive today and reduces the cost of giving capital to theagent up front. This dynamic tradeoff leads to history-dependent distortions in the agent’saccess to capital. After good outcomes the agent’s access to capital improves, which allowshim to keep growing rapidly; after bad outcomes he gets starved for capital and stagnates.The flip side is that the agent’s consumption is somewhat insured on the downside, and

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he is punished instead with lower consumption growth. This hurts the agent, but hiddensavings can’t help him get around the incentive scheme.

The presence of hidden savings has important implications for the types of financialfrictions facing the agent. The optimal contract can be implemented with a simple capitalstructure subject to equity constraints and leverage constraints. The agent can issue equityand debt to buy assets, but he must retain an equity stake and the fund/firm leverage(assets over total equity) is capped. The equity constraint is related to the hidden actioncomponent of the agency problem; i.e. the agent must keep some “skin in the game” todeter fund diversion. The leverage constraint is related to hidden savings; the optimalcontract without hidden savings only requires a constant retained equity constraint. Afuture leverage constraint, which tightens after bad outcomes, restricts the agent’s futureaccess to capital and allows the principal to relax the equity constraint today. Intuitively,a binding leverage constraint makes the marginal value of inside equity larger than themarginal value of hidden savings, because an extra dollar in inside equity allows the agentto also get more capital. In addition, after bad outcomes the agent is punished with tighterfinancial frictions. As a result of both effects, a smaller retained equity stake is enough todeter misbehavior. Thus, the leverage constraint in our model arises from the presence ofhidden savings and reflects a completely different logic than that of models with limitedcommitment such as Hart and Moore (1994) and Kiyotaki and Moore (1997). The leverageconstraint exists not because the agent can walk away, but because it weakens the agent’sprecautionary saving motive and allows the principal to relax the equity constraint.

Since the principal uses the agent’s access to capital to provide incentives, it is natural toask how hidden investment affects results. If the agent can secretly invest his hidden savingsin risky capital (for example by secretly investing more than indicated by the contract),the principal finds it harder to provide incentives. However, there are still gains from tradebecause the principal can provide some risk sharing for the capital invested through thecontract, while the agent must bear all the risk on the capital he invests on his own. Asa result, the optimal contract with hidden investment has the same qualitative features,and can also be implemented with retained equity and leverage constraints. We also addobservable market risk that commands a premium into our setting, and allow the agentto also invest his hidden savings in the market. Our agency model can therefore be fullyembedded within the standard setting of continuous-time dynamic asset pricing theory andmacro-finance.

A potential concern is that the optimal contract requires commitment. The principal

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relaxes the agent’s precautionary saving motive by promising an inefficiently low amountof capital in the future, especially after bad outcomes. It is therefore tempting at thatpoint to renegotiate and start over. To address this issue, we also characterize the optimalrenegotiation-proof contract. This leads to a stationary leverage constraint that restrictsthe agent’s access to capital to reduce his precautionary saving motive in a uniform way.

One of the main methodological contributions of this paper is to provide an analyticalverification of the validity of the first order approach. Contractual environments where theagent has access to hidden savings are often difficult to analyze because we need to ensureincentive compatibility with respect to double deviations. Dealing with single deviationsis relatively straightforward using the first-order approach, introduced to the problem ofhidden savings by Werning (2001). We can deter the agent from diverting funds and im-mediately consuming them by giving him some skin in the game. Likewise, we can ensurethat he will not secretly save his recommended consumption for later by incorporating hisEuler equation as a constraint on the contract design. But what if the agent diverts returnsand saves them for later? Since diverting funds makes bad outcomes more likely, the agentexpects to be punished with lower consumption (high marginal utility) in the future, sothis could potentially be an attractive double deviation. Establishing global incentive com-patibility is a difficult problem, as the characterization of the agent’s deviation incentiveshas one more dimension than the recursive structure of the contract on path. Kocherlakota(2004) provides a well-known example in which double deviations are profitable (and thefirst-order approach fails) when cost of effort is linear. To establish the validity of thefirst-order approach, the existing literature has often pursued the numerical option (e.g.see Farhi and Werning (2013)).

We prove the validity of the first order approach analytically by establishing an upperbound on the agent’s continuation utility for any deviation, after any history. Doubledeviations are not profitable under the sufficient condition that agent’s precautionary savingmotive becomes weaker after bad outcomes. Intuitively, if the contract becomes less riskyand the precautionary saving motive gets weaker after bad outcomes (produced by thefund diversion), then hidden savings become less valuable to the agent. We show that thisis sufficient to rule out double deviations. It is important to note that this verificationargument applies beyond the optimal contract, to any contract in which the precautionarysaving motive weakens after bad outcomes. If the agent has access to hidden investment,verifying global incentive compatibility is potentially more difficult, as the agent has evenmore valid deviations. However, the same sufficient condition is valid for the case with

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hidden investment.

Literature Review. This paper fits within the literature on dynamic agency problems.It builds upon the standard recursive techniques, but adds the problem of persistent privateinformation. In our case, about the agent’s hidden savings. There is extensive literature thatuses recursive methods to characterize optimal contracts, including Spear and Srivastava(1987), Phelan and Townsend (1991), Sannikov (2008), He (2011), Biais et al. (2007), andHopenhayn and Clementi (2006). The agency problem we study is one of cash flow diversion,as in DeMarzo and Sannikov (2006) and DeMarzo et al. (2012), but unlike their modelswe have CRRA rather than risk neutral preferences. With risk-neutral preferences, theoptimal contracts with and without hidden savings are the same. Once concave preferencesare introduced, the principal has incentives to front load consumption in order to reducethe private benefit of cash diversion and allow better risk sharing. This opens the door topotential distortions to control the agent’s incentives to save, but also presents the problemof double deviations. In some settings distortions do not arise, e.g. the CARA settings,such as He (2011), and Williams (2013), where the ratio of the agent’s current utility tocontinuation utility is invariant to contract design.1 However, when distortions do arise, itis difficult to characterize the specific form they take, and the first-order approach may fail.We are able to provide a sharp characterization of the optimal contract and the distortionsgenerated by the presence of hidden savings, and provide a verification theorem for globalincentive compatibility which is valid for a wide class of contracts.

Our paper is related to the literature on persistent private information, since the agenthas private information about savings. The growing literature in this area includes thefundamental approach of Fernandes and Phelan (2000), who propose to keep track of theagent’s entire off-equilibrium value function, and the first-order approach, such as He etal (2015) and DeMarzo and Sannikov (2016), who use a recursive structure that includesthe agent’s “information rent,” i.e. the derivative of the agent’s payoff with respect toprivate information. In our case, information rent is the marginal utility of consumption(of an extra unit of savings). While the connection to our work may appear subtle, it isactually quite direct - key issues are (1) the way that information rents enter the incentiveconstraint, (2) distortions that arise from this interaction and (3) forces that affect thevalidity of the first-order approach. In our case, we can verify the validity of the first-order

1Likewise, the dynamic incentive accounts of Edmans et al. (2011) exhibit no distortions either, as hiddenaction enters multiplicatively and project size is fixed.

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approach by characterizing an analytic upper bound, related to the CRRA utility function,on the agent’s payoff after deviations. The bound coincides with the agent’s utility on path(hence the first-order approach is valid), and its derivative with respect to hidden savingsis the agent’s “information rent." It is also possible to find the agent’s value function afterdeviations explicitly, and hence verify the first-order approach, numerically. Farhi andWerning (2013) do this in the context of insurance with unobservable skill shocks. Otherpapers that study the problems of persistent private information via a recursive structurethat includes information rents include Garrett and Pavan (2015), Cisternas (2014), Kapička(2013), and Williams (2011).

We use a classic portfolio-investment environment widely used in macroeconomic andfinancial applications. Our model provides a unified account of equity and leverage con-straints, which are two of the most commonly used financial frictions in the macro-financeliterature in the tradition of Bernanke and Gertler (1989) and Kiyotaki and Moore (1997).2

Di Tella (2014) adopts a version of our setting without hidden savings to study optimalfinancial regulation policy in a general equilibrium environment. Our paper is also relatedto models of incomplete idiosyncratic risk sharing, such as Aiyagari (1994) and Krusell andSmith (1998). Here the focus is on risky capital income, as in Angeletos (2006) or Christianoet al. (2014) (rather than risky labor income). This affords us a degree of scale invariancethat allows us to provide a sharp characterization of the optimal contract. In our setting,after good performance, the agent does not need to be retired nor outgrow moral hazardas in Sannikov (2008) or Hopenhayn and Clementi (2006) respectively. Likewise, since theproject can be scaled down, neither will the agent retire after sufficiently bad outcomes asin DeMarzo and Sannikov (2006). Rather, the optimal contract dynamically scales the sizeof the agent’s fund with performance, taking into account his precautionary saving motive.Access to capital provides the principal with an important incentive tool. He is able to re-lax the incentive constraints and improve risk sharing by committing to distort project sizebelow optimum over time and after bad performance. This result stands in contrast to Coleand Kocherlakota (2001), where project scale is fixed and the optimal contract is risk-freedebt. We recover the result of Cole and Kocherlakota (2001) only in the special case whenthe agent can secretly invest on his own just as efficiently as through the principal, so the

2See Bernanke and Gertler (1989), Kiyotaki and Moore (1997), Bernanke et al. (1999), Iacoviello (2005),Lorenzoni (2008), Gertler et al. (2010), Gertler and Karadi (2011), Adrian and Boyarchenko (2012), Brun-nermeier and Sannikov (2014), He and Krishnamurthy (2012), He and Krishnamurthy (2013), Di Tella(2013), Moll (2014).

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principal cannot control the scale of investment at all.This paper is organized as follows. Section 2 presents the model in the absence of

the stock market. Section 3 solves for the optimal contract and provides a suitable suffi-cient condition to verify the validity of the first-order approach. Section 4 discusses theimplementation of the optimal contract as a portfolio problem subject to equity and lever-age constraints. Section 5 compares the optimal contract with several simple benchmarkcontracts, in order to better understand dynamics and financial frictions. Section 6 incorpo-rates both aggregate risk and hidden investment into the setting, and Section 7 introducesrenegotiation. Section 8 concludes.

2 The model

Let (Ω, P,F) be a complete probability space equipped with filtration F generated by aBrownian motion Z, with the usual conditions. Throughout, all stochastic processes areadapted to F. There is a complete financial market with equivalent martingale measure Q.The risk-free interest rate is r > 0 and Z is idiosyncratic risk and therefore not priced bythe market. In the baseline setting there is no aggregate risk so Q = P , but later we willallow them to differ.

The agent can manage capital to obtain a risky return that exceeds the required returnof r, but he may also get a private monetary benefit by diverting returns. If the diversionrate is at, the observed return per dollar invested in capital is

dRt = (r + α− at) dt+ σ dZt

where α > 0 is the excess return and σ > 0 is the volatility. Z is agent-specific idiosyncraticrisk. If we think of the agent as a fund manager, it represents the outcome of his particularinvestment/trading activity.3 If we take the agent to be an entrepreneur it represents theoutcome of his particular project.

If capital is kt ≥ 0, diversion of at ≥ 0 gives the agent a flow of φatkt. For each stolendollar, the agent keeps only fraction φ ∈ (0, 1). If the agent also receives payments ct ≥ 0

3If we give $1 to invest to two fund managers, they will obtain different returns depending on exactlywhich assets they buy or sell, and the exact timing and price of their trades.

7

from the principal and consumes ct ≥ 0 then his hidden savings ht ≥ 0 evolve according to4

dht = (rht + ct − ct + φktat) dt.

The agent invests his hidden savings at the risk-free rate r. Later we will introduce hiddeninvestment, and allow the agent to also invest his hidden savings in risky capital.

The agent wants to raise funds and share risk with the market, which we refer to as theprincipal. The principal observes returns R but not the agent’s diversion a, consumptionc, or hidden savings h. The principal can commit to a fully history-dependent contractC = (c, k) that specifies payments to the agent ct and capital kt as a function of the historyof realized returns R up to time t. After signing the contract C the agent can choose astrategy (c, a) that specifies ct and at, also as a history of returns up to time t.

The agent has CRRA preferences. Given contract C, under strategy (c, a) the agentgets utility

U c,a0 = E

[ˆ ∞0

e−ρtc1−γt

1− γdt

](1)

Given contract C, we say a strategy (c, a) is feasible if 1) there is a finite utility U c,a0 , and2) ht ≥ 0 always. Let S(C) be the set of feasible strategies (c, a) given contract C.

The principal pays for the agent’s consumption, but keeps the excess return α on thecapital that the agent manages. He tries to minimize the cost of delivering utility u0 to theagent

J0 = EQ[ˆ ∞

0e−rt (ct − ktα) dt

](2)

A standard argument in this setting implies that the optimal contract must implement nostealing, i.e. a = 0. In addition, without loss of generality and for analytic convenience, wecan restrict attention to contracts in which h = 0 and c = c, i.e. the principal saves forthe agent.5 Of course, the optimal contract has many equivalent and more natural forms,in which the agent maintains savings, but all these forms can be deduced easily from theoptimal contract with c = c.

4We don’t allow negative hidden savings, ht ≥ 0. This is without loss of generality if the contract canexhaust the agent’s credit capacity.

5Lemma 19 establishes this in the more general setting of Section 6, with both aggregate risk and hiddeninvestment.

8

We say a contract C = (c, k) is admissible if 1) there is a finite utility U c,00 , and 2)6

EQ[ˆ ∞

0e−rt|ct + ktα|dt

]<∞ (3)

We say an admissible contract C is incentive compatible if

(c, 0) ∈ arg max(c,a)∈S(C)

U c,a0

Let IC be the set of incentive compatible contracts. For an initial utility u0 for the agent, anincentive compatible contract is optimal if it minimizes the cost of delivering initial utilityu0 to the agent

v0 = min(c,k)

J0

st : U c,00 ≥ u0

(c, k) ∈ IC

By changing u0 we can trace the Pareto frontier for this problem. To make the problemwell defined and avoid infinite profits/utility, we assume throughout that ρ > r(1− γ), and

α ≤ α ≡ φσγ√

2√1 + γ

√ρ− r(1− γ)

γ

3 Solving the model

We solve the model as follows. We first derive necessary first-order incentive-compatibilityconditions for the agent’s effort and savings choice, using two appropriate state variables:the agent’s continuation utility and consumption level. This allows us to formulate theprincipal’s relaxed problem, minimizing the cost subject to only first-order conditions, as acontrol problem. We use the HJB equation to solve this problem.

We then derive a sufficient condition for global incentive compatibility (against all de-viations, not just local), which uses the same two state variables. The condition is on-path,i.e. for a particular recommended strategy of the agent, but it is sufficient because it al-

6This assumption plays the role of a no-Ponzi condition, making sure the principal’s objective functionis well defined. It rules out exploding strategies where the present value of both consumption and capitalis infinity.

9

lows us to bound the agent’s payoff off-path after arbitrary deviations. We show that thesolution to the relaxed problem satisfies the sufficient condition, thereby proving it is theoptimal contract. More generally, the sufficient condition identifies a whole class of globallyincentive compatible contracts, and is useful in a broader context as we show in the nextsection.

Incentive compatibility

We use the continuation utility of the agent as a state variable for the contract

U c,0t = Et

[ˆ ∞t

e−ρ(s−t)c1−γs

1− γds

]

First we obtain the law of motion for the agent’s continuation utility.

Lemma 1. For any admissible contract C = (c, k), the agent’s continuation utility U c,0

satisfies

dU c,0t =(ρU c,0t −

c1−γt

1− γ

)dt+ ∆t (dRt − (α+ r) dt)︸ ︷︷ ︸

σ dZt if at=0

(4)

for some stochastic process ∆.

Faced with this contract, the agent might consider stealing and immediately consumingthe proceeds, i.e. following a strategy (c + φka, a) for some a, which results in savingsh = 0. The agent adds φktat to his consumption, but reduces the observed returns dRt, andtherefore his continuation utility U c,0t by ∆tat. Incentive compatibility therefore requires

0 ∈ arg maxa≥0

(ct + φkta)1−γ

1− γ−∆ta (5)

Taking FOC yields∆t ≥ c−γt φkt (6)

which is positive. We need to give the agent some “skin in game”, which exposes him torisk. This is costly because the principal is risk-neutral with respect to Z so he would liketo provide full insurance to the agent.

Notice how the private benefit of the hidden action depends on the marginal utility ofconsumption c−γt , so the principal would like to front load the agent’s consumption to relax

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the risk-sharing constraint. With hidden savings this is not possible. If the principal triesto front load the agent’s consumption, he will secretly save and consume when his marginalutility is higher. The optimal contract must therefore respect the agent’s Euler equation:the discounted marginal utility e(r−ρ)tc−γt must be a supermartingale.7

The following lemma summarizes all the necessary conditions for incentive compatibil-ity and their implication on the path of the agent’s consumption. We present sufficientconditions in the next subsection.

Lemma 2. If C = (c, k) is an incentive compatible contract, then (6) must hold and theagent’s consumption must satisfy

dctct

=

(r − ργ

+1 + γ

2(σct )

2

)dt+ σct

1

σ(dRt − (α+ r) dt)︸ ︷︷ ︸

dZt if at=0

+dLt (7)

for some σc and a weakly increasing process L.

Equation (7) imposes a lower bound on the growth rate of the agent’s consumption.The first term r−ρ

γ captures the benefit of postponing consumption without risk, given bythe risk-free rate r, the discount rate ρ, and the elasticity of intertemporal substitution1/γ. The second term 1+γ

2 (σct )2 captures the agent’s precautionary saving motive. A risky

consumption profile induces the agent to postpone consumption to self-insure, resulting ina steeper consumption profile.

State space

It is convenient to work with the following transformation of the state variables

xt =(

(1− γ)U c,0t

) 11−γ

> 0

ct =ctxt≥ 0

Variable x is just a monotone transformation of continuation utility, but it is measuredin consumption units (up to a constant). As a result, c measures how front loaded theagent’s consumption is. The state c is related to the agent’s precautionary saving motive

7The agent can expect lower marginal utility in the future, because he can’t borrow. If the agent couldhave hidden debt, then e(r−ρ)tc−γt would have to be a proper martingale. As it turns out, this is the casein the optimal contract.

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for saving. If the agent faces risk looking forward, he will want to postpone consumptionin an attempt to self insure (low c). As a result, while xt can take any positive value, cthas an upper bound.

Lemma 3. For any incentive compatible contract C, at all times t, ct ∈ (0, ch], where

ch ≡(ρ− r(1− γ)

γ

) 11−γ

> 0 (8)

If ever ct = ch, then the continuation contract satisfies kt+s = 0 and ct+s = ch at allfuture times t+ s and gives the agent a unique deterministic consumption path with growth(r − ρ)/γ. The contract has cost vhxt to the principal, where vh ≡ cγh.

This upper bound has a simple interpretation. The contract that minimizes the agent’sprecautionary saving motive and maximizes c is the fully safe contract, which gives theagent no capital to manage and lets consumption grow at the deterministic growth rateof r−ρ

γ . This corresponds to ch. A lower c means the agent expects to manage capitaland be exposed to risk in the future. The safe contract minimizes the cost of the agent’sconsumption, but is very costly because it doesn’t give any capital to the agent and sodoesn’t take advantage of the excess return α > 0.

The relaxed problem

Using Ito’s lemma we can obtain laws of motion for xt and ct from (4) and (7). Using thenormalization ∆tσ/U

c,0t = (1− γ)σxt , we obtain

dxtxt

=(ρ− c1−γt

1− γ+γ

2(σxt )2

)︸ ︷︷ ︸

µxt

dt+ σxt dZt (9)

and

dctct

=(r − ρ

γ+c1−γt − ρ

1− γ+

(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2)

︸ ︷︷ ︸µct

dt+ σct dZt + dLt (10)

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for some σct = σct − σxt . The constraint (6) can be rewritten as

σxt ≥ c−γt ktφσ, (11)

where kt = ktxt. It will always be binding because conditional on σxt it is always better to

give the agent more capital to manage. The cost flow for the principal is

ct − ktα = xt

(ct − cγt

α

φσσxt

)(12)

We obtain a stochastic control problem with laws of motion of the state variables (9) and(10) with absorption at ch, controls σxt and σct , and cost flow (12). This is the relaxedproblem.

The principal would like to give capital to the agent because it yields an excess returnα. However, he must expose the agent to risk to provide incentives. This is costly for tworeasons. First, since the agent is risk averse he must be compensated for the exposure to riskwith higher utility in the future. This is reflected in the drift of xt in equation (9). Second,because of hidden savings, exposing the agent to risk induces a precautionary saving motive- a low c - that distorts the optimal intertemporal consumption profile. In fact, these twocosts interact because a low c further increases the incentives to steal through c−γ in (11).

Here’s how this works in the recursive formulation. While x0 is determined by theinitial utility u0, the principal must choose an initial c0 ≤ ch. A lower ct means that theagent expects to be given capital and therefore be exposed to risk in the future, so we caninterpret ch − ct as the principal’s “budget” for risk exposure. If the principal exposes theagent to risk today with σxt , he must expose him to less risk in the future. This is reflectedin equation (10) where the drift of c is increasing in σxt : exposing the agent to risk todayeats up the budget for risk exposure in the future. A higher ct creates distortions becauseit reduces the principal’s ability to give capital to the agent. Indeed, if ct ever reached theupper bound ch, the principal would have to give him the perfectly safe contract withoutany capital. The principal must therefore choose the initial c0 weighting the benefit ofgiving capital to the agent against the distortions in intertemporal consumption smoothingand risk sharing. After that, he manages his budget for risk exposure using σx and σc.

