Optimal Asset Management Contracts with Hidden

Savings

Sebastian Di Tella and Yuliy Sannikov∗

Stanford University

June 2020

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 dynamically distorts the agent’saccess to capital to manipulate his precautionary saving motive and reduce incentivesfor misbehavior. We provide a sufficient condition for the validity of the first-order ap-proach which holds in the optimal contract: global incentive compatibility is ensuredif the agent’s precautionary saving motive weakens after bad outcomes. We extend ourresults to incorporate market risk, hidden investment, and renegotiation.

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. To alignincentives, agents must retain a risky stake in their investment activity. However, hiddensavings pose a significant challenge. Agents can save to self insure against bad outcomes,undoing the incentive scheme. In this paper we characterize optimal dynamic asset man-agement contracts when the agent has access to hidden savings.

This paper has two main contributions. First, we build a model of delegated asset man-agement based on a classic environment of portfolio investment. An agent with constantrelative risk aversion (CRRA) continuously invests in risky assets, but can secretly divertreturns and has access to hidden savings. These assumptions lead to a tractable charac-terization of the optimal contract and the dynamic distortions induced by hidden savings.Second, we provide a general verification theorem for the validity of the first-order approach∗sditella@stanford.edu and sannikov@gmail.com. We would like to thank Alex Bloedel and Erik Madsen

for outstanding research assistance.

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that is valid for a wide class of contracts. Global incentive compatibility is ensured as longas the agent’s precautionary saving motive weakens after bad outcomes. This conditionholds in the candidate optimal contract, proving the validity of the first-order approach inour setting.

When the agent has access to hidden savings, his incentives to divert funds depend onhis precautionary motive. If the agent expects a risky consumption stream in the future,hidden savings that allow him to self insure are very valuable. This makes fund diversionto build up savings attractive. The optimal contract must therefore manage the agent’sprecautionary saving motive by committing to limit his future risk exposure, especially afterbad outcomes. This is accomplished by restricting the agent’s access to capital, since givingthe agent capital to manage requires exposing him to risk in order to align incentives. Bypromising a future amount of capital that is ex-post inefficiently low, the principal makesfund diversion less attractive today and reduces the cost of giving capital to the agent upfront. This dynamic tradeoff leads to history-dependent distortions in the agent’s accessto capital and a skewed compensation scheme. After good outcomes his access to capitalimproves, which allows him to keep growing rapidly. After bad outcomes he gets starvedfor capital and stagnates. The flip side is that the agent’s consumption is somewhat insuredon the downside, and he is punished instead with lower consumption growth.

One of the main methodological contributions of this paper is to provide an analyticalverification of the validity of the first-order approach. This is an old, hard problem in thetheory of dynamic contracts because the agent has such a rich action space.1 The first-orderapproach, introduced to the problem of hidden savings by Werning (2001), incorporatesthe agent’s Euler equation as a constraint in the contract design problem. This ensuresthe agent does not have a profitable deviation with savings alone. However, the agent mayfind it attractive to divert funds and save for when he is punished for poor performance inthe future. We prove the validity of the first order approach analytically by establishingan upper bound on the agent’s continuation utility off-path, for any deviation and afterany history. Global incentive compatibility is ensured if the agent’s precautionary savingmotive becomes weaker after bad outcomes. Intuitively, if the contract becomes less riskyafter bad outcomes, hidden savings become less valuable to the agent after diverting funds.This condition is guaranteed to hold in the candidate optimal contract, proving the validityof the first-order approach. But it’s worth stressing that this verification argument appliesbeyond the optimal contract, to any contract in which the precautionary saving motiveweakens after bad outcomes.

Since the principal uses dynamic distortions to the agent’s access to capital to provideincentives, it is natural to ask how hidden investment and renegotiation affect results.

1Kocherlakota (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 the first-order approach,the existing literature has often pursued the numerical option (e.g. see Farhi and Werning (2013)).

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Fortunately, both issues can be easily handled. If the agent is able to secretly invest inrisky capital, the principal is limited in his ability to restrict his access to capital. Theconstraint becomes binding only after sufficiently bad outcomes, when the principal wouldlike to reduce capital managed by the agent so much that the agent would find it attractiveto invest on his own (i.e. without sharing risk with the principal). We can add to our modelobservable market risk that commands a premium, and allow the agent to also invest hishidden savings in the market. Our agency model can therefore be fully embedded withinthe standard setting of continuous-time portfolio choice.

A second concern is that the optimal contract requires commitment. The principalrelaxes the agent’s precautionary saving motive by promising an inefficiently low amountof capital in the future. It is therefore tempting at that point to renegotiate and start over.To address this issue, we also characterize the optimal renegotiation-proof contract. Thisleads to a stationary contract that restricts the agent’s access to capital in a uniform way.

Finally, we map our optimal contract into a classic portfolio-consumption problem witha dynamic leverage constraint that limits the agent’s risk exposure. This constraint reducesthe agent’s precautionary motive and allows the agent’s incentive constraints to hold with alower retained equity stake, improving risk sharing. Hidden savings therefore link retainedequity constraints and leverage constraints, which are widely used in applied macro-financework. The logic is completely different than that of models with limited commitment suchas Hart and Moore (1994) and Kiyotaki and Moore (1997). The leverage constraint existsnot because the agent can walk away, but because it limits risk and therefore helps relaxthe equity constraint.

Literature Review. This paper fits within the literature on dynamic agency problems.There is extensive literature that uses recursive methods to characterize optimal contracts,including Spear and Srivastava (1987), Phelan and Townsend (1991), Sannikov (2008), He(2011), Biais et al. (2007), and Clementi and Hopenhayn (2006). Our paper builds uponthese standard recursive techniques, but adds the problem of persistent private information.In our case, about the agent’s hidden savings.

Our environment is based on the classic portfolio-investment problem, widely used inmacroeconomic and financial applications. Two salient features of this environment areconcave preferences (CRRA in our case), and endogenous capital under management. Herewe would like to explain the role of each feature and put them in context within theliterature, keeping in mind that the focus of our paper is the role of hidden savings.

First, concave preferences create a role for hidden savings. The agency problem westudy is one of cash flow diversion, as in DeMarzo and Fishman (2007), DeMarzo andSannikov (2006) and DeMarzo et al. (2012), but unlike their models we have CRRA ratherthan risk neutral preferences. With risk-neutral preferences, the optimal contracts with

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and without hidden savings are the same. Once concave preferences are introduced, hiddensavings become binding. Rogerson (1985) shows that the inverse Euler equation, whichcharacterizes the optimal contract without hidden savings, implies that the agent wouldsave if he could. This opens the door to potential distortions to control the agent’s incentivesto save, but also presents the problem of double deviations. In some settings distortionsdo not arise, e.g. the CARA settings, such as He (2011), and Williams (2013), where theratio of the agent’s current utility to continuation utility is invariant to contract design.2

However, when distortions do arise, it is difficult to characterize the specific form they take,and the first-order approach may fail. We are able to provide a sharp characterization ofthe optimal contract and the distortions generated by the presence of hidden savings, andprovide a verification theorem for global incentive compatibility which is valid for a wideclass of contracts.

Second, the classic portfolio-investment problem where capital can be costlessly adjustedaffords us a degree of scale invariance. In our setting, after good performance, the agentdoes not need to be retired nor outgrow moral hazard as in Sannikov (2008) or Clementi andHopenhayn (2006) respectively. Likewise, since the project can be scaled down, neither willthe agent retire after sufficiently bad outcomes as in DeMarzo and Sannikov (2006). Rather,the optimal contract dynamically scales the size of the agent’s fund with performance, takinginto account his precautionary saving motive.

To understand the link between concave preferences and endogenous capital better, itis useful to contrast our paper to DeMarzo et al. (2012), which is closely connected to ours.In that paper agents are risk neutral, so hidden savings are not binding (the Euler andinverse Euler equations coincide). With concave preferences, hidden savings are binding.But concave preferences play another important role. Since the principal can give theagent an unboundedly large amount of capital to manage, if the agent is able to obtainan excess return over the market, he can obtain an unboundedly large utility if he is riskneutral. He effectively has access to an arbitrage opportunity. DeMarzo et al. (2012)introduce adjustment costs to capital to eliminate this issue while maintaining risk neutralpreferences. Our environment does not have adjustment costs to capital, but the agent’srisk aversion eliminates the arbitrage opportunity and makes the problem well defined.

Our model provides a unified account of equity and leverage constraints, which are twoof the most commonly used financial frictions in the macro-finance literature in the traditionof Bernanke and Gertler (1989) and Kiyotaki and Moore (1997).3 Di Tella (2016) adopts aversion of our setting without hidden savings to study optimal financial regulation policy

2Likewise, the dynamic incentive accounts of Edmans et al. (2011) exhibit no distortions either, as hiddenaction enters multiplicatively and project size is fixed.

3See 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(2017), Moll (2014).

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in a general equilibrium environment. Our paper is also related to models of incompleteidiosyncratic risk sharing, such as Aiyagari (1994) and Krusell and Smith (1998). Here thefocus is on risky capital income, as in Angeletos (2006) or Christiano et al. (2014), ratherthan risky labor income.

Access to capital provides the principal with an important incentive tool. He is able torelax the incentive constraints and improve risk sharing by committing to distort projectsize below optimum over time and after bad performance. This result stands in contrast toCole and Kocherlakota (2001), where project scale is fixed and the optimal contract is risk-free debt. We recover the result of Cole and Kocherlakota (2001) only in the special casewhen the agent can secretly invest on his own just as efficiently as through the principal,so the principal cannot control the scale of investment at all.

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 et al.(2017) and DeMarzo and Sannikov (2016) who use a recursive structure that includes theagent’s “information rent,” i.e. the derivative of the agent’s payoff with respect to privateinformation. In all cases, in designing the contract we have to evaluate the agent’s payoffoff-path, after actions which the agent is not supposed to take, in order to ensure that theagent’s incentives to refrain from those actions are designed properly. In our case, sincewe want to control the agent’s incentives to save secretly, the agent’s information rent ismarginal utility of consumption (of an extra unit of savings). Key common issues are (1)the way that information rents enter the incentive constraint, (2) distortions that arise fromthis interaction and (3) forces that affect the validity of the first-order approach. In ourcase, we verify the validity of the first-order approach by characterizing an analytic upperbound, related to the CRRA utility function, on the agent’s payoff after deviations. Thebound coincides with the agent’s utility on path (hence the first-order approach is valid),and its derivative with respect to hidden savings is the agent’s “information rent.” Farhi andWerning (2013) have verified the first-order approach numerically by computing the agent’svalue function after deviations explicitly, in the context of insurance with unobservableskill shocks. Other papers that study the problems of persistent private information viaa recursive structure that includes information rents include Garrett and Pavan (2015),Cisternas (2014), Kapička (2013), and Williams (2011).

2 The model

We consider a setting in which an agent manages risky capital and contracts with a set ofoutside investors to raise funds and share risk. We may think of these investors collectively

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as “the principal”, whose objective is to maximize the market value of their payoff, so wemay also refer to them as “the market”.4

The agent manages an amount of capital kt ≥ 0, determined by the contract, to obtaina risky return per dollar invested of

dRt = (r + α− at) dt+ σ dZt. (1)

In this expression, r > 0 is the risk-free rate, which is also the required market return oninvestment with idiosyncratic risk σ dZ, α > 0 is the excess return, and at ≥ 0 is a hiddenaction the agent takes to divert returns for private monetary benefit. The Brownian motionZ, defined on a complete probability space (Ω, P,F) with filtration F = Ft, representsagent-specific idiosyncratic risk. If the agent is a fund manager, Z represents the outcomeof his particular investment/trading activity. If he is an entrepreneur, Z represents theoutcome of his particular project.

The principal can commit to any fully history-dependent contract C = (c, k), whichspecifies payments to the agent ct > 0 and capital kt ≥ 0 as a function of the observedhistory of returns R up to time t. Payments ct may differ from the agent’s true consumptionct > 0, which is also a hidden action. The principal does not observe hidden actions at orct, and so the contract C depends only on observed returns R. After signing the contract C,the agent chooses a strategy (c, a), that specifies ct and at, also as functions of the historyof returns R.

Diversion of at ≥ 0 gives the agent a flow of funds φatkt. For each stolen dollar, theagent keeps only a fraction φ ∈ (0, 1). Given the agent’s payments ct, consumption ct anddiversion action at, his hidden savings ht evolve according to

dht = (rht + ct − ct + φktat) dt, (2)

and must remain non-negative, ht ≥ 0.5 The agent invests his hidden savings at the risk-free rate r. Later we will introduce hidden investment, and allow the agent to also investhis hidden savings in risky capital.

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

U c,a0 = E

[∫ ∞0

e−ρtc1−γt

1− γdt

], (3)

4Even if there isn’t a single principal, the agent himself would like to commit to a contract to then sellshares to dispersed outside investors.

5We do not allow negative hidden savings, ht ≥ 0. If the agent can secretly borrow up to some limit,ht ≥ h ∈ (−∞, 0], the principal can first ask the agent to borrow up to the limit and transfer the funds tothe principal to invest on his behalf, so we are back to no hidden borrowing. Also, at ≥ 0 means that theagent cannot secretly use his hidden savings to improve observed returns. Section 8 discusses the role ofthis assumption in detail.

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with ρ and γ strictly positive and γ 6= 1.6 Given contract C, we say a strategy (c, a) isfeasible if (1) utility U c,a0 is finite and (2) ht ≥ 0 always. Let S(C) be the set of feasiblestrategies (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. The principal can access a complete financial market withequivalent martingale measure Q where idiosyncratic risk is not priced, so Q = P . In theOnline Appendix we allow for aggregate risk with a market price, such that Q 6= P . Theprincipal tries to minimize the cost of delivering utility u0 to the agent

J0 = EQ[∫ ∞

0e−rt (ct − ktα) dt

]. (4)

A standard argument in this setting implies that the optimal contract must implementno stealing, i.e. a = 0. In addition, without loss of generality and for analytic convenience,we can restrict attention to contracts in which h = 0 and c = c, i.e. the principal saves forthe agent.7 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.

We say a contract C = (c, k) is admissible if (1) utility U c,00 is finite, and (2)8

EQ[∫ ∞

0e−rtctdt

]<∞, EQ

[∫ ∞0

e−rtktdt

]<∞. (5)

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,an incentive compatible contract is optimal if it minimizes the cost of delivering utility u0

to the agentv0 = min

(c,k)J0 (6)

st : U c,00 ≥ u0

(c, k) ∈ IC.

We need several parameter restrictions to make the problem well defined and avoid6As long as γ 6= 1, utility in (3) is always well defined, but could take values ∞ or −∞. We do not

explicitly analyze the log utility case with γ = 1, but it can be treated as the limit γ → 1.7Lemma O.1 in the Online Appendix establishes this in a more general setting with both aggregate risk

and hidden investment.8This assumption ensures the principal’s objective function is well defined.

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infinite profits/utility. We assume throughout that

ρ > r(1− γ). (7)

If this condition fails, which can happen only if γ < 1, the agent can obtain infinite utilitysimply by investing at the risk-free rate r > 0. Also, if α is too large, the principal’s profitcan be infinite. We assume the following necessary condition throughout

α ≤ α ≡ φσγ√

2

1 + γ

√ρ− r(1− γ)

γ. (8)

If this fails, the principal can get an infinite profit through simple stationary contractsdescribed in Section 3.6. The exact bound on α to guarantee that the profit from the optimaldynamic contract is finite can be found only numerically, but the following conditions aresufficient,9

α ≤ α ≡

φσγ√

12γ

√ρ−r(1−γ)

γ if γ ≥ 1/2,

φσγ√

2(1− γ)√

ρ−r(1−γ)γ if γ ≤ 1/2,

(9)

where α < α for all γ.

Remark. Notice that we build the model with the aim of tractability, and to highlightour main object of interest, hidden savings. CRRA preferences together with a scalableinvestment technology give us scale invariance. In particular, if contract (c, k) gives utilityu0 to the agent and has cost v0 to the principal, then the scaled version (λc, λk) hasutility λ1−γu0 and cost λv0. If the former contract is optimal for utility u0, then the latteris optimal for utility λ1−γu0. The scale invariance property reduces the problem by onedimension—the utility dimension—and allows us to highlight the dimension central to ourinterest—the dimension of the precautionary savings motive, as the analysis of the nextsection shows.

3 Solving the model

We solve the model as follows. We first derive necessary first-order incentive-compatibilityconditions for the agent’s fund diversion and savings choices, using two appropriate statevariables: the agent’s continuation utility and consumption level. We introduce a changeof variables that takes advantage of the homothetic structure of the problem, and provide asufficient condition for global incentive compatibility in terms of these transformed variables.It is a condition on how these variables depend on returns under the agent’s recommendedstrategy (c, 0), i.e. it is an on-path condition, but it allows us to bound the agent’s payoffsoff-path, after arbitrary deviations.

9See Lemma O.8 and Lemma O.5 in the Online Appendix for details.

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We then solve the principal’s relaxed problem, minimizing cost subject to only first-orderconditions, as a control problem, and characterize the solution with an HJB equation. Weshow that the solution to the relaxed problem satisfies the sufficient condition for globalincentive compatibility, so we obtain the optimal contract. More generally, the sufficientcondition identifies a whole class of globally incentive compatible contracts, and is usefulin a broader context.

3.1 Local incentive compatibility

We can express the local incentive compatibility conditions in terms of two state variables:the agent’s continuation utility and his consumption. Continuation utility is defined as

U c,0t = Et

[∫ ∞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

(10)

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,and therefore his continuation utility U c,0t , by ∆tat. For at = 0 to be optimal, we need

0 ∈ arg maxa≥0

(ct + φkta)1−γ

1− γ−∆ta (11)

everywhere. Taking the first-order condition yields

∆t ≥ c−γt φkt. (12)

We need to give the agent some “skin in game” by exposing him to risk. This is costlybecause the principal is risk-neutral with respect to Z so he would like to provide fullinsurance to the agent.

For hidden savings, the first-order condition to make sure the agent does not wantto save is the Euler equation: the condition that the agent’s discounted marginal utilitye(r−ρ)tc−γt is a supermartingale.10

10Notice also that if ct = 0 were allowed, then c−γt = ∞, hence this possibility can only arise at the

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The following lemma summarizes the necessary local incentive conditions, expressed interms of variables U c,0t and ct. We present sufficient incentive compatibility conditions insubsection 3.3.11

Lemma 2. If C = (c, k) is an incentive compatible contract, then (12) 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 (13)

for some stochastic process σc and a weakly increasing stochastic process L.