In addition to σxt the principal can use a negative σct to relax the agent’s precautionarysaving motive. The agent is concerned the most about risk exposure after bad outcomes,when his utility is low. Thus, the principal can reduce the agent’s precautionary saving

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motive by giving him less risk - raising ct - in the event that xt goes down. This is capturedby the drift of ct in equation (10), which can be reduced with a negative σct . As it turns outthe principal will also prefer to raise ct after bad outcomes for dynamic hedging reasons: hewould rather restrict his ability to give capital to the agent (with a high ct) when he mustdeliver less utility xt to the agent, and relax it (with a low ct) when he must deliver moreutility xt.

It turns out that relaxing the agent’s precautionary saving motive after bad outcomes,σct ≤ 0, is also a sufficient condition for the contract to be globally incentive compatible ifthe necessary conditions of Lemma 2 hold. While our paper is the first to our knowledgeto prove this form of a result analytically, the intuition is simple. The marginal value ofhidden savings rises in the agent’s precautionary saving motive, i.e. savings become morevaluable when the agent is exposed to risk going forward and therefore ct is lower. Therefore,whenever a contract reduces the precautionary saving motive after bad outcomes - theseoutcomes become more likely when the agent steals - stealing and saving is an unattractivedeviation. The agent expects to have hidden savings when they are less valuable to him.This intuition is formalized below in Theorem 3.

The HJB equation

Because preferences are homothetic and the principal’s objective is linear, we know theprincipal’s cost function takes the form v(x, c) = v(ct)xt with v(ch) = vh > 0. Notice thatsince we could always raise ct using dLt, we know that v(c) must be weakly increasing. Infact, we show below that v(c) increases strictly over the interval in which ct stays over thecourse of the optimal contract, so we can drop the term dLt from what follows. We willalso sometimes write vt = v(ct), and v instead of v(c).

The HJB equation associated with this problem is

rvx = minσx,σc

(c− kα

)x+ EQt [d (vtxt)]

subject to (9), (10), and (11), and k ≥ 0. Using Ito’s lemma and canceling the x on bothsides, we get

rv = minσx,σc

c− σxcγ αφσ

+ v

(ρ− c1−γ

1− γ+γ

2(σx)2

)(13)

+v′c

(r − ργ

+(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2 − ρ− c1−γ

1− γ

)+v′′

2c2(σc)2

14

Even though we have two state variables, c and x, the HJB equation boils down to a secondorder ODE in c. This is a feature of homothetic preferences and linear technology thatmakes the problem more tractable. The following lemma characterizes the shape of theprincipal’s cost function and the range of ct under the optimal contract.

Theorem 1. The principal’s cost function v(c) has a flat portion on [0, cl] and a strictlyincreasing portion on [cl, ch], for some cl ∈ (0, ch). The HJB equation (13) holds withequality for c ≥ cl. For c < cl, we have v(c) = v(cl) ≡ vl and the HJB holds as aninequality

A(c, vl) ≡ minσx

c− σxcγ αφσ− rvl + vl

(ρ− c1−γ

1− γ+γ

2(σx)2

)> 0 ∀c < cl. (14)

At cl the cost function v(c) satisfies v′(cl) = 0, v′′(cl) > 0, and A(cl, vl) = 0.The optimal contract starts at c0 = cl, where σx0 is chosen without taking into account

its effect on the agent’s precautionary saving motive, to maximize

σxcγα

φσ− v(c)

γ

2(σx)2 (15)

At cl we have µc(cl) > 0 and σc(cl) = 0. For all t > 0, ct ∈ [cl, ch], σct ≤ 0 and σxt ≥ 0.

Figures 1 and 2 show the cost function and the drift and volatility of state variables xand c for a numerical solution. To understand why the cost function has a flat portion andan increasing portion, consider the problem of the optimal choice of c0. Choosing c0 = ch

is suboptimal, because then the principal cannot give the agent any capital. It is beneficialto give the agent capital and expose him to risk because capital generates an excess returnof α. However, risk exposure is costly because the agent is risk averse. In addition, withhidden savings risk exposure also generates a precautionary saving motive which lowersc, distorting the intertemporal consumption profile and further tightening the incentiveconstraint. Eventually, the costs of risk exposure outweigh the benefits. Under these trade-offs, we denote by cl the value that minimizes the cost of delivering utility to the agent,indicated with a blue dot in Figure 1. For c < cl, the principal has the option to raisec to cl immediately using the process dLt. Hence, the cost function is flat over [0, cl]. Forc > cl, he must give the agent an inefficiently low risk exposure to reduce his precautionarysaving motive. As a result the cost function v(c) is increasing over [cl, ch], and the optimalcontract starts at c0 = cl.

15

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.10

0.12

0.14

0.16

0.18

0.20

0.22

v

vh

chcl

Figure 1: The cost function v(c) solid in blue for the optimal contract, and dashed in redfor stationary contracts. The the starting point of the optimal contract is indicated by theblue dot, the optimal stationary contract by the red dot, the optimal portfolio plan withφ = φ is indicated the black dot, and the optimal contract without hidden savings by thegreen dot. The dashed black curve is the locus A(c, v) = 0. Parameters: ρ = r = 5%,α = 1.7%, γ = 1/3, φσ = 0.2.

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.000

0.005

0.010

0.015

μc^

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.00

0.01

0.02

0.03

0.04

μx

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

σc^

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.0

0.1

0.2

0.3

0.4

σx

chcl

Figure 2: The drift, µc and µx, and volatility, σc and σx, of the state variables c andx. The starting point of the optimal contract is indicated by the blue dot, the optimalstationary contract by the red dot, the optimal portfolio plan with φ = φ is indicated theblack dot, and the optimal contract without hidden savings by the green dot. Parameters:ρ = r = 5%, α = 1.7%, γ = 1/3, φσ = 0.2.

16

The optimal contract starts at the optimal cl but becomes less risky over time - c driftsup to a steady state, as show in Figure 2. Why does ct not stay at the cost-minimizinglevel cl forever? By promising a less risky contract in the future (higher c), the principalcan weaken the agent’s precautionary saving motive and relax his incentives to steal today.To see this in more detail, consider the first-order condition8 for σx

α = γ(vc−γφσσx

)︸ ︷︷ ︸risk sharing

+φσv′c1−γ(

(1 + γ)σc + σx)

︸ ︷︷ ︸intertemporal tradeoff

(16)

The left hand side and the first term on the right capture the tradeoff between excessreturn α and exposure to risk σx. The second term captures an intertemporal tradeoff.The principal can give more capital to the agent today (exposing him to more risk σx

today) without a larger precautionary saving motive (lower c) if he promises a less riskycontract in the future (the drift of c is increasing in σx). Since less risky contracts requirean inefficiently low level of capital - v(c) is strictly increasing in c as described above -this is a potentially costly tradeoff for the principal. At the optimal point cl, however, theprincipal is indifferent about small changes in c because v′(cl) = 0, so the intertemporaltrade-off vanishes. The principal picks σx to myopically maximize (15), and doesn’t carethat he is promising less risk in the future. Intuitively, distorting future σxt relaxes theincentive problem at t = 0, but distorting σx0 does not help relax the incentive problem inthe future. As a result, the drift of c is initially positive. As c moves up from cl, the costof the contract goes up. The principal would benefit from reducing c - giving the agent amore risky contract - but he cannot do so because he must keep his promise to the agent.The principal then chooses σx taking into account this intertemporal tradeoff.

The principal can reduce the cost by setting σc < 0, as shown in Figure 2, i.e. after badoutcomes the contract becomes less risky for the agent. This has two benefits, as shown bythe FOC for σc

v′(γσx + (1 + γ)σc

)︸ ︷︷ ︸

relax precautionary motive

+(v′σx + v′′cσc

)︸ ︷︷ ︸

hedging

= 0 (17)

The first term says that by setting σc < 0 the principal can reduce the agent’s precautionary8This first-order condition characterizes a minimum (rather than a maximum) because vγ + v′c > 0,

since v > 0 and v′ ≥ 0.

17

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0

2000

4000

6000

8000

10000

ch

Figure 3: The stationary distribution of c. Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3,φσ = 0.2.

saving motive. The principal promises the agent less risk - a higher c - after bad outcomes,when it matters most to him. What is going on is that the agent’s consumption ct = ctxt

is somewhat insured because c and x move in opposite directions. As a result, the agent’sprecautionary saving motive is weaker and the drift of c lower, which is attractive to theprincipal because v(c) is increasing. The second term in the FOC captures a hedging motivefor the principal: he also prefers to use the relatively costly contracts with high c when hemust deliver less utility x, because then he can use the relatively less costly contracts withlow c when he must deliver more utility x. Both motives induce a negative σc.

If the agent’s consumption is somewhat insured and he faces less risk after bad out-comes, how exactly is he being punished? The answer is that he faces a lower growthrate in his consumption. After bad outcomes the agent’s access to capital is restricted,and he is given an inefficiently safe contract with low consumption growth. This hurts theagent, but he can’t use hidden savings to get around it. The principal could punish theagent by proportionally scaling down his capital and consumption (keeping c constant), butthis would be too costly: it would make the agent’s consumption too risky and give hima precautionary saving motive for savings that would distort intertemporal consumptionsmoothing and increase his incentives to steal. Instead, the principal uses the agent’s accessto capital to manipulate his precautionary saving motive.

Long-run behavior. The contract is scale invariant with respect to x. After a very goodhistory, the agent’s continuation utility x will be large, and he will get a large amount of

18

capital and consumption. Conversely, after a bad history his continuation utility x will besmall and he will get a small amount of capital and consumption.

The behavior of c, however, creates non-trivial dynamics. While the contract starts atthe lower end of the domain c0 = cl, with high growth and risk, in the absence of shocksthe drift of c takes the contract to a “steady state” css ∈ (cl, ch), as shown in Figure 2. Butthis steady state can be a misleading guide to the long-run behavior of the contract. If thevolatility σc is high near the steady state, the contract might spend very little time there.Figure 3 shows the stationary distribution of c. For this numerical solution, the contractspends most of the time near the upper bound ch, with low growth and risk, where boththe drift and volatility of c are small.

Verification theorems

We know that the principal’s cost function in the relaxed problem satisfies the HJB equation,but how do we know that the equation has no other solutions? And how do we know thatwe have identified the true (non-relaxed) optimal contract? The following theorem showsthat if an appropriate solution has been found, e.g. numerically, then it must be the truecost function, and we can use it to build the optimal contract.

Theorem 2 (Verification Theorem). Let v(c) : [cl, ch] → [vl, vh] be a strictly increasingC2 solution to the HJB equation (13) for some cl ∈ (0, ch), such that vl ≡ v(cl) ∈ (0, vh],

v′(cl) = 0, v′′(cl) > 0 and v(ch) = vh. If γ < 12 , we also need to check that

1− vl(c−γl + c2γ−1l α2(φσ)−2v−2l

)≤ 0 (18)

Then,

1) For any incentive compatible contract C = (c, k) that delivers at least utility u0 to theagent, we have v(cl) ((1− γ)u0)

11−γ ≤ J0(C).

2) Let C∗ be a contract generated by the policy functions of the HJB. Specifically, thestate variables x∗ and c∗ are solutions to (9) and (10) (with potential absorption at ch),with initial values x∗0 = ((1− γ)u0)

11−γ and c∗0 = cl. If C∗ is admissible, and σc∗ is bounded,

then C∗ is an optimal contract, with cost J0(C∗) = v(cl) ((1− γ)u0)1

1−γ .

The HJB equation can be solved as an ODE by plugging in the FOCs. We only need

19

to verify condition (18) in case γ < 12 , and that the contract generated by the HJB C∗ is

admissible. The following sufficient condition can be useful.

Lemma 4. If the candidate contract C∗ constructed in Theorem 2 has µx∗ < r, then C∗ isadmissible and delivers utility u0 to the agent.

Global incentive compatibility. To finish the section, we provide sufficient conditionsfor global incentive compatibility of any contract that satisfies the local constraints onsavings (10) and effort (11). This result is used in Theorem 2 to verify that the candidateoptimal contract C∗ is incentive compatible. However, it is more general than that and canbe used to check incentive compatibility of many other contracts of interest (see Section 4).

While (10) and (11) ensure that neither stealing and immediately consuming, nor se-cretly saving without stealing are attractive on their own, they leave open the possibilitythat a double deviation (stealing and saving the proceeds for later) could be attractiveto the agent. To see how this can happen, notice that since stealing makes bad out-comes more likely, it increases the expected marginal utility of consumption in the futureEat[e(r−ρ)uc−γt+u

].9 Saving the stolen funds for consumption later could therefore be very

attractive. However, hidden savings have decreasing marginal value (the first dollar yieldsc−γt , the second one less than that), which depends on the agent’s precautionary savingmotive. This observation allows us to derive a sufficient condition to rule out profitabledouble deviations.

Theorem 3. Let C = (c, k) be an admissible contract with associated processes x andc satisfying (9) and (10) and (11), with bounded µx, µc, and σc, and with c uniformlybounded away from zero and bounded above by ch. Suppose that the contract satisfies thefollowing property

σct ≤ 0 (19)

Then for any feasible strategy (c, a), with associated hidden savings h, we have the followingupper bound on the agent’s utility, after any history

U c,at ≤(

1 +htxtc−γt

)1−γU c,0t (20)

In particular, since h0 = 0, for any feasible strategy U c,a0 ≤ U c,00 , and the contract C is

9In other words, even if e(r−ρ)tc−γt is a martingale under P , it might be a submartingale under P a.

20

therefore incentive compatible.

Theorem 3 shows that condition (19) is sufficient by providing a closed-form explicitupper bound (20) on the agent’s off-equilibrium payoff for any savings level ht ≥ 0. Ac-cording to (20), if the agent does not have any hidden savings, ht = 0, the most utility hecould get is U c,0t , i.e. the utility level he obtains from “good behavior” (c, 0). Hence, goodbehavior is incentive compatible. However, if the agent had somehow accumulated hiddensavings in the past, he would want to deviate from (c, 0) in the future, at the very least toincrease his consumption, and attain a greater utility. Inequality (20) bounds the utilitythe agent can get, and the bound tightens as ct rises and the agent’s precautionary savingmotive decreases. The bound is consistent with the intuition that the marginal value of hid-den savings becomes lower as the agent’s precautionary saving motive decreases (however,remember that this is just an upper bound on achievable utility).

The sufficient condition σct ≤ 0 can be understood as follows. Hidden savings becomemore valuable when the agent faces more risk, i.e. has a higher precautionary savingmotive. With σct ≤ 0 the contract becomes less risky for the agent after bad outcomes(hidden savings becomes less valuable). Since stealing makes bad outcomes more likely,if the agent steals and saves for later, he expects to have a hidden dollar when it is leastvaluable to him. This makes double deviations unprofitable.

As shown above, the optimal contract has the property that σct ≤ 0 because the principalwants to contain the agent’s precautionary saving motive and the most efficient way to dothis is by reducing the agent’s risk exposure after bad outcomes. As it happens it is thesame property that is sufficient for global incentive compatibility.

We would like emphasize once more that Theorem 3 identifies σct ≤ 0 as a generalsufficient condition for incentive compatibility, without even assuming that the contract isrecursive in variables x and c. These variables are well-defined for an arbitrary contract,and their laws of motion do not need to be Markov for Theorem 3 to apply. Conditionσct ≤ 0 can be used to verify global incentive compatibility of other contracts. For example,it implies that stationary contracts that we discuss below are also all globally incentivecompatible.

21

4 Implementation of the Optimal Contract

In this section we derive an implementation of the optimal contract. A capital structureimplements a contract if it leads to exactly the same map from histories of portfolio returnsR to the agent’s compensation c and funds invested k as the contract.

Consider a family of capital structures, in which the fund’s assets kt are financed bydebt dt and outside equity et held by the investors, as well as inside equity nt, the agent’slegitimate wealth, so that

kt︸︷︷︸assets

= nt + et︸ ︷︷ ︸equity

+ dt︸︷︷︸debt

. (21)

Suppose that the stakes held by the the principal, debt, and outside equity earn the requiredreturn of r in expectation. This restriction on capital structure ensures that investors arealways willing to provide the required fund inflows and outflows. Suppose that inside andoutside equity absorb risks proportional to et and nt, respectively. Subject to these rules,any capital structure within this family can specify the evolution of the stakes nt, et anddt as functions of the history of returns. We call capital structures of this form standard.

Let us introduce some terminology about this family of capital structures. Denote byφt = nt

nt+etthe fraction of equity held by the agent and by λt = kt

nt+etthe fund/firm

leverage ratio. These two quantities determine the inflows and outflows of the fund, giventhe requirements about the division of risk and return. Those requirements imply that thereturns of debt and outside equity are then r dt and r dt+λtσ dZt, respectively. The returnof inside equity nt has to be, therefore, (r+λtα/φt)dt+λtσdZt.10 Given the payout rate ofthe agent’s equity of ct/nt = θt, the agent’s inside equity nt follows the budget constraintof

dntnt

=(r + λtα/φt − θt

)︸ ︷︷ ︸

µnt

dt+ λtσ︸︷︷︸σnt

dZt. (22)

It is convenient to specify standard capital structures through the triplet of processes S =

(λ, φ, θ), all contingent on the history of returns R, with an associated process for agent’sequity given by (22). A standard capital structure S = (λ, φ, θ) implements contractC = (c, k) if ct = θt × nt and kt = λt(nt + et) = λt/φt × nt.

By matching appropriate terms, we obtain the unique implementation of the optimal10Notice that leverage λt gets the agent an excess return α/φt because while he retains only a fraction

φt of the equity, he appropriates all the excess returns αkt = αλt(nt + et) = α(λt/φt)nt.

22

contract with a standard capital structure. In order to proceed, notice first that the value ofthe stake of outside investors has to be dt+et = kt−xtv(ct), since the cost of compensatingthe agent under the optimal contract is nt = xtv(ct). In order to obtain φt, we measure therisk of the agent’s equity of xtv(ct), which is

(σxt xtv(ct) + xtv

′(ct)σct ct)dZt, against total

capital risk of σkt dZt. Since kt =xtσxt c

γt

φσ , it follows that

φt =σxt xtvt + xtv

′tσct ct

σkt= φ

(vtcγt

+v′tσ

ct ct

σxt cγt

). (23)

We obtain the fund leverage from

λt =σxt c

γt

φσvt︸ ︷︷ ︸kt/nt

φt =σxtσ

+v′tσ

ct ct

vtσ(24)

and the payout policy is simplyθt = ct/vt. (25)

This spells out the triplet of processes S = (λ, φ, θ) as functions of ct, which itself isdetermined by the history of observed returns. The following Lemma summarizes theimplementation.

Lemma 5. The optimal contract has a unique implementation with a standard capitalstructure, with inside equity share φt = nt

nt+etgiven by (23), leverage λt = kt

nt+etgiven by

(24), a payout policy θt = ctnt

given by (25) and assets given by (21).

The agent obtains utility in consumption units xt = v(ct)−1nt, so the marginal value of

equity nt is v(ct)−1. The marginal value of hidden savings (also measured in consumption

units xt) is c−γt , and is always less than the marginal value of inside equity, v(ct)

−1 > c−γt .

This wedge plays an important role for the agent’s incentives. In particular, (23) impliesthat φt ≤ φ always. Figure 4 shows the implementation for the optimal contract, and a setof relevant benchmarks which will be discussed below.

Lemma 6. In the implementation of the optimal contract with a standard capital structure,for all c ∈ (cl, ch) the marginal value of inside equity is larger than the marginal value ofhidden savings, v(c)−1 > c−γ, and the inside equity share φ(c) < φ.

The optimal contract with hidden savings has two important features. First, there is aleverage constraint that restricts the agent’s access to capital. Second, the capital structure

23

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.10

0.12

0.14

0.16

0.18

0.20

0.22

v

vh

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.0

0.1

0.2

0.3

0.4

λ

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.00

0.02

0.04

0.06

0.08

θ

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.10

0.12

0.14

0.16

0.18

0.20

ϕ

chcl

ϕ

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.00

0.01

0.02

0.03

0.04

0.05

μn

chcl

0.005 0.006 0.007 0.008 0.009 0.010 0.011c0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

σn

chcl

Figure 4: The implementation of the optimal contract. The starting point of the optimalcontract is indicated by the blue dot, the optimal stationary contract by the red dot, theoptimal portfolio plan with φ = φ is indicated the black dot, and the optimal contractwithout hidden savings by the green dot. Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3,φσ = 0.2.

24

depends on the history of returns R. As Figure 4 shows, leverage λ decreases with c, soit goes down over time (up to a steady state) and especially after bad outcomes. Bothfeatures allow the contract to relax the agent’s equity share, φt < φ, which also dependson the history of returns R. Figure 4 shows that the agent’s inside equity share φ isnon-monotonic in c.

To understand how it is possible to provide incentives against fund diversion with anequity share φt < φ, notice that because of the leverage constraint, the marginal value ofinside equity v(ct)

−1 is larger than the marginal value of hidden savings c−γt (both measuredin consumption units x). Intuitively, an extra dollar in inside equity gives the agent notonly more consumption, but also allows him to access more capital. Hidden savings incontrast can only be used to consume. Because of the leverage constraint the agent has toolittle capital, so he values inside equity more than hidden savings,11 v(ct)

−1 > c−γt – theagent is nonetheless willing to consume as indicated because with low leverage his exposureto risk and precautionary saving motive are small, so the marginal value of hidden savingsis low. As a result, the losses in inside equity after stealing are worth more than the gainsin hidden savings, so incentive compatibility is satisfied with a smaller equity share. Thisis captured by the first term of (23), φvtc

−γt < φ.

The dynamic incentive scheme helps further reduce the inside equity share φt. First,because the marginal value of inside equity v(ct)

−1 is forward-looking, we can relax the eq-uity share today by reducing leverage in the future, but not the other way. This asymmetrymeans we want to back-load distortions. Leverage starts high and then falls as ct increases(up to its steady state). Second, leverage falls after bad outcomes (σc ≤ 0), which reducesthe marginal value of inside equity v(ct)

−1. This allows us to further relax the equity shareφt. To the extent that we punish the agent with less access to capital after bad outcomes,we don’t need to punish him with less equity. This is captured by the second term of (23),φv′tσ

ct ct(σ

xt cγt )−1 < 0.

5 Discussion: The Dynamics of Financial Frictions.

In this section, we discuss the implications of hidden savings on financial frictions. Theoptimal contract without hidden savings has a stationary capital structure where the agentmust retain an equity stake for incentive reasons, but leverage is chosen optimally condi-

11If the contract leverage was too high, this would reverse: the value of hidden savings would be largerthan the value of inside equity.

25

tional on the equity constraint. In contrast, the optimal contract with hidden savings hasa capital structure with history-dependent leverage constraints and retained equity stake.Hidden savings therefore link leverage and equity constraints, and introduce a dynamicbehavior into an otherwise stationary environment. The agent starts with high leverage,large exposure to risk and high growth expectations. Over time, and especially after badoutcomes, his leverage is reduced, he faces less risk, but gives up growth (c rises). This isreversed after good outcomes, as leverage, risk and growth rise (c falls). As a result, agentswho are initially lucky and have good outcomes will have access to capital and high growthrates, while those that are initially less lucky get starved for capital and languish.

To understand the role of history-dependent financial frictions, it is useful to consider asimple benchmark where the agent keeps a fraction φ of the equity but is otherwise free tochoose leverage and payouts. Starting from this benchmark, the cost of reducing leverageis second-order because it was being chosen optimally. However, lower leverage reduces theagent’s exposure to risk and his precautionary saving motive, driving a wedge between themarginal value of inside equity and hidden savings. This allows us to relax the agent’s insideequity share φ < φ. Since the equity constraint was binding the benefit is first-order, so weimprove the contract. Thus, even stationary contracts that use a restriction on leverage,together with an appropriate incentive-compatible inside equity share φ < φ improve uponthis simple incentive scheme.

Non-stationary elements can improve the contract further. Distortions to the agent’srisk exposure further in the future reduce the agent’s precautionary saving motive, relaxingthe incentive constraint at all times leading to those future histories. Thus, it makes sensethat under the optimal contract leverage declines over time. Furthermore, distortions tothe agent’s risk exposure have the most impact on the precautionary saving motive, andare least costly, if imposed after bad performance. Thus, in the optimal contract leveragedeclines after bad performance and may rise after good performance.

Of course, if the agent didn’t have access to hidden savings, we could do even better. Theprincipal could simply front-load the agents’ consumption to reduce the private benefit ofstealing, which allows him to reduce the agent’s retained equity without distorting leverage.Without the need to manage precautionary saving motive dynamically, the optimal contractwithout hidden savings is stationary.

We illustrate this logic by designing several benchmark contracts in sequence. Westart by presenting the contract that solves the agent’s optimal portfolio choice problemconditional on retaining a fraction φ of equity, without any leverage constraint. Leverage

26

constraints reduce the agent’s precautionary saving motive, so we embed the solution ofthis portfolio choice problem within the family of stationary contracts. We compare allthese contracts to the optimal contract, which manages leverage and the agent’s sharedynamically. We finish this section by considering optimal contracting without hiddensavings.

Benchmark #1: Optimal Portfolio with an Equity Constraint

Consider the reasonable incentive scheme where that the agent must retain a fraction φ ofequity, but can otherwise choose the leverage and consumption ratio freely at any momentof time. His budget constraint is then

dntnt

=(r + λtα/φ− θt

)dt+ λtσdZt (26)

This is a classic optimal portfolio choice problem, in which the risky asset has the Sharperatio of α/(φσ). The solution is well-known and has leverage, consumption ratio and valuefunction given by

λp =α

γφσ2θp =

ρ− r(1− γ)

γ− 1− γ

2

(α

γφσ

)2

, and vp = θγ

1−γp . (27)

The resulting contract Cp = (cp, kp) corresponds to the black dot in Figures 1, 2, and 4. It

has a constant cp = θpvp = θ1

1−γp , and is therefore incentive compatible.

Lemma 7. The contract implemented with a constant equity constraint Cp is an incentivecompatible contract.

To understand the intuition, suppose the agent diverts a dollar. The agent absorbsfraction φ of the loss, i.e. his inside equity nt drops by φ. At the same time, the agent getsextra φ dollars in hidden savings. Because the consumption rate θp is chosen optimally by

the agent, the marginal value of inside equity v−1p = θ−γ1−γp and hidden savings c−γp = θ

−γ1−γp

is the same. In addition, since the contract is stationary, the marginal value of equity v−1pis not affected by bad performance, so the inside equity share φ = φ given by (23) ensuresincentive compatibility.

Contract Cp is suboptimal. It is possible to improve upon Cp even through stationarycontracts that combine a higher payout ratio θ, lower leverage λ, and a lower incentive-compatible equity share φ < φ. The very fact that an equity share φ < φ can be incentive

27

compatible may appear surprising. Simple stationary contracts illustrate how this is possi-ble.

Benchmark #2: Stationary Contracts

Consider now the wider family of stationary contracts with a constant c. We can chose anyc, and set σc = 0 and

σx = σxr (c) ≡√

2

√ρ− rγ

+ρ− c1−γ

1− γ(28)

so that µc = 0 in (10). If we initiate the contract with x0 = ((1 − γ)u0)1

1−γ we get astationary contract that gives the agent utility u0. Stationary contracts are indicated bythe red dashed line in Figures 1, 2, and 4.

Equation (28) highlights the trade-off between the risk exposure and precautionarysaving motive, since σx increases from 0 while c decreases from ch. Higher permanent riskexposure σx corresponds to a greater precautionary saving motive and a lower c. Theexpected growth rate µxt given by (9) is increasing in the risk exposure σx of the stationarycontract. To guarantee admissibility, we have to restrict c to be in the range (c∗, ch], wherec∗ ∈ (0, ch) is given by

c∗ =

(2γ

1 + γ

ρ− r(1− γ)

γ

) 11−γ

< ch (29)

so that µx < r and the No-Ponzi condition (3) is satisfied. Other than that, Theorem 3 isgeneral enough to ensure that stationary contracts are globally incentive compatible. TheHJB equation (13) yields the cost of the stationary contract. Thus, we obtain the followingcharacterization.

Lemma 8. For any c ∈ (c∗, ch], the corresponding stationary contract with σc = 0 andσx = σxr (c) given by (28) is globally incentive compatible and has cost vr(c)x0, where

vr(c) =c− α

φσ cγσxr (c)

2r − ρ− (1 + γ)ρ−c1−γ

1−γ

. (30)

Since stationary contracts are incentive compatible, we have v(c) ≤ vr(c).12

Figures 1 shows that the cost function vr(c) is non-monotonic in c (or σx). Startingfrom σx = 0 and ch, raising σx and increasing capital is attractive at first because capital

12Notice that vr(ch) = vh = cγh, which makes sense because at ch the stationary contract has σx = 0.

28

yields an excess return α, and the distortions to risk-sharing and intertemporal consumptionare second order. However, at some point these distortions become too costly. The beststationary contract with minimum cost Cminr is indicated by the red point at (cminr , vminr ),where cminr ≡ arg maxc∈(c∗,ch] vr(c) and vminr ≡ vr(cminr ).

The best stationary contract Cminr is better than the optimal portfolio contract with anequity constraint Cp, which is also an incentive compatible stationary contract indicatedby the black dot. To understand the improvement, consider a small reduction in leveragestarting from Cp. Since leverage was being chosen optimally, the effect on the agent’s utilityis second-order. However, the effect on the precautionary saving motive is first-order.Incentive-compatible payouts go up, and a first-order drop in the agent’s marginal utility ofconsumption leads it to drop below the marginal value of net worth (on which the leverageconstraint has a second-order effect). The wedge leads to a lower incentive-compatibleequity share φ and a first-order positive effect on the agent’s utility.13

Lemma 9. Any stationary contract can be implemented with a constant capital structurewith

φr(c) = φvr(c)

cγ, λr(c) =

σxr (c)

σand θr(c) =

c

vr(c)(31)

The contract implemented with a constant equity constraint Cp is the stationary contractcorresponding to cp, but is not optimal. The best stationary contract Cminr is less risky forthe agent, i.e. we have c∗ < cp < cminr . For all c ∈ (cp, ch) the marginal value of equity islarger than the marginal value of hidden savings, v−1r (c) > c−γ, and the inside equity shareφr(c) < φ.

For stationary contracts with c ∈ (cp, ch) it is worth making several observations. First,since there is a wedge between the marginal utility of inside equity and the marginal utilityof consumption, vr(c)−1 > c−γ , we can relax the inside equity share, φr(c) < φ. Second,the fund must mandate the payout rate θr(c) since the agent would prefer to postponeconsumption. However, the agent is willing to consume as indicated, rather than save,because the payout rate corresponds to his precautionary saving motive. Third, hiddensavings create a leverage constraint for a completely different reason from models of limitedcommitment, as in Hart and Moore (1994) and Kiyotaki and Moore (1997), where theagent can “run away” with some resources. Here the leverage constraint reduces the agent’sprecautionary saving motive, leading to a lower incentive compatible risk exposure.

13Of course, lower φ implies even lower precautionary saving motive, so φ can be lowered even further,until a fixed point.

29

Benchmark #3: The Optimal Contract

The best stationary contract illustrates how restricting the agent’s access to capital througha leverage constraint can help provide incentives. The optimal contract does even better byadding a dynamic dimension to the financial frictions. The principal front loads the agent’saccess to capital and punishes him after bad outcomes by further restricting his access tocapital, rather than taking away equity or consumption (he gives the agent more capitalafter good outcomes). This improves risk sharing and relaxes the agent’s precautionarysaving motive, making stealing less attractive, and allows the principal to use the moreefficient continuation contracts when he must deliver more utility to the agent (after goodoutcomes).

Lemma 10. For any c ∈ [cl, ch], the optimal contract has a lower equity constraint thanthe corresponding stationary contract, i.e.

φ(c) ≡ φ(v(ct)

cγt+v′(ct)σ

ct ct

σxt cγt

)≤ φr(c). (32)

Of course, if the agent did not have access to hidden savings, we could do even better.Let us now turn to the optimal contract without hidden savings.

Benchmark #4: No hidden savings

If the agent had no access to hidden savings, the principal can control the agent’s consump-tion directly. Hence, the principal specifies the agent’s consumption and capital to maximizeprofit and subject to incentive compatibility with respect to diversion of returns, withoutregard to the agent’s precautionary saving motive. While we present it as a benchmarkhere, optimal contracting without hidden savings is a classic problem that is interesting inits own right. The solution we derive here, in particular, is of special interest because it hasa closed form and a stationary structure. The convenient form of the solution should provehelpful in applications. In fact, Di Tella (2014) adopts a generalized version of this modelin the context of financial regulation, and provides important characterizations, some ofwhich we cite here.

Without hidden savings, (6) is still the incentive compatibility constraint for returndiversion and (9) is still the law of motion of xt, but the agent’s consumption ct and ratioct = ct/xt are no longer bound by the Euler equation (7) or (10). Rather, ct is now the

30

principal’s control rather than a state variable. We can write the HJB equation for theproblem without hidden savings by taking (13), removing terms that contain v′ and v′′,

and replacing control σc with c. We obtain the HJB equation

rvn = minσx,c

c− σxcγ αφσ

+ vn

(ρ− c1−γ

1− γ+γ

2(σx)2

), (33)

where vn(x) = vnx is the principal’s cost function.The optimal contract takes a simple stationary form, with constant volatility σx and

ratio c and hence constant growth µx. The solution exists only if γ ≤ 1/2 and only if α issufficiently low; otherwise the principal’s value function becomes infinite. More generally,Di Tella (2014) shows that with Epstein-Zin preferences, the elasticity of intertemporalsubstitution must be greater than 2 in order for the solution to exist. Intuitively, if the EISis low a principal who can control the agent’s consumption has a lot of power over him andcan effectively eliminate the moral hazard problem and obtain an arbitrage for any α > 0.An example of a solution for γ = 1/3 is indicated with a green dot in Figures 1, 2, and 4.

The agent’s consumption in the optimal contract is determined without regard to theagent’s precautionary saving motive. The first-order condition for optimal consumption is

1 = vnc−γ + vnγ

2(σx)2c−1 (34)

The left-hand side is the marginal cost of paying the agent, one. The two terms on theright-hand side capture the benefits of paying the agent. First, the principal reduces theagent’s continuation utility by paying today. Second, according to (6), higher consumptiontoday reduces the agent’s incentives to divert returns, and hence reduces the required riskexposure given any level of capital under management. Hence, the agent’s consumption inthe optimal contract becomes front-loaded. This can be expressed in several ways. TheEuler equation does not hold, i.e. e−(r−ρ)tc−γt is a strict submartingale and so, if theagent had access to hidden savings, he would want to save.14 Equivalently, given choiceof σxn in the optimal contract, the consumption rate that reflects the precautionary saving

14It is interesting to note that the standard inverse Euler equation of agency problems without hiddensavings does not hold in our setting either. The inverse Euler equation requires that e−(r−ρ)tcγt is amartingale, where c−γt is the marginal cost of delivering utility to the agent, or inverse marginal utility.If the inverse Euler equation holds, then the agent wants to save if he could, i.e. consumption is morefront-loaded relative to the hidden-savings case. In our problem, because consumption enters the agent’sincentive constraint, it is even more front-loaded than the profile of the inverse Euler equation.

31

motive, given by (28), is lower than the consumption rate cn of the optimal contract. Torepeat the main intuition, front-loaded consumption is beneficial because it reduces theagent’s marginal utility from consumption, and therefore makes stealing and immediatelyconsuming less attractive. This relaxes the incentive constraint.

We can implement the contract by following the derivations of Section 4. By stationarity,the optimal contract has fixed leverage, equity shares held by the agent and principal, andpayout rate. We have analogous formulas as (31),

φn = φvncγn, λn =

σxnσ

and θn =cnvn. (35)

We summarize important properties of the implementation in the following lemma.

Lemma 11. The optimal contract without hidden savings is implemented with a constantcapital structure given by (35). The marginal value of equity is larger than the marginalvalue of consumption, v−1n > c−γn , and the inside equity share φn < φ. There is no leverageconstraint; leverage is chosen optimally conditional on φn, λn = α

γφnσ.

For a long time, researchers have been wondering to what extent the possibility ofhidden savings matters. The view that the savings constraint matters very little leadsto a justification of models without savings, especially given the technical difficulties thathidden savings pose. Without solving for the optimal contract, this question is difficultto address, although Farhi and Werning (2008) provide important assessments based oningenious explorations of the Euler and inverse Euler equations. In our setting, given fullcharacterization of optimal contracts, we can answer the question of how much hiddensavings matter more definitively. The answer depends on parameters, but potentially thedifference between the optimal contracts with and without hidden savings can be very large.In fact, for some parameters the principal’s value function without savings becomes infinite,while it is finite with hidden savings.

6 Aggregate risk and hidden investment

The previous results show how the principal can use the agent’s access to capital to ma-nipulate his precautionary saving motive. This suggests that hidden investment could bean important constraint in this setting. To study the role of hidden investment, it is usefulto also introduce aggregate risk, and allow the agent to invest his hidden savings in it.

32

Fortunately, we can easily incorporate hidden investment into our setting and extend ourtools to study what role they play. Appendix B has all the formal results. Here we willfocus on the main economic insights.

The return on capital is now

dRt = (r + πσ + α− at) dt+ σ dZt + σ dZt

Where Z is an independent Brownian motion that represents aggregate risk, with marketprice π. Capital has a loading σ on aggregate risk, so the excess return on capital for theagent is α, as in the baseline. Let Q be the associated martingale measure. Now the agentcan invest his hidden savings (besides the risk-free rate). We always allow the agent toinvest in aggregate risk, in the same way the principal would be able to. This is reallyan extension of the notion of hidden savings to an environment with aggregate risk. Inaddition, the agent may be able to invest his hidden savings in his own private technology.His hidden savings therefore follow the law of motion

dht = (rht + ztht(α+ πσ) + zthtπ + ct − ct + φktat) dt+ ztht

(σdZt + σdZt

)+ zthtdZt

where z is the portfolio weight on his own private technology, and z the weight on aggregaterisk. While the agent can choose any position on aggregate risk, zt ∈ R, for his hiddenprivate investment we consider two cases: 1) no hidden private investment, zt ∈ H = 0,and 2) hidden private investment, zt ∈ H = R+.15 A valid interpretation for hiddeninvestment in the private technology zt is that the principal can give the agent an amountof capital kt that he monitors, but the agent can secretly invest more.

A contract C = (c, k) specifies the contractible payments c and capital k, contingent notonly on returns R but also on the observable aggregate shock Z. After signing the contractthe agent can choose a strategy (c, a, z, z) to maximize his utility. The agent’s utility andthe principal’s objective function are still given by (1) and (2). As in the baseline setting,it is without loss of generality to look for a contract where the agent does not steal, has nohidden savings, and no hidden investment. A contract is therefore incentive compatible ifthe agent’s optimal strategy is (c, 0, 0, 0), or (c, 0) for short.

15We can also study other cases where the agent may not be able to invest in aggregate risk, or only takea positive position, which requires small modifications to the relevant incentive compatibility constraints.We focus on the economically most relevant case, where the agent can always invest his hidden savings inthe market in the same way the principal would.

33

Since there is now aggregate risk that pays a premium, we need to slightly modify theparameter restrictions

(ρ− r(1− γ)

γ− 1− γ

2

(πγ

)2) 11−γ

> 0 (36)

and

α < α ≡ φσγ√

2√1 + γ

√ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2 (37)

We can check that with π = 0 we recover the formulas without aggregate risk.Since the contract can depend on the history of aggregate shocks Z, so can his continu-

ation utility U c,0 and his consumption c. However, because the agent is not responsible foraggregate shocks, incentive compatibility does not place any constraints on his exposure toaggregate risk. On the other hand, since the agent can invest his hidden savings in aggre-gate risk, his Euler equation needs to be modified appropriately. His discounted marginalutility must be a supermartingale under any valid trading strategy. Otherwise, the agentcould save a dollar, invest it in aggregate risk or his private technology, and consume itlater. As a result, the laws of motion for the state variables x and c need to be modified to

dxtxt

=

(ρ− c1−γ

1− γ+γ

2(σxt )2 +

γ

2(σxt )2

)dt+ σxt dZt + σxt dZt (38)

dctct

=(r − ρ

γ− ρ− c1−γt

1− γ+

(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2 (39)

+(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2)dt+ σct dZt + σct dZt + dLt

The “skin in the game” IC constraint is unchanged

σxt = c−γt φktσ (40)

Since the agent can invest his hidden savings in aggregate risk (both long and short) wehave a new IC constraint

σct + σxt =π

γ(41)

If the agent can also invest his hidden savings in his private technology, H = R+, we also

34

have another IC constraintσxt + σct ≥

α

σγ(42)

The interpretation for (41) and (42) is simple: for the agent, aggregate risk has a premiumπ and his idiosyncratic risk a premium α

σ . The principal must allow him to invest in thesesources of risk through the contract, or he will do it on his own. This restricts the principal’sability to provide incentives. In particular, the principal cannot promise to give the agent aperfectly deterministic consumption at some point in the future. If he tried to do this, theagent would just take risk on his own. This is reflected in a lower upper bound ch, which iscostly because the principal would like to use a promise of future safety to relax the agent’sprecautionary saving motive. If the agent cannot invest in his private technology, we have

ch =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ

. This is the c corresponding to the optimal portfolio

strategy when the agent can invest only in aggregate risk. Notice that if we take π = 0

we recover expression (8) in the baseline setting, corresponding to the optimal portfolio ifthe agent can only invest in a risk-free asset. If the agent can also invest in his private

technology, then ch =

(ρ−r(1−γ)

γ − 1−γ2

(ασγ

)2− 1−γ

2

(πγ

)2) 11−γ

, which is even lower and

corresponds to the c in an optimal portfolio strategy when the agent can invest in bothaggregate risk and in his private technology on his own (so he can’t get any risk sharing).16

We can now characterize the relaxed problem with an HJB equation and appropriateconstraints

0 = minσx,σc,σx,σc

c− rv − σxcγ αφσ

+ v

(ρ− c1−γ

1− γ+γ

2(σx)2 +

γ

2(σx)2 − πσx

)(43)

+v′c

(r − ργ− ρ− c1−γ

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2 +

(σx)2

2

+(1 + γ)σxσc +1 + γ

2

(σc)2 − σcπ)+

v′′

2c2(

(σc)2 +(σc)2)

subject to σx ≥ 0 and (41) and (42).Figure 5 shows the optimal contract with hidden investment. The optimal contract

has the same features as in the baseline setting. Hidden investment, however, restrictsthe principal’s ability to manipulate the agent’s precautionary saving motive by promising

16The parameter restrictions (36) and (37) ensure ch is positive in both cases.