Equation (13) gives 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 expected consumption profile. The process L in equation (13) allows consumptionto grow at a higher rate than the one that makes e(r−ρ)tc−γt a martingale. This is okaybecause the agent cannot borrow against future consumption. However, we shall see thatin the optimal contract dL = 0, so the Euler equation holds with equality.

There is an important interaction between the precautionary motive and incentives todivert funds. Notice how the private benefit of fund diversion depends on the marginalutility of consumption, c−γt . The agent’s consumption, in turn, depends on future risk.Exposing the agent to risk today though ∆t helps provide incentives against diversion. Butexposing the agent to risk in his consumption in the future, σct+s, creates a precautionarymotive that reduces ct, which tightens (12) and makes fund diversion today more attractive.The intuition is that when faced with future risk, the agent places a large value on hiddensavings he can use to self-insure. Important dynamic properties of the optimal contract ariseout of the principal’s attempt to manipulate the agent’s precautionary motive in order torelax the IC constraints.

Without hidden savings, the principal could control the agent’s consumption withoutworrying about the agent’s precautionary motive. In this case, ct would be the principal’scontrol rather than a state. The principal would choose ct to minimize the cost of com-pensating the agent, also taking into account that at any time higher consumption relaxes

beginning of the contract until a stopping time, and only with zero capital according to (12). Thus, givingthe agent ct = 0 at the beginning is equivalent to delaying the contract, which can never be optimal. Hence,our assumption that ct > 0 at all times is without loss of generality.

11It can be shown that condition (12) is necessary and sufficient to deter any deviations (c, a) with justfund diversion, and condition (13) is necessary and sufficient to deter any deviation (c, 0) with hiddensavings. We could make these statements formal, but this still leaves us concerned about double deviations,which we can rule out with an additional sufficient condition.

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the constraint (12). The optimal consumption path without hidden savings would satisfythe inverse Euler equation, i.e. e(ρ−r)tcγt would be a martingale, and ct would have a lowerdrift than that required by condition (13).

3.2 A change of variables

It is convenient to work with the following transformation of variables that exploits thesetting’s homothetic structure. For any admissible contract we can compute

xt =(

(1− γ)U c,0t

) 11−γ

> 0,

ct =ctxt> 0.

Variable xt is just a monotone transformation of continuation utility, but it is measured inconsumption units (up to a constant). As a result, ct measures how front loaded the agent’sconsumption is. The state ct is related to the agent’s precautionary saving motive. If theagent faces risk looking forward, he will want to postpone consumption in an attempt toself insure, leading to lower ct.While xt can take any positive value, ct has an upper bound.

Lemma 3. For any incentive compatible contract C = (c, k), we have for all t

ct ≤ ch ≡(ρ− r(1− γ)

γ

) 11−γ

> 0. (14)

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 continuation contract has cost vhxt to the principal, where vh ≡ cγh.

Under sufficient technical conditions (e.g. if σc is bounded), we have

ct ≤ EPt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcu)2

)duds

]− 11−γ

, (15)

with equality if Lt = 0 always, where P is an equivalent measure such that Zt−∫ t

0 (1−γ)σcsds

is a P -martingale.

Remark. In the optimal contract Lt = 0 and (15) holds as an equality. This expres-sion captures exactly how future risk exposure translates to precautionary motive andto a level of ct below ch. In the special case where σct = σc is constant, we have ct =(ρ−r(1−γ)

γ − 1−γ2 (σc)2

) 11−γ which is decreasing in σc.

The upper bound ch corresponds to autarky, where the agent saves and consumes on hisown without managing any capital. It can also be interpreted as the point of terminationor the agent’s retirement. The corresponding contract minimizes the agent’s precautionary

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motive because it is fully safe. At the same time, this contract is very costly because itdoes not give any capital to the agent and so does not take advantage of the excess returnα > 0. A value ct < ch means the agent expects to manage capital and be exposed to riskin the future, and we may think of ct as indexing how risky the continuation contract is forthe agent.

Using Ito’s lemma we can obtain the laws of motion for xt and ct from (10) and (13).Using the normalization ∆tσ/U

c,0t = (1− γ)σxt , we obtain

dxtxt

=(ρ− c1−γ

t

1− γ+γ

2(σxt )2

)︸ ︷︷ ︸

µxt

dt+ σxt dZt (16)

and

dctct

=( c1−γ

t − c1−γh

1− γ+

(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2)

︸ ︷︷ ︸µct

dt+ σct dZt + dLt, dLt ≥ 0, (17)

with σct = σct − σxt . The constraint (12) can be rewritten as

σxt ≥ c−γt ktφσ, (18)

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 IC constraint (18) establishes a link betweenthe agent’s exposure to risk σxt and the amount of capital he manages kt, mediated by themarginal utility c−γt , which captures the precautionary savings motive.

In terms of variables xt, ct and kt, we call any admissible contract locally incentivecompatible if (17) and (18) hold and ct ≤ ch. Notice that equation (16) follows automaticallyfrom the definition of xt.

In the other direction, given a pair of processes xt > 0 and ct > 0 satisfying (16) and(17) with ct ≤ ch and σxt ≥ 0, we can build a locally incentive compatible contract (c, k),with ct = ctxt > 0 and kt = xtσ

xt cγt /(φσ) ≥ 0. Under technical conditions, the contract

(c, k) is admissible and delivers utility U c,0t =x1−γt

1−γ under good behavior. It is thereforelocally incentive compatible.

Lemma 4. Let x > 0 and c > 0 be stochastic processes satisfying (16) and (17) withbounded volatility σx ≥ 0, and with c bounded away from zero and above by ch.

Then the contract C = (c, k) with ct = ctxt > 0 and kt = xtσxt cγt /(φσ) ≥ 0 delivers

utility U c,0t =x1−γt

1−γ if the agent follows strategy (c, 0). The contract C is admissible, andtherefore locally incentive compatible, if and only if E[

∫∞0 e−rtxtdt] <∞.

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3.3 Sufficient conditions for global incentive compatibility

Every incentive compatible contract is locally incentive compatible. Here we provide suf-ficient conditions for incentive compatibility for any locally incentive compatible contract,that is, an admissible contract that satisfies the local IC constraints (17), (18), and ct ≤ ch.We shall see that these sufficient conditions are guaranteed to hold in the relaxed optimalcontract, which proves the validity of the first-order approach, but are more general thanthat and can be used to check incentive compatibility of many other contracts of interest.

While (17) and (18) 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

].12 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 1. Let C = (c, k) be a locally incentive compatible contract with c bounded awayfrom zero and bounded volatilities σx and σc. Suppose that the contract satisfies the followingproperty

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 istherefore incentive compatible.

Theorem 1 shows that condition (19) is sufficient for global incentive compatibility byproviding a closed-form explicit upper bound (20) on the agent’s off-equilibrium payoff forany savings level ht ≥ 0. According to (20), if the agent does not have any hidden savings,ht = 0, the most utility he could get is U c,0t , i.e. the utility level he obtains from “goodbehavior” (c, 0). Hence, the contract is incentive compatible. However, if the agent hadsomehow accumulated hidden savings in the past, he would want to deviate from (c, 0) in thefuture, at the very least to increase his consumption and attain a greater utility. Inequality

12In other words, even if e(r−ρ)tc−γt is a martingale under P , it might be a submartingale under P a.

13

(20) bounds the utility the agent can get, and the bound tightens as ct rises and the agent’sprecautionary saving motive decreases. The bound is consistent with the intuition that thevalue of hidden 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 become less valuable). Since stealing makes bad outcomes more likely, ifthe 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 we show below, the sufficient condition σct ≤ 0 is guaranteed to hold in the relaxedoptimal contract. The principal wants to constrain the agent’s precautionary saving motiveand the most efficient way to do this is by reducing the agent’s risk exposure after badoutcomes. As it happens this property is sufficient for global incentive compatibility, whichshows the validity of the first-order approach. But we would like to emphasize that Theorem1 identifies σct ≤ 0 as a general sufficient condition for incentive compatibility, without evenassuming that the contract is recursive in variables x and c. These variables are well-definedfor any admissible contract, and their laws of motion do not need to be Markovian forTheorem 1 to apply. Condition σct ≤ 0 can be used to verify global incentive compatibilityof suboptimal locally incentive compatible contracts of interest. For example, it impliesthat stationary contracts that we discuss below are also globally incentive compatible.

3.4 The relaxed problem

The relaxed problem consists of minimizing cost within the class of locally incentive com-patible contracts, and we call a solution to the relaxed problem a relaxed optimal contract.

We approach the relaxed problem as an optimal control problem with states xt > 0

and ct > 0 satisfying (16), (17) and ct ≤ ch, controls σxt ≥ 0, σct , and dLt ≥ 0, and initialcondition x0 = ((1− γ)u0)

11−γ . The objective function is

EQ[∫ ∞

0e−rtxt

(ct − cγt

α

φσσxt

)dt

],

where the cost flow already incorporates the binding IC constraint (18).To understand the relaxed problem, notice that the initial value of x0 is determined by

the initial utility u0, but there is no corresponding constraint for c0—the principal can set c0

freely to minimize the cost. A low ct means the agent expects a very risky consumption inthe future, as captured by (15), so ct determines the principal’s “budget for risk exposure”. Ifct ever reached the upper bound ch, the principal would have to give the agent the perfectlysafe autarky contract without any capital, which is a costly way to deliver utility. On the

14

other hand, a low ct means the agent has large incentives to divert funds, as captured bythe IC constraint (18). Hence, the choice of c0 takes into account the cost of committingto lower risk in the future and the benefit of improved incentives now.

After c0 is set the principal chooses σxt and σct dynamically. Higher σxt today, i.e.exposing the agent to more risk, allows the principal to give more capital to the agentand earn the excess return α. However, this is costly because (a) the agent is risk averseand must be compensated with more utility in the future—the drift of xt is increasing inσxt—and (b) it eats up the budget for risk exposure in the future—the term (σxt )2/2 raisesthe drift of ct.

The principal can choose σct to mitigate the latter effect: when σxt > 0, setting σct < 0

reduces the drift of ct, preserving the budget for risk exposure. The intuition is that alower volatility of consumption, σct = σxt + σct , weakens the agent’s precautionary motive.This is the first of two forces that guide the dynamic choice of σct . The second force is thatthe benefit of a larger budget of risk exposure (lower ct) is proportional to xt due to scaleinvariance, so the principal prefers to have a lower ct when xt goes up. This also impliesσct < 0.

As we show below analytically, σct < 0 is a key property of the relaxed optimal contract:it implies that the contract gets safer after poor outcomes. Theorem 1 shows that σct ≤ 0

is a sufficient condition that ensures global incentive compatibility, so the solution of therelaxed problem is indeed the optimal contract.

3.5 The solution to the relaxed problem gives the optimal contract

In this subsection we characterize the relaxed optimal contract, and use Theorem 1 to showit is globally incentive compatible and therefore an optimal contract.

Because preferences are homothetic and the principal’s objective is linear, we know theprincipal’s cost function in the relaxed problem takes the form v(x, c) = v(c)x. We willsometimes write vt = v(ct), and v instead of v(c).

Notice that since we could always raise ct using dLt > 0, we know that v(c) must beweakly increasing. In fact, we show below that the cost function v(c) has a flat region(0, cl) where the optimal contract does not spend any time, followed by a strictly increasingregion (cl, ch) with dLt = 0 in which ct stays over the course of the optimal contract.

The HJB equation in the strictly increasing region with v′(c) > 0 is

rv = minσx≥0,σc

c− σxcγ αφσ

+ v

(ρ− c1−γ

1− γ+γ

2(σx)2

)(21)

+v′c

(c1−γ − c1−γ

h

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2

)+v′′

2c2(σc)2.

Even though we have two state variables, c and x, the HJB equation boils down to a second

15

order ODE in c. This is a feature of homothetic preferences and linear technology thatmakes the problem more tractable.

In the flat region with v′(c) = 0, the HJB equation (21) must hold as a ≤ inequality,

A(c, v) ≡ minσx≥0

c− σxcγ αφσ− rv + v

(ρ− c1−γ

1− γ+γ

2(σx)2

)> 0. (22)

The optimal contract does not spend any time in this region.The following theorem characterizes the solution to the relaxed problem and shows that

it is globally incentive compatible and therefore an optimal contract.

Theorem 2. The relaxed problem has the following properties:

(1) The cost function v(c) has a flat portion on (0, cl) and a strictly increasing C2

portion on (cl, ch), for some cl ∈ (0, ch). The HJB equation (21) holds with equality forc ∈ (cl, ch). For c < cl, we have v(c) = v(cl) ≡ vl and the HJB holds as an inequality,A(c, vl) > 0.(2) At cl, we have v′(cl) = 0, v′′+(cl) > 0, and A(cl, vl) = 0. The cost function satisfies

v(c) < cγ for all c ∈ [cl, ch), with v(ch) = vh = cγh.(3) The state variables xt and ct follow the laws of motion (16) and (17) with bounded

volatilities σxt > 0 and σct < 0 for all t > 0, and dLt = 0 always, so the Euler equation holdsas an equality. The state ct starts at c0 = cl , with µc0 > 0 and σc0 = 0, and immediatelymoves into the interior of the domain never reaching either boundary, that is, ct ∈ (cl, ch)

for all t > 0.

(4) ct does not have a stationary distribution. In the long-run, ct spends almost all thetime near ch,

1

t

∫ t

01c>ch−ε(cs)ds→ 1 a.s. ∀ε > 0.

However, ch is never reached, even in infinite time, Pct → ch = 0.(5) Since the relaxed optimal contract satisfies the sufficient condition in Theorem 1, it is

incentive compatible and therefore an optimal contract.

Let us now highlight some features of the cost function and the relaxed optimal con-tract from Theorem 2. Figures 1 and 2 illustrate these properties in a numerical example,discussed below in Section 7. The shape of the cost function, with a flat portion up to clfollowed by an increasing portion, reflects the optimal choice of c0 = cl. It would be subop-timal to run the contract with ct < cl that makes incentive provision excessively difficult.Yet, since the principal can raise c to the optimal level of cl using the process dLt ≥ 0, thecost function is not decreasing but flat over (0, cl). As ct moves above cl, the contract mustgive the agent an inefficiently low risk exposure, with the end point of ch that corresponds

16

0.005 0.006 0.007 0.008 0.009 0.010 0.011c

0.12

0.14

0.16

0.18

0.20

0.22

v

chcl

vh

Figure 1: The cost function v(c) of the optimal contract. The starting point of the optimalcontract is indicated by the blue dot. Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3,φσ = 0.2.

0.005 0.007 0.009 0.011c0.000

0.005

0.010

0.015

μc^

chcl

0.005 0.007 0.009 0.011c0.00

0.01

0.02

0.03

0.04

μx

chcl

0.005 0.007 0.009 0.011c

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

σc^

chcl

0.005 0.007 0.009 0.011c0.00

0.05

0.10

0.15

0.20

0.25

0.30

σx

chcl

Figure 2: The drift, µc and µx, and volatility, σc and σx, of the state variables c and xunder the optimal contract. The starting point of the optimal contract is indicated by theblue dot. Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3, φσ = 0.2.

17

to the safe autarky contract with no capital. As a result, the cost function v(c) is increas-ing over (cl, ch). The marginal cost of delivering utility, ∂v(x, c)/∂U = v(c)xγ , is below theinverse marginal utility of consumption, cγ , i.e. v < cγ , because higher c not only deliversutility to the agent, but also relaxes the IC constraint.

An important feature of the relaxed optimal contract is that σct < 0, which can beseen in the bottom left panel of Figure 2. After bad performance ct rises, so the agentexpects a less risky consumption in the future. There are two reasons to set σct < 0. First,it reduces the volatility of the agent’s consumption, σct = σxt + σct , and therefore helpsrelax the precautionary motive. Second, since the cost is proportional to promised utility,v(ct)xt, the principal prefers to use the least costly continuation contracts with low ct aftergood shocks when he must deliver more utility to the agent. This also implies σct < 0. Itfollows, using Theorem 1, that the relaxed optimal contract is fully incentive compatible,and therefore an optimal contract.

The top left panel of Figure 2 shows the drift of c, which is positive at cl. This means thatwhile the contract starts at the optimal c0 = cl, it immediately moves into the interior of thedomain. The reason for this is that, since the Euler equation is forward-looking, promisingan inefficiently safe contract at a future time t (higher ct) relaxes the IC constraint for everyperiod s < t, so it makes sense to backload distortions. This intertemporal tradeoff allowsthe principal to relax the IC constraints today at the cost of an inefficiently safe contractin the future.

The optimal contract starts at c0 = cl and moves up over time and after bad returns.Given the behavior of µc, one may suspect that there is a stationary distribution of cconcentrated near the point of zero drift. It turns out that this is not the case. Eventhough we can show analytically that the drift of ct is negative near ch, the asymptoticproperties of µc and σc near ch are such that the process c slows down but never reachesch. As a result, there is no stationary distribution. The principal backloads distortions inorder to relax the agent’s IC constraints, so the contract starts at cl and in the long-runspends most of the time near the upper boundary corresponding to autarky ch. However,Pct → ch = 0. No matter how close any path gets to ch, it will always leave thatneighborhood at some point in the future, but this happens progressively less and less oftenfor any path. Part (4) of Theorem 2 formalizes this asymptotic result.

Finally, to understand the choice of σx, look at the FOC for σx. There is a myopic anda dynamic motive for exposing the agent to risk through σx,

σx =α

γφσ

cγ

v︸ ︷︷ ︸myopic

− v′

v

c

γ

((1 + γ)σc + σx

)︸ ︷︷ ︸

dynamic

. (23)

The myopic motive trades off the excess return on capital, α > 0, against the higher future

18

0.006 0.007 0.008 0.009 0.010 0.011c

200

400

600

800

1000

1200

1400

density

10 20 30 40 50t

0.0055

0.0060

0.0065

0.0070

0.0075

0.0080

c

0.006 0.007 0.008 0.009 0.010 0.011c

0.01

0.02

0.03

0.04

0.05

0.06

μc, μk

0.006 0.007 0.008 0.009 0.010 0.011c

0.05

0.10

0.15

0.20

0.25

0.30

σc, σk

t=10

t-20

t=50

50%

75%

25%

μc

μk

σc

σk

Figure 3: The distribution of ct at different time points (top left panel), a sample pathof ct (top right), drifts µc(c) and µk(c), and volatilities σc(c) and σk(c) (bottom panels).Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3, φσ = 0.2.

utility required to compensate the agent for the risk. The dynamic component takes intoaccount that higher σx eats up the budget for risk exposure (increases the drift of c). Thisis costly because the cost v(c) is increasing in c. At t = 0 we know that v′ = 0 because cl ischosen optimally, so the dynamic motive disappears and we choose σx myopically withouttaking into account the effect on the agent’s precautionary motive.