35

4.1 4.2 4.3 4.4c

65

70

75

80

85

v

Figure 5: The cost function v(c) solid in blue without hidden investment, green withhidden investment, and dashed in red for stationary contracts. The the starting pointwithout hidden investment is indicated by the blue dot, with hidden investment by thegreen dot, the optimal stationary contract by the red dot, the optimal portfolio plan withφ = φ by the black dot. The black dashed line is the locus A(c, v) = 0. Parameters:ρ = r = 5%, α = 6%, γ = 3, φσ = 0.2, π = 4%.

4.2 4.3 4.4c

-0.001

0.001

0.002

0.003

0.004

0.005

μc^

4.2 4.3 4.4c

0.005

0.010

0.015

0.020

0.025

μx

4.2 4.3 4.4c

-0.005

-0.004

-0.003

-0.002

-0.001

σc^

4.2 4.3 4.4c

0.02

0.04

0.06

0.08

0.10

0.12

σx

Figure 6: The drift, µc and µx, and volatility, σc and σx, of the state variables c and x,without hidden investment (blue) and with hidden investment (green). The black dashedlines indicate cl and ch with hidden investment, and cl without hidden investment. Param-eters: ρ = r = 5%, α = 6%, γ = 3, φσ = 0.2, π = 4%.36

less risk in the future and after bad outcomes, and thus leads to a higher cost. Thecontract starts at some cl and then moves immediately into the interior of the domain[cl, ch]. Because the principal cannot promise a completely deterministic contract in thefuture (ch is lower with hidden investment), the agent’s precautionary saving motive isstronger, and the contract starts at a lower cl.

Turn now to the contract’s dynamic behavior. First, using (41) to eliminate σc, andtaking FOC for σx, we obtain

σx =π

γσc = 0

This is the first best exposure to aggregate risk. The principal and the agent don’t haveany conflict about aggregate risk, and the principal cannot use it to relax the moral hazardproblem, so they implement the first best aggregate risk sharing.17 The presence of thisinvestment opportunity still affects the optimal contract, however, by limiting the principal’sability to manipulate the agent’s precautionary saving motive (it lowers ch).

In contrast, the FOC for σx and σc depend on whether the agent can invest his hiddensavings in his private technology, as shown in Figure 6. Without hidden investment, theFOCs are the same as in the baseline, (16) and (17), and capture the intertemporal andhedging motives described in Section 3. With hidden investment, the IC constraint (42)can be binding in some region of the state space. This is the case in Figure 6 near theupper bound of the hidden investment contract ch. It is still true, however, that µc(cl) > 0

and σc ≤ 0 always, so the contract dynamics are the same as in the baseline: the agent getsless capital and less risk over time (up to a “steady state”) and after bad outcomes.

The optimal contract can be implemented with a capital structure S = (λ, φ, θ, σn)

given by (23), (24), (25), and σn = π/γ, with the law of motion for the agent’s insideequity

dntnt

=(r + λtα/φt + σnt π − θt

)dt+ λtσ dZt + σnt dZ

There is still a leverage constraint that allows the principal to relax the agent’s inside equitystake φ < φ. How come the principal is able to impose a leverage constraint if the agent caninvest on his own with his hidden savings? If the agent invests on his own he is the residualclaimant and must bear the whole risk, while if he invests through the principal he gets toshare some risk and keep only a fraction φt < 1. The leverage constraint really limits the

17If the agent didn’t have access to hidden investment in aggregate risk, and the agent’s private technologyis exposed to aggregate risk σ 6= 0, then the principal could potentially use the agent’s exposure to aggregaterisk to relax the moral hazard problem.

37

amount of capital the principal will agree to share risk on. As in the setting without hiddeninvestment, the optimal contract starts with a non-binding leverage constraint. It is worthpointing out that although ch corresponds to the consumption profile under autarky, thereare still gains from trade if the agent is able to invest in his private technology, so that thecost for the principal at this point is lower v(ch) < cγh. To see why, notice that the principalcan give the agent the same consumption process that he would get in autarky, but morecapital. Since φ < φ < 1, the agent can manage more capital than under autarky and yethave the same exposure to risk. Since he keeps the total exposure to risk corresponding toautarky, which is relatively low given that the agent must bear the whole risk, there is abinding leverage constraint here. In other words, if the agent could invest more capital andkeep only a fraction φ < 1 he would like to invest more, but not if he has to keep all therisk.

It is useful to ask under what conditions the gains from trade are completely exhausted,and the optimal contract corresponds to letting the agent invest on his own without anyrisk sharing. This is true in the special (but salient) case with φ = 1, where we have apure misreporting problem. Without hidden investment, the are still gains from trade, andthe optimal contract has the same qualitative features described in Section 3. While theagent can save on his own, the principal can still control his access to capital to provideincentives, and can therefore provide some risk sharing. However, with hidden investment,the optimal contract coincides with autarky. Intuitively, the agent can both save and investon his own, so the principal cannot provide any risk sharing in an incentive compatibleway. We can see this case as a limit in Figure 5. If we let φ→ 1, while also adjusting σ sothat φσ is constant, all the curves corresponding to case with no hidden investment remainunchanged. The optimal contract with hidden investment, however, becomes progressivelyworse, because the agent finds investing on his own more attractive (the green curve shifts

up). This is reflected in a falling ch cp =

(ρ−r(1−γ)

γ − 1−γ2

(αγφσ

)2− 1−γ

2

(πγ

)2), where

cp corresponds to the optimal portfolio with an equity constraint φ = φ (which in the limitis 1).

Verifying global incentive compatibility with hidden investment

The agent’s ability to invest his hidden savings makes the verification of global incentivecompatibility potentially more difficult. The agent could find it attractive to steal andsave the proceeds for later, while investing them in aggregate risk or even his own pri-

38

vate technology. However, we can extend the results of Theorem 3 to deal with hiddeninvestment.

Theorem 4. Let C = (c, k) be an admissible contract with associated processes x and c

satisfying (62) and (63), and (64), (65), and (66), with bounded µx, µc, σc, and σc, andwith c uniformly bounded away for zero and bounded above by ch. Suppose that the contractsatisfies the following property

σct ≤ 0

Then for any feasible strategy (c, a, z, z), with associated hidden savings h, we have thefollowing upper bound on the agent’s utility, after any history

U c,a,z,zt ≤(

1 +htxtc−γt

)1−γU c,0t

In particular, since h0 = 0, for any feasible strategy U c,a,z,z0 ≤ U c,00 , and the contract C istherefore incentive compatible.

7 Renegotiation

The optimal contract requires commitment. The principal can relax the agent’s precau-tionary saving motive by promising an inefficiently low amount of capital and risk overtime and after bad outcomes. This suggests that the agent and principal could be temptedto renegotiate the contract and “start over”, undoing the whole incentive scheme. Herewe formalize a notion of renegotiation, and characterize the optimal renegotiation-proofcontract. As it turns out, the optimal renegotiation-proof contract is the best stationarycontract, so even without commitment we are able to do better than letting the agent investwhile keeping a fraction φ of his risk, and a leverage constraint can be used to relax theequity constraint. This section is consistent with the presence of aggregate risk and hiddeninvestment introduce in Section 6.

After signing an incentive compatible contract C = (c, k), the principal can at any timeoffer a new continuation contract that leaves the agent at least as well off (the offer is“take it or leave it”). The question is what kind of contracts can he offer - what is a valid“challenger” to the original contract? Here we explore a notion of internal consistency.If C is renegotiation proof, then surely appropriately scaled parts of it should be a validchallenger. This means that at any stopping time τ , the principal can renegotiate and get

39

a continuation cost xτ × inf v(ω, t). With this in mind, we say that an incentive compatiblecontract is renegotiation-proof (RP) if

∞ = arg minτ

EQ[ˆ τ

0e−rt(c− ktα)dt+ e−rτxτ inf v

]The optimal contract with hidden savings is not renegotiation proof, because after anyhistory vt > vl = inf v(ω, s) , so the principal is always tempted to “start over”. In fact, itis easy to see that RP contracts must have a constant vt. The converse it also true.

Lemma 12. An incentive compatible contract C is renegotiation proof if and only if thecontinuation cost v is constant.

Stationary contracts have a constant v, because c is constant. However, those contractswere built using dLt = 0. There are other contracts with a constant c that use dLt > 0, i.e.the drift of c would be negative without dLt. In addition, there could be non-stationarycontracts with a constant cost v(c) for all c in the domain. The next Lemma shows theyare all worse than the best stationary contract Cminr , with cost vminr = minc∈(c∗,ch] vr(c).

Theorem 5. The optimal renegotiation-proof contract is the optimal stationary contractCminr with cost vminr .

Remark. It is possible that cminr = ch if the agent can invest his hidden savings and φ is closeenough to 1. In the special case with hidden investment and φ = 1, we have cminr = cp = ch,as show in Lemma 28.

This result shows that even without full commitment, hidden savings generate a leverageconstraint. The principal can do better than just giving the agent an equity constraint φ = φ

by restricting leverage λ = kn+e .

8 Conclusions

We study the role of hidden savings in a classic portfolio-investment problem with funddiversion, which can be embedded in macroeconomic and financial environments. With-out hidden savings, the principal front-loads the agent’s consumption to reduce the privatebenefit of fund diversion. The agent must keep an equity stake to provide incentives, buthe is otherwise free to choose his leverage. With hidden savings, the principal cannot con-trol the agent’s consumption, and must use the agent’s access to capital to manipulate his

40

precautionary saving motive. By reducing access to capital over time, and especially afterbad outcomes, the principal reduces incentives to divert funds and secretly save, reducingthe cost of providing incentives. As a result, the optimal contract requires a leverage con-straint in addition to the equity constraint. Hidden savings generate a leverage constraintfor completely different reason than models of limited commitment such as Hart and Moore(1994) and Kiyotaki and Moore (1997) – a binding leverage constraint reduces the agent’sprecautionary saving motive and allows the principal to relax the equity constraint. Ourresults are robust to introducing market risk, hidden investment, and renegotiation.

An important methodological contribution is that we provide a sufficient analyticalcondition for the validity of the first-order approach. If the agent’s precautionary savingmotive is weaker after bad outcomes, the contract is globally incentive compatible. Thiscondition holds in the optimal contract and in a wider class of contracts beyond the optimalone. In fact, the sufficient condition does not even require a recursive structure, and it isvalid even with aggregate risk and hidden investment.

41

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44

Appendix A - Omitted Proofs

Lemma 1

Consider

Yt = Et[ˆ ∞

0

e−ρsc1−γs

1− γ ds]

=

ˆ t

0

e−ρsc1−γs

1− γ ds+ e−ρtUc,0t

Since Y is an F-adapted P -martingale, and F is generated by Z, we can apply a martingale representationtheorem to obtain

dYt = e−ρtc1−γt

1− γ dt+ e−ρtdUc,0t − ρe−ρtUc,0t dt = e−ρt∆tσ dZt

for some stochastic process ∆ also adapted to F. Dividing by e−ρt and rearranging we get (4).

Lemma 2

First, take the strategy (c+ φka, a). Here a is zero where (5) is satisfied and a = a∗t ∧ a where a∗t achieves

the maximum in (5) and a > 0 is an arbitrary bound. Because the objective in (5) is concave and it’s c1−γt1−γ

for a = 0, then for a∗t ∧ a it is strictly greater when (5) fails. Notice ht = 0 by construction, and we canfollow strategy (cn, an) until some stopping time τn and then revert to good behavior, where the stoppingtime τn →∞ a.s., ensures it’s feasible and reduces the stochastic integral. We can compare the utility fromthis strategy U c

n,an with the utility from good behavior Uc,0:

U cn,an − Uc,00 = Ea0

[ˆ τn

0

e−ρt(

(ct + φktat)1−γ

1− γ − c1−γt

1− γ −∆tat

)dt

](44)

where we have used Zat = Zt +´ t

0asσds to express Uc,00 as an expected integral under P a, and also the fact

that U cn,an

τn = Uc,0τn . Taking n→∞ and using the monotone convergence theorem (the integrand is alwayspositive), if (5) fails we get U c

n,an

0 > Uc,00 for some large n, and C is not incentive compatible. (6) is simplythe FOC necessary condition for (5) (and sufficient because of concavity).

For (7) this is the result of e−(ρ−r)tc−γt being a supermartingale, which is a standard necessary conditionin a savings/consumption problem. Note this doesn’t involve stealing a, just hidden consumption c. Wecan then write it e−(ρ−r)tc−γt = Mt − At, where Mt =

´ t0σMt dZt is a local martingale, and A a weakly

increasing process. Using Ito’s lemma, we get the desired expression.

Lemma 3

For the bound, since both ct ≥ 0 and xt ≥ 0, we only need to show that ct ≤ ch. Marginal utility of

consumption is mt = c−γt and the utility flow c1−γt1−γ = 1

1−γmγ−1γ

t . This is a convex and decreasing functionof mt. From Lemma 2, we have by (7) that Et [mt+u] ≤ e(ρ−r)umt. Given any m0 we have by Jensen’sinequality

E[c1−γt

1− γ

]≥ 1

1− γE [mt]γ−1γ ≥ 1

1− γmγ−1γ

0 e(ρ−r) γ−1

γt

=c1−γ0

1− γ e(ρ−r) γ−1

γt

45

with equality only if c is deterministic. So

Uc,0t = Et[ˆ ∞

t

e−ρ(u−t)c1−γu

1− γ du]

≥ˆ ∞t

e−ρ(u−t)c1−γt

1− γ e(ρ−r) γ−1

γ(u−t)

ds =c1−γt

1− γγ

ρ− r(1− γ)

where the second equality uses the fact that the termination contract must deliver U to the agent. Thisbound implies

xt =((1− γ)Uc,0t

) 11−γ ≥ ct

(γ

ρ− r(1− γ)

) 11−γ

So ct = ct/xt has an upper bound

ct ≤(ρ− r(1− γ)

γ

) 11−γ

= ch

In addition, since the upper bound can only be achieved with a deterministic consumption, Uc,0 is deter-ministic too. This implies that both ct and xt grow at rate r−ρ

γ, so ch is an absorbing state. In light of (6),

we must have kt+u = 0 in the continuation contract, so we have the “retirement” contract with cost vhxt.This completes the proof.

Theorem 1

The proof is split into parts.

1) The cost function must be bounded above by vh since we can always just give consumption to theagent without any capital, and obtain cost vh. It must be strictly positive because if v(c) = 0 for anyc ∈ [0, ch], then we can scale up the contract and give infinite utility to the agent at zero cost, or elseachieve infinite profits.

2) Because we can always move c up using dLt, we know that v must be weakly increasing. It is usefulto write the function A(c; v) from (14) as

A(c, v) ≡ c− rv − 1

2

(αcγ

φσ

)2

vγ+ v

ρ− c1−γ

1− γ (45)

which is the HJB equation when v(c) is flat. The region where the HJB equation holds cannot have anyflat parts, because this would mean that A(c, v(c)) = 0. We know however from Lemma 13 (with π = 0)that this function can have at most two roots, so v must be strictly increasing in the region where the HJBholds.

The only way the HJB could not hold is if the optimal contract never spends any time there, i.e. if weever found ourselves there, it would be optimal to immediately jump out using dLt. The value in that regionthen must be constant and equal to the value at the destination point (the upper end of the flat region).The HJB should hold as an inequality with A(c; v(c)) ≥ 0, since otherwise we could improve by lingeringin the flat region for a while before jumping. Since v(c) is strictly increasing when the HJB equation holds,

46

the contract will start with c0 at the upper end of a flat region. It cannot be that c0 = 0 because of Inadaconditions, so we must have at least one flat interval [0, cl].

3) We would like to show that this is the only flat interval, and the HJB equation holds as an equalityin the strictly increasing region [cl, ch], and both regions are connected with smooth pasting, i.e. v′(cl) = 0.Suppose then that there is a region [c1, c2] ⊂ (0, ch) where the cost function is flat, and it’s strictly increasingimmediately above it (possibly, c1 = 0). Let’s show that as c c2, A(c, v(c)) → 0 and v′(c) → 0 (i.e. wehave smooth pasting). Towards contradiction, imagine there is a kink at c2, i.e. the right-derivative of v(c)

is strictly positive. This can only happen if σc(c+ ε)→ 0 and µc(c2 + ε) ≥ 0 as ε→ 0, since otherwise wewould cross into the flat region where the HJB doesn’t hold (with v′(c) > 0 we must have dL = 0). Firstconsider the case with lim inf µc(c2 + ε) > 0. We can contemplate the following deviation: start at c2 − δ,with the same σx and σc = 0: by continuity µct > 0 along this plan, so after some time we will end up at c2and we can go back to the optimal contract and obtain continuation cost v(c2). The value of this strategyfor c < c2 extends the cost function v(c) below c2 and satisfies the HJB equation (with σc = 0, so it’s afirst order ODE with boundary condition given by v(c2) at c2). However, because v′(c2) > 0, we obtain alower cost at c2 − δ. This is therefore an attractive deviation, which is a contradiction, so µc(c2 + ε)→ 0 isthe only option left. In this case, since we also have σc(c+ ε)→ 0, it means we have a stationary contract,and because we are at a local minimum of the cost function, we must also be at a local minimum of thecost of stationary contracts vr(c) given by (30), implying v′r(c2) = 0. However, because v′+(c2) > 0, we getv(c+ ε) > vr(c+ ε), which cannot be. We conclude that v′(c2) = 0.

We still need to show that A(c, v(c)) → 0 as c c2. To see this it’s useful to use the FOC for σx

conditional on σc to obtain

σx =

αcγ

φσ− v′c(1 + γ)σc

vγ + v′c(46)

and plug it into the HJB equation. We can then re-write the HJB

0 = minσc

A+ Bσc +1

2C(σc)2

with

A = c− rv − 1

2

(αcγ

φσ

)2

vγ + v′c+ v

(ρ− c1−γ

1− γ

)− v′c

(ρ− rγ

+ρ− c1−γ

1− γ

)(47)

B = v′c(1 + γ)

αcγ

φσ

vγ + v′c≥ 0 (48)

C = γv′c(1 + γ)v − v′cvγ + v′c

+ v′′c2 (49)

For the HJB to have a minimum, it must be that C ≥ 0. If C = 0 we can only have a minimum if B = 0

as well, in which case A = 0, and with v′(c2) = 0 we get A(c, v(c)) → 0 as c c2 as desired. If insteadC > 0, then

σc = − BC≤ 0 (50)

With v′(c2) = 0 we must have v′′(c + ε) ≥ 0 for small ε (or else v′(c + ε) < 0). We can then show thatCB2 → ∞ as c c2, which implies B2

C→ 0 and therefore A → 0, and therefore A(c, v(c)) → 0 as c c2,

47

as desired. Since this is true in particular when c1 = 0 and c2 = cl, we have proven the smooth pastingcondition.

Now since at c2 we have A(c, v(c)) = 0, this is a root of A. Below that we have A(c, v(c)) ≥ 0. FromLemma 13 we know that this can only be the case if c2 is the first root of A(c, v(c2)). We would like to showthat it cannot be the case that for c < c1 < c2 the cost function is lower, i.e. v(c1 − δ) < v(c1) = v(c2).To see this, imagine the same problem with a smaller α′ < α, and cost function vα′(c) ≥ v(c). We canpick α′ small enough that vα′(cα

′l ) = v(c2), where cα

′l is the upper end of the first flat region for the

contract with α′ (and it minimizes vα′(c)). We can do this because vα′(c) is continuously increasing inα′ for any c, and has vα′(c) = vh for all c ≤ ch if α′ = 0. It must be that cα

′l ≤ c2, because to the

right of c2 vα′(c) ≥ v(c) > v(c2) for all c > c2. However, looking at A(c, v(c2)) we notice it is decreasingin α, so Aα′(c

α′l , v(c2)) > A(cα

′l , v(c2)) ≥ 0. But the previous argument shows that Aα′(cα

′l , v(c2)) =

Aα′(cα′l , vα′(c

α′l )) = 0 because cα

′l is the upper end of the first flat region of the optimal contract for α′.

This is a contradiction, so we cannot have v(c1 − δ) < v(c1) = v(c2).Putting all of this together, we only have one flat region, [0, cl), where the HJB equation holds as an

inequality A(c, v(c)) > 0 (the inequality is strict because of Lemma 13 and the fact that cl must be thefirst root), and a strictly increasing region [cl, ch] where the HJB equation holds with equality. In the flatregion we have v(c) = v(cl) ≡ vl for all c ≤ cl. At cl we have v′(cl) = 0 and A(cl; vl) = 0.

4) Now we want to show that v′′(cl) > 0. First suppose the A(c, v(cl)) is strictly positive below cl andstrictly negative above it, so that Ac(cl, vl) < 0 (we will show that this must indeed be the case below).Consider the first order ODE that results from fixing σc = 0 and σx = α

σγ

cγl

vlφin the HJB equation:

c− σxcγ αφσ

+ vfo

(ρ− c1−γ

1− γ +γ

2(σx)2

)︸ ︷︷ ︸

A(c,v)

+v′foc

(r − ργ

+(σx)2

2− ρ− c1−γ

1− γ

)= 0

and consider the solution with boundary condition vfo(cl) = v(cl). Since we already know that A(cl, v(cl)) =

0, we must have v′fo(cl) = 0 because the term in parenthesis is µc if σc = 0, and Lemma 15 shows thisis strictly positive under these conditions. Furthermore, we must have v′′fo(cl) ≤ v′′(cl). To see this, ifv′′fo(cl) > v′′(cl) ≥ 0, then v′fo(cl + ε) > v′(cl + ε), while vfo(cl + ε) = v(cl + ε) + o(ε) (because both havefirst derivatives equal to zero) for some small ε. By continuity, it will still be the case that µc > 0 forcl + ε, so starting at cl + ε − δ we will eventually get to cl + ε. We can then solve the first order ODEbackwards with boundary condition vfo(cl + ε) = v(cl + ε) and we will obtain a lower cost for cl + ε − δ,i.e. vfo(cl + ε − δ) < v(cl + ε − δ), because by continuity v′fo(cl + ε) > v′(cl + ε) for this solution as well.This cannot be, so we must have v′′fo(cl) ≤ v′′(cl).

Differentiating the first order ODE with respect to c we obtain

0 = ∂cA(c, vl)|c=cl + v′′fo(cl)cl

(r − ργ

+(σx)2

2−ρ− c1−γl

1− γ

)

where we have used v′fo(cl) = 0 and the envelope theorem to compute the derivative ∂cA(c, vl)|c=cl . It

follows that since Ac(cl, vl) < 0, then either v′′fo(cl) > 0 and r−ργ

+ (σx)2

2− ρ−c1−γ

l1−γ > 0, or both are strictly

negative. But we already know that r−ργ

+ (σx)2

2− ρ−c1−γ

l1−γ > 0, so this leaves only v′′(cl) ≥ v′′fo(cl) > 0.

48

It only remains to rule out Ac(cl, v(cl)) = 0 which means A(c, v(cl)) > 0 for all c 6= cl (from Lemma 13).Since A(c, v) = 0 is the HJB equation if v is flat, by a martingale verification argument as in Theorem 2 weobtain that the cost of any incentive compatible contract C is strictly larger than v(cl), unless it has ct = cl

always so that A(ct, vl) = 0 always and σx = ασγ

cγl

vlφ. This requires µc = σc = 0. But Lemma 15 shows

that for σc = 0 and σx = ασγ

cγl

vlφ, and A(cl, vl) = 0 we get µc(cl) > 0. Since the optimal contract must be

incentive compatible and achieve cost v(cl), Ac(cl, v(cl)) = 0 cannot be. So we have v′′(cl) > 0.

5) v′′(cl) > 0 implies σc(cl) = 0 and µc(cl) > 0. This in turn proves that ct > c0 = cl. From (50)we get σc(c) ≤ 0, and (46) implies σx(c) ≥ 0. It also implies that at cl , since v′(cl) = 0, the choice of σx

maximizes (15).

Theorem 2

First let’s extend the function v(c) as described above, with v(c) = vl ≡ v(cl) for all c < cl (we always havec ∈ [0, ch]). The HJB holds as an equality for c ≥ cl, but we need to check that it holds as an inequalityfor c < cl. Using the version of A(c, v) in (45), notice A(cl, vl) = 0, so cl is a root of A(c; vl). If γ ≥ 1

2,

Lemma 13 (with π = 0) says that it can have at most one root, and it’s positive for small c, so A(c; vl) ≥ 0

for all c < cl. For γ < 12, it’s convex and can have at most two roots. Condition (18) guarantees that the

derivative is negative, so cl is the smaller root, and we also have A(c; vl) ≥ 0 for all c < cl. Notice that weonly need to check (18) if γ < 1

2.

Consider any incentive compatible contract C = (c, k) that delivers utility of at least u0 to the agent,with associated state variables x and c. Because v′(cl) = 0 we can use Ito’s lemma18 and the HJB equationto obtain

e−rτn

v(cτn)xτn ≥ v(c0)x0 −ˆ τn

0

e−rt(ct − ktα

)xtdt

+

ˆ τn

0

e−rtv(ct)xt

(v′(ct)

v(ct)ctσ

ct + σxt

)dZt

for the localizing sequence of stopping times τnn∈N:

τn = inf

T ≥ 0 :

ˆ T

0

∣∣∣∣e−rtv(ct)xt

(v′(ct)

v(ct)ctσ

ct + σxt

)∣∣∣∣2 dt ≥ n

The stopped stochastic integrals are therefore martingales, so take expectations under Q to obtain

EQ0[e−rτ

n

v(cτn)xτn]≥ v(c0)x0 − EQ0

[ˆ τn

0

e−rt (ct − ktα) dt

](51)

Now we would like to take the limit n→∞, but we need to use the dominated convergence theorem. First,∣∣∣∣∣ˆ τn

0

e−rt(ct − ktα)dt

∣∣∣∣∣ ≤ˆ ∞

0

e−rt |ct − ktα| dt ≤ˆ ∞

0

e−rt(|ct + ktα|)dt

18Notice v′′ is discontinuous at cl, but this doesn’t change Ito’s formula.

49

which is integrable because the contract is admissible. Second, for an admissible contract

0 ≤ limn→∞

EQ0[e−rτ

n

v(cτn)xτn]≤ limn→∞

EQ0[e−rτ

n

vhxτn]

= 0

To see why the last equality holds, notice that since vhx is the cheapest way of delivering utility to theagent without capital, the cost of consumption on the contract is

∞ > EQ[ˆ ∞

0

e−rtctdt

]≥ EQ

[ˆ τn

0

e−rtctdt+ e−rτn

vhxτn

]

Taking the limit n→∞ and using the monotone convergence theorem, we obtain 0 ≤ limn→∞ EQ[e−rτ

n

vhxτn]≤

0.Upon taking the limit n→∞ in (51), we obtain τn →∞ a.s. and therefore

EQ0[ˆ ∞

0

e−rt (ct − ktα) dt

]≥ v(c0)x0

Using v(cl) ≤ v(c0) and x0 ≥ ((1− γ)u0)1

1−γ we obtain the first result.For the second part, first let’s show that C∗ is incentive compatible. We already know that for the HJB

to have a solution with well defined policy functions it must be the case that B ≥ 0 and C > 0, defined in(48) and (49), and therefore σc

∗≤ 0 and σx

∗≥ 0. If c∗t ∈ [cl, ch], because σc

∗is bounded, so is σx

∗and

therefore so is µx∗and µc

∗. We can then use Lemma 3 and the fact that h0 = 0 to show that

U c,a0 ≤ Uc∗,0

0 = u0

for any feasible strategy (c, a), and therefore C∗ is indeed incentive compatible. To show that the cost of thecontract is vlx∗0, we can use the HJB. If c∗t ∈ [cl, ch] always, where the HJB holds, then the same argumentas in the first part shows the cost is vlx∗0. Notice that with v′(cl) = 0 and v′′(cl) > 0 we get that as c cl,B → 0 and C → Cl > 0, so σc

∗→ 0, σx

∗→ α

σγ

cγl

vlφ, and A(cl; vl) = 0. From the first part we know v(c) is

weakly below the true cost function, which is also bounded above by vh, so there is a finite cost functionand vl is weakly below it, and therefore vl ≤ vp from equation (27). Lemma 15 shows that under theseconditions µc(cl) > 0 and therefore c∗t ∈ [cl, ch]. Notice that the candidate contract does indeed deliverutility u0 to the agent. To see this let U∗ = (x∗)1−γ

1−γ , so using the law of motion of x∗, (9), we get

U∗0 = E

[ˆ τn

0

e−ρtc1−γt

1− γ dt+ e−ρτn

U∗τn

]

with some sequence τn →∞ a.s. Use the monotone convergence theorem and notice that

limn→∞

E[e−ρτ

n

U∗τn]

= 0

because ρ − (1 − γ)(µx∗− γ

2(σx∗)2) = c1−γ ≥ min

c1−γl , c1−γh

> 0. We then get that Uc

∗,00 = U∗0 = u0.

This completes the proof.

50

Lemma 4

We know from Lemma 15 that c∗t ∈ [cl, ch] and recall that cl > 0. Then an upper bounded µx∗< r implies

a bounded 0 ≤ σx∗≤ σX . Then

EQ[ˆ ∞

0

e−rt(|c∗t |+ |k∗tα|)dt]≤ 2 max

ch,

σX cγh

φσα

EQ[ˆ ∞

0

e−rtx∗t dt

]<∞

where the last inequality follows from µx∗< r. Let U∗ = (x∗)1−γ

1−γ , so using the law of motion of x∗, (9), weget

U∗0 = E

[ˆ τn

0

e−ρtc1−γt

1− γ dt+ e−ρτn

U∗τn

]with τn →∞ a.s. Use the monotone convergence theorem and notice that

limn→∞

E[e−ρτ

n

U∗τn]

= 0

because ρ − (1 − γ)(µx∗− γ

2(σx∗)2) = c1−γ ≥ min

c1−γl , c1−γh

> 0. We then get that Uc

∗,00 = U∗0 = u0.

We conclude that the contract is indeed admissible.

Theorem 3

We will focus on the case with dLt = 0, but the proof can be extended quite naturally. Let F (h, c) =(1 + hc−γ

)1−γwhere h = h/x, and let Ft = F (ht, ct), and likewise for derivatives, e.g. Fc,t = ∂cF (ht, ct).

Also, define ˆc = c/x. Write

e−ρt(U c,at − F (ht, ct)U

c,0t

)= Eat

[ˆ τn

t

e−ρuc1−γu

1− γ du+

ˆ τn

t

d(e−ρuFuU

c,0u

)+e−ρτ

n(U c,aτn − FτnU

c,0τn

)]for a localizing sequence τnn∈N with τn →∞ a.s. We will show that the rhs is non-positive. First writethe integral part

Eat

[ˆ τn

t

e−ρuc1−γu

1− γ du+

ˆ τn

t

d(e−ρuFuU

c,0u

)]= Eat

[ˆ τn

t

e−ρu(1− γ)Uc,0u Yudu

](52)

with

Yt =ˆc1−γt

1− γ +−ρFt + ρFt − c1−γt Ft

1− γ

+Fc,t

(1− γ)ct

(r − ργ

+1 + γ

2(σxt + σct )

2 − ρ− c1−γ

1− γ − γ

2(σxt )2 − σxt σct

)+

Fc,t(1− γ)

(1− γ)σxt σct ct

+Fh,t

(1− γ)ht

(r +

ct − ˆct

ht− ρ− c1−γt

1− γ − γ

2(σxt )2 + (σxt )2

)−

Fh,t(1− γ)

(1− γ)(σxt )2ht

51

+(σct )

2c2tFcc,t − 2σctσxt cthtFch,t + (σxt )2h2

tFhh,t2(1− γ)

+1

(1− γ)

(−(1− γ)σxt Ft − Fc,tσct ct + (σxt ht + ktφσ)Fh,t

)︸ ︷︷ ︸

At

atσ

whereFc,t = γ(γ − 1)htc

−γ−1t

(1 + htc

−γt

)−γFh,t = (1− γ)c−γt

(1 + htc

−γt

)−γFcc,t = −γ(γ − 1)htc

−2−γ(

1 + htc−γt

)−γ− γ2(γ − 1)htc

−γ−2t

(1 + htc

−γt

)−γ−1 (1− ht(γ − 1)c−γt

)Fch = γ(γ − 1)c−γ−1

t

(1 + htc

−γt

)−γ− γ2(γ − 1)htc

−2γ−1t

(1 + htc

−γt

)−γ−1

Fhh,t = −γc−2γt (1− γ)

(1 + htc

−γt

)−γ−1

We know e−rt(1 − γ)Uc,0t > 0, and we will show that Yt ≤ 0. To do this, we will split the expression intothree parts, Yt = At +Bt + Ct, and show each one is non-positive. First take the terms multiplying st

At =1

(1− γ)

(−(1− γ)σxt

(1 + htc

−γt

)1−γ

−γ(γ − 1)htc−γ−1t

(1 + htc

−γt

)−γσct ct + (σxt ht + ktφσ)(1− γ)c−γt

(1 + htc

−γt

)−γ)At =

(1 + htc

−γt

)−γ (−σxt

(1 + htc

−γt

)+ γhtc

−γ−1t σct ct + (σxt ht + ktφσ)c−γt

)At ≤

(1 + htc

−γt

)−γγhtc

−γt σct ≤ 0

where we have used σx ≥ ktφσc−γ in the first inequality, and ht ≥ 0 and σct ≤ 0 in the second. Here is theonly place where σct ≤ 0 is used.

Second, all the remaining terms that have σxt or σct are

Bt =Fc,t

(1− γ)ct

(1 + γ

2(σxt + σct )

2 − γ

2(σxt )2 − σxt σct

)+

Fc,t(1− γ)

(1− γ)σxt σct ct

+Fh,t

(1− γ)ht(−γ

2(σxt )2 + (σxt )2

)−

Fh,t(1− γ)

(1− γ)(σxt )2ht +(σct )

2c2tFcc,t − 2σctσxt cthtFch,t + (σxt )2h2

tFhh,t2(1− γ)

Bt =Fc,t

(1− γ)ct

(1 + γ

2(σxt + σct )

2 − γ

2(σxt )2 − γσxt σct

)+

Fh,t(1− γ)

ht(−γ

2(σxt )2 + γ(σxt )2

)+

(σct )2c2tFcc,t − 2σctσ

xt cthtFch,t + (σxt )2h2

tFhh,t2(1− γ)

Bt =(σxt )2

2

(Fc,t

1− γ ct +Fh,t

1− γ htγ +Fhh,t1− γ h

2t

)+

(σct )2

2

(Fc,t

1− γ (1 + γ)ct +Fcc,t1− γ c

2t

)

52

−σxt σct(− Fc,t

1− γ ct +Fch,t1− γ ctht

)which plugging in the F ’s gets us

Bt =(σxt )2

2

γ(γ − 1)htc−γ−1t

(1 + htc

−γt

)−γ1− γ ct

+(1− γ)c−γt

(1 + htc

−γt

)−γ1− γ htγ +

−γc−2γt (1− γ)

(1 + htc

−γt

)−γ−1

1− γ h2t

+(σct )

2

2

γ(γ − 1)htc−γ−1t

(1 + htc

−γt

)−γ1− γ (1 + γ)ct+

−γ(γ − 1)htc−2−γ

(1 + htc

−γt

)−γ− γ2(γ − 1)htc

−γ−2t

(1 + htc

−γt

)−γ−1 (1− ht(γ − 1)c−γt

)1− γ c2t

−σxt σct

−γ(γ − 1)htc−γ−1t

(1 + htc

−γt

)−γ1− γ ct

+γ(γ − 1)c−γ−1

t

(1 + htc

−γt

)−γ− γ2(γ − 1)htc

−2γ−1t

(1 + htc

−γt

)−γ−1

1− γ ctht

which simplifies to:

Bt =(σxt )2

2

(−γhtc−γ−1

t

(1 + htc

−γt

)−γct + c−γt

(1 + htc

−γt

)−γhtγ − γc−2γ

t

(1 + htc

−γt

)−γ−1

h2t

)

+(σct )

2

2

(−γhtc−γ−1

t

(1 + htc

−γt

)−γ(1 + γ)ct + γhtc

−2−γ(

1 + htc−γt

)−γc2t

+γ2htc−γ−2t

(1 + htc

−γt

)−γ−1 (1− ht(γ − 1)c−γt

)c2t

)−σxt σct

(γhtc

−γ−1t

(1 + htc

−γt

)−γct − γc−γ−1

t

(1 + htc

−γt

)−γctht + γ2htc

−2γ−1t

(1 + htc

−γt

)−γ−1

ctht

)and

Bt = − (σxt )2

2

(1 + htc

−γt

)−γ−1

c−γ−1t γc1−γt h2

t

− (σct )2

2

(1 + htc

−γt

)−γ−1

c−γ−1t h2

tγ3c1−γ

−σxt σct(

1 + htc−γt

)−γ−1

c−γ−1t γ2h2

t c1−γt

Bt = − (σxt + σctγ)2

2γh2

t c1−γ

(1 + htc

−γt

)−γ−1

c−γ−1t ≤ 0

53

And finally all the remaining terms that don’t involve σxt or σct are

Ct =ˆc1−γt

1− γ −c1−γt Ft(1− γ)

+γ(γ − 1)htc

−γ−1t

(1 + htc

−γt

)−γ(1− γ)

ct

(r − ργ− ρ− c1−γt

1− γ

)

+(1− γ)c−γt

(1 + htc

−γt

)−γ(1− γ)

ht

(r +

ct − ˆct

ht− ρ− c1−γt

1− γ

)which is maximized for ˆct = ct + htc

1−γt . Plugging this in yields

Ct ≤c1−γt

(1 + htc

−γt

)1−γ

1− γ − c1−γt Ft(1− γ)

+γ(γ − 1)htc

−γ−1t

(1 + htc

−γt

)−γ(1− γ)

ct

(r − ργ− ρ− c1−γt

1− γ

)

+(1− γ)c−γt

(1 + htc

−γt

)−γ(1− γ)

ht

r +ct −

(1 + htc

−γt

)ct

ht− ρ− c1−γt

1− γ

Ct ≤ −γhtc−γt

(1 + htc

−γt

)−γ (r − ργ− ρ− c1−γt

1− γ

)

+c−γt

(1 + htc

−γt

)−γht

r +ct −

(1 + htc

−γt

)ct

ht− ρ− c1−γt

1− γ

Ct ≤

(1 + htc

−γt

)−γc−γt ht

ρ− r +

ρ− c1−γt

1− γ γ + r − c1−γt − ρ− c1−γt

1− γ

Ct ≤

(1 + htc

−γt

)−γc−γt ht

ρ−

(ρ− c1−γt

)− c1−γt

= 0

Since Au, Bu, Cu ≤ 0 we conclude that Yt ≤ 0.Now for the last term, since the agent’s strategy (c, a) is feasible, we have that

limn→∞

Eat[e−ρτ

n

U c,aτn]

= 0

For γ < 1 we use Fatou’s lemma to show

limn→∞

Eat[e−ρτ

n

FτnUc,0τn

]≥ 0

As a result, when we take n→∞ we get Ea[e−ρt

(U c,at − F (ht, ct)U

c,0t

)]≤ 0.

For γ > 1, if limn→∞ Eat[e−ρτ

n

FτnUc,0τn

]= 0 for any feasible strategy (c, a), then we are done. To show

this, let Nt = xt + htc−γt , so that FtUc,0t =

N1−γt

1−γ . The law of motion of N satisfies

dNt ≤ (λ1Nt − λ2ct)dt+ σNt Nt dZat

where λ2 = c−γh > 0, and λ1 = µx + r+ c1−γ + γ|µc|+ 12γ(1 + γ)(σc)2. Here’s where we use the assumption

that µx, µc, and σc are bounded and c ≤ ch is uniformly bounded away from 0; µx, µc, c, and σc areappropriate bounds on each process. Notice that stealing only reduces the drift of Nt, since the change in

54

the drift of x and h cancel out, and it increases the drift of c. Since Nt > 0 always, Lemma 14 and it’scorollary ensure the desired limit.

Lemma 5

Start with the optimal contract C = (c, k), with associated processes x, c, and k. Define nt = vtxt. Wehave from Ito’s lemma the geometric volatility

σnt = σxt +v′tvtctσ

ct = c−γt ktφσ +

v′

vtctσ

ct

=

(c−γt φ+

v′tvtct(σkt)

−1σct

)ktσ

Use kt =σxt c

γt

φσand multiply and divide by vt to obtain:

σnt =(vtc−γt φ+ v′tc

1−γt (σxt )−1σctφ

) ktntσ

Using the definition of φt in (23) and of of λt in (24), we obtain

σnt = φtktntσ = λtσ

The net worth nt is the principal’s continuation cost, so it satisfies the HJB equation (13). This immediatelyyields geometric drift

µnt = r +ktntα− ct

nt

using the definition of θt in (25) and λt in (24) we obtain the budget constraint (22). Since this is a linearSDE with locally bounded stochastic coefficients, if we fix the capital structure S = (λ, φ, θ), it has a uniquesolution nt = xtvt . We get ct = θtnt and kt = λt/φt × nt, so S implements the optimal contract C.

Lemma 6

Lemma 18 establishes v(c) < cγ for all c ∈ (cl, ch), and together with v′ ≥ 0, σx ≥ 0, and σc ≤ 0 fromTheorem 1, and the definition of φ in (23), we obtain φ(c) < φ.

Lemma 7

The optimal portfolio plan generates a stationary contract with cp = θ1

1−γp . Lemma 17 ensures that

cp ∈ (c∗, ch] and therefore from Lemma 8 we know we have an incentive compatible contract.

Lemma 8

We use the HJB equation (13) with µc = σc = 0. Lemma 16 ensures that v(c) > 0 for all c ∈ (c∗, ch].The same argument as in Theorem 2 shows that vr(c) from (30) is the cost corresponding to the stationary

55

contract with c and σx given by (28), as long as the contract is indeed admissible and delivers utility u0

to the agent. We can check that µx < r for the stationary contract if and only if c > c∗, where c∗ is givenby (29). In this case, since µx < r arguing as in the proof of Lemma 4 we can show that the stationarycontract is admissible and delivers utility u0 to the agent if and only if c > c∗. Since the contract satisfies(9), (10), and (11) by construction, Theorem 3 then ensures that it is incentive compatible.

Lemma 9

The capital structure in (31) is a special case of the implementation in (23), (24), and (25). The sameargument as in Lemma 5 works here. We already know that the contract implemented with a constantequity stake φ, Cp is stationary. Lemma 17 ensures that cp ∈ (c∗, ch] and Lemma 8 that it is incentivecompatible.

The best stationary contract Cminr has cminr > cp > c∗ from part 1) of Lemma 17. So Cp is not the beststationary contract. Part 2) of Lemma 17 shows that v−1

r (c) > c−γ for all c ∈ (cp, ch), with equality at cpand ch. The expression for φr(c) in (31) then shows that φr(c) < φ in this region.