What happens to the drift and volatility of x, σx and µx, when c increases? The topand bottom right panels of Figure 2 show their behavior in the numerical solution. Whilewe do not have a proof, the following properties hold across all solutions we computed, andhelp understand the behavior of the optimal contract. The agent’s risk exposure σx declineswith c. This is intuitive: the principal exposes the agent to less risk when the “budget forrisk" exposure gets depleted. Lower risk σx, in combination with higher consumption c,imply a lower expected growth in the agent’s utility xt, so µx declines in c. Thus, sinceσc < 0, the agent is punished after bad outcomes in part by slower growth.

According to our original definition, the contract has to specify the pair (ct, kt) for anyhistory of output. We have ct = ctxt and kt = σxt

cγtφσxt, so ct determines the drift and

volatility of these variables. These are shown in the bottom panels of Figure 3. Variable ctreflects the principal’s commitment to make the contract more secure, and the top left panelshows the distributions of values of ct at three different time points. The top right panelshows a sample path of ct starting from cl, relative to the whole probability distribution.Initially consumption and capital grow fast, and the agent faces a large amount of risk in

19

consumption and capital. Over time, expected growth and risk decline as ct moves up. Thevolatility of consumption is lower than that of capital. After poor returns, capital declinesfaster than consumption, because the agent faces a safer contract going forward.

We close this section with a verification theorem. Theorem 2 shows that the principal’scost function satisfies the HJB equation, but how do we know the converse? The followingtheorem shows that if an appropriate solution to the HJB has been found, e.g. numerically,then it must be the true cost function and we can use it to build a globally incentivecompatible optimal contract.

Given an appropriate solution to the HJB equation, we can identify controls σx and σc

as functions of c, and use those to build a candidate optimal contract C∗. Specifically, let x∗

and c∗ be solutions to (16) and (17) with σx∗t = σx(c∗t ), σc∗t = σc(c∗t ) and dLt = 0, starting

from initial values x∗0 = ((1− γ)u0)1

1−γ and c∗0 = cl. We then construct the candidatecontract C∗ = (c∗, k∗) with c∗ = c∗x∗ and k∗ = σx

∗ (c∗)γ

φσ x∗.

Theorem 3 (Verification Theorem). Let v(c) : [cl, ch] → [vl, vh] be a strictly increasingC2 solution to the HJB equation (21) for some cl ∈ (0, ch), such that vl ≡ v(cl) ∈ (0, vh],

v′(cl) = 0, v′′(cl) > 0 and v(ch) = vh. Assume that for c < cl the HJB equation holds as aninequality, A(c, vl) > 0. Then,

(1) For any locally incentive compatible contract C = (c, k) that delivers at least utilityu0 to the agent, we have v(cl) ((1− γ)u0)

11−γ ≤ J0(C).

(2) Let C∗ be a candidate contract generated by the policy functions of the HJB asdescribed above. If C∗ is admissible, then C∗ is an optimal contract with cost J0(C∗) =

v(cl) ((1− γ)u0)1

1−γ .

Remark. Part (2) requires checking that the candidate optimal contract is admissible.Lemma 4 shows that E[

∫∞0 e−rtxtdt] <∞ is a necessary and sufficient condition.

3.6 Discussion: hidden savings and dynamic distortions

In the optimal contract, the principal commits to a level of safety summarized by c0, andthen allocates the agent’s risk across future states to minimize the cost of compensation.Initially, c0 minimizes the cost of compensating the agent, and so v′(c0) = 0. We can saythat at time 0, there are no distortions. Afterwards, ct > c0 and v′(ct) > 0, reflectingthe principal’s costly commitment to make the contract safer. This commitment benefitsincentives before time t, and distorts the principal’s choice of risk after time t.

The optimal contract resolves two trade-offs. First, the standard risk-return trade-off:the principal earns α at the cost of exposing the agent to risk. This is also the trade-off ofthe standard portfolio choice problem. In addition to this trade-off, after time 0 we haveadditional distortions because v′(ct) > 0. When exposing the agent to risk, to respect the

20

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

cγ

Stat. Euler

myopic

optimal

optimal w/o hidden savings

Figure 4: The cost function v(c) solid in blue for the optimal contract, dashed in red forstationary contracts vs(c), and dashed in black for contracts with myopic optimization overσx, vm(c). Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3, φσ = 0.2.

law of motion (17), the principal has to raise ct. This carries the marginal cost of v′(ct) > 0.

To sum up, we have the risk-return trade-off as well as distortions to manage the agent’sprecautionary motive after time 0.

It is useful to compare the optimal contract with hidden savings to a series of bench-mark contracts to better understand these trade-offs. The benchmark contracts are alsointeresting in their own right.

First, the reader may wonder what happens without hidden savings. In this case, theprincipal determines σx purely from the risk-return trade-off, as the choice of c is free andunconstrained by the agent’s precautionary motive. For a fixed level of c, optimizing overσx subject to the law of motion of the agent’s utility (16) and the IC constraint (18), weobtain the cost function vm(c) that satisfies HJB equation

rvm(c) = minσx≥0

c− σxcγ αφσ

+ vm(c)

(ρ− c1−γ

1− γ+γ

2(σx)2

). (24)

See the black dashed line in Figure 4. The optimal choice of σx satisfies the first-ordercondition

σx =α

γφσ

cγ

vm(c).

This condition reflects the pure risk-return trade-off, taking into account how the cost ofrisk impacts the principal’s value function (hence the ratio cγ/vm(c)). We call such choiceof σx myopic, i.e. without taking into account the impact on the agent’s precautionarymotive (which is irrelevant without hidden savings).

21

The optimal contract without hidden savings corresponds to the minimum point ofvm(c), indicated by a yellow dot in Figure 4.13 The choice of c without hidden savingssatisfies the inverse Euler equation, i.e. e(ρ−r)tcγt must be a martingale, which implies thatthe agent would like to save if he could (see Lemma O.6).

Once we introduce hidden savings, we are constrained by the Euler equation. Fora sharper contrast to the case without hidden savings, it is useful to consider stationarycontracts with a fixed choice of c, with ct, kt, and xt following Geometric Brownian Motions,and dLt = 0. To keep c constant while respecting the Euler equation, we must set σc = 0

and pick σx to satisfy

σx = σxs (c) ≡√

2

√c1−γh − c1−γ

1− γ, (25)

so that µc = 0 in (17). According to (25), if we want to convince the agent to consumemore (i.e. choose higher c), we need to expose him to less risk (i.e. lower σx) in order toweaken the precautionary motive. The red dashed line in Figure 4 shows the cost vs(c) ofstationary contracts, which satisfy the HJB equation

rvs(c) = c− σxs (c)cγα

φσ+ vs(c)

(ρ− c1−γ

1− γ+γ

2σxs (c)2

). (26)

The optimal stationary contract is indicated by the red dot at cr, with cost vr = vs(cr).It is interesting that the curves vm and vs meet at a point where the Euler equation

holds and σx satisfies the risk-return trade-off condition (25). These are the two conditionsfor the classic consumption and portfolio choice problem (with Sharpe ratio α/(σφ)), andthey give the black dot in Figure 4. Here we do not take into account the effect of riskexposure on the agent’s precautionary motive, and how a reduction in risk can raise c andrelax the agent’s incentive constraint. Hence, the minimal incentive-compatible share of riskthe agent must retain is given by parameter φ, and we get the Sharpe ratio of α/(σφ). (Seealso Section 6, where we discuss in more detail the division of risk between the principaland agent through fund capital structure).

For the record, the solution of the classic consumption-portfolio choice problem, in which13The optimal contract without hidden savings only exists if γ < 1/2. If γ > 1/2 the principal can obtain

infinite profits for any α > 0. See Di Tella (2016). Figure 4 shows results for γ < 1/2, but hidden savingsconstrain the principal and make the problem well defined for any γ.

22

the agent retains a fraction φ of the risk, is given by the well-known formula14

cp =

(c1−γh − 1− γ

2

(α

γφσ

)2) 1

1−γ

, σxp =α

γφσ, vp = cγp . (27)

The curves vm and vs are tangent at point cp because for any given c the cost vm from theoptimal unconstrained choice of σx must be weakly below the cost vs that is constrainedby the Euler equation.

Relative to the classic portfolio solution, the optimal stationary contract has lower riskexposure, i.e. σxr < σxp and cr > cp. To see why, suppose we start at the classic portfoliosolution and reduce σx in a uniform way, i.e. within the family of stationary contracts thatrespect the Euler equation. This change raises c, which creates a first-order benefit relaxingthe IC constraint (18). The optimal stationary contract takes into account this effect of therisk exposure σx on the stationary level of the agent’s precautionary motive.

The optimal dynamic contract with hidden savings does even better, by taking intoaccount how risk exposure σxt at any time t affects the agent’s precautionary motive andincentives along the entire path leading to time t. At time 0, the contract features myopicrisk-return optimization, as the optimal contract without hidden savings. Since we chosec0 = cl optimally, we have v′(cl) = 0, so the HJB equation (21) reduces to (24) and theoptimal contract therefore starts on the curve vm. In contrast the choice of σxt at time t > 0

has to be made taking into account the precautionary motive and the tightening of theincentive constraints that this choice creates leading to time t. In the optimal contract, thebenefits that a reduction in risk σxt would have on past incentives must equal the cost ofdistortions going forward, captured by the derivative v′(ct). Distortions will be greater, i.e.ct and v′(ct) must be higher, when the history prior to t is longer (larger t) and when fundsize prior to time t had been higher. Hence, ct tends to rise in the optimal contract overtime and after poor outcomes. As distortions build up, the contract looks more and morelike a stationary contract with ct = ch (see also Section 4, where the right boundary ch canbe any other point on the stationary contract curve vs).

4 Hidden investment

In this section we consider the possibility that the agent can secretly save not only in therisk-free asset, but also in his private technology. This may not be a concern when the agentis already exposed to significant risk through the contract: the excess return the agent earns

14Formally, the problem is

max(c,k)

U(c) s.t. dwt = (rwt + αkt − ct)dt+ kt(φσ)dZt, wt ≥ 0

for a given wealth w0.

23

from his technology would not justify the incremental risk. However, hidden investmentbecomes a concern in future states, in which the principal would like to significantly reducethe agent’s risk exposure to control ex-ante precautionary motive. In other words, thepossibility of hidden investment may raise the agent’s incentives to save if in some futurestates he can benefit by investing these savings in his private technology. In this section,we analyze exactly how this plays out. To deal with hidden investment, the principal hasto give the agent some minimal level of risk exposure. This constraint binds in some futurestates. Here we focus on the main economic insights. The Online Appendix has all theformal results and also extends the setting to incorporate aggregate risk.

If the agent can secretly invest in his private technology, his hidden savings follow thelaw of motion

dht = (rht + zthtα+ ct − ct + φktat) dt+ zthtσdZt. (28)

The new term zt ≥ 0 is the portfolio weight on his private technology in his hidden savings.If he invests in his private technology he takes on risk to earn the excess return α. Aninterpretation for hidden investment is that the principal can give the agent an amount ofcapital kt, but the agent can secretly invest more.

A contract C = (c, k) specifies the contractible payments c and capital k, contingent onreturns R. After signing the contract the agent can choose a strategy (c, a, z) to maximizehis utility. The agent’s utility and the principal’s objective function are still given by (3) and(4). As in the baseline setting, it is without loss of generality to look for a contract wherethe agent does not steal, has no hidden savings, and no hidden investment. A contract istherefore incentive compatible if the agent’s optimal strategy is (c, 0, 0).

The laws of motion of the state variables xt and ct, (16) and (17), as well as the “skin inthe game” constraint (18) remain unchanged. The no-savings constraint is that the agent’sdiscounted marginal utility e−ρtc−γt times the return of any trading strategy available tothe agent must be a supermartingale. Otherwise, the agent can benefit by saving in sometrading strategy to consume later. Condition dLt ≥ 0 in the constraint (17) ensures thatthe agent cannot benefit through risk-free savings, and we have a new IC constraint for theagent’s private technology,

σxt + σct ≥α

γσ. (29)

The interpretation (29) is simple. The agent can obtain a premium ασ for his idiosyncratic

risk. If his exposure to risk were σct <αγσ , he would benefit from deviating by secretly

investing and taking on risk. The excess return α would more than compensate for theextra risk exposure at the margin.

Hidden investment restricts the principal’s ability to provide incentives. In particular,the principal cannot promise to give the agent a perfectly deterministic consumption stream.If the principal tried to do this, the agent would just secretly take on risk by investing

24

on his own. This is costly because the principal would like to promise future safety torelax the agent’s precautionary saving motive. The extra constraint implies a lower upperbound ch. In the baseline where the agent cannot invest in his private technology, we have

ch =(ρ−r(1−γ)

γ

) 11−γ , corresponding to the c in autarky where the agent cannot invest in

capital. If the agent can invest in his private technology the upper bound is

ch =

(ρ− r(1− γ)

γ− 1− γ

2

(α

σγ

)2) 1

1−γ

,

which is lower and corresponds to the c in autarky where the agent invests in his privatetechnology on his own (so he cannot get any risk sharing).15

The HJB equation is the same as in the baseline setting, but we have an extra constraint.In the region where v(c) is increasing, the HJB equation is

0 = minσx,σc

c− rv − σxcγ αφσ

+ v

(ρ− c1−γ

1− γ+γ

2(σx)2

)(30)

+v′c

(c1−γ − c1−γ

h

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2

)+v′′

2c2(σc)2

subject to σx ≥ 0 and (29).Figure 5 shows the optimal contract with hidden investment. As in the baseline setting,

the contract starts at some cl and then moves immediately into the interior of the domain(cl, ch). While the new IC constraint (29) is not binding, the FOC for σx and σc arethe same as in the baseline setting. But with hidden investment the IC constraint (29)can be binding in some region of the state space. In Figure 6, the IC constraint (29) isbinding near the upper bound ch. The property σct < 0 holds for all t > 0 both when thehidden investment constraint (29) binds, and when it does not. When the constraint binds,σx + σc = α

σγ with σxt >ασγ . Contract dynamics are therefore qualitatively the same as in

the baseline. The contract starts at cl with myopic optimization over σx and becomes lessrisky over time and after bad outcomes, never revisiting cl or reaching autarky ch.16

Just as in the baseline setting without hidden savings, the marginal cost of the agent’sutility is lower than the inverse of the marginal utility of consumption, i.e. v(c)xγ < cγ ⇔v(c) < cγ , and the optimal contract optimizes myopically over σx at t = 0. After that,the principal distorts the agent’s access to capital to manipulate his precautionary motive.How can the principal do this, if the agent has access to hidden investment? The answeris that if the agent invests on his own he is the residual claimant and must bear the wholerisk, while if he invests through the principal he gets to share some risk because φ < 1.

15The parameter restrictions (7) and (8) ensure ch is positive in both cases.16There is one subtle difference. The optimal contract without hidden investment does not have a

stationary distribution, but instead limits in the long-run to autarky in the sense of Theorem 2. Withhidden investment, there is a stationary distribution ψ(c). See Theorem O.2 in the Online Appendix.

25

Optimal without

hidden investment

Optimal with

hidden investment

Optimal

renegotiation-proof

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

Stationary

cγ

myopic

autarky

Figure 5: The cost function v(c) without hidden investment is solid in blue, with hiddeninvestment, dotted in green. For reference, the cost of stationary contracts is dashed in red,and myopic optimization A(c, v) = 0, dashed in black. The starting point without hiddeninvestment is indicated by the blue dot, with hidden investment by the green dot, themyopic stationary contract by the black dot, autarky by the purple dot, and the optimalrenegotiaton-proof contract by the red dot (see Section 5). Parameters: ρ = r = 5%,α = 1.7%, γ = 1/3, φ = 0.8, σ = 0.2/0.8.

0.0055 0.006 0.0065 0.007c

0.002

0.004

0.006

0.008

0.010

0.012

0.014

μc^

chcl

0.0055 0.006 0.0065 0.007c

0.01

0.02

0.03

0.04

0.05

μx

chcl

0.0055 0.006 0.0065 0.007c

-0.025

-0.020

-0.015

-0.010

-0.005

σc^

chcl

0.0055 0.006 0.0065 0.007c

0.05

0.10

0.15

0.20

0.25

0.30

σx

chcl

Figure 6: The drift, µc and µx, and volatility, σc and σx, of the state variables c and xwith hidden investment. The dotted part is where the hidden investment IC constraint isbinding. The black dashed lines indicate cl and ch with hidden investment. Parameters:ρ = r = 5%, α = 1.7%, γ = 1/3, φ = 0.8, σ = 0.2/0.8.

26

The principal is really restricting the amount of capital he is willing to share risk on. Theagent would like to invest more if he could share the risk on the extra capital with theprincipal, but not if he must bear the whole risk on his own. For the same reason, althoughch corresponds to the consumption profile under autarky, there are still gains from tradeat that point. Since φ < 1, the principal can give the agent the same consumption processthat he would get in autarky, but more capital. As a result, the cost to the principal islower than what the agent could get in autarky, v(ch) = vs(ch) < cγh. Figure 5 shows thecost of autarky as a purple dot.

It is useful to ask under what conditions the gains from trade are completely exhausted,and the optimal contract corresponds to autarky. Lemma O.9 in the Online Appendixshows this is true in the special (but salient) case with hidden investment and φ = 1.In this case, the optimal contract, the myopic contract, and autarky coincide. Withouthidden investment there would still be gains from trade. While the agent can save onhis own, the principal can still control his access to capital to provide incentives, and cantherefore provide some risk sharing. However, with hidden investment, the optimal contractcoincides with autarky. Intuitively, the agent can both save and invest on his own, so theprincipal cannot provide any risk sharing in an incentive compatible way. We can see thiscase as a limit in Figure 5. If we let φ→ 1, while also adjusting σ so that φσ = ς is constant,all the curves corresponding to the case with no hidden investment remain unchanged. Theoptimal contract with hidden investment, however, becomes progressively worse (the dottedgreen curve shifts up), because the agent finds investing on his own more attractive, andthe upper bound ch shrinks. In the limit as φ → 1 while φσ = φσ = ς remains constant,

the upper bound ch converges to cp =

(ρ−r(1−γ)

γ − 1−γ2

(αγς

)2)

from above.

We close this section by considering the validity of the first-order approach with hiddeninvestment. The agent’s ability to invest his hidden savings makes the verification of globalincentive compatibility potentially more difficult. The agent has a greater space of potentialdeviations with his savings. Fortunately, we can extend the results of Theorem 1 to deal withhidden investment. Theorem O.1 in the Online Appendix shows that σc ≤ 0 is still sufficientto ensure global incentive compatibility even when the agent has hidden investment in bothhis private technology and aggregate risk. Since σc ≤ 0 holds in the optimal contract withhidden investment, the first-order approach remains valid. This is formalized in TheoremO.2 in the Online Appendix.

5 Renegotiation

The optimal contract requires commitment. The principal can relax the agent’s precaution-ary saving motive by promising an inefficiently low amount of capital and risk over timeand after bad outcomes. This suggests that the agent and principal could be tempted to

27

renegotiate the contract and “start over”, undoing the whole incentive scheme. Here we for-malize a notion of renegotiation, and characterize the optimal renegotiation-proof contract.As it turns out, the optimal renegotiation-proof contract is the best stationary contract, soeven without commitment we are able to do better than under myopic optimization. Thisresult is consistent with the presence of hidden investment and aggregate risk introducedin Section 4 and the Online Appendix.

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 use a notion of internal consistency. If C isrenegotiation proof, then surely appropriately scaled parts of it should be a valid challenger.At any point in time, vt = Jt/xt is the continuation cost per unit of promised utility xt.Define v = inf v(ω, t), where the minimization is across both ω ∈ Ω and t ∈ R+. At anystopping time τ , the principal can renegotiate and get a continuation cost xτ × v whiledelivering utility xτ to the agent. With this in mind, we say that an incentive compatiblecontract is renegotiation-proof (RP) if

∞ ∈ arg minτ

EQ[∫ τ

0e−rt(ct − ktα)dt+ e−rτxτ v

].

The optimal contract with hidden savings is not renegotiation proof, because after anyhistory at any t > 0, vt > vl = v, so the principal is always tempted to “start over”.

It is easy to see that an incentive compatible contract is renegotiation-proof if andonly if the continuation cost vt is constant. In Section 3.6 we define a class of stationarycontracts in which ct is constant and ct, kt and xt follow proportional Geometric BrownianMotions and dLt = 0. These contracts have constant vt, hence they are renegotiation proof.However, these are not the only renegotiation-proof contracts. There are contracts in whichdLt > 0, and ct, kt and xt follow proportional Geometric Brownian Motions; they are alsorenegotiation proof. In addition, there could be non-stationary contracts with a constantcost vt.

However, Theorem O.4 in the Online Appendix shows that the optimal renegotiation-proof contract is the optimal stationary contract, with cost vr, shown in Figure 5 with thered dot. This result shows that even without full commitment, the optimal contract withhidden savings distorts how much capital the agent receives. That is, the principal can dobetter than the myopic contract given by (27).

Remark. It is possible that cr = 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 cr = cp = ch,as shown in Lemma O.9.

28

6 Fund Capital Structure

In this section, we show how dynamic distortions become reflected in fund leverage, payoutrate and the division of risk between the agent and principal (i.e. outside investors). Thisperspective is useful because these quantities are of applied interest. Also, they connectour results to the classic consumption-portfolio problem.

As a starting point, notice that without a long-term contract, the agent could simplyshift fraction 1−φ of risk to the outside market and allocate his wealth between the risk-freeasset and capital, as in the standard optimal consumption and portfolio choice problem.This risk sharing is incentive compatible because insurance does not cover more than thecost of cash diversion, hence the agent does not want to steal. The resulting contract isthe myopic contract in (27), which satisfies the Euler equation and myopic optimizationover risky capital (taking into account that only a fraction φ of the risk is retained by theagent).

Relative to this simple benchmark, distortions in the optimal contract take the form ofdynamic deviations from myopic optimization for capital: the contract restricts the fundsinvested in capital (i.e. the ratio kt/xt), particularly after bad outcomes. That is, if theagent could increase the investment in capital, while passing on some risk to the principalin an incentive compatible way, he would. It is important that the optimal contract tiesthe agent’s hands dynamically with respect to capital investment kt relative to xt.

These distortions allow the principal to improve risk sharing. That is, the agent can shifta greater fraction of risk than 1−φ to the principal and still respect incentive compatibility.The agent would like to invest more but the contract limits the investment, so an extra dollarof returns relaxes the limit. Thus, the restricted access to capital makes the marginal valueof funds invested in capital greater than the marginal utility of consumption, which relaxesthe incentive constraint. As we move away from the simple optimal portfolio benchmark,distorting the allocation to capital has a second order negative effect on welfare, but theincentive and improved risk sharing effect of this distortion has a first order positive effect.

To see this in more detail, let us be more explicit about how the optimal contractdeviates from the myopic consumption-portfolio solution. At any time, there is capital ktin the fund. Since the cost of compensating the agent is Jt = Et

[∫∞t e−r(s−t)(cs − αks)ds

],

the value of the principal’s stake in the fund is kt − Jt. This gives us a specific division ofthe value of capital kt: the principal’s stake is kt − Jt and the agent’s stake is Jt. Recallthat the agent needs to contribute J0 at time 0 in order for the principal to break even.

The principal has to earn the required return of r on his stake in the fund. This meansthe agent’s stake Jt earns the α and some portion of risk, i.e. Jt follows the law of motion

dJt = (rJt − ct + αkt) dt+ φtσkt dZt (31)

29

0.007 0.009 0.011c0.0

0.5

1.0

1.5

k/J

chcl

0.005 0.007 0.009 0.011c0.00

0.01

0.02

0.03

0.04

0.05

c/J

chcl

0.007 0.009 0.011c

0.14

0.15

0.16

0.17

0.18

0.19

0.20

ϕ˜

chcl

ϕ

Figure 7: Fund leverage k/J , consumption rate c/J , and retained equity φ for the op-timal contract. Dashed line indicates the simple benchmark portfolio (myopic contract).Parameters: ρ = r = 5%, α = 1.7%, γ = 1/3, φ = 0.8, σ = 0.2/0.8.

for some stochastic process φt. This equation is a classic dynamic budget equation, withthe allocation of funds to the risk-free asset, capital (of which risk 1 − φt is insured) andconsumption ct. We call φt the agent’s retained equity stake. Ideally, we would set φt = 0

to obtain perfect risk-sharing, but the retained equity stake is important to give the agentsome “skin in the game” and ensure incentive compatibility.

For any contract that is recursive in ct and scale invariant in xt, with a cost functionJt = v(ct)xt, we can match terms to obtain

kt/Jt =σxt

vtc−γt φσ

, φt = (1 + σvt /σxt )(vtc

−γt )φ, ct/Jt =

ctvt, (32)

where σv = v′(ct)/v(ct) σct ct is the volatility of vt.

The optimal contract distorts the agent’s portfolio-weight on capital, or leverage, k/J , toweaken the agent’s incentives to steal. These distortions imply a lower incentive compatibleretained equity stake

φt = (1 + σvt /σxt )︸ ︷︷ ︸

<1

(vtc−γt )︸ ︷︷ ︸

<1

φ < φ.

The retained equity stake φt is below φ for two reasons. First, the marginal value ofconsumption is below the marginal value of funds that can be invested in capital, i.e.vtc−γt < 1. Second, after bad outcomes the agent is punished not only with less wealth, but

also with tighter distortions, σvt /σxt < 0. Facing less risk, the agent’s consumption rate outof wealth, ct/Jt, is above the level corresponding to the simple benchmark consumption-portfolio problem (the myopic contract).

Figure 7 shows these quantities in our optimal contract. Fund leverage kt/Jt is decreas-ing with ct, the agent’s share of risk φt is below φ, and the consumption rate ct/Jt is abovethe level of the optimal portfolio benchmark. Notice that the optimal contract uses futuredistortions to relax the retained equity stake, φ0 < φ, but optimizes myopically over capitalat t = 0, so fund leverage is higher at t = 0 than what it would have been under the simpleportfolio benchmark.

30

7 Numerical application

We computed a numerical example to illustrate our results and show how our theoremsare applied. The interest rate and impatience rate of the agent are r = ρ = 5%, andrisk aversion γ = 1/3. The excess return the agent obtains from investing in capital isα = 1.7%. The efficiency of stealing φ and the idiosyncratic risk σ enter together, andwe set φσ = 20%. These parameter values seem reasonable and satisfy the constraints inSection 2, but there isn’t a specific calibration target.

For the baseline case without hidden investment or renegotiation, we solved the HJBnumerically as an ODE, plugging in the FOCs. The resulting µx, σx, µc and σc are boundedand µx(c) is bounded below away from r, so Lemma 4 ensures that the resulting contract isadmissible and delivers utility u0 to the agent. The verification Theorem 3 shows we havean optimal contract. Figures 1 and 2 show the cost function and the laws of motion of thestates, and Figure 3 shows a simplified simulated path for the main variables of interest.

Lemma 4 and Theorem 1 can also be used to verify local and global incentive compati-bility for the suboptimal contracts introduced in subsection 3.6. We can get a quantitativesense of the importance of the incentive mechanism in the optimal contract by comparing itscost with the cost of the myopic stationary contract, which corresponds to a simple portfolioproblem where the agent invests and consumes on his own, subject only to retaining fractionφ of the equity. The myopic contract has a cost of vp = 0.168, compared to v(cl) = 0.151

for the optimal contract. This means that switching from the myopic stationary contract tothe optimal contract reduces the cost of delivering utility x0 by roughly 10%. Equivalently,switching to the optimal contract allows us to leave the principal indifferent and improvethe agent’s utility by an amount equivalent to a proportional 10% increase in consumption.To put this in context, the optimal contract without hidden savings has a cost vn = 0.138.Eliminating the agent’s access to hidden savings would therefore reduce costs by a further8.4%. Switching from the myopic stationary contract to the optimal contract with hiddensavings closes roughly half the gap in cost between the myopic stationary contract and theoptimal contract without hidden savings.

We can also quantitatively explore the cost of introducing renegotiation and hiddeninvestment into the environment. The optimal contract with renegotiation corresponds tothe best stationary contract, with cost vr = 0.156. This is roughly 3.75% more costly thanthe optimal contract without renegotiation. Renegotiation eliminates roughly a third of thegain from going from the myopic stationary contract (the simple portfolio problem) to theoptimal contract with full commitment.

We solve for the optimal contract with hidden investment using φ = 0.8 and keepingφσ = 20% unchanged. We solve the appropriate HJB equation (O.20) in the Online Ap-pendix numerically, and we use the more general Lemma O.4 and Theorem O.3 to verifywe have constructed an optimal contract. Figure 5 shows the optimal contract with hidden

31

investment. The cost is v(cl) = 0.152, or roughly 1% more costly than the optimal contractwithout hidden investment. Introducing hidden investment eliminates 1/10 of the gain fromgoing from the myopic stationary contract to the optimal contract without hidden invest-ment. We know, however, that as φ → 1, while φσ remains fixed, the cost of the optimalcontract with hidden investment converges to the cost of the myopic stationary contract,while the cost of the optimal contract without hidden investment is unchanged. In thatlimit introducing hidden investment eliminates all of the gains of the optimal contract overthe myopic stationary contract.

8 Discussion: dynamic private information

Our model is related to a whole class of problems with the broad title “dynamic adverseselection." These are dynamic environments where the agent has private information. Inour case, information about savings. In these environments the agent’s deviation payoffplays a central role, also called his information rent. This term originates from the staticauction environment of Myerson (1981), where agents whose type (valuation) is higher canmimic lower types, and their higher valuation earns them higher utility or rents. Althoughthe two problems may appear unrelated, at the core of both lie the distortions that theprincipal uses to control the payoff of a deviating type.

In the static Myerson (1981) setting the distortion can take the form of a reserve pricewhich reduces the rents of high valuation bidders and helps the principal screen. In dynamicmodels of adverse selection distortions are more complex.17 In our setting the principaldistorts the agent’s access to capital ex-post to reduce his precautionary motive. Thislowers the value of hidden savings and improves ex-ante incentives against stealing.

Generally, the agent’s incentives depend on his entire off-path value function. Thisis the approach in Fernandes and Phelan (2000), which is challenging due to an infinite-dimensional state space. Instead, we use the first-order approach, which focuses on localincentives, and verify global incentive compatibility using an upper bound of the agent’s off-path value function. In terms of transformed utility, this bound takes the form xt + c−γt ht.

The agent’s deviation payoff would take this exact form if we had φ = 1 and the agent canchoose any real at (so the agent can move funds between returns and hidden savings withoutfriction), and if we took a contract with σct = 0. The stationary contracts of Section 3.6have this property, as well as contracts in which the path for ct is deterministic. Indeed, theproof of Theorem 1, which shows that xt + c−γt ht is an upper bound on the agent’s off-pathvalue function in our setting, extends to the possibility that at ≤ 0 if also σct = 0. To seewhy in this case the bound is attainable, consider the utility he would get if he immediately

17Pavan et al. (2014), Battaglini (2005), DeMarzo and Sannikov (2016), He et al. (2017).

32

put all his hidden savings back into returns. In general, this utility is

xt +

∫ h

0c(h′)−γdh′, (33)

where c(h′) is the level of c that results after the agent uses h′ of savings to boost returns,because c−γ is the marginal benefit of a dollar of returns from incentive compatibility. Thisexpression resembles the bound used in DeMarzo and Sannikov (2016) and Pavan et al.(2014). With deterministic contracts, c(h′) = ct for all h′ because ct does not depend onthe history of observed returns, so the agent obtains utility xt + c−γt ht, achieving the upperbound. So in this special case we actually know exactly what the agent’s off-path utilityis, and we can therefore show that contracts with σct = 0 are globally incentive compatibleeven if we allow at ≤ 0.

In our model xt+c−γt ht is only an upper bound. The assumption that at ≥ 0 is necessary

for this result, since otherwise the agent can obtain a higher utility of (33) by immediatelyusing all his hidden savings to boost returns. Indeed, when ct declines with reported returns(i.e. when σct < 0), this utility is greater than the bound, xt +

∫ h0 c(h

′)−γdh′ > xt + c−γt ht.The restriction at ≥ 0 implies that xt + c−γt ht is an upper bound on the agent’s deviationpayoff, which allows us to prove global incentive compatibility for all contracts with σct ≤ 0.

It is interesting what would happen if we allowed at ≤ 0. We do not know in thiscase if the solution to the relaxed problem remains incentive compatible, i.e. whether thefirst-order approach is valid. We know for sure that in that contract the agent’s deviationpayoff is above xt + c−γt ht : it is at least (33), and if it is higher than that, the first-orderapproach fails. If this is so, the principal can always reduce the agent’s off-equilibriumutility to our bound, xt + c−γt ht, by using deterministic contracts. This means that solvingthe relaxed problem with the extra constraint σct = 0 provides an upper bound on the costof the optimal contract, while the solution to the relaxed problem with free σct provides alower bound. The optimal contract may be in between, and may use dynamic distortionsto bound the agent’s deviation payoff at savings levels ht > 0.

9 Conclusions

We study the role of hidden savings in a classic portfolio-investment problem with funddiversion. The agent’s precautionary saving motive plays central role in his incentives todivert funds. If the agent expects a large exposure to risk in the future, he places a largevalue on hidden savings that he can use to self insure. As a result, the principal mustmanipulate the agent’s precautionary saving motive by committing to limit his exposureto risk in the future, especially after bad outcomes. Since giving capital to the agentrequires exposing him to risk to align incentives, this leads to dynamic distortions in the

33

agent’s access to capital and a skewed compensation scheme. After good outcomes theagent’s access to capital improves, allowing his fund to keep growing rapidly. After badoutcomes his access to capital is restricted and he stagnates. In exchange, his consumptionis somewhat insured on the downside, and he is punished instead with lower growth. Wealso extend our environment to incorporate hidden investment and renegotiation, and showthat the optimal contract can be mapped into a consumption-portfolio allocation with aretained equity constraint and a leverage constraint.

An important methodological contribution is to provide a sufficient analytical conditionfor the validity of the first-order approach. If the agent’s precautionary saving motive isweaker after bad outcomes, the contract is globally incentive compatible. This conditionholds in the optimal contract and in a wider class of contracts beyond the optimal one. Infact, the sufficient condition does not even require a recursive structure, and it is valid evenwith aggregate risk and hidden investment.

34

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38

Appendix

Lemma 1

Consider the strategy (c, 0). Define

Yt = Et

[∫ ∞0

e−ρsc1−γs

1− γds

]=

∫ t

0e−ρs

c1−γs

1− γds+ e−ρtU c,0t .

Since Y is a P -martingale adapted to the filtration generated by Brownian Motion Z, wecan write

dYt = e−ρtc1−γt

1− γdt+ e−ρtdU c,0t − ρe−ρtU

c,0t dt = e−ρt∆tσ dZt

for some stochastic process ∆, also adapted to the same filtration. Dividing by e−ρt andrearranging we get (10).

Lemma 2

We will show that if (11) fails then the agent has a profitable deviation with stealing.Equation (12) is simply the first-order condition necessary for (11). After that, we showthat (13) follows from the standard Euler equation, which must necessarily hold to rule outdeviations with savings (without stealing).

First, suppose that (11) fails on a set of positive measure. Consider the stealing processa where at is the action that achieves the maximum of (11) on [0, a], and a is an arbitrarybound. Since the objective in (11) is concave, we have that at > 0 whenever (11) fails.Consider the resulting strategy (c+ φka, a), which implies zero savings, ht = 0.

Define the process

Vt ≡∫ t

0e−ρs

(cs + φksas)1−γ

1− γds+ e−ρtU c,0t ,

which corresponds to the utility of following this strategy until time t and then revertingto (c, 0). From the representation (10), under the strategy (c + φka, a) the process Vt hasdrift

e−ρt

((ct + φktat)

1−γ

1− γ− ρU c,0t + ρU c,0t −

c1−γt

1− γ−∆tat

),

since dRt− (α+ r) dt has drift −at. By our construction of a, the drift is non-negative andpositive on a set of positive measure, so Vt is a local submartingale, and we can pick a largeenough stopping time τ such that

U c,00 = V0 < Ea0[Vτ ] = Ea0[∫ τ

0e−ρs

(cs + φksas)1−γ

1− γds+ e−ρτU c,0τ

],

39

The last expectation is the agent’s payoff from following (c+ φka, a) until time τ and thenreverting to (c, 0), and this strategy is a profitable deviation over following (c, 0) throughout.