Lemma 10

For any given c ∈ [cl, ch], we have v(c) ≤ vr(c), because stationary contracts are incentive compatible andalways available as continuation contracts. Furthermore, v′(c) ≥ 0 and σc(c) ≤ 0. Using the definition ofφ(c) in (23) and φr(c) in (31), we obtain φ(c) ≤ φr(c).

Lemma 11

The same argument as in Lemma 5 shows that the capital structure in (35) implements the optimal contractwithout hidden savings. From the FOC for c (34) we obtain vn < cγn, and therefore φn < φ. Finally, theFOC for k in the HJB can be written

σxn =1

γ

α

φσ

cγ

vn

=⇒ σxn =α

γφnσ=⇒ λn =

α

γφnσ2

which is precisely the leverage the agent would choose on his own. This completes the proof.

Lemma 12

Proof. If ever vt > inf v(ω, s), then renegotiating at that point is better than never renegotiating andobtaining v0. In the other direction, if v is constant, any stopping time τ yields the same value to theprincipal, so τ =∞ is an optimal choice.

56

Theorem 5

Proof. Since the optimal stationary contract is incentive compatible and has a constant v, we only needto show that any incentive compatible contract with constant v has v ≥ vminr . This is clearly true for allstationary contracts as defined in Lemma 8, or Lemma 25 with aggregate risk.

There could also be stationary contracts with a constant c but dLt > 0. For these contracts the driftµc < 0 in the absence of dLt. Consider the optimization problem

0 = minσx

c− rv − σxcγ αφσ

+ v

(ρ− c1−γ

1− γ +γ

2(σx)2 − γ

2

(π

γ

)2)

st :r − ργ− ρ− c1−γ

1− γ +1

2(σx)2 +

1

2

(π

γ

)2

≤ 0

If the constraint is binding, we get the stationary contracts with dLt = 0, so v = vr. We want to showthat it must be binding. Towards contradiction, if the constraint is not binding we have σx = α

σγ

cγl

vlφand

therefore we have A(c, v) = 0, where A is defined as in Lemma 13. If v ≤ vminr then v ≤ vp, becausethe portfolio plan with φ = φ is always one of the incentive compatible stationary contracts (cp ≤ ch for

any valid hidden investment). Then Lemma 30 ensures that r−ργ− ρ−c1−γ

1−γ + 12(σx)2 + 1

2

(πγ

)2

> 0, which

violates the constraint. This means that v ≥ vminr .Finally, if we have a non-stationary contract with a constant v < vminr , the domain of c must have an

upper bound c∗ ≤ ch because otherwise they would have a lower cost than the optimal contract near ch,and this cannot be for an IC contract. For the upper bound c∗ we must have σc = 0 and µc ≤ 0. But thisis the same situation with stationary contracts with dLt > 0, and we know their cost is above vminr .

Intermediate lemmas

Lemma 13. Define the function

A(c, v) ≡ c− rv − 1

2

(cγαφσ

)2vγ

+ v

(ρ− c1−γ

1− γ− 1

2

π2

γ

)

For any v ∈ (0, (c∗)γ), we have A(c; v) > 0 for c near 0, where c∗ =(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ .

In addition, if γ ≥ 12 then A(c; v) has at most one root in [0, c∗]. If instead γ < 1

2 , A(c; v)

is convex and has at most two roots.

Proof. First, for γ < 1 limc→0 A(c; v) = v γ1−γ

(ρ−r(1−γ)

γ− 1−γ

2(πγ

)2)> 0. For γ > 1, limc→0 A(c; v) =∞.

For γ ≥ 1/2, to show that A(c; v) has at most one root in [0, c∗] for any v ∈ (0, vh), we will show thatA′c(c; v) = 0 =⇒ A(c; v) > 0 for all c < c∗. Compute the derivative (dropping the arguments to avoid

57

clutter)

A′c = 1− vc−γ − c2γ−1

(α

φσ

)21

v

So

A′c = 0 =⇒ c− vc1−γ = c2γ(α

φσ

)21

v

Plug this into the formula for A to get

A = c− rv + vρ− c1−γ

1− γ − c2γ

2vγ

(α

φσ

)2

− v

2

π2

γ

A = c− rv + vρ− c1−γ

1− γ − 1

2γ

(c− vc1−γ

)− v

2

π2

γ

=2γ − 1

2γc+

1− 3γ

2γvc1−γ

1− γ + vρ− r(1− γ)

1− γ − v

2

π2

γ≡ B(c, v)

B(c, v) is convex in c because 1− 3γ < 0 for γ ≥ 12, so it’s minimized in c when B′c = 0:

2γ − 1

3γ − 1= vc−γ (53)

and it is strictly decreasing before this point. Now we have two possible cases:CASE 1: The minimum of B is achieved for c ≥ c∗, so in the relevant range, it is minimized at ch. So

let’s plug in c∗ into B(c, v):

2γB(c∗, v) = (2γ − 1) c∗ +v

1− γ((ρ− r(1− γ))2γ + (1− 3γ)(c∗)1−γ)− vπ2

= (2γ − 1) c∗ +v

1− γ(ρ− r(1− γ))

γ

(2γ2 + (1− 3γ)

)− v

1− γ (1− 3γ)1

2(1− γ)

(πγ

)2 − vπ2

= (2γ − 1) c∗ + v

((ρ− r(1− γ))

γ

)(1− 2γ)− 1

2(1− γ)

(πγ

)2v

(1− 3γ

1− γ +2γ2

1− γ

)= (2γ − 1) c∗ + v

((ρ− r(1− γ))

γ

)(1− 2γ)− 1

2(1− γ)

(πγ

)2v(1− 2γ)

(2γ − 1)

(c∗ − v

(ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2)) ≥ 0

and the inequality is strict if v < (c∗)γ . So A(c, v) = B(c, v) > B(c∗, v) ≥ 0 for any c < c∗.CASE 2: If the minimum is achieved for cm ∈ [0, c∗) it must be that γ > 1/2. Then plugging in (53)

into B:

B(c, v) ≥ 2γ − 1

2γcm −

2γ − 1

2γ

cm1− γ + v

ρ− r(1− γ)

1− γ − v

2

π2

γ

=1− 2γ

2

cm1− γ + v

ρ− r(1− γ)

1− γ − v

2

π2

γ

=1− 2γ

2

cm1− γ +

2γ − 1

3γ − 1cγm

(ρ− r(1− γ)

1− γ − 1

2

π2

γ

)

58

and dividing throughout by 2γ − 1 > 0

= −1

2

cm1− γ +

cγm3γ − 1

(ρ− r(1− γ)

1− γ − 1

2

π2

γ

)

and multiplying by c−γm > 0 and using c1−γm1−γ < (c∗)1−γ

1−γ :

> −1

2

(c∗)1−γ

1− γ +1

3γ − 1

(ρ− r(1− γ)

1− γ − 1

2

π2

γ

)=

(ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2)(−1

2

1

1− γ +γ

(3γ − 1)(1− γ)

)(ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2) 1− 3γ + 2γ

(1− γ)(3γ − 1)2=

(ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2) 1

(3γ − 1)2> 0

So A(c; v) ≥ B(c, v) > 0 for all c ∈ [0, c∗].For the case with γ < 1

2, the second derivative of A is

A′′c = γvc−γ−1 − (2γ − 1)c2γ−2

(α

φσ

)21

v> 0

So A(c; v) is strictly convex and so can have at most two roots.

Lemma 14. Assume there are some constants λ1, λ2, λ3 ∈ R, and λ4 > 0 such that for anyfeasible strategy (c, a, z, z) there is a non-negative process N with

dNt ≤ ((λ1 + λ2σNt + λ3σ

Nt )Nt − λ4ct)dt+ σNt NtdZ

at + σNt NtdZt

for some processes σN and σN , which can depend on the strategy. Then there is a constantλ5 > 0 such that for any T > 0 and any feasible strategy (c, a, z, z)

Ea[ˆ T

0e−ρt

c1−γt

1− γdt

]≤ λ5

N1−γ0

1− γ

Proof. First define nt as the solution to the SDE

dnt =(

(λ1 + λ2σNt + λ3σ

Nt )nt − ct

)dt+ σNt ntdZ

at + σNt ntdZt

and n0 = N0λ4

. It follows that nt ≥ Ntλ4≥ 0. Now define ζ as

dζtζt

= −λ1dt− λ2dZat − λ3dZt, ζ0 = 1

59

and

nt =

ˆ t

0

ζscsds+ ζtnt

We can check that nt is a local martingale under P a. Since ζt > 0 and nt ≥ 0 it follows that

Ea[ˆ τm∧T

0

ζscsds

]≤ Ea

[ˆ τm∧T

0

ζscsds+ ζτm∧Tnτm∧T

]= n0

where τm reduces the stochastic integral and has limm→∞ τm = ∞ a.s. Taking m → ∞ and using the

monotone convergence theorem we obtain

Ea[ˆ T

0

ζscsds

]≤ n0

Now we want to maximize Ea[´ T

0e−ρt

c1−γt1−γ dt

]subject to this budget constraint. Notice that a appears

both in the budget constraint and objective function, but does not affect the law of motion of ζ under P a,so we can ignore it since we are choosing c. The candidate solution c has

e−ρtc−γt = ζtµ

where µ > 0 is the Lagrange multiplier and is chosen so that the budget constraint holds with equality. Forany c that satisfies the budget constraint we have

Ea[ˆ T

0

e−ρtc1−γt

1− γ dt]≤ Ea

[ˆ T

0

e−ρt(c1−γt

1− γ + c−γt (ct − ct))dt

]

= Ea[ˆ T

0

e−ρtc1−γt

1− γ dt]

+ µEa[ˆ T

0

ζt(ct − ct)dt]≤ Ea

[ˆ T

0

e−ρtc1−γt

1− γ dt]

Now since ct = (ζtµ)− 1γ e− ργt it follows a geometric Brownian motion so Ea

[´ T0e−ρt

c1−γt1−γ dt

]is finite.

Because of homothetic preferences, we know that Ea[´ T

0e−ρt

c1−γt1−γ dt

]= λ5

n1−γ0

1−γ = λ5N

1−γ0

1−γ for some

λ5 > 0.

Corollary. For γ > 1, limn→∞ Eat[e−ρτ

n N1−γτn

1−γ

]= 0 for any feasible strategy (c, a, z, z).

Proof. The continuation utility at any stopping time τn <∞ has

U c,aτn = Eaτn

[ˆ τn+T

τne−ρ(t−τ

n) c1−γt

1− γ dt+ e−ρ(T−τn)U c,aτn+T

]

≤ Eaτn

[ˆ τn+T

τne−ρ(t−τ

n) c1−γt

1− γ dt

]≤ λ5

N1−γτn

1− γ

60

So at t = 0 we get

U c,a0 = Ea[ˆ τn

0

e−ρtc1−γt

1− γ dt+ e−ρτn

U c,aτn

]≤ Ea

[ˆ τn

0

e−ρtc1−γt

1− γ dt+ e−ρτn

λ5N1−γτn

1− γ

]

Take limits n→∞ and use the monotone convergence theorem on the first term on the right hand side to

get 0 ≥ limn→∞ Eat[e−ρτ

n N1−γτn

1−γ

]≥ 0.

Lemma 15. Let cl ∈ (0, ch) and vl ≤ vp for any c ∈ (0, ch]. If σc = 0 and σx = ασγ

cγlvlφ

andA(cl, vl) = 0, then

µc =r − ργ−ρ− c1−γl

1− γ+

(σx)2

2> 0

Proof. Looking at (10), with σc = 0 we get for the drift

µc =r − ργ

+1

2(σx)2 − ρ− c1−γ

1− γ

So for any c, µc > 0 implies1

2(σx)2 >

ρ− rγ

+ρ− c1−γ

1− γSince we also want A(c; v) = 0, we get

0 = c− rv + v

(ρ− c1−γ

1− γ − γ

2(σx)2

)< c− vc1−γ ≡M

Notice that if v = cγ we have M = 0. If v > cγ we have M < 0 and if v < cγ we have M > 0. So forA(c; v) = 0 and µc > 0 we need v < cγ . In fact, if v = cγ and in addition

1

2

(α

φσγ

)2

=ρ− c1−γ

1− γ +ρ− rγ

(54)

then we have A = 0 and µc = 0. In this case, because we have µc = 0 we therefore have the value of astationary contract, i.e. v = vr(c). This point corresponds to the optimal portfolio with φ = φ, (cp, vp).We know from Lemma 17 that cp ∈ [c∗, ch]. By assumption, vl ≤ vp.

First we will show that µc ≥ 0, and then make the inequality strict. Towards contradiction, supposeµc < 0 at cl. Then it must be the case that vl > cγl because we have A(cl, vl) = 0. We will show thatA(cl, vl) > 0 and get a contradiction. First take the derivative of A:

A′c(cl, vl) = 1− vl

(c−γl + c2γ−1

l

(α

φσ

)21

v2l

)< 0

61

where the inequality holds for all c < v1γ

l . So A(cl, vl) > A(v1γ

l , vl). Letting cm = v1γ

l we get

A(cl, vl) > cm − rvl + vl

(ρ− c1−γm

1− γ − 1

2

(α

φσ

)21

γ

)

= cm − rvl + vl

(ρ− c1−γm

1− γ − γρ− c1−γp

1− γ − (ρ− r))

=⇒ A(cl, vl) > cm + vlγc1−γp − c1−γm

1− γ = cγmγc1−γp − c1−γm

1− γ ≥ 0

where the last equality uses vl = cγm and the last inequality uses cm = v1γ

l ≤ v1γp = cp. This is a contradiction,

and therefore it must be the case that µc ≥ 0 at cl.It’s clear from the previous argument that µc(cl) = 0 only if (cl, vl) = (cp, vp). We will show this

cannot be the case because α > 0. First, note that (cp, vp) is a tangency point where vr(c) touches thelocus vb(c) defined by A(c; vb(c)) = 0. If (cl, vl) = (cp, vp) then this must be the minimum point for vr(c),so the derivative of both vr(c) and vb(c) must be zero. This means that A′c(cl, vl) = 0. However,

1− vl

(c−γl + c2γ−1

l

(α

φσ

)21

v2l

)< 0

where the inequality follows from vl = vp = cγl (note that cl > 0 because as Lemma 13 shows A(c, vl) isstrictly positive for c near 0). This can’t be a minimum of vr(c). Therefore (cl, vl) 6= (cp, vp) and µc(cl) > 0.This completes the proof.

Lemma 16. The cost function of stationary contracts vr(c) defined by (30) is strictlypositive for all c ∈ (c∗, ch] if and only if

α < α ≡ φσγ√

2√1 + γ

√ρ− r(1− γ)

γ

Proof. We need to check the numerator in (30), since the denominator is positive for all c ≥ c∗:

c− α

φσcγ√

2

√√√√( ρ−r(1−γ)γ

)− c1−γ

1− γ

The rest of the proof consists of evaluating this expression at c = c∗ and showing it is non-positive iff thebound is violated. We get c times

1− α

φσ

√2√

1 + γ1

2γ

√(ρ− r(1− γ)

γ

)−1

So if α ≥ α the numerator is non-positive, and if α < α then it’s strictly positive. This completes the proof.

62

Lemma 17. Let

cp ≡

(ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)2

− 1− γ2

(π

γ

)2) 1

1−γ

vp ≡ cγp

be the c and v corresponding to the optimal portfolio plan with φ = φ. We have the followingproperties

1) c∗ < cp < cminr ≤ ch, for any valid hidden investment setting

2) cγ intersects vr(c) only at cp and c∗ =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ

in [0, c∗]. Further-

more, cγ ≥ vr(c) for all c ∈ [cp, c∗], and cγ ≤ vr(c) for all c ∈ [c∗, cp], with strict inequality

in the interior of each region.3) A(c, cγ) = 0 only at c = 0 and cp. Furthermore, A(c, cγ) ≤ 0 for all c ∈ [cp, ch] and

A(c, cγ) ≥ 0 for all c ∈ [0, cp], and ∂1A(c, cγ) < 0 for all c ∈ (0, ch].

Proof. First let’s show that cp ∈ (c∗, ch). Clearly, cp < ch for any type of valid hidden investment, becauseφ < 1. Now write cp

cp =

(ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)2

− 1− γ2

(π

γ

)2) 1

1−γ

>

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2) 1

1−γ (1− 1− γ

1 + γ

) 11−γ

where the inequality comes from α < α = φσγ√

2√1+γ

√ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2

. Notice 1 − 1−γ1+γ

= 2γ1+γ

and use

the definition of c∗:

c∗ =

(2γ

1 + γ

) 11−γ

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2) 1

1−γ

to conclude that c∗ < cp. The cost of this portfolio contract is vp = cγp .Now go to 2). We are looking for roots of vr(c) = cγ :

c− α

φσcγ√

2

√√√√√(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ = cγ(

2r − ρ− 1 + γ

1− γ ρ+ γ

(π

γ

)2

+c1−γ

1− γ (1 + γ)

)

Divide throughout by cγ > 0 and reorganize the right hand side

c1−γ

1− γ (1− γ)− α

φσ

√2

√√√√√(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ = −2γ

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)

1− γ +c1−γ

1− γ (1 + γ)

63

− α

φσ

√2

√√√√√(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ = −2γ

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)

1− γ +c1−γ

1− γ 2γ

α

φσ

√2

√√√√√(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ = 2γ

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ

If c =

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2) 1

1−γwe have a root. If not, then we can write

α

φσγ=√

2

√√√√√(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)− c1−γ

1− γ

c =

(ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)− 1− γ

2

(π

γ

)2) 1

1−γ

= cp <

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2) 1

1−γ

We know that at c = 0, cγ = 0, while vr(c) is always positive above c∗ and diverges to infinity as c c∗.So we know that cp is the first time they intersect and therefore cγ intersectsvr(c) from below. Since theywon’t intersect again until c∗, we get the other inequality.

Back to 1), consider the locus vb(c) defined by A(c, vb(c)) = 0. Since A(c, v) minimizes over σx, it isalways below vr(c). At (cp, vp) we have vb(c) = vr(c) by part 3) below, which means this is a tangencypoint of vb and vr. We can now show that A′c(cp, vp) < 0 and A′v(cp, vp) < 0, so that vb(cp) = vr(cp) < 0

which means that the Cp is not the optimal stationary contract, since cp < ch. Write

A′c(cp, vp) = 1− vp

(c−γp + c2γ−1

p

(α

φσ

)21

v2p

)= −cγ−1

p

(α

φσ

)< 0

A′v(cp, vp) =1

1− γ

(γ

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2)− c1−γp

)+

1

2

(cγpα

φσ

)2

v2pγ

=1

1− γ

(γ

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2)−

(ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)2

− 1− γ2

(π

γ

)2))

+γ

2

(α

φσγ

)2

=1

1− γ

((γ − 1)

(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2))

+1 + γ

2

(α

φσγ

)2

< 0

where the last inequalities follows from the bound on α < α ≡ φσγ√

2√1+γ

√ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2

. From Lemma

9 or Lemma 74, the minimum stationary contract can be implemented with a constant capital structureS = (λ, φ, θ, σn). We know it has a lower cost than vp, which is optimal subject to the equity share φ, sothis means the agent must keep a smaller equity share φminr < φ. Since (σx)minr = φkminr (cminr )−γσ, we get

φminr = vminr (cminr )−γφ < φ

which implies vminr × (cminr )−γ < 1 and therefore vminr < (cminr )γ . From 2) this means cminr > cp. We know

64

cminr ≤ ch from the definition of cminr .For 3), we are looking for roots of

c− rcγ − 1

2

(αcγ

φσ

)2

cγγ+ cγ

(ρ− c1−γ

1− γ − 1

2

π2

γ

)= 0

This works for c = 0. Otherwise, divide by cγ

c1−γ

1− γ (1− γ)− r − 1

2

(αφσ

)2

γ+ρ− c1−γ

1− γ − 1

2

π2

γ= 0

ρ− r(1− γ)

1− γ − γ

2

(α

φσγ

)2

− γ

2

(π

γ

)2

=c1−γ

1− γ γ

ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)2

− 1− γ2

(π

γ

)2

= c1−γ

c =

(ρ− r(1− γ)

γ− 1− γ

2

(α

φσγ

)2

− 1− γ2

(π

γ

)2) 1

1−γ

= cp

So we have only cp and c = 0 as roots. This argument also shows that A(c, cγ) ≤ 0 for c ∈ [cp, ch], andA(c, cγ) ≥ 0 for c ∈ [0, cp]. Also, evaluating the derivative ∂1A(c, cγ)

∂1A(c, cγ) = 1− cγ c−γ − c2γ−1

(α

φσ

)21

cγ

∂1A(c, cγ) = 1− 1− cγ−1

(α

φσ

)2

= −cγ−1

(α

φσ

)2

< 0

for all c ∈ (0, ch].

Lemma 18. For all c ∈ [cl, ch)

v(c) < cγ

If ch =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ

then v(ch) = cγh. If ch <(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ

then v(ch) < cγ.

Proof. We already know from the proof of Lemma 30 that at cl we have v(cl) < cγl . We also knowthat v(c) ≤ vr(c), so from Lemma 17 if ever v(c) = cγ for some c > cl, it must be either that c =

c∗ =

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2) 1

1−γ≥ ch ; or that c ≤ cp and therefore A(c, cγ) = A(c, v(c)) ≥ 0 and

∂1A(c, v(c)) < 0 (because c ≥ cl > 0). From Lemma 13 we know that A(c, v) is positive near 0 and eitherhas one root in c if γ ≥ 1/2, or is convex with at most two roots if c ≤ 1/2. This means that A(c−δ, cγ) > 0

for all δ ∈ (0, c].