For (13), notice that e−(ρ−r)tc−γt must be a supermartingale to ensure that the agent doesnot want to save. Then we can use the Doob-Meyer decomposition to write e−(ρ−r)tc−γt =

Mt − At, where Mt is a local martingale and A a weakly increasing process. Since M isadapted to the filtration generated by Z, we can write Mt =

∫ t0 σ

Mt dZt for some process

σMt . Define σct by σMt = −γσcte−(ρ−r)tc−γt . Then using Ito’s lemma we obtain (13) with aweakly increasing process L.

Lemma 3

To establish the bound ct ≤ ch, use the fact that in an incentive compatible contract

mt = e(r−ρ)tc−γt is a supermartingale. Let yt =c1−γt1−γ = e

((γ−1)(ρ−r)

γ)t m

γ−1γ

t1−γ . We can write,

for s > t,

Et [ys] ≥ e((γ−1)(ρ−r)

γ)sEt [ms]

γ−1γ

1− γ≥ e(

(γ−1)(ρ−r)γ

)s e(r−ρ)

(γ−1)γ

tc1−γt

1− γ,

where the first inequality follows from Jensen’s inequality (with equality only if ct is deter-ministic), and the second inequality from mt being a supermartingale (with equality if it’sa martingale). Now we can write,

Ut = Et[∫ ∞

te−ρ(s−t)ysds

]≥ Et

∫ ∞t

e−ρ(s−t)e(

(γ−1)(ρ−r)γ

)s e(r−ρ)

(γ−1)γ

tc1−γt

1− γds

=c1−γt

1− γEt[∫ ∞

te− ρ−r(1−γ)

γ(s−t)

ds

]=

c1−γt

1− γ

(ρ− r(1− γ)

γ

)−1

.

If follows that

xt = ((1− γ)Ut)1

1−γ ≥ ct(ρ− r(1− γ)

γ

)− 11−γ

=⇒ ct =ctxt≤(ρ− r(1− γ)

γ

) 11−γ

= ch.

In addition, this upper bound can only be achieved with deterministic consumption, so U c,0tmust be deterministic too in that case. This implies that both ct and xt grow at rate r−ρ

γ ,so ch is an absorbing state. In light of (12), we must have kt+u = 0 in the continuationcontract, so we have the autarky contract with cost vhxt.

For expression (15), use the law of motion of ct in equation (13) to compute

dyt = (1− γ)yt

(r − ργ

+1 + γ

2(σct )

2 − γ

2(σct )

2

)dt+ (1− γ)ytσ

ctdZt + dAt

40

dyt = yt(1− γ)

(r − ργ

+1

2(σct )

2

)dt+ (1− γ)ytσ

ctdZt + dAt,

where A is a weakly increasing process coming from the term dLt in (13). This is a linearSDE with solution

ys ≥ yte∫ st (1−γ)

(r−ργ

+ 12

(σcu)2)due∫ st [(1−γ)σcudZu− 1

2(1−γ)2(σcu)2du] (34)

for s > t, with equality if Lt = 0.Suppose that exp

(∫ t0 (1− γ)σcudZu −

∫ t0

12(1− γ)2(σcu)2du

)is a proper martingale. A

sufficient condition for this is that σc be bounded. It then defines a probability measureP (σc) such that Zt = Zt −

∫ t0 (1− γ)σcsds is a P -martingale. Now write

Ut = Et[∫ ∞

te−ρsysds

]≥ ytEt

[∫ ∞t

e−ρse∫ st (1−γ)

(r−ργ

+ 12

(σcu)2)due∫ st (1−γ)σcudZu− 1

2(1−γ)2(σcu)2duds

]

= ytEt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcu)2

)due∫ st (1−γ)σcudZu− 1

2(1−γ)2(σcu)2duds

],

and therefore

Ut ≥ ytEPt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcs)

2)duds

].

Now compute as before

xt = ((1− γ)Ut)1

1−γ ≥ ctEPt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcs)

2)duds

] 11−γ

=⇒ ctxt≤ EPt

[∫ ∞t

e−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcs)

2)duds

]− 11−γ

. (35)

With Lt = 0 we have equality in (34), so we have equality in (35).

Lemma 4

First we show the contract delivers utility x1−γt

1−γ <∞ if the agent follows strategy (c, 0). Let

Yt =x1−γt

1−γ , and using the law of motion of x, (16), we get

dYt = Yt(1− γ)(µxt −γ

2(σxt )2)︸ ︷︷ ︸

ρYt−c1−γt1−γ

dt+ Yt(1− γ)σxt dZt. (36)

41

Integrating we obtain

Y0 = E

[∫ τn

0e−ρs

c1−γs

1− γds+ e−ρτ

nYτn

]

for an increasing sequence of bounded stopping times with τn → ∞ a.s. Take the limitn→∞, using the monotone convergence theorem on the first term to get

Y0 = E

[∫ ∞0

e−ρsc1−γs

1− γds

]︸ ︷︷ ︸

Uc,00

+ limn→∞

E[e−ρτ

nYτn].

We will now show that the last term is zero,

limn→∞

E[e−ρτ

nYτn]

= 0.

Since σx is bounded and c bounded away from zero and above by ch, µx is bounded too,and so is therefore the growth rate, (1− γ)(µxt −

γ2 (σxt )2), and volatility, (1− γ)σxt , of Yt in

(36). Furthermore, the growth rate of Yt in (36), is bounded away below ρ,

(1− γ)(µxt −γ

2(σxt )2)− ρ = −c1−γ

t ≤ max−c1−γ , −c1−γ

h

< 0,

where c is a lower bound on ct. We then get that limn→∞ E[e−ρτ

nYτn]

= 0 and therefore

U c,00 = Y0 =x1−γ

01−γ <∞. The same reasoning yields U c,0t =

x1−γt

1−γ <∞ for all t.Now we show that the resulting contract C is admissible if and only if EQ

[∫∞0 e−rtxtdt

]<

∞. To show sufficiency, notice that since c is bounded above by ch and σx bounded, wecan write

EQ[∫ ∞

0e−rt(ct + kt)dt

]≤ 2 max

ch,

σxcγhφσ

EQ[∫ ∞

0e−rtxtdt

]<∞,

where σx is an upper bound on σxt . For necessity, since ct is bounded away from zero,ct ≥ c > 0,

∞ > EQ[∫ ∞

0e−rtctdt

]≥ c× EQ

[∫ ∞0

e−rtxtdt

].

Since the contract is admissible and satisfies (17) and (18), and ct ≤ ch, it is locally incentivecompatible.

Theorem 1

We’ll do the proof for dLt = 0 to keep it simple; it can be easily generalized for dLt 6= 0.First write the bound

(1 + htc

−γt

)1−γU c,0t =

x1−γt

1−γ , where xt = xt + htc−γt , and define

42

ct = ct/xt.For any feasible strategy (c, a), we can write the difference between the utility of that

strategy, U c,at , and the bound x1−γt

1−γ as

e−ρt

(U c,at −

x1−γt

1− γ

)= Eat

[∫ τn

te−ρu

c1−γu

1− γdu+

∫ τn

td

(e−ρu

x1−γu

1− γ

)+ e−ρτ

n(U c,aτn −

x1−γτn

1− γ

)],

for a localizing sequence τn with τn →∞ a.s.. We would like to show that this is alwaysnon-positive.

First, take the first two terms on the rhs and write them:18

Eat

[∫ τn

te−ρu

c1−γu

1− γdu+

∫ τn

td

(e−ρu

x1−γu

1− γ

)](37)

= Eat

[∫ τn

te−ρux−γu

(c1−γu − ρ1− γ

xu + xuµxu − xu

γ

2(σxu)2

)du

],

where µx and σx are the geometric drift and volatility of x, and satisfy

xtµxt = xt

(ρ− c1−γ

t

1− γ+γ

2(σxt )2 − σxt

σat

)+ c−γt (rht + ct − ct + φktat)

+ htc−γt

(ρ− r − γ c

1−γ − ρ1− γ

− γ

2(σxt )2 − γ2σxt σ

ct + γσctat

)xtσ

xt = σxt xt − htc

−γt γσct .

We will show that the rhs of (37) is non-positive. To do this, we will split the integrandinto three parts,

c1−γu − ρ1− γ

xu + xuµxu − xu

γ

2(σxu)2 = Auau +Bu + Cu,

and we will show that At, Bt, and Ct are always non-positive.The term At collects the terms that multiply at:

At = c−γt φkt − xtσxtσ︸ ︷︷ ︸

≤0 by IC

+htc−γt γσct︸ ︷︷ ︸≤0

≤ 0,

where the second term is non-positive because ht ≥ 0 and σct ≤ 0. Since at ≥ 0, we canwlog take at = 0.19

18If dLt 6= 0 we would get an extra negative term on the rhs; dLt > 0 pushes xt down because ht ≥ 0.Since we want to prove that the rhs is non-positive, we can take dLt = 0 wlog.

19Notice that if φ = 1 the expression for At is valid also if we allow at ∈ R. As a result, in this casecontracts with σct = 0 are IC even if we allow at ∈ R.

43

The term Bt collects the remaining terms that do not have volatilities

Bt = xtρ− c1−γ

1− γ+ c−γt (rht + ctxt − ctxt) + htc

−γt (ρ− r − γ c

1−γ − ρ1− γ

) + xtc1−γu − ρ1− γ

.

This expression is maximized for ct = ct, and then it simplifies to

Bt ≤ c−γt (−cthtc−γt ) + htc−γt (−γ c

1−γ

1− γ) + htc

−γt

c1−γt

1− γ= 0.

Finally, Ct collects the terms involving volatilities

Ct = xtγ

2(σxt )2 + htc

−γt (−γ

2(σxt )2 − γ2σxt σ

ct )− xt

γ

2(σxt )2

Ct =γ

2

(xt(σ

xt )2 − htc−γt (σxt )2 − htc−γt 2γσxt σ

ct −

1

xt(xtσ

xt )2

)Ct =

γ

2

(xt(σ

xt )2 − htc−γt (σxt )2 − htc−γt 2γσxt σ

ct −

1

xt

(x2t (σ

xt )2 + (htc

−γt γσct )

2 − xtσxt htc−γt γσct

))

Ct =γ

2

1

xt

(x2t (σ

xt )2 + htc

−γt xt(σ

xt )2 − xthtc−γt (σxt )2 − (htc

−γt )2(σxt )2 − xthtc−γt 2γσxt σ

ct

− (htc−γt )22γσxt σ

ct −

(x2t (σ

xt )2 + (htc

−γt γσct )

2 − 2xtσxt htc

−γt γσct

))

Ct =γ

2

1

xt

(−(htc

−γt )2(σxt )2 − (htc

−γt )22γσxt σ

ct − (htc

−γt γσct )

2)

Ct = −γ2

(htc−γt )2

xt

((σxt )2 + 2γσxt σ

ct + (γσct )

2)

= −γ2

(htc−γt )2

xt

(σxt + γσct

)2≤ 0.

Putting this together and plugging into (37), we get

e−ρt

(U c,at −

x1−γt

1− γ

)≤ Eat

[e−ρτ

n(U c,aτn −

x1−γτn

1− γ

)].

Taking the limit n→∞, it only remains to show that the tail term limn→∞ Eat[e−ρτ

n(U c,aτn −

x1−γτn

1−γ

)]≤

0.Since the agent’s strategy (c, a) is feasible, we have that

limn→∞

Eat[e−ρτ

nU c,aτn

]= 0.

44

For γ < 1 we have x1−γt

1−γ ≥ 0, so it follows that

limn→∞

Eat

[e−ρτ

n x1−γτn

1− γ

]≥ 0.

As a result, when we take n → ∞ we get limn→∞ Eat[e−ρτ

n(U c,aτn −

x1−γτn

1−γ

)]≤ 0, and

therefore U c,at ≤ x1−γt

1−γ as desired.

For γ > 1, if limn→∞ Eat[e−ρτ

n x1−γτn

1−γ

]= 0 for any feasible strategy (c, a), then we are

done. To show this, notice that the law of motion of x satisfies

dxt ≤ (λ1txt − λ2ct)dt+ σxt xt 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 σx and σc are bounded and c bounded below ch and away abovezero. They imply that the drifts µx and µc are also bounded, and we let µx, µc, σc, and cbe appropriate bounds. Notice that stealing only reduces the drift of xt, since the changein the drift of xt and ht cancel out, and it increases the drift of c. Since xt > 0 always,Lemma O.13 and it’s corollary ensure the desired limit.

Theorem 2

The proof is split into parts (which do not correspond to the numbers in the statement ofthe theorem).

(1) The cost function must be bounded above by vh since we can always just giveconsumption to the agent without any capital, and obtain cost vh. It must be strictlypositive because if v(c) = 0 for any c ∈ [0, ch], then we can scale up the contract and giveinfinite utility to the agent at zero cost, or else achieve infinite profits. Because we canalways move c up using dLt, we know that v must be weakly increasing.

(2) Lemma O.11 in the Online Appendix shows that v has (1) a flat portion on (0, cl)

for some cl ∈ (0, ch), where the HJB holds as an inequality; and (2) a strictly increasing,C2 portion (cl, ch) with v′(c) > 0 where the HJB equation holds. At cl we have the smoothpasting condition v′(cl) = 0. To understand this, it is useful to write the function A(c; v)

from (22) as

A(c, v) ≡ c− rv − 1

2

(αcγ

φσ

)2

vγ+ v

ρ− c1−γ

1− γ, (38)

which is the HJB equation when v(c) is flat. Lemma O.11 says that in the flat region (0, cl)

we have A(c, v(c)) > 0. This means that any contract that spends time in this region has

45

cost strictly greater than one that immediately jumps up to cl and attains cost v(c) = v(cl).If the smooth pasting condition didn’t hold, v′(cl) > 0, we would get a kink at cl, whichLemma O.11 rules out.

(3) Now let’s show that A(cl, v(cl)) = 0. 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, (39)

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

)(40)

B = v′c(1 + γ)

αcγ

φσ

vγ + v′c> 0 (41)

C = γv′c(1 + γ)v − v′cvγ + v′c

+ v′′c2. (42)

For the HJB to have a minimum, it must be that C > 0, and hence

σc = −BC< 0 (43)

for all c ∈ (cl, ch). Since v′(cl + ε) > 0 and v′(cl) = 0, we must have v′′(cl + ε) ≥ 0. We canthen show that C

B2 →∞ as c cl, which implies B2

C→ 0 and therefore A→ 0. So we get

that A(c, v(c))→ 0 as c cl, as desired. Now since at cl we have A(c, v(c)) = 0, this is aroot of A(c, v(cl)). For c < cl we have A(c, v(cl)) ≥ 0. From Lemma O.12 we know thatthis can only be the case if cl is the first root of A(c, v(cl)).

(4) Now we want to show that v′′+(cl) > 0. Since we know cl is the first root of A(c, v(cl)),we know that A′1(cl, vl) ≤ 0. Consider the first order ODE

c− rf − σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2

)︸ ︷︷ ︸

A(c,f)

+f ′c

(c1−γ − c1−γ

h

1− γ+

(σx)2

2

)= 0

that fixes σc = 0 and σx = ασγ

cγlvlφ

in the HJB equation. Consider the solution withboundary condition f(cl) = v(cl). Since we already know that A(cl, v(cl)) = 0, we musthave f ′(cl) = 0 because the term in parenthesis is the drift µc if σc = 0, and LemmaO.14 shows this is strictly positive under these conditions. Furthermore, we must have

46

f ′′(cl) ≤ v′′+(cl). To see this, if f ′′(cl) > v′′+(cl) ≥ 0, then f(cl + ε) > v(cl + ε), sinceboth have equal first derivative. We can then slightly lower f(cl) < v(cl). Since the ODEis locally Lipschitz continuous, the solution f changes continuously so it must intersectv above cl, and the drift µc is still strictly positive. The cost f(cl) < v(cl) is thereforeattainable with such a deterministic contract that starts at cl, drifts up, and then revertsto the optimal contract. This cannot be, so we must have f ′′(cl) ≤ v′′+(cl).

Differentiating the first order ODE with respect to c we obtain

0 = A′1(cl, vl) + f ′′(cl)cl

(c1−γ − c1−γ

h

1− γ+

(σx)2

2

)︸ ︷︷ ︸

>0

,

where we have used f ′(cl) = 0 and the envelope theorem to compute the derivativeA′1(cl, vl). If A′1(cl, vl) < 0, it follows that v′′+(cl) ≥ f ′′(cl) > 0. It only remains torule out A′1(cl, v(cl)) = 0 which means A(c, v(cl)) > 0 for all c 6= cl (from Lemma O.12).Since A(c, v) = 0 is the HJB equation if v is flat, by a martingale verification argumentas in Theorem 3 we obtain that the cost of any incentive compatible contract C is strictlylarger than v(cl), unless it has ct = cl always so that A(ct, vl) = 0 always and σx = α

σγ

cγlvlφ

.

This requires µc = σc = 0. But Lemma O.14 shows that for σc = 0 and σx = ασγ

cγlvlφ

, andA(cl, vl) = 0 we get µc(cl) > 0. Since the optimal contract must be incentive compatibleand achieve cost v(cl), A′1(cl, v(cl)) = 0 cannot be. So we have v′′+(cl) > 0.

(5) Now we can study the properties of σc, σx, and µc. First, v′′+(cl) > 0 and v′(cl) = 0

implies σc(cl) = 0 and therefore with Lemma O.14, we have µc(cl) > 0. From (43) we getthat for c ∈ (cl, ch), we have σc(c) < 0, and (39) implies σx(c) > 0. Since v is C2, σc(c)and σx(c) are bounded in any compact subset of (cl, ch). The argument in part (7) belowshows they are also bounded near the boundaries cl and ch, which are inaccessible from theinterior, so σct and σxt are bounded.

(6) Now we can show that v < cγ for all c ∈ [cl, ch), with v(ch) = cγh. We already knowfrom Lemma O.14 that at cl we have v(cl) < cγl . We also know that v(c) ≤ vs(c) for c > ca

(defined in (O.31)), so from Lemma O.15 if ever v(c) = cγ for some c > cl, it must be eitherthat c = 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 O.12 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 meansthat A(c− δ, cγ) > 0 for all δ ∈ (0, c].

Now consider a distortion of the problem, changing α. We can pick an α′ < α so thatvα′(c

α′l ) = v(c), and cα′l < c, because vα′(c) is decreasing in α′. However, Aα(c, cγ) is de-

creasing 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.

47

We have v(ch) = cγh. The only way to avoid reaching ch and becoming absorbed there isto set σc ≈ 0 and σx ≈ 0 near ch, so µc ≈ 0. This means the contract spends an arbitrarilylong time near ch with kt arbitrarily close to zero, so the cost is v(ch) ≥ vh. So we concludethat v(c) < cγ for all c ∈ [cl, ch) and v(ch) = cγh.

(7) We want to show that cl and ch are inaccessible from (cl, ch), so that ct ∈ (cl, ch)

for all t > 0, Lt = 0 always, and the Euler equation holds as an equality. First note thatwe already have proven that σc(c) < 0 in the interior, so ct is a regular diffusion. For cl,use (43) and replace v′ ≈ v′′+ε to obtain σc(cl + ε) ≈ Kε, with K = −σx(cl)(1 + γ)c−1

l < 0,and recall µc(cl)cl = µ > 0. Compute the scale function20

S(c) =

∫ c

exp

(−∫ y 2µc(z)z

(σc(z)z)2dz

)dy.

Using the approximation near cl we obtain S(c) = −∞, so cl is inaccessible, Pτcl <∞ = 0, and non-attracting, Pct → cl = 0. The speed function is

m(c) =1

(σc(c)c)2exp

(∫ c 2µc(z)z

(σc(z)z)2dz

),

and using the approximation we evaluate∫cl

S(c)m(c)dc <∞.

So we conclude that cl is an entrance boundary: ct starts at cl and immediately moves intothe interior and never returns.

To show that ch is inaccessible, we use the approximation in Lemma O.16, µcc =

(4γ−6(1+γ)2)c−γh (ch− c)2 and (σcc)2 = 8(1+γ)2c−γh (ch− c)3. Plug into the expression forthe scale function to obtain S(c) = const×

(−1K+1

)(ch− c)K+1, where K = γ

(1+γ)2 − 32 < −1

for any γ > 0. We take c → ch and obtain S(ch) = ∞. This means ch is inaccessible andnon-attracting too. In fact, we can also show that ch is a natural boundary because∫ ch

S(c)m(c)dc =∞.

(8) Now we show that the optimal contract of the relaxed problem does not have astationary distribution. In the long-run, it spends almost all the time near ch,

1

t

∫ t

01c>ch−ε(cs)ds→ 1 a.s. ∀ε > 0.

20See Karlin and Taylor (1981) Chapter 15, part 6.

48

Using the same approximation near ch in Lemma O.16,

µcc = (4γ − 6(1 + γ)2)c−γh (ch − c)2

(σcc)2 = 8(1 + γ)2c−γh (ch − c)3,

we compute the speed measure

m(c) = (σc(c)c)−2 exp

(∫ c 2µc(z)z

(σc(z)z)2dz

)= const× (ch − c)−K−3,

where K = γ(1+γ)2 − 3

2 < −1, so that −(K + 3) = −(

γ(1+γ)2 + 3

2

)< −3/2. If there is a

stationary distribution is must be proportional to m(c) and integrate to 1. But∫ c

m(y)dy =1

−K − 2(ch − c)−K−2.

When we take c → ch we find that∫ chm(y)dy = ∞ because −(K + 2) < 0, which means

there cannot be a stationary distribution.The same computation near cl shows that

∫ cl+εcl

m(c)dc <∞. This means that

1

t

∫ t

01c<ch−ε(cs)ds→

∫ ch−εcl

m(c)dc∫ chclm(c)dc

= 0 a.s. ∀ε > 0,

which in turn implies that

1

t

∫ t

01c>ch−ε(cs)ds→ 1 a.s. ∀ε > 0.

Pct → ch = 0 follows from S(ch) =∞.

(9) Finally, we show the relaxed optimal contract is globally incentive compatible andtherefore an optimal contract. We know 0 < cl ≤ ct ≤ ch for all t, and σx and σc arebounded. Most importantly, σct ≤ 0 always, so we can use Theorem 1 to obtain our result.

Theorem 3

Consider any locally incentive compatible contract C = (c, k) that delivers utility of at leastu0 to the agent, with associated state variables x and c. Because v′(cl) = 0 we can use Ito’slemma21 and the HJB equation to obtain

e−rτnv(cτn)xτn ≥ v(c0)x0 −

∫ τn

0e−rt

(ct − ktα

)xtdt

21Notice v′′ is discontinuous at cl, but this doesn’t change Ito’s formula. See Proposition 4.12 in Harrison(2013).

49

+

∫ τn

0e−rtv(ct)xt

(v′(ct)

v(ct)ctσ

ct + σxt

)dZt,

for localizing sequence of stopping times τn → ∞ a.s. such that the last term has expec-tation zero. Take expectations to obtain

EQ0[e−rτ

nv(cτn)xτn

]≥ v(c0)x0 − EQ0

[∫ τn

0e−rt (ct − ktα) dt

]. (44)

Now we would like to take the limit n → ∞. For the second term on the right we canuse the dominated convergence theorem, using the fact that locally incentive compatiblecontracts are admissible.

Also, for an admissible contract we have

0 ≤ limn→∞

EQ0[e−rτ

nv(cτn)xτn

]≤ lim

n→∞EQ0[e−rτ

nvhxτn

]= 0.

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

∞ > EQ[∫ ∞

0e−rtctdt

]≥ EQ

[∫ τn

0e−rtctdt+ e−rτ

nvhxτn

].

Taking the limit n → ∞ and using the dominated convergence theorem, we obtain 0 ≤limn→∞ EQ

[e−rτ

nvhxτn

]≤ 0.

Upon taking the limit n→∞ in (44), we obtain

EQ0

[∫ ∞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 we use the same steps as in the proof of Theorem 2 to show

that c∗t ∈ (cl, ch) for all t > 0, and that σc∗t < 0, σx∗t > 0, µx∗t , and µc∗t are all bounded. Wealso know that, given x0 > 0, any solution x∗ to (16) is strictly positive. Lemma 4 thenensures C∗ delivers utility U c

∗,0t = (x∗t )

1−γ/(1− γ), with U c∗,0

0 = u0. Since C∗ is admissible,it is locally incentive compatible. The same argument as in the first part shows the cost ofC∗ is vlx∗0. C∗ is therefore a relaxed optimal contract, which implies it is also an optimalcontract.

50

51

Online AppendixThis Online Appendix extends the results of Di Tella and Sannikov (2016) to incorporatehidden investment, aggregate risk, and renegotiation. The case with no hidden investmentand price of aggregate risk π = 0 yields the expressions in the paper.

A Aggregate risk and hidden investment

We introduce aggregate risk and hidden investment into the baseline setting in the paper.The observed return is:

dRt = (r + πσ + α− at) dt+ σ dZt + σ dZt, (O.1)

where Z and Z are independent Brownian motions that represents idiosyncratic and aggre-gate risk. There is a complete financial market with equivalent martingale measure Q. Therisk-free rate is r, aggregate risk has market price π, and idiosyncratic risk is not priced.Capital has a loading σ on idiosyncratic risk and σ on aggregate risk, so the excess returnon capital for the agent is α, as in the baseline.

The agent receives cumulative payments I from the principal and manages capital kfor him. Payments I can be any semimartingale (it could be decreasing if the agent mustpay the principal). This nests the relevant case where the contract gives the agent onlywhat he will consume, i.e. dIt = ctdt. As in the baseline setting, the agent can stealfrom the principal at rate at ≥ 0 and decide when to consume ct > 0. He can invest hishidden savings in the same way the principal would, not only in a risk-free asset, but alsoin aggregate risk Z. In addition, the agent may be able to invest his hidden savings in hisprivate 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+.22

The agent’s utility is

U0 = E

[∫ ∞0

e−ρtc1−γt

1− γdt

]22We 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.

52

and the cost to the principal is

J0 = EQ[∫ ∞

0e−rt (dIt − (α− at)kt) dt

].

A contract C = (I, k, c, a, z, z) specifies the contractible payments I and capital k, andrecommends the hidden action (c, a, z, z), all contingent on the history of observed returnsR and the 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) utility U c,a,z,z0 is finite, and (2)hidden savings ht ≥ 0 always. Since the agent can secretly invest in his private technology,we also impose the regularity condition (3) EQ

[∫∞0 e−rt (ct + αztht) dt

]<∞. Let S(C) be

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

EQ[∫ ∞

0e−rtdIt

]<∞, EQ

[∫ ∞0

e−rtktdt

]<∞, EQ

[∫ ∞0

e−rtatktdt

]<∞. (O.2)

An admissible contract C = (I, k, c, a, z, z) is incentive compatible if the agent’s optimalfeasible strategy given C is (c, a, z, z), as recommended by the principal. Let IC be theset of incentive compatible contracts. An incentive compatible contract is optimal if itminimizes the principal’s cost

v0 = minCJ0(C) ≡ EQ

[∫ ∞0

e−rt (dIt − (α− at)kt) dt]

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

.

A.1 No stealing or hidden savings in the optimal contract

Lemma O.1. 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 hidden

53

investment.

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

∫ t0 csds and k

′ = k(Ra) + kh (to simplify notation, we suppress dependence on Z).If the agent now chooses c′ = dI ′, a = z = z = 0, he gets hidden savings h′ = 0 and

consumption c, so he gets the same utility as under the original contract and this strategy istherefore feasible under the new contract. If instead he chooses a different feasible strategy(c′, a′, z′, z′), he gets the utility associated with c′. We will show that he could achievethis 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 strategyis feasible under the original contract we only need to show that hidden savings remainnon-negative always

h′t =

∫ t

0er(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 twonon-negative terms. First, the hidden savings under the original contract, following theoriginal feasible strategy, had Ra′ been the true return

At =

∫ t

0er(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 =

∫ t

0er(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, becausedRt − 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 utilitythat he could have obtained - but chose not to - under the original contract C.

54

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

J ′0 = EQ[∫ τn

0e−rt

(ct − α(kt(R

a) + kht ))dt+ e−rτ

nJ ′τn

]=

EQ[∫ τn

0e−rt (dIt(R

a)− (α− at)kt(Ra)dt) + e−rτnJτn

]−EQ

[∫ τn

0e−rtatkt(R

a)(1− φ)dt

]

−EQ[∫ τn

0e−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 thedestruction produced by stealing under the original contract, which is non-negative; andthe third term is EQ

[e−rτ

nhτn]≥ 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[∫ τn

0d(e−rtht)

]= EQ

[∫ τn

0e−rt

(dIt(R

a)− ctdt+ φkt(Ra)atdt+ kht αdt

)]= EQ

[e−rτ

nhτn − 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

nJτn]

= 0. Since in ad-dition the agent’s response was feasible, the new contract is also admissible, and we getlimn→∞ EQ

[e−rτ

nJ ′τn]

= 0 as well. This shows the new contract is admissible, and the costfor the principal is not greater than under the old contract. 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.

55

A.2 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.

Lemma O.2. If C = (c, k) is an incentive compatible contract, the agent’s continuationutility U c,0 and consumption c satisfy the laws of motion

dU c,0t =

(ρU c,0t −

c1−γt

1− γ

)dt+ ∆tσdZt + σut dZt, (O.3)

dctct

=

(r − ργ

+1 + γ

2(σct )

2 +1 + γ

2(σct )

2

)dt+ σctdZt + σctdZt + dLt. (O.4)

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

∆t ≥ c−γt φkt (O.5)

z(α− σctσγ) ≤ 0 ∀z ∈ H (O.6)

σct =π

γ. (O.7)

Proof. The proof of (O.3) and (O.5) are similar to Lemma 1 and 2, where the dZ termappears because the contract can depend on the history of aggregate shocks. For (O.4), theproof is analogous to Lemma 2, but now 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 (O.8)

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

dctct

= µctdt+ σctdZt + σctdZt + dLt.

Since the finite variation part of expression (O.8) must be non-increasing, we get(r − ρ− γµct +

γ

2((1 + γ)σct )

2 +γ

2((1 + γ)σct )

2)dt (O.9)

56

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

Taking z = z = 0, which are always allowed, we obtain wlog the expression for µc in (O.4),and L weakly increasing. Once we plug this into (O.9), and using that zt can be bothpositive or negative, we get (O.7). Condition (O.6) is therefore necessary to ensure (O.9)holds.

The IC constraint (O.6) depends on whether the agent is allowed to have a hiddeninvestment in his own private technology. If hidden investment in the agent’s private tech-nology is not allowed, H = 0 so condition (O.6) drops out. If instead hidden investmentin the agent’s private technology is allowed, H = R+, so condition (O.6) reduces to σct ≥ α

σγ .

A.3 Change of variables

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, (O.10)

dctct

=(r − ρ

γ− ρ− c1−γ

t

1− γ+

(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2 (O.11)

+(σxt )2

2+ γσxt σ

ct +

1 + γ

2(σct )

2)dt+ σct dZt + σct dZt + dLt,

dLt ≥ 0

and the incentive compatibility constraints can be written

σxt ≤ c−γt φktσ (O.12)

z(α− (σct + σxt )σγ) ≤ 0 ∀z ∈ H (O.13)

σct + σxt =π

γ. (O.14)

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 O.3. For any incentive compatible contract C = (c, k), we have for all t

ct ≤ ch, (O.15)

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

(O.16)

57

st : z(α− σxσγ) ≤ 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 (O.10), where σx is the optimizingchoice in (O.16) and σx = π

γ . Let vh be the cost of this continuation contract:

vh =ch − α

φσ cγhσ

x

r − ρ−c1−γ1−γ −

γ2 (σx)2 + γ

2 (πγ )2.

Under sufficient technical conditions (e.g. if σc and σc are bounded), we have

ct ≤ EPt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcu)2− 1−γ

2(σcu)2

)duds

]− 11−γ

, (O.17)

with equality if Lt = 0 always, where P is an equivalent measure such that Zt−∫ t

0 (1−γ)σcsds

and Zt −∫ t

0 (1− γ)σcsds are P -martingales.

Proof. The same reasoning as in Lemma 3 yields ct ≤(ρ−r(1−γ)

γ

) 11−γ . For any c between

ch and(ρ−r(1−γ)

γ

) 11−γ , preventing ct from crossing above c requires σc = σc = 0 at that

point and µc ≤ 0. But σc = σc = 0 implies the drift of ct is

µct =r − ργ− ρ− c1−γ

t

1− γ+

(σxt )2

2+

(σxt )2

2.

If ever ct > ch, the drift is strictly positive for any σxt and σxt satisfying (O.13) and (O.14),so we must have ct ≤ ch at all times, and if ever ct = ch, it must remain absorbed thereforever. Using the IC constraint (O.12) and the law of motion of x we obtain σxt and σxt inthe continuation contract, and from (O.12) we get kt. The cost of the continuation contractwith ct+s = ch and kt =

σxcγhφσ for all future times t + s can be obtained from the HJB

equation with σc = σc = µc = 0, or simply applying the formula (O.32) for the cost ofstationary contracts at c = ch.

For (O.17), the same reasoning as in Lemma 3 gives us

ct ≤ EPt[∫ ∞

te−∫ st

(ρ−r(1−γ)

γ− 1−γ

2(σcu)2− 1−γ

2(σcu)2

)duds

]− 11−γ

,

with equality if Lt = 0 always, where P is an equivalent measure such that Zt−∫ t

0 (1−γ)σcsds

and Zt −∫ t

0 (1− γ)σcsds are P -martingales.

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 σc = 0 and we get

58

ch =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 1

1−γ. Notice that if π = 0 this boils down to expression (14)

in the baseline setting without aggregate risk or hidden investment. If the agent can alsoinvest his hidden savings in his own private technology, H = R+, then the maximizing

choice is σc = ασγ , and ch =

(ρ−r(1−γ)

γ − 1−γ2

(ασγ

)2− 1−γ

2

(πγ

)2) 1

1−γis lower.

We call any admissible contract locally incentive compatible if ct ≤ ch, and (O.11)-(O.14)hold. Notice that equation (O.10) follows automatically from the definition of xt.

We can build a locally incentive compatible contract from processes x > 0 and c > 0

satisfying (O.10), (O.11), (O.13), and (O.14), with σxt ≥ 0 and ct ≤ ch. Define the contract(c, k) by ct = ctxt > 0 and kt = xtσ

xt cγt /(φσ) ≥ 0. Then, under technical conditions, the

contract (c, k) is admissible as defined in Section 2 and delivers utility U c,0t =x1−γt

1−γ undergood behavior. It is locally incentive compatible by definition.

Lemma O.4. Let x > 0 and c > 0 be stochastic processes satisfying (O.10), (O.11), (O.13),and (O.14), bounded volatilities σx ≥ 0 and σx, and with c bounded away from zero andabove by ch.

Then the contract C = (c, k) with ct = ct× xt > 0 and kt = σxt cγt /(φσ)× xt ≥ 0 delivers

utility U c,0t =x1−γt

1−γ if the agent follows strategy (c, 0). The contract C is admissible andtherefore locally incentive compatible if and only if EQ

[∫∞0 e−rtxtdt

]<∞.

Proof. The proof is analogous to Lemma 4 in the main body of the paper. First we showthe contract delivers utility x1−γ

t1−γ < ∞ if the agent follows strategy (c, 0). Let Yt =

x1−γt

1−γ ,and using the law of motion of x, (16), we get

dYt = Yt(1− γ)(µxt −γ

2(σxt )2 − γ

2(σxt )2)︸ ︷︷ ︸

ρYt−c1−γt1−γ

dt+ Yt(1− γ)σxt dZt + Yt(1− γ)σxt dZt. (O.18)

Integrating we obtain

Y0 = E

[∫ τn

0e−ρs

c1−γs

1− γds+ e−ρτ

nYτn

]

for an increasing sequence of bounded stopping times with τn → ∞ a.s. Take the limitn→∞, using the monotone convergence theorem on the first term to get

Y0 = E

[∫ ∞0

e−ρsc1−γs

1− γds

]︸ ︷︷ ︸

Uc,00

+ limn→∞

E[e−ρτ

nYτn].

59

We will now show that the last term is zero,

limn→∞

E[e−ρτ

nYτn]

= 0.