65

Now we’ll use the same reasoning as in Theorem 1. We can pick an α′ < α so that vα′(cα′l ) = v(c), and

cα′l < c, because vα′(c) is decreasing in α′. However, Aα(c, cγ) is decreasing in α, so we get Aα′(cα

′l , c

γ) >

A(cα′l , c

γ) > 0 , which contradicts Aα′(cα′l , vα′(c

α′l )) = 0.

So we conclude that v(c) < cγ for all c ∈ [cl, ch). If ch = c∗ then we have v(ch) = cγh, but if ch < c∗

then we must also have v(ch) < cγh.

Corollary. For all c ∈ [cl, ch]

φ(c) = (vc−γ)︸ ︷︷ ︸∈(0,1)

φ+ v′ck−1σ−1σc︸ ︷︷ ︸≤0

≤ φ

66

Appendix B: aggregate risk and hidden investment

This appendix formally introduces aggregate risk and hidden investment into the baselineenvironment. The return on capital is now

dRt = (r + πσ + α− at) dt+ σ dZt + σ dZt

Where Z is an independent Brownian motion that represents aggregate risk, with marketprice π. Capital has a loading σ on aggregate risk, so the excess return on capital for theagent is α, as in the baseline. Let Q be the associated martingale measure.

The agent receives cumulative payments I from the principal and manages capital k forhim. Payments I can be any semimartingale (it could be decreasing if the agent must paythe principal). This nests the relevant case where the contract gives the agent only whathe will consume, i.e. dIt = ctdt. As in the baseline setting, the agent can steal from theprincipal at rate a ≥ 0 and decide when to consume c ≥ 0. He can invest his hidden savingsin the same way the principal would, not only in a risk-free asset, but also in aggregate riskZ. In addition, the agent may be able to invest his hidden savings in his private technology.His hidden savings follow the law of motion

dht = dIt + (rht + ztht(α+ πσ) + zthtπ − ct + φktat) dt+ ztht

(σdZt + σdZt

)+ zthtdZt

where z is the portfolio weight on his own private technology, and z the weight on aggregaterisk. While the agent can chose any position on aggregate risk, zt ∈ R, for his hidden privateinvestment we consider two cases: 1) no hidden private investment, zt ∈ H = 0, and 2)hidden private investment, zt ∈ H = R+.19

The cost to the principal is

J0 = EQ[ˆ ∞

0e−rt (dIt(R

a)− (α− at)kt(Ra)) dt]

where Ra the observed return that results from the agent’s stealing activity a.A contract C = (I, k, c, a, z, z) specifies the contractible payments I and capital k, and

recommends the hidden action (c, a, z, z), all contingent not only on returns R but also on19We can also study other cases where the agent may not be able to invest in aggregate risk, or only take

a positive position, which requires small modifications to the relevant incentive compatibility constraints.We focus on the economically most relevant case, where the agent can always invest his hidden savings inthe market in the same way the principal would.

67

the observable aggregate shock Z. After signing the contract the agent can choose a strategy(c, a, z, z) to maximize his utility (potentially different from the one recommended by theprincipal). Given contract C, a strategy is feasible if 1) there is a finite utility U c,a,z,z, and 2)hidden savings ht ≥ 0 always. Since the agent can secretly invest in his private technology,we also impose the No-Ponzi condition on him 3) EQ

[´∞0 e−rt (ct + αztht) dt

]<∞.

A contract C = (I, k, c, a, z, z) is admissible if 1) (c, a, z, z) is feasible given C, and 2)

EQ[ˆ ∞

0e−rt (dIt(R

a) + kt(Ra)αdt+ atkt(R

a)dt)

]<∞

An admissible contract C = (I, k, c, a, z, z) is incentive compatible if the agent’s optimalstrategy is (c, a, z, z) as recommended by the principal. An incentive compatible contractis optimal if it minimizes the principal’s cost

v0 = minCJ0(C)

st : U c,a,z,z0 ≥ u0

C ∈ IC

To incorporate aggregate risk into the setting, we need to slightly modify the parameterrestrictions. We assume throughout that

ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2

> 0

α < α ≡ φσγ√

2√1 + γ

√ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2

No stealing or hidden savings in the optimal contract

Just like in the baseline setting, it is without loss of generality to look only at contractswhich induce no stealing, no hidden savings, and no hidden investment.

Lemma 19. It is without loss of generality to look only at contracts that induce no stealinga = 0, no hidden savings, h = 0, and no hidden investment, z = z = 0.

Remark. This lemma is also valid for the baseline setting without aggregate risk or hiddeninvestment.

68

Proof. Imagine the principal is offering contract C = (I, k, c, a, z, z) with associated hidden savings h.Let kh = zh and kh = zh be the agent’s absolute hidden positions in his private technology and aggregaterisk respectively. We will show that we can offer a new contract C′ = (I ′, k′, dI ′, 0, 0, 0) under which it isoptimal for the agent to choose not to steal, no hidden savings, and no hidden investment, i.e. c′ = dI ′,a = z = z = 0. The new contract has I ′ =

´ t0ctdt and k′ = k(Ra) + kh.

If the agent now chooses c′ = dI ′, a = z = z = 0, he gets hidden savings h′ = 0 and consumption c, sohe gets the same utility as under the original contract and this strategy is therefore feasible under the newcontract. If instead he chooses a different feasible strategy (c′, a′, z′, z′), he gets the utility associated withc′. We will show that he could achieve this utility under the original contract by picking consumption c′,stealing dR − dRa(Ra

′), hidden investment in private technology kh(Ra

′) + (kh)′, and hidden investment

in aggregate risk kh(Ra′) + (kh)′. Since the strategy (c′, a′, z′, z′) is feasible under the new contract C′, and

(c, a, z, z) feasible under the old contract C, then in order to ensure the new strategy is feasible under theoriginal contract we only need to show that hidden savings remain non-negative always

h′t =

ˆ t

0

er(t−s)(dIt(R

a(Ra′))− c′tdt+ φkt(R

a(Ra′))(dRt − dRat (Ra

′))

+(kht (Ra′) + (kh)′t)dRt + (kht (Ra

′) + (kh)′t)(πdt+ dZt)

)To show this is always non-negative, we will show it’s greater or equal to the sum of two non-negative terms.First, the hidden savings under the original contract, following the original feasible strategy, had Ra

′been

the true return

At =

ˆ t

0

er(t−s)(dIt(R

a(Ra′))− ct(Ra

′)dt+ φkt(R

a(Ra′))(dRa

′t − dRat (Ra

′))

+kht (Ra′)dRa

′t + kht (Ra

′)(πdt+ dZt)

)≥ 0

Second, hidden savings under the new contract, following the feasible new strategy

Bt =

ˆ t

0

er(t−s)(ct(R

a′)dt− c′tdt+ φ(kt(Ra(Ra

′)) + kht (Ra

′))(dRt − dRa

′t )

+(kh)′tdRt + (kh)′t(πdt+ dZt))≥ 0

If φ = 1 then h′t = At + Bt ≥ 0. With φ < 1, we have h′t ≥ At + Bt ≥ 0, because dRt − dRa′t = a′dt ≥ 0

and kh(Ra′) ≥ 0. This means that c′ = c′, a = z = z = 0 is the agent’s optimal choice under the new

contract C′, since any other choice delivers an utility that he could have obtained - but chose not to - underthe original contract C.

We can now compute the principal’s cost under the new contract

J ′0 = EQ[ˆ τn

0

e−rt(ct − α(kt(R

a) + kht ))dt+ e−rτ

n

J ′τn

]=

EQ[ˆ τn

0

e−rt (dIt(Ra)− (α− at)kt(Ra)dt) + e−rτ

n

Jτn

]− EQ

[ˆ τn

0

e−rtatkt(Ra)(1− φ)dt

]

69

−EQ[ˆ τn

0

e−rt(dIt(R

a)− ctdt+ φkt(Ra)atdt+ kht αdt

)]+ EQ

[e−rτ

n (J ′τn − Jτn

)]On the rhs, the first term is the cost under the original contract; the second term the destruction producedby stealing under the original contract, which is non-negative; and the third term is EQ

[e−rτ

n

hτn]≥ 0,

where h is the agent’s hidden savings under the original contract. To see this, write

dht = htrdt+ dIt(Ra)− ctdt+ φkt(R

a)atdt+ kht ((α+ πσ)dt+ σdZt + σdZt) + kht (πdt+ dZt)

Sod(e−rtht) = e−rtdht − re−rthtdt

= e−rt(dIt(R

a)− ctdt+ φkt(Ra)atdt+ kht ((α+ πσ)dt+ σdZt + σdZt) + kht (πdt+ dZt)

)Now take expectations under Q, choosing the localizing process appropriately to get

EQ[ˆ τn

0

d(e−rtht)

]= EQ

[ˆ τn

0

e−rt(dIt(R

a)− ctdt+ φkt(Ra)atdt+ kht αdt

)]= EQ

[e−rτ

n

hτn − h0

]≥ 0

Given these inequalities, we can write:

J ′0 − J0 ≤ EQ[e−rτ

n (J ′τn − Jτn

)]Because the original contract was admissible, limn→∞ EQ

[e−rτ

n

Jτn]

= 0. Since in addition the agent’s

response was feasible, the new contract is also admissible, and we get limn→∞ EQ[e−rτ

n

J ′τn]

= 0 as well.This shows the new contract is admissible, and the cost for the principal is not greater than under the oldcontract. This completes the proof.

We can then simplify the contract to C = (c, k), and say an admissible contract isincentive compatible if the agent’s optimal strategy is (c, 0, 0, 0), or (c, 0) for short.

Incentive compatibility

Since the contract can depend on the history of aggregate shocks Z, so can his continuationutility U c,0 and his consumption c. However, because the agent is not responsible foraggregate shocks, incentive compatibility does not place any constraints on his exposureto aggregate risk. On the other hand, since the agent can invest his hidden savings, hisEuler equation needs to be modified appropriately. The discounted marginal utility of ahidden dollar must be a supermartingale under any feasible hidden investment strategy,since otherwise the agent could save a dollar instead of consuming it, invest it in aggregaterisk and his private technology, and consume it later when the marginal utility is expectedto be higher.

70

Lemma 20. Take the agent’s hidden investment possibility set H as given. If C = (c, k)

is an incentive compatible contract, the agent’s continuation utility U c,0 and consumption csatisfy the laws of motion

dU c,0t =

(ρU c,0t −

c1−γt

1− γ

)dt+ ∆tσdZt + σut dZt (55)

dctct

=

(r − ργ

+1 + γ

2(σct )

2 +1 + γ

2(σct )

2

)dt+ σctdZt + σctdZt + dLt (56)

for some ∆, σu, σc, σc, and a weakly increasing processes L, such that

∆t ≥ c−γt φkt (57)

z(α− σctσγ) ≤ 0 ∀z ∈ H (58)

σc =π

γ(59)

Proof. The proof of (55) and (57) are similar to Lemma 4 and 2, where the dZ term appears because thecontract can depend on the history of aggregate shocks. For (56), the proof is analogous to Lemma 2, butnow we need the discounted marginal utility

Yt = e´ t0 r−ρ+zs(α+πσ)+πszs− 1

2(zsσ)2− 1

2(zsσ+zs)2ds+

´ t0 (zsσ)dZs+

´ t0 (zsσ+zs)dZsc−γt (60)

to be a supermartingale for any investment strategy zt ∈ R and zt ∈ H. Using the Doob-Meyer decompo-sition, the Martingale Representation theorem, and Ito’s lemma, we can write

dctct

= µctdt+ σctdZt + σctdZt + dLt

Since the drift of expression (60) must be weakly negative, we get(r − ρ− γµct +

γ

2((1 + γ)σct )

2 +γ

2((1 + γ)σct )

2)dt (61)

+ (z(α+ πσ) + πz − γσct zσ − γσct (zσ + z)) dt− γdLt ≤ 0

Taking z = z = 0, which are always allowed, we obtain wlog the expression for µc in (56), and L weaklyincreasing. Once we plug this into (61), and using that zt can be both positive or negative, we get (59).Condition (58) is therefore necessary to ensure (61) holds.

The IC constraint (58) depends on whether the agent is allowed to have a hiddeninvestment in his own private technology. If hidden investment in the agent’s private

71

technology is not allowed,H = 0 so condition (58) drops out. If instead hidden investmentin the agent’s private technology is allowed, H = R+, so condition (58) reduces to σc ≥ α

σγ .

State Space

We can still use the the state variables x and c. Their laws of motion are

dxtxt

=

(ρ− c1−γ

1− γ+γ

2(σxt )2 +

γ

2(σxt )2

)dt+ σxt dZt + σxt dZt (62)

dctct

=(r − ρ

γ− ρ− c1−γt

1− γ+

(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2 (63)

+(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2)dt+ σct dZt + σct dZt + dLt

and the incentive compatibility constraints can be written

σxt = c−γt φktσ (64)

z(α− (σct + σxt )σγ) ≤ 0 ∀z ∈ H (65)

σct + σxt =π

γ(66)

As before, c has an upper bound ch, which must be modified to take into account that it isnot incentive compatible to give the agent a perfectly safe consumption stream.

Lemma 21. Take the agent’s hidden investment possibility set H as given. For any incen-tive compatible contract C, ct ∈ (0, ch] at all times, where ch is given by

ch ≡ maxσx≥0

(ρ− r(1− γ)

γ− 1− γ

2(σx)2 − 1− γ

2

(π

γ

)2) 1

1−γ

(67)

st : z(α− σxt σγ) ≤ 0 ∀z ∈ H

If ever ct = ch, then the continuation contract satisfies ct+s = ch and kt =σxcγhφσ for all

future times t+ s, and xt follows the law of motion (62), where σx is the optimizing choicein (67) and σx = π

γ . The cost of this continuation contract is vh.

72

Proof. We know from Lemma 3 that c has an upper bound for any incentive compatible contract. Thisrequires σc = σc = 0, µc ≤ 0, and dLt = 0 and that point. Using the law of motion of c we get

r − ργ− ρ− c1−γ

1− γ +(σx)2

2+

(σx)2

2≤ 0

where σx = πγ. If we choose σx to minimize the rhs of this expression, subject to the IC constraints imposed

by hidden investment, we can then solve for the largest c that can be attained by an incentive compatiblecontract. Since these choices are independent of c, this is equivalent to the maximization problem inexpression (67). It follows that ch is an absorbing boundary. Using the IC constraint (64) and the law ofmotion of x we obtain the desired result. This completes the proof.

The upper bound ch restricts the principal’s ability to promise safety in the future.Even if the agent cannot invest his hidden savings in his private technology, H = 0, hecan still invest in aggregate risk. In this case the maximizing choice is σx = 0 and we

get ch =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 11−γ

. Notice that if π = 0 this boils down to expression

(8) in the baseline setting without aggregate risk or hidden investment. If the agent canalso invest his hidden savings in his own private technology, H = R+, then the maximizing

choice is σx = ασγ , and ch =

(ρ−r(1−γ)

γ − 1−γ2

(ασγ

)2− 1−γ

2

(πγ

)2) 11−γ

is lower.

The HJB equation

The optimal contract can be characterized with the same HJB equation as in the casewithout hidden investment, appropriately extended to incorporate aggregate risk and thenew incentive compatibility constraints.

0 = minσx,σc,σx,σc

c− rv − σxcγ αφσ

+ v

(ρ− c1−γ

1− γ+γ

2(σx)2 +

γ

2(σx)2 − πσx

)(68)

+v′c

(r − ργ− ρ− c1−γ

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2 +

(σx)2

2

+(1 + γ)σxσc +1 + γ

2

(σc)2 − σcπ)+

v′′

2c2(

(σc)2 +(σc)2)

subject to σx ≥ 0 and (65) and (66).

73

Using (66) to eliminate σc, and taking FOC for σx, we obtain

σx =π

γσc = 0

This is the first best exposure to aggregate risk. The principal and the agent don’t haveany conflict about aggregate risk, and the principal cannot use it to relax the moral hazardproblem, so they implement the first best aggregate risk sharing.20

The FOC for σx and σc depend on whether the agent can invest his hidden savingsin his private technology. Without hidden investment, the FOCs are the same as in thebaseline, (16) and (17). With hidden investment, the IC constraint (65) could be bindingin some region of the state space. The shape of the contract, however, is the same as in thebaseline without hidden investment.

Theorem 6. Take the agent’s hidden investment possibility set H as given. The principal’scost function v(c) has a flat portion on [0, cl] and a strictly increasing portion on [cl, ch],for some cl ∈ (0, ch). The HJB equation (68) holds with equality for c ≥ cl. For c < cl, wehave v(c) = v(cl) ≡ vl and the HJB holds as an inequality

A(c, vl) ≡ minσx

c− σxcγ αφσ− rvl + vl

(ρ− c1−γ

1− γ+γ

2(σx)2 − γ

2

(πγ

)2) ∀c < cl. (69)

At cl the cost function v(c) satisfies v′(cl) = 0, v′′(cl) > 0, and A(cl, vl) = 0.The optimal contract has σxt = π

γ and σct = 0 always. It starts at c0 = cl, where σx0 ischosen without taking into account its effect on the agent’s precautionary saving motive, tomaximize

σxcγlα

φσ− v(cl)

γ

2(σx)2 (70)

At cl we have µc(cl) > 0 and σc(cl) = 0. For all t > 0, we have ct ≥ cl, σct ≤ 0 and σxt ≥ 0.

Proof. The proof is similar to Theorem 1, except we use the more general definition of A(c, v)

A(c, v) = c− rv − 1

2

(cγαφσ

)vγ

+ v

(ρ− c1−γ

1− γ − 1

2

π2

γ

)where A(c, v) = 0 is the HJB equation if v′ = v′′ = 0 and the IC constraint (65) is not binding. Notice wealready know from the FOCs that σx = π/γ and σc = 0.

20If the agent didn’t have access to hidden investment in aggregate risk, and the agent’s private technologyis exposed to aggregate risk σ 6= 0, then the principal could potentially use the agent’s exposure to aggregaterisk to relax the moral hazard problem.

74

Part 1 goes through without any change. In part 2, to show that the region where the HJB holdscannot have flat parts, we notice that with v′ = v′′ = 0, σc drops out of the HJB, and can be used to ensurethat the IC constraint (65) holds for any choice of σx. Therefore at a flat part A(c, v) = 0 should hold,which we know cannot be the case.

In part 3, the proof that v′(cl) = 0 goes through with small modifications, as well as v′′(cl + ε) ≥ 0.Indeed, if at the upper end of a flat region c2 we have a kink, with the right derivative v′(c2) > 0, we musthave σc(c2 + ε) → 0 as ε → 0, since otherwise we would cross into the flat region where the HJB doesn’thold. If the drift is strictly positive there, we can start at c2 − δ, with the same σx that we were using atc2. The IC constraint (65) does not depend on c so it is still satisfied. Since this extends the solution andv′(c2) > 0, we would get a lower cost, which contradicts the flat region immediately below c2. The prooffor the case with zero drift is unchanged.

With v′(c2) = 0, we must have v′′(c2 + ε) ≥ 0, or else v′(c + ε) < 0 for small ε. At c2 we must haveA(c2 − ε, v(c2)) ≥ 0 because otherwise we could do better by lingering below c2 before jumping up to c2.We must also have A(c2, v(c2)) ≤ 0, because at c2 we have v′ = 0, and v′′ ≥ 0, the RHS of the HJB will beat least as large as A(c2, v(c2)) (it could be strictly larger if the IC constraint (65) is binding). Since A iscontinuous in c, we must have A(c2, v(c2)) = 0. Since this is true in particular with c1 = 0 and c2 = cl, wehave proven v′(cl) = 0 and A(cl, vl) = 0. The rest of part 3 goes through without changes.

Part 4 goes through with natural modifications. In the case Ac(cl, vl) < 0, we consider setting σc = 0,and σx = α

σγ

cγlvl. This is consistent with IC constraint (65) because cγl ≥ vl from Lemma β. We then obtain

the first order ODE

A(c, vfo) + v′foc

r − ργ− ρ− c1−γ

1− γ +

(ασγ

cγ

vl

)2

2+

1

2

π2

γ2

︸ ︷︷ ︸

µc

= 0

and the rest of the proof goes through, with the only exception that we use the more general Lemma 30 toestablish µc > 0. Se we get v′′(cl) > 0.

Part 5 goes through and ignoring the IC constraint (65) we obtain σc(cl) = 0 and σx(cl) = ασγ

cγlvl,

which maximize (70). These satisfy the IC constraint (65) because vl ≤ cγl . Lemma 30 implies µc(cl) > 0.It only remains to show that σct ≤ 0 always. We already know this is the case when the IC constraint (65)is not binding. If it is binding, then σc = α

σγ− σx, and the FOC for σx yields:

σx =cγ α

φσ+ v′′c2 α

σγ

γ(v − v′c) + v′′c2≥ α

σγ

which implies σc ≤ 0. To see this inequality, multiply both sides by the denominator, which must bepositive (second order condition) to get

cγα

φσ+ v′′c2

α

σγ≥ α

σγ

(γ(v − v′c) + v′′c2

)cγ

α

φσ≥ α

σ(v − v′c)

75

1 ≥ φ

cγ(v − v′c) ≥ φ v

cγ

where the next to last inequality uses v′ ≥ 0, and the last inequality uses v ≤ cγ . We conclude that σct ≤ 0

always. The rest of the proof is unchanged.