Since σx and σx are bounded and c bounded away from zero and above by ch, µx is boundedtoo, and so is therefore the growth rate, (1 − γ)(µxt −

γ2 (σxt )2 − γ

2 (σxt )2), and volatilities,(1− γ)σxt and (1− γ)σxt , of Yt in (O.18). Furthermore, the growth rate of Yt in (O.18), isbounded away below ρ,

(1− γ)(µxt −γ

2(σxt )2 − γ

2(σxt )2)− ρ = −c1−γ

t ≤ max−c1−γ , −c1−γ

h

< 0,

where c is a lower bound on ct. We then get that limn→∞ E[e−ρτ

nYτn]

= 0 and therefore

U c,00 = Y0 =x1−γ

01−γ <∞. The same reasoning yields U c,0t =

x1−γt

1−γ <∞ for all t.Now we show that the resulting contract C is admissible if and only if EQ

[∫∞0 e−rtxtdt

]<

∞. To show sufficiency, notice that since c is bounded above by ch and σx bounded, wecan write

EQ[∫ ∞

0e−rt(ct + kt)dt

]≤ 2 max

ch,

σxcγhφσ

EQ[∫ ∞

0e−rtxtdt

]<∞,

where σx is an upper bound on σxt . For necessity, since ct is bounded away from zero,ct ≥ c > 0,

∞ > EQ[∫ ∞

0e−rtctdt

]≥ c× EQ

[∫ ∞0

e−rtxtdt

].

Since the contract is admissible and satisfies (O.11)-(O.14), and ct ≤ ch, it is locally incen-tive compatible.

A.4 Sufficient conditions for global incentive compatibility

Incentive compatible contracts are locally incentive compatible. Here we provide sufficientconditions for a locally incentive compatible contract to be incentive compatible. We canextend Theorem 1 to verify global incentive compatibility.

Theorem O.1. Let C = (c, k) be locally incentive compatible contract with c bounded awayfor zero and bounded volatilities σx, σx, σc and σc. Suppose that the contract satisfies thefollowing 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 .

60

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

Proof. Focus on the simple case with dLt = 0; the proof can be easily generalized fordLt ≥ 0. Following the same steps as in the proof of Theorem 1, we obtain

e−ρt

(U c,at −

x1−γt

1− γ

)= Eat

[∫ τn

te−ρu

c1−γu

1− γdu+

∫ τn

td

(e−ρu

x1−γu

1− γ

)+ e−ρτ

n(U c,aτn −

x1−γτn

1− γ

)],

where xt = xt + htc−γt , and ct = ct/xt. The first two terms can be written

Eat

[∫ τn

te−ρu

c1−γu

1− γdu+

∫ τn

td

(e−ρu

x1−γu

1− γ

)](O.19)

= Eat

[∫ τn

te−ρux−γu

(c1−γu − ρ1− γ

xu + xuµxu − xu

γ

2(σxu)2 − xu

γ

2(σxu)2

)du

],

where

xtµxt = xt

(ρ− c1−γ

t

1− γ+γ

2(σxt )2 +

γ

2(σxt )2 − σxt

σat

)+ c−γt (rht + ztht(α+ πσ) + zthtπ + ct − ct + φktat)

+ htc−γt

(ρ− r − γ c

1−γ − ρ1− γ

− γ

2(σxt )2 − γ2σxt σ

ct −

γ

2(σxt )2 − γ2σxt σ

ct + γσctat

)− htc−γt γσctztσ − htc

−γt γσct (ztσ + zt)

xtσxt = σxt xt − htc

−γt γσct + htc

−γt ztσ

xtσxt = σxt − htc

−γt γσct + htc

−γt (ztσ + zt).

Following the proof of Theorem 1, we can write the integrand as the sum of four parts

c1−γu − ρ1− γ

xu + xuµxu − xu

γ

2(σxu)2 − xu

γ

2(σxu)2 = Auau +Bu + Cu + Cu.

At and Bt are unchanged and we know they are non-positive:

At = c−γt φkt − xtσxtσ

+ htc−γt γσct ≤ 0

Bt = xtρ− c1−γ

1− γ+ c−γt (rht + ctxt − ctxt) + htc

−γt (ρ− r − γ c

1−γ − ρ1− γ

) + xtc1−γu − ρ1− γ

≤ 0.

Ct needs to be modified to account for hidden investment, and the new term Ct collects

61

the terms dealing with aggregate risk.

Ct = xtγ

2(σxt )2 + htc

−γt

(ztα− γσctztσ −

γ

2(σxt )2 − γ2σxt σ

ct

)− xt

γ

2(σxt )2

Ct =γ

2

(xt(σ

xt )2 + 2htc

−γt zt(

α

γ− σctσ)− htc−γt (σxt )2 − htc−γt 2γσxt σ

ct −

1

xt(xtσ

xt )2

).

Notice that we included the zthtα term here. We will include π(zthtσ + ztht) in Ct.Expand (xtσ

xt )2 :

Ct =γ

2

(xt(σ

xt )2 + 2htc

−γt zt(

α

γ− σctσ)− htc−γt (σxt )2 − htc−γt 2γσxt σ

ct

− 1

xt

(x2t (σ

xt )2 + (htc

−γt (ztσ − γσct ))2 + 2xtσ

xt htc

−γt (ztσ − γσct )

)).

Take the 1/xt out of the parenthesis:

Ct =γ

2

1

xt

(x2t (σ

xt )2 + htc

−γt xt(σ

xt )2 + 2xthtc

−γt zt(

α

γ− σctσ) + 2

(htc−γt

)2zt(α

γ− σctσ)

− xthtc−γt (σxt )2 −(htc−γt

)2(σxt )2 − xthtc−γt 2γσxt σ

ct −

(htc−γt

)22γσxt σ

ct

−(x2t (σ

xt )2 + (htc

−γt (ztσ − γσct ))2 + 2xtσ

xt htc

−γt (ztσ − γσct )

)).

Cancel some terms:

Ct =γ

2

1

xt

(2xthtc

−γt zt(

α

γ− σctσ) + 2

(htc−γt

)2zt(α

γ− σctσ)−

(htc−γt

)2(σxt )2

−(htc−γt

)22γσxt σ

ct − (htc

−γt (ztσ − γσct ))2 − 2xtσ

xt htc

−γt ztσ

).

And group the remaining ones to form a square:

Ct = −γ2

(htc−γt

)2

xt

((σxt )2 + (γσct − ztσ)2 + 2σxt (γσct − ztσ)

)+γ

2

1

xt

(2xthtc

−γt zt(

α

γ− (σxt + σct )σ) + 2

(htc−γt

)2zt(α

γ− (σxt + σct )σ)

)

62

Ct = −γ2

(htc−γt

)2

xt

(σxt + (γσct − ztσ)

)2+

1

xt

(xt + htc

−γt

)htc−γt zt(α− γ(σxt + σct )σ).

Rearrange:

Ct = −γ2

(htc−γt

)2

xt

(σxt + (γσct − ztσ)

)2+ htc

−γt ztσ(

α

σ− γ(σxt + σct )) ≤ 0.

where the last inequality uses the IC constraint for hidden investment (O.13).The term Ct collects the terms dealing with aggregate risk:

Ct = xtγ

2(σxt )2 + htc

−γt

((ztσ + zt)π − γσct (ztσ + zt)−

γ

2(σxt )2 − γ2σxt σ

ct

)− xt

γ

2(σxt )2.

This term is the same as Ct except that α/σ is replaced with π and ztσ is replaced withztσ+ zt, and all the volatilities are with respect to the aggregate shock. The same steps asabove, therefore, lead to

Ct = −γ2

(htc−γt

)2

xt

(σxt + (γσct − (ztσ + zt))

)2+ htc

−γt (ztσ + zt)(π − γ(σxt + σct )) ≤ 0,

where the last inequality follows from the IC constraint for hidden investment in aggregaterisk (O.14).

The rest of the proof dealing with the terminal term follows the same steps as in Theorem1 using Lemma O.13.

A.5 The solution to the relaxed problem gives the optimal contract

The relaxed problem minimizes cost within the class of locally incentive compatible con-tracts, and we call a solution to the relaxed problem a relaxed optimal contract. As inthe baseline setting, the relaxed optimal contract is in fact globally incentive compatible,and therefore an optimal contract. The solution to the relaxed problem can be character-ized with the same HJB equation as in the case without hidden investment, appropriatelyextended to incorporate aggregate risk and the new 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

)(O.20)

+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)

,

63

subject to σx ≥ 0 and (O.13) and (O.14).Using (O.14) to eliminate σc, and taking FOC for σx, we obtain

σx =π

γ, σc = 0.

This is the first best exposure to aggregate risk. The principal and the agent do not 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.23

The FOC for σx and σc depend on whether the agent can invest his hidden savings in hisprivate technology. Without hidden investment, the FOCs are the same as in the baseline.With hidden investment, the IC constraint (O.13) could be binding in some region of thestate space. The shape of the contract, however, is the same as in the baseline withouthidden investment.

It is useful to define

A(c, v) ≡ minσx≥0

c− σxcγ αφσ− rv + v

(ρ− c1−γ

1− γ+γ

2(σx)2 − γ

2

(πγ

)2). (O.21)

The HJB equation when v′ = v′′ = 0 and (O.13) is not binding is A(c, v) = 0.

Theorem O.2. The relaxed problem has the following properties:

(1) The cost function v(c) has a flat portion on (0, cl) and a strictly increasing C2

portion on (cl, ch), for some cl ∈ (0, ch). The HJB equation (O.20) holds with equality inc ∈ (cl, ch). For c < cl, we have v(c) = v(cl) ≡ vl and the HJB holds as an inequality,A(c, vl) > 0.(2) At cl, we have v′(cl) = 0, v′′+(cl) > 0, and A(cl, vl) = 0. The cost function satisfies

v(c) < cγ for all c ∈ [cl, ch), with v(ch) = vh.(3) The state variables xt and ct follow the laws of motion (O.10) and (O.11) with bounded

σxt > 0, σct < 0, σxt = πγ , and σct = 0 for all t > 0, and dLt = 0 always, so the Euler

equation holds as an equality. The state ct starts at c0 = cl, with µc0 > 0 and σc0 = σc0 = 0,and immediately moves into the interior of the domain never reaching either boundary, thatis, ct ∈ (cl, ch) for all t > 0.(4) Without hidden investment, the optimal contract in the relaxed problem does not have

a stationary distribution:

1

t

∫ t

01ct>ch−ε(cs)ds→ 1 a.s. ∀ε > 0,

23If 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.

64

but Pct → ch = 0. With hidden investment, the optimal contract in the relaxed problemhas a stationary distribution with density proportional to m(c) = 1

σc(c)cexp

(∫ c 2µc(z)z(σc(z)z)2dz

),

which spikes near ch, i.e. m(c)→∞ as c→ ch.

(5) Since the relaxed optimal contract satisfies the sufficient condition in Theorem O.1, itis incentive compatible and therefore an optimal contract.

Proof. The proof is similar to Theorem 2, except we use the more general definition ofA(c, v) in (O.21), which once we optimize over σx can be written

A(c, v) = c− rv − 1

2

(cγαφσ

)vγ

+ v

(ρ− c1−γ

1− γ− 1

2

π2

γ

).

Notice we already know from the FOCs that σx = π/γ and σc = 0.Part (1) and part (2) go through without modifications.In Part (3), the proof that A(cl, v(cl)) = 0 requires that we consider the possibility

that the hidden investment constraint is binding as we approach cl. Since the right-handside of the HJB can only be greater if the hidden investment constraint is binding, we getthat A(cl, v(cl)) ≤ 0. But we know that A(c, v(cl)) ≥ 0 for c < cl from Lemma O.11, andsince A is continuous in c, we get that A(cl, v(cl)) = 0, as desired.

Part 4 goes through with natural modifications. The first-order ODE has the naturalmodification

c− rf − σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)︸ ︷︷ ︸

A(c,f)

+f ′c

(c1−γ − (cu)1−γ

1− γ+

(σx)2

2

)= 0

where we are fixing σc = 0 and σx = ασγ

cγlvlφ

. This is consistent with the hidden investmentconstraint because cγl ≥ vl from Lemma O.14. The rest of the proof goes though and weget v′′+(cl) > 0.

Part (5) and (6) go though with appropriate modifications, noticing that σc(cl) = 0

and σx(cl) = ασγ

cγlvl

satisfy the hidden investment constraint because vl ≤ cγl , from lemmaO.14. The proof that v(c) < cγ for c ∈ (cl, ch) is unchanged, except replacing ch with cuwhere appropriate. We only need to check that σc(c) < 0 and σx(c) > 0 for all c ∈ (cl, ch)

in the case where the hidden investment constraint is binding. In that case σc = ασγ − σ

x,and the FOC for σx yields:

σx =cγ α

φσ + v′′c2 ασγ

γ(v − v′c) + v′′c2>

α

σγ,

65

which implies σc < 0. To see this inequality, use v < cγ to write

φc−γ v < 1,

and use v′ ≥ 0 to get

φc−γ(v − v′c) < 1

α

σ(v − v′c) < cγ

α

φσ

α

γσ

(γ(v − v′c) + v′′c2

)< cγ

α

φσ+ v′′c2 α

γσ.

Finally, divide throughout by γ(v− v′c)+ v′′c2 which must be strictly positive (second ordercondition for optimality)

α

γσ<

cγ αφσ + v′′c2 α

γσ

γ(v − v′c) + v′′c2= σx.

Part (7) is unchanged for the behavior near cl. For ch we need to consider two cases.Without hidden investment, Lemma O.16 shows that

µcc ≈ (4γ − 6(1 + γ)2)c−γh ε

σcc ≈ −√

22(1 + γ)c−γ/2h ε3/2,

and the same analysis as in Theorem 2 shows that ch is inaccessible. With hidden in-vestment, the IC constraint will be binding near the upper boundary. Lemma O.16 showsthat

µcc ≈ (η − 2)1

2

(α

σγ

)2( γ

1− η

)2

(ch − c) < 0

σcc ≈ −(α

σγ

)γ

1− η(ch − c),

for some η ∈ (0, 1). We can compute the scale function

S(c) =

∫ c

exp

(−∫ y 2µ

σ2

1

ch − zdz

)dy = − 1

2µ/σ2 + 1(ch − c)

2µ

σ2 +1,

where µ = (η−2)12

(ασγ

)2 (γ

1−η

)2< 0 and σ2 =

(ασγ

)2 (γ

1−η

)2, so that 2µ/σ2 = η−2 < −1.

So S(ch) =∞, which means that ch is inaccessible and non-attracting (Pct → ch = 0).For Part (8), without hidden investment the proof is unchanged. For the case with

66

hidden investment, for the behavior near ch we must compute the speed measure

m(c) =1

σ2(ch − c)2exp

(∫ c 2µ

σ2

1

ch − zdz

).

Using the approximation,

µcc ≈ (η − 2)1

2

(α

σγ

)2( γ

1− η

)2

(ch − c) < 0

σcc ≈ −(α

σγ

)γ

1− η(ch − c),

we get that near chm(c) ≈ 1

σ2(ch − c)−

2µ

σ2−2,

where −2µσ2 − 2 = 2 − η − 2 = −η < 0. This means that m(c) → ∞ as c → ch. But the

integral of m(c) is

M(c) =

∫ c 1

σ2(ch − z)−ηdz =

1

1− η1

σ2(ch − c)1−η,

which is finite as c→ ch. Since cl is an entrance boundary, we have a stationary distribution,

ψ(c) =m(c)∫ ch

clm(z)dz

,

with a spike near ch.Part (9) uses the more general Theorem O.1.

For a given solution to the HJB equation, we can identify controls σx and σc as functionsof c, and use those to build a candidate optimal contract C∗. Specifically, let x∗ and c∗ bethe solutions to (O.10) and (O.11) with σx∗t = σx(c∗t ), σ

c∗t = σc(c∗t ) and dLt = 0, starting

from initial values x∗0 = ((1− γ)u0)1

1−γ and c∗0 = cl. We then construct the candidatecontract C∗ = (c∗, k∗) with c∗ = c∗x∗ and k∗ = σx

∗ (c∗)γ

φσ x∗.

Theorem O.3. Let v(c) : [cl, ch]→ [vl, vh] be a strictly increasing C2 solution to the HJBequation (O.20) for some cl ∈ (0, ch), such that vl ≡ v(cl) ∈ (0, vh], v′(cl) = 0, v′′(cl) > 0

and v(ch) = vh. Assume that for c < cl the HJB equation holds as an inequality, A(c, vl) > 0,and that v(c) ≤ cγ for c ∈ [cl, ch]. Then,

(1) For any locally incentive compatible contract C = (c, k) that delivers at least utilityu0 to the agent, we have v(cl) ((1− γ)u0)

11−γ ≤ J0(C).

(2) Let C∗ be a candidate optimal contract generated by the policy functions of the HJBas described above. If C∗ is admissible then C∗ is an optimal contract with cost J0(C∗) =

67

v(cl) ((1− γ)u0)1

1−γ .

Proof. The proof is very similar to Theorem 3, except we use the more general Lemma O.4and Theorem O.2.

The following Lemma is useful to ensure the existence of an optimal contract.

Lemma O.5. When γ ≥ 1/2, if α ≤ φσ√γ√

2

√ρ−r(1−γ)

γ − 1−γ2

(πγ

)2then v ≥ cγh/2 > 0.

When γ ≤ 1/2, if α ≤ φσγ√

2(1− γ)

√ρ−r(1−γ)

γ − 1−γ2

(πγ

)2then v ≥ (1−γ)(2γ)

γ1−γ cγh > 0.

Proof. For the case γ ≥ 1/2. We will show that vl = cγh/2 is a lower bound on the costfunction. To do this, it’s sufficient to show that A(c, vl) ≥ 0 for any c ∈ (0, ch).

A(c, vl) = c+ vlγc1−γh − c1−γ

1− γ− γ

2

c2γ(αφσ

)2

vlγ2

≥ c+cγh2

γc1−γh − c1−γ

1− γ− 1

2c2γ c1−2γ

h =c1+γh c−γ

2

(2y1+γ +

γyγ − y1− γ

− y3γ

),

where y = c/ch ∈ (0, 1). Since y3γ < y1+γ the expression in parenthesis is greater or equalto

y1+γ +γyγ − y1− γ

.

Here we have three powers of y, and the middle coefficient is always negative, while theoutside coefficients are positive (this is true both if γ < 1 and γ > 1). Moreover, the sumof the coefficients is 0 and the weighted sum (with weights equal to the powers) is

1 + γ +γ2 − 1

1− γ= 0.

Hence, by Jensen’s inequality, the expression is positive.For the case γ ≤ 1/2. We will show that vl = (1− γ)cγm is a lower bound, where cm is

defined by 2γc1−γh = c1−γ

m . We have

A(c, vl) = c+ vlγc1−γh − c1−γ

1− γ− γ

2

c2γ(αφσ

)2

vlγ2

≥ c+ cγm(c1−γm /2− c1−γ)− 1

2

c2γ c1−γm

cγm

= c+ cm/2− c1−γ cγm −1

2c2γ c1−2γ

m

= (cm2

(1 + (c/cm)γ)− c1−γ cγm)(1− (c/cm)γ).

68

If c/cm < 1 then c1−γ cγm < cγ c1−γm < cm and therefore the expression is positive. If

c/cm > 1, then c1−γ cγm > cγ c1−γm > cm and also the expression is positive. This completes

the proof.

A.6 Benchmark contracts and autarky limit

We can extend the benchmark contracts in Section 3 to incorporate aggregate risk. Inaddition, we can find conditions under which the gains from trade are exhausted and theoptimal contract coincides with autarky, as mentioned in Section 4 in the paper.

Without hidden savings

The optimal contract without hidden savings is characterized by the HJB equation:

rvn = minσx,c,σx

c− σxcγ αφσ

+ vn

(ρ− c1−γ

1− γ+γ

2(σx)2 +

γ

2(σx)2 − σxπ

), (O.22)

where vn(x) = vnx is the principal’s cost function. The FOC are:

σx =α

γ(vnc−γn φ)σ

(O.23)

1 = vnc−γ + vnγ

2(σx)2c−1 (O.24)

σx =π

γ. (O.25)

The optimal contract exists only if γ ≤ 1/2 and only if α is sufficiently low; otherwise theprincipal’s value function becomes infinite.

The inverse Euler equation says that e(r−ρ)tcγt is a Q-martingale. If the contract hasconstant c, it requires

σx =

√(cu)1−γ − c1−γ

1− γ2

1− 2γ, (O.26)

where

cu =

(ρ− r(1− γ

γ− (1− γ)

1

2(π

γ)2

) 11−γ

(O.27)

coincides with ch without hidden investment.

Lemma O.6. The optimal contract without hidden savings satisfies the inverse Euler equa-tion, i.e. e(r−ρ)tcγt is a Q-martingale, and myopic optimization over σx, i.e. (O.23). Themarginal cost of utility is lower than the inverse of the marginal utility of consumption,vn < cγn.

Proof. Myopic optimization follows from the FOC (O.23), and vn < cγn from FOC (O.24).

69

Given stationarity, the inverse Euler equation is equivalent to:

µx =r − ργ

+ (1− γ)1

2(σx)2 + (1 + γ)

1

2(σx)2.

Using the FOC for c we can write

=⇒ vc1−γ = c− vγ2(σx)2. (O.28)

Plug into the HJB (O.22) along with the FOC for σx, (O.23), to obtain

rv = c− (σx)2vγ +1

2(σx)2vγ + v

ρ− c1−γ

1− γ− vγ (σx)2

2.

Divide by v and use (O.28), to obtain

c1−γ =

(ρ− r(1− γ)

γ− (1− γ)

(σx)2

2

)− (σx)2

2(1− 2γ)(1− γ). (O.29)

And now compute µx :

µx =ρ− c1−γ

1− γ+γ

2(σx)2 +

γ

2(σx)2.

After some algebra we obtain the inverse Euler equation

µx =r − ργ

+ (1− γ)1

2(σx)2 + (1 + γ)

1

2(σx)2.

Stationary contracts and the myopic contract

Stationary contracts have a constant c and are obtained by setting σc = σc = 0, σx = π/γ,and σx to satisfy

σx = σxs (c) ≡√

2

√(cu)1−γ − c1−γ

1− γ, (O.30)

so that µc = 0 in (17). To ensure admissibility, we must restrict

c > ca ≡ cu(

2γ

1 + γ

) 11−γ

< cu, (O.31)

so that µx − π2

γ < r. Theorem O.1 is general enough to ensure that stationary contractsare globally incentive compatible. The HJB equation (21) yields the cost of the stationary

70

contract,

vs(c) =c− α

φσ cγσxs (c)

2r − ρ− (1 + γ)ρ−c1−γ

1−γ + γ(π/γ)2. (O.32)

A special stationary contract corresponds to myopic optimization,

σxs (cp) =α

γ(vs(cp)c−γp φ)σ

,

which yields

cp =

(c1−γh − (1− γ)

1

2

(α

γφσ

)2

− (1− γ)1

2(π/γ)2

) 11−γ

, σxp =α

γφσ, vp = cγp .

(O.33)The best stationary contract minimizes the cost, cr ≡ arg minc∈(ca,ch] vs(c) and vr ≡ vs(cr).

Lemma O.7. For any c ∈ (ca, ch], the corresponding stationary contract is globally incentivecompatible and has cost vs(c) given by (O.32). Since stationary contracts are incentivecompatible, we have v(c) ≤ vs(c).

The myopic stationary contract is an incentive compatible stationary contract corre-sponding to cp, and the marginal cost of utility is equal to the inverse of the marginal utilityof consumption, vs(cp) = cγp . The best stationary contract is less risky for the agent, i.e. wehave ca < cp < cr and σxs (cr) < σxp . For all c ∈ (cp, ch) the marginal cost of utility is belowthe inverse of the marginal utility of consumption vs(c) < cγ, and we depart from myopicoptimization, σxs (c) < α

γ(vs(c)c−γφ)σ.

Proof. First, using α < α, we can verify that 0 ≤ ch ≤ cu, regardless of whether the agentcan invest in his hidden savings. Second, vs(c) > 0 for all c ∈ (ca, ch) from Lemma O.8. Thesame argument as in Theorem O.3 shows that vs(c) from (O.32) is the cost correspondingto the stationary contract with c and σx given by (O.30), as long as the contract is indeedadmissible and delivers utility u0 to the agent. We can check that µx < r + π2

γ for thestationary contract if and only if c > ca. In this case, we can use Lemma O.4 to show thatthe stationary contract is admissible and delivers utility u0 to the agent if c > ca. Sincethe contract satisfies (O.10), (O.11), and (O.12), and (O.14) by construction, we only needto check that (O.13) holds too. It’s easy to see this is the case because c ≤ ch. TheoremO.1 then ensures that it is incentive compatible.

The myopic stationary contract has c = cp given by (O.33). Lemma O.15 ensures thatcp ∈ (ca, ch] and therefore by the argument above, it is an incentive compatible contract.The best stationary contract has cr > cp > ca from part 1) of Lemma O.15. From (O.30) itfollows that σxp > σxs (cr). Part 2) of Lemma O.15 shows that vs(c) < cγ for all c ∈ (cp, ch),

71

with equality at cp and ch. Therefore,

σxs (c) < σxp =α

γφσ<

α

γ(vs(c)c−γφ)σ.

Lemma O.8. The cost function of stationary contracts vs(c) defined by (O.32) is strictlypositive for all c ∈ (ca, ch] if and only if α < α.

Proof. We need to check the numerator in (O.32), since the denominator is positive for allc ≥ ca:

c

1− α

φσ

√2

√√√√√cγ−1

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)cγ−1 − 1

1− γ

.

The rest of the proof consists of evaluating this expression at c = ca and showing it isnon-positive iff the bound 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. Thiscompletes the proof.

Lemma O.9. If the agent has access to hidden investment, H = R+ and φ = 1, the optimalcontract is the myopic stationary contract characterized in (O.33).

Proof. The myopic stationary contract is both admissible and incentive compatible bylemma O.7. Since in this case

ch = cp =

(ρ− r(1− γ)

γ− 1− γ

2

(α

γσ

)2

− 1− γ2

(π

γ

)2),

we can use the same verification argument as in Theorem O.3, using the flat value functionv(c) = vp for all c ∈ (0, ch). For the argument to go through, it must be the case that theHJB holds as an inequality for all c < 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 O.15 we know that ∂1A(cp, cγp) < 0. From

Lemma O.12 we know that A(c, v) is positive near 0 and either has one root in c if γ ≥ 1/2,

72

or is convex with at most two roots if c ≤ 1/2. This means that A(c, cγp) > 0 for allc ∈ (0, cp).

A.7 Renegotiation

Here we provide technical details for Section 5 of the paper. This section is consistentwith the presence of aggregate risk and hidden investment introduced in Section 4 and theOnline Appendix.

We say that an incentive compatible contract C = (c, k) is renegotiation-proof (RP) if

∞ ∈ arg minτ

EQ[∫ τ

0e−rt(c− ktα)dt+ e−rτxτ v

],

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

Lemma O.10. An incentive compatible contract C is renegotiation proof if and only if thecontinuation cost v is constant.

Proof. If ever vt > v, then renegotiating at that point is better than never renegotiatingand obtaining v0. In the other direction, if v is constant, any stopping time τ yields thesame value to the principal, so τ =∞ is an optimal choice.

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 Cr, with cost vr = minc∈(ca,ch] vs(c).

Theorem O.4. The optimal renegotiation-proof contract is the optimal stationary contractCr with cost vr.

Proof. Since the optimal stationary contract is incentive compatible and has a constant v,we only need to show that any incentive compatible contract with constant v has v ≥ vr.This is clearly true for all stationary contracts as defined in Lemma O.7 with aggregaterisk.

There could also be stationary contracts with a constant c but dLt > 0. For thesecontracts the drift µc < 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)

73

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 = vs.We want to show that it must be binding. Towards contradiction, if the constraint is notbinding we have σx = α

σγ

cγlvlφ

and therefore we have A(c, v) = 0, where A is defined as inLemma O.12. If v ≤ vr then v ≤ vp, because the myopic stationary contract is incentivecompatible (cp ≤ ch for any valid hidden investment). Then Lemma O.14 ensures thatr−ργ −

ρ−c1−γ1−γ + 1

2(σx)2 + 12

(πγ

)2> 0, which violates the constraint. This means that v ≥ vr.

Finally, if we have a non-stationary contract with a constant v < vr, the domain of cmust have an upper bound c ≤ ch because otherwise they would have a lower cost than theoptimal 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 this is the same situation with stationary contractswith dLt > 0, and we know their cost is above vr.

Remark. It is possible that cr = 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 cr = cp = ch,as show in Lemma O.9.

A.8 Intermediate results

Lemma O.11. The cost function v is flat on (0, cl), v(c) = v(cl), and the HJB equationholds as an inequality in that region, A(c, v(c)) ≥ 0. For c ∈ (cl, ch), the cost function isC2, strictly increasing with v′(c) > 0, and satisfies the HJB equation. At cl we have thesmooth pasting condition v′(cl) = 0.

Proof. Denote v the true cost function. We will use f to denote test functions, and some-times use f , f ′, and f ′′ to denote its value and derivatives at a point c. Because the dLt ≥ 0

term in the law of motion of ct allows it to go up at any time, v must be non-decreasingand it can have a flat region (0, cl) where ct would jump up to cl, so v(c) = v(cl) for allc ∈ (0, cl). cl > 0 because c = 0 requires not giving the agent any capital, so it’s justdelaying the start of the contract, which is not optimal because ρ > r(1− γ).

Recall the HJB equation, for a generic test function f ,

0 = minσx≥0,σc

c− rf − σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)(O.34)

+f ′c

(c1−γ − (cu)1−γ

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2

)+f ′′

2c2(

(σc)2).

If the agent has access to hidden investment, the minimization is subject to the constraintσx +σc ≥ α/(σγ). Notice we already plugged in σx = π/γ and σc = 0. Recall cu is definedin (O.27).

74

Before going into the proof, let us review a few facts. A C2 function f is called asupersolution of (O.34) if instead of equality, it satisfies the inequality

0 ≥ minσx,σc

. . .

For a supersolution f, if it is possible to attain points c− and c+ at cost less than orequal to f(c−) and f(c+), respectively, then the contract that satisfies the inequality aboveattains any point c ∈ [c−, c+] with a cost of less or equal to f(c), as long as the contract isadmissible.

A C2 subsolution satisfies0 ≤ min

σx,σc. . .

If the cost of attaining points c− and c+ is greater than or equal to f(c−) and f(c+), thenthe cost of attaining c ∈ [c−, c+] is greater than f(c).We call functions strict super and sub-solutions if the corresponding inequality is strict. If f is locally a strict supersolution, thena perturbation of f, e.g. a small translation or rotation, is also locally a strict supersolution(and a similar statement holds for strict subsolutions). We will use super and subsolutionsas test functions around the true cost v to prove properties of v (such as differentiability).

Next fact, equation (O.34) implies a value of f ′′ only for some triples (c, f, f ′). Let uselaborate. Let us write the HJB equation for deterministic contracts, in which we mustchoose σc = 0, as

A(c, f, f ′) = 0, (O.35)

where

A(c, f, f ′) ≡ minσx≥0

c− σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)− rf

+f ′c

(c1−γ − (cu)1−γ

1− γ+

(σx)2

2

),

subject to σx ≥ α/(φσ) if the agent has access to hidden investment.Notice that A(c, f, f ′) is concave in f ′ as the minimum of linear functions, and that A

goes to −∞ as f ′ goes to ∞ or −γf/c. For f ′ ∈ (−γf/c,∞), the optimal choice of σx is

σx = cγα

φσ

1

γf + f ′c

if the hidden investment constraint is not binding, and when this leads to

(σx)2

2=

(cu)1−γ − c1−γ

1− γ,

the hidden investment constraint is not binding (because c ≤ ch) and function A(c, f, f ′)

75

achieves its maximum in variable f ′, because then

A3(c, f, f ′) = 0.

Thus, we have

maxf ′

A(c, f, f ′) = c− σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)− rf,

σx =

√2

(cu)1−γ − c1−γ

1− γ.

When maxf ′ A(c, f, f ′) > 0 then the equation A(c, f, f ′) = 0 has two roots f ′L andf ′R > f ′L, in the range f ′ = (−γf/c,∞). Then at point (c, f) it is possible to solve thedeterministic equation as a first-order ODE with slopes f ′(c) = f ′L and f ′R. The formersolution has positive drift at c, points on the solution to the left of c are attainable if (c, f)

is attainable. The latter solution has negative drift at c, and points on the solution to theright of c are attainable if (c, f) is attainable. When maxf ′ A(c, f, f ′) = 0, we can say thatf ′L = f ′R is the unique root.

In fact, maxf ′ A(c, f, f ′) ≥ 0 if and only if f < vs(c) on (ca, ch) and if and only iff > vs(c) on (0, ca). Recall that ca is defined in (O.31), and recall that the curve f = vs(c)

is defined by

c− σxcγ αφσ

+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)− rf = 0,

σx =

√2

(cu)1−γ − c1−γ

1− γ.

For c ∈ (ca, ch), we have ρ−c1−γ1−γ + γ

2 (σx)2− 12π2

γ < r, hence maxf ′ A(c, f, f ′) ≥ 0 if and onlyif f < vs(c). For c < ca,

ρ−c1−γ1−γ + γ

2 (σx)2 − 12π2

γ > r, hence maxf ′ A(c, f, f ′) ≥ 0 if and onlyif f > vs(c). We have maxf ′ A(c, f, f ′) = 0 if and only if f = vs(c).

Whenever maxf ′ A(c, f, f ′) > 0, if f ′ ∈ (f ′L, f′R) and f ′ > 0, equation (O.34) implies

a unique value of f ′′, and can be solved locally as a second-order ordinary differentialequation (ODE). To see this, notice that with f ′′ =∞, the right-hand side becomes equalto A(c, f, f ′) > 0, and is increasing in f ′′. As f ′′ → −γ(f − f ′c)c−1 (1+γ)f ′

fγ+f ′c from above,the objective function diverges to −∞ if the hidden investment constraint is not bindingas we approach the limit. If the hidden investment constraint is binding, then as f ′′ →−γ(f−f ′c)c−1 the objective function diverges to −∞ if γ(f−f ′c)c ασγ < cγ α

φσ ; if not, then itmeans we hit the σx ≥ 0 constraint and we must set σc = α/(σγ) and the objective functiondiverges to −∞ as f ′′ → −∞. So we have a unique f ′′ that solves (O.34). The equation is

76

locally Lipschitz-continuous, as long as we stay in the region where maxf ′ A(c, v(c), f ′) >

0, which implies all the usual properties (existence, uniqueness, and continuity in initialconditions).

Notice that also if maxf ′ A(c, f, f ′) ≤ 0, and f ′ > 0, then any C2 function is locally astrict supersolution, no matter how high f ′′ is. Indeed, if we set σc = 0 and minimize overσx, we get the inequality ≥ in equation (O.34), and we can make the inequality strict usingσc. Also, for any triple (c, f, f ′) with f ′ > 0, a sufficiently concave C2 function is locally astrict supersolution.

Now, let us prove some regularity properties of function v. First, left and right derivativesv′−(c) and v′+(c) exist. If not, for example if

lim infcn→c

v(c)− v(cn)

c− cn< lim sup

cn→c

v(c)− v(cn)

c− cn

for some sequence cn converging to c from below, then we can take a local strict super-solution f with f(c) = v(c) and f ′(c) between these two bounds. Points (cn, v(cn)) abovef for sufficiently large n can be improved upon by a contract based on the solution f.

Second, we have v′−(c) ≤ v′+(c). If not, i.e. v′−(c) > v′+(c), then a local supersolution fwith f(c) = v(c) and f ′(c) = (v′−(c) + v′+(c))/2 can be used to improve upon the optimalcontract with value v(c). Indeed, if we slightly lower f(c) = v(c) − ε, the solution is stilla local strict supersolution that goes above v on both sides of c, and the correspondingcontract has cost less than or equal to v(c)− ε at c.

Third, we have maxf ′ A(c, v(c), f ′) ≥ 0. For c ∈ (ca, ch), this follows immediately be-cause v(c) ≤ vs(c), because stationary contracts provide an upper bound on the cost func-tion from the optimal contract. For c ∈ (0, ca), the argument is a bit more involved.Consider the time-varying version of the HJB equations with choices σc = 0 and

σx =

√2

(cu)1−γ − c1−γ

1− γ,

which satisfies the hidden investment constraint for all c ≤ ch,

∂f

∂t+ c− σxcγ α

φσ+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)− rf = 0.

For c ∈ (0, ca) and f < vs(c), this equation implies ∂f∂t > 0, i.e. this choice of controlsleads to f drifting straight up. This means that if v(c) < vs(c) on (0, ca), this contractallows us to achieve lower cost.

Fourth, let