We also have a verification theorem for the HJB equation.

Theorem 7. Take the agent’s hidden investment possibility set H as given. Let v(c) :

[cl, ch] → [vl, vh] be a strictly increasing C2 solution to the HJB equation (68) for somecl ∈ (0, ch), such that vl ≡ v(cl) ∈ (0, vh], v′(cl) = 0, v′′(cl) > 0 and v(ch) = vh. Assumealso that v(c) ≤ cγ for c ∈ [cl, ch], and, if γ < 1

2 that

1− vl(c−γl + c2γ−1l α2(φσ)−2v−2l

)≤ 0

Then,1) For any incentive compatible contract C = (c, k) that delivers at least utility u0 to the

agent, we have v(cl)((1− γ)u0)1

1−γ ≤ J0(C).2) Let C∗ be a contract generated by the policy functions of the HJB. Specifically, the

state variables x∗ and c∗ are solutions to (62) and (63) (with potential absorption at ch),with initial values x∗0 = ((1 − γ)u0)

11−γ and c∗0 = cl. If C∗ is admissible and σc∗ bounded,

then C∗ is an optimal contract, with cost J0(C∗) = v(cl)((1− γ)u0)1

1−γ .

Proof. The proof is very similar to Theorem 2, but we use the definition

A(c, v) = c− rv − 1

2

(cγαφσ

)vγ

+ v

(ρ− c1−γ

1− γ − 1

2

π2

γ

)and use it to show that the HJB holds as an inequality below cl. Notice that the optimal policies in theHJB imply σx = π/γ and σc = 0 throughout. With v′(cl) = 0 and v′′(cl) > 0, the IC constraint (65) isnot binding at cl, because v(cl) ≤ cγ , so we get σx(cl) = α

σγφ

cγlvl

and σc(cl) = 0 and therefore A(cl, vl) = 0.Lemma 30 then shows that µc(cl) > 0. We also want to show that σx ≥ 0 and σc ≤ 0 for all c ∈ [cl, ch].When know this is true when the IC constraint (65) is not binding. Using v(c) ≤ cγ we can show this is alsothe case if the constraint is binding, as in Part 5 of Theorem 6. We can then use Theorem 4 to establishthe global incentive compatibility of the candidate optimal contract. The rest of the proof is unchanged.

The following result is useful to verify admissibility.

Lemma 22. If the candidate contract C∗ constructed in Theorem 7 has µx∗ < r + π2/γ,then C∗ is admissible and delivers utility u0 to the agent.

76

Proof. We know that c∗t ∈ [cl, ch] and recall that cl > 0. Then an upper bounded µx∗< r + π2/γ implies

a bounded 0 ≤ σx∗≤ σX , and we also know that σx = π/γ. Then

EQ[ˆ ∞

0

e−rt(|c∗t |+ |k∗tα|)dt]≤ 2 max

ch,

σX cγh

φσα

EQ[ˆ ∞

0

e−rtx∗t dt

]<∞

where the last inequality follows from µx∗< r + π2/γ (notice the expectations is taken under Q). Let

U∗ = (x∗)1−γ

1−γ , so using the law of motion of x∗, (9), we get

U∗0 = E

[ˆ τn

0

e−ρtc1−γt

1− γ dt+ e−ρτn

U∗τn

]

with τn →∞ a.s. Use the monotone convergence theorem and notice that

limn→∞

E[e−ρτ

n

U∗τn]

= 0

because ρ − (1 − γ)(µx∗− γ

2(σx∗)2 − γ

2(πγ

)2) = c1−γ ≥ minc1−γl , c1−γh

> 0. We then get that Uc

∗,00 =

U∗0 = u0. We conclude that the contract is indeed admissible.

Verifying global incentive compatibility

We can extend Theorem 3 to verify global incentive compatibility.

Theorem 4. Let C = (c, k) be an admissible contract with associated processes x and csatisfying (38) and (39), and (40), (41), and in the case of hidden investment (42), withbounded µx, µc, σc, and σc, and with c uniformly bounded away for zero and bounded aboveby ch. Suppose that the contract satisfies the following property

σct ≤ 0

Then for any feasible strategy (c, a, z, z), with associated hidden savings h, we have thefollowing upper bound on the agent’s utility, after any history

U c,a,z,zt ≤(

1 +htxtc−γt

)1−γU c,0t

In particular, since h0 = 0, for any feasible strategy U c,a,z,z0 ≤ U c,00 , and the contract C istherefore incentive compatible.

77

Proof. As in the proof of Lemma 3, we get

e−ρt(U c,a,z,zt − F (ht, ct)U

c,0t

)= Eat

[ˆ τn

t

e−ρu(1− γ)Uc,0u Yud+ e−ρτn(U c,a,z,zτn − FτnUc,0τn

)]

where Yt = At + Bt + Bt + Ct must be modified to 1) incorporate hidden investment in term Bt, and 2)incorporate aggregate risk in term Bt. Terms At and Ct are not changed, and we already know they arenon-positive from the proof of Theorem 3. We only need to show that Bt + Bt ≤ 0 as well.

Start with Bt. Because the agent can now invest his hidden savings, we have the following expression:

Bt =Fc,t

(1− γ)ct

(1 + γ

2(σxt + σct )

2 − γ

2(σxt )2 − σxt σct

)+

Fc,t(1− γ)

(1− γ)σxt σct ct

+Fh,t

(1− γ)ht(ztα−

γ

2(σxt )2 − σxt (σht − σxt )

)+

Fh,t(1− γ)

(1− γ)σx(ztσ − σx)ht

+(σct )

2c2tFcc,t + 2σct (ztσ − σxt )cthtFch,t + (ztσ − σxt )2h2tFhh,t

2(1− γ)

Notice that when the agent invests in his private technology, we gets an expected return zt(α + πσ), butwe have only included the term ztα in the expression for Bt. We will include the premium for aggregaterisk πσ in the term Bt, which takes an analogous form:

Bt =Fc,t

(1− γ)ct

(1 + γ

2(σxt + σct )

2 − γ

2(σxt )2 − σxt σct

)+

Fc,t(1− γ)

(1− γ)σxt σct ct

+Fh,t

(1− γ)ht(

(ztσ + zt)π −γ

2(σxt )2 − σxt (σht − σxt )

)+

Fh,t(1− γ)

(1− γ)σxt (ztσ + zt − σxt )ht

+(σct )

2c2tFcc,t + 2σct (ztσ + zt − σxt )cthtFch,t + (ztσ + zt − σxt )2h2tFhh,t

2(1− γ)

The two terms Bt and Bt are very similar. The only difference is that instead of ztσ - the exposure toidiosyncratic risk Z induced by hidden investment in the private technology - we have ztσ+ zt - the exposureto aggregate risk from investment in both private technology and aggregate risk; and instead of ztσ ασ wehave (ztσ + zt)π. In addition, all the volatilities are with respect to aggregate risk Z instead of Z.

Now let’s re-write these terms in a more convenient form. Take Bt and simplify to obtain:

Bt =Fc,t

(1− γ)ct

(1 + γ

2(σxt + σct )

2 − γ

2(σxt )2 − γσxt σct

)+

Fh,t(1− γ)

ht(ztα+

γ

2(σxt )2 − γσxt ztσ

)

+(σct )

2c2tFcc,t + 2σct (ztσ − σxt )cthtFch,t + (ztσ − σxt )2h2tFhh,t

2(1− γ)

We know from the proof of Theorem 3 that with zt = 0 this will be

B0,t = − (σxt + σctγ)2

2γh2

t c1−γ

(1 + htc

−γt

)−γ−1

c−γ−1t ≤ 0

78

So let’s gather all the terms with zt into B1,t so that Bt = B0,t +B1,t

B1,t =Fh,t

(1− γ)ht (ztα− γσxt ztσ) +

2σct ztσcthtFch,t + ((ztσ)2 − 2ztσσx)h2

tFhh,t2(1− γ)

Now plug in the formula for Fh, Fch, and Fhh to obtain:

B1,t = c−γt

(1 + htc

−γt

)−γhtzt (α− γσσxt )

+1

2

(2σct ztσctht

(−γc−γ−1

t

(1 + htc

−γt

)−γ+ γ2htc

−2γ−1t

(1 + htc

−γt

)−γ−1))

−1

2((ztσ)2 − 2ztσσ

x)h2tγc−2γt

(1 + htc

−γt

)−γ−1

Reorganize to get:

B1,t = c−γt htzt(α− γσ(σxt + σct )

)(1 + htc

−γt

)−γ+

1

2

(2σct ztσcthtγ

2htc−2γ−1t − ((ztσ)2 − 2ztσσ

xt )h2

tγc−2γt

)(1 + htc

−γt

)−γ−1

B1,t = c−γt htzt(α− γσ(σxt + σct )

)(1 + htc

−γt

)−γ+

1

2

(2σct ztσγ − ((ztσ)2 − 2ztσσ

xt ))h2tγc−2γt

(1 + htc

−γt

)−γ−1

Adding the two terms back together we get

Bt = B0,t +B1,t = c−γt htzt(α− γσ(σxt + σct )

)(1 + htc

−γt

)−γ+h2

tγc−2γt

(1 + htc

−γt

)−γ−1(

1

2

(2σct ztσγ − ((ztσ)2 − 2ztσσ

xt ))− (σxt + σctγ)2

2

)= c−γt htztσ

(ασ− γ(σxt + σct )

)(1 + htc

−γt

)−γ− h2

tγc−2γt

(1 + htc

−γt

)−γ−1 1

2

(σxt + σctγ − ztσ

)2

The second term is always positive, so we only need to concern ourselves with the first one. Performing thesame algebraic steps on Bt, and taking advantage of the structural similarities, we obtain

Bt = c−γt ht(ztσ + zt)(π − γ(σxt + σct )

)(1 + htc

−γt

)−γ−h2

tγc−2γt

(1 + htc

−γt

)−γ−1 1

2

(σxt + σctγ − (ztσ + zt)

)2

where the second term is also positive. Finally, we can add the two first terms

c−γt ht(

1 + htc−γt

)−γ [zt(α− γσ(σxt + σct )

)+ (ztσ + zt)

(π − γ(σxt + σct )

)]= c−γt ht

(1 + htc

−γt

)−γ [zt(α+ πσ) + πzt − γ(σct + σxt )ztσ − γ(σct + σxt )(ztσ + zt)

]≤ 0

where the last inequality follows from the incentive compatibility conditions (65) and (66). Notice thisargument works for any contract where the discounted marginal utility of consumption is a supermartingaleunder any valid trading strategy. The rest of the proof dealing with the terminal term follows the same

79

steps as in Theorem 3.

Implementation

The implementation as a portfolio problem is still valid, but now the agent also had exposureσn to aggregate risk Z

dntnt

=(r + λtα/φt + σnt π − θt

)dt+ λtσ dZt + σnt dZ. (71)

The capital structure S = (λ, φ, θ, σn) with associated process for n that solves (71) imple-ments the contract C = (c, k) if ct = θt× nt and kt = λt(nt + et) = λt/φt× nt. Expressions(23), (24), and (25) are still valid, and we have in addition σn = π/γ.

Lemma 23. The optimal contract has a unique implementation in the family of standardcapital structures, with inside equity share φt = nt

nt+etgiven by (23), leverage λt = kt

nt+et

given by (24), a payout policy θt = ctnt

given by (25), exposure to aggregate risk σn = π/γ andassets given by (21). The marginal value of inside equity is larger than the marginal valueof hidden savings, v(c)−1 > c−γ, and the inside equity share φ(c) < φ for all c ∈ [cl, ch).

Proof. The proof is the same as in Lemma 5 and 6, but using the HJB equation (68), and with the additionthat σn = σx = π/γ.

Optimal portfolio with φ = φ. The optimal portfolio with φ = φ can be adapted tothe presence of aggregate risk.

λp =α

γφσ2θp =

ρ− r(1− γ)

γ−1− γ

2

(α

φσγ

)2

−1− γ2

(π

γ

)σn = π/γ and vp = θ

γ1−γp .

The resulting contract Cp = (cp, kp) corresponds to the black dot in Figures 5, 6. It has a

constant cp = θ1

1−γp and vp = θ

γ1−γp , and is therefore incentive compatible by Theorem 4.

Lemma 24. The contract implemented with a constant equity constraint Cp is an incentivecompatible contract.

80

Proof. The proof follows the same lines as that of Lemma 7, but uses Lemma 25 to establish the incentivecompatibility of the resulting contract.

Stationary contracts Set µc = σc = σc = 0. To do this we need to set

σx = σxr (c) ≡√

2

√ρ− c1−γ

1− γ+ρ− rγ− 1

2(π

γ)2 (72)

and σx = π/γ which implies σc = 0. We can find a stationary contract for any c ∈ (c∗, ch],where

c∗ =

(2γ

1 + γ

) 11−γ(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2) 1

1−γ

∈ (0, ch)

Lemma 25. For any c ∈ (c∗, ch], the corresponding stationary contract with σc = 0 andσx = σxr (c) given by (72) is globally incentive compatible and has cost vr(c)x0, where

vr(c) =c− α

φσ cγσxr (c)

2r − ρ− (1 + γ)ρ−c1−γ

1−γ + γ(π/γ)2. (73)

Since stationary contracts are incentive compatible, we have v(c) ≤ vr(c).Proof. The proof is analogous to that of Lemma 8, using the HJB (68). First, using α < α, we can verifythat 0 ≤ c∗ ≤ ch, regardless of whether the agent can invest in his hidden savings. Second, vr(c) > 0 forall c ∈ (c∗, ch) from Lemma 29. The same argument as in Theorem 2 shows that vr(c) from (73) is thecost corresponding to the stationary contract with c and σx given by (72), as long as the contract is indeedadmissible and delivers utility u0 to the agent. We can check that µx < r + π2

γfor the stationary contract

if and only if c > c∗. In this case, since µx < r + π2

γarguing as in the proof of Lemma 22 we can show

that the stationary contract is admissible and delivers utility u0 to the agent if and only if c > c∗. Sincethe contract satisfies (62), (63), and (64), and (66) by construction, we only need to check that (65) holdstoo. It’s east to see this is the case because c ≤ ch. Lemma 4 then ensures that it is incentive compatible.This completes the proof.

Lemma 26. Any stationary contract can be implemented with a constant capital structurewith

φr(c) = φvr(c)

cγ, λr(c) =

σxr (c)

σθr(c) =

c

vr(c)and σn(c) = π/γ (74)

81

The contract implemented with a constant equity constraint Cp is the stationary contractcorresponding to cp, but is not optimal. The best stationary contract Cminr is less risky forthe agent, i.e. we have c∗ < cp < cminr . For all c ∈ (cp, ch) the marginal value of equity islarger than the marginal value of hidden savings, v−1r (c) > c−γ, and the inside equity shareφr(c) < φ.

Proof. The capital structure in (74) is a special case of the implementation in (23), (24), (25), andσn = π/γ. The same argument as in Lemma 23 works here. We already know that the contract implementedwith a constant equity stake φ, Cp is stationary. Lemma 17 ensures that cp ∈ [c∗, ch] and Lemma 25 thatit is incentive compatible.

The best stationary contract Cminr has cminr > cp > c∗ from part 1) of Lemma 17. So Cp is not the beststationary contract. Part 2) of Lemma 17 shows that v−1

r (c) > c−γ for all c ∈ (cp, ch), with equality at cp

and(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2) 1

1−γ> ch. The expression for φr(c) in (74) then shows that φr(c) < φ in this

region.

Lemma 27. For any c ∈ [cl, ch], the optimal contract has a lower equity constraint thanthe corresponding stationary contract, i.e.

φ(c) ≡ φ(v(ct)

cγt+v′(ct)σ

ct ct

σxt cγt

)≤ φr(c). (75)

For any given c ∈ [cl, ch], we have v(c) ≤ vr(c), because stationary contracts are incentive compatible andalways available as continuation contracts. Furthermore, v′(c) ≥ 0 and σc(c) ≤ 0. Using the definition ofφ(c) in (23) and φr(c) in (74), we obtain φ(c) ≤ φr(c).

Lemma 28. If the agent has access to hidden investment, H = R+ and φ = 1, the optimalcontract is the optimal portfolio with an equity constraint φ = φ, Cp.

Proof. Cp is both admissible and incentive compatible by lemma 24. Since ch = cp =

(ρ−r(1−γ)

γ− 1−γ

2

(αγσ

)2

− 1−γ2

(πγ

)2),

we can use the same verification argument as in Theorem 7, using the flat value function v(c) = vp for allc ∈ (0, ch). For the argument to go through, it must be the case that the HJB holds as an inequality for allc < cp:

A(c, vp) = c− rvp −1

2

(cγαφσ

)vpγ

+ vp

(ρ− c1−γ

1− γ − 1

2

π2

γ

)> 0

This is true because vp = cγp , and from lemma 17 we know that ∂1A(cp, cγp) < 0. From Lemma 13 we know

that A(c, v) is positive near 0 and either has one root in c if γ ≥ 1/2, or is convex with at most two rootsif c ≤ 1/2. This means that A(c, cγp) > 0 for all c ∈ (0, cp).

82

Intermediate results

Lemma 29. The cost function of stationary contracts vr(c) defined by (73) is strictlypositive for all c ∈ (c∗, ch] if and only if α < α.

Proof. We need to check the numerator in (73), since the denominator is positive for all c ≥ c∗:

c

1− α

φσ

√2

√√√√√cγ−1

(ρ−r(1−γ)

γ− 1−γ

2

(πγ

)2)cγ−1 − 1

1− γ

The rest of the proof consists of evaluating this expression at c = c∗ and showing it is non-positive iff thebound is violated, since the expression is increasing in c. We get c times

1− α

φσ

√2√

1 + γ1

2γ

√√√√(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2)−1

So if α ≥ α the numerator is non-positive, and if α < α then it’s strictly positive. This completes the proof.

Lemma 30. Let cl ∈ (0, ch) and vl ≤ vp. If σc = σc = 0 , σx = ασγ

cγlvlφ

, and σx = π/γ, andA(cl, vl) = 0, where

A(c, v) ≡ c− rv − 1

2

(cγαφσ

)2vγ

+ v

(ρ− c1−γ

1− γ− 1

2

π2

γ

)then

µc =r − ργ−ρ− c1−γl

1− γ+

1

2(σx)2 +

1

2(π

γ)2 > 0

Proof. Looking at (63), with σc = σc = 0 we get for the drift

µc =r − ργ

+1

2(σx)2 +

1

2(σx)2 − ρ− c1−γ

1− γ

So µc > 0 implies1

2(σx)2 +

1

2(σx)2 >

ρ− rγ

+ρ− c1−γ

1− γSince we also want A(c; v) = 0, we get

0 = c− rv + v

(ρ− c1−γ

1− γ − γ

2(σx)2 − γ

2

(π

γ

)2)

< c− vc1−γ ≡M

83

Notice that if v = cγ we have M = 0. If v > cγ we have M < 0 and if v < cγ we have M > 0. So forA(c; v) = 0 and µc > 0 we need v < cγ . In fact, if v = cγ and in addition

1

2

(α

φσγ

)2

+1

2

(π

γ

)2

=ρ− c1−γ

1− γ +ρ− rγ

(76)

then we have A = 0 and µc = 0. In this case, because we have µc = 0 we therefore have the value of astationary contract, i.e. v = vr(c) given by (73). This point corresponds to the optimal portfolio withφ = φ, (cp, vp). We know from Lemma 17 that cp ∈ [c∗, ch]. By assumption, vl ≤ vp.

First we will show that µc ≥ 0, and then make the inequality strict. Towards contradiction, supposeµc < 0 at cl. Then it must be the case that vl > cγl because we have A(cl, vl) = 0. We will show thatA(cl, vl) > 0 and get a contradiction. First take the derivative of A:

A′c(cl, vl) = 1− vl

(c−γl + c2γ−1

l

(α

φσ

)21

v2l

)< 0

where the inequality holds for all c < v1γ

l . So A(cl, vl) > A(v1γ

l , vl). Letting cm = v1γ

l we get

A(cl, vl) > cm − rvl + vl

(ρ− c1−γm

1− γ − 1

2

(α

φσ

)21

γ− γ

2

(π

γ

)2)

= cm − rvl + vl

(ρ− c1−γm

1− γ − γρ− c1−γp

1− γ − (ρ− r))

=⇒ A(cl, vl) > cm + vlγc1−γp − c1−γm

1− γ = cγmγc1−γp − c1−γm

1− γ ≥ 0

where the last equality uses vl = cγm and the last inequality uses cm = v1γ

l ≤ v1γp = cp. This is a contradiction,

and therefore it must be the case that µc ≥ 0 at cl.It’s clear from the previous argument that µc(cl) = 0 only if (cl, vl) = (cp, vp). We will show this

cannot be the case because α > 0. First, note that (cp, vp) is a tangency point where vr(c) touches thelocus vb(c) defined by A(c; vb(c)) = 0. If (cl, vl) = (cp, vp) then this must be the minimum point for vr(c),so the derivative of both vr(c) and vb(c) must be zero. This means that A′c(cl, vl) = 0. However,

1− vl

(c−γl + c2γ−1

l

(α

φσ

)21

v2l

)< 0

where the inequality follows from vl = vp = cγl (note that cl > 0 because as Lemma 13 shows A(c, vl) isstrictly positive for c near 0). This can’t be a minimum of vr(c). Therefore (cl, vl) 6= (cp, vp) and µc(cl) > 0.This completes the proof.

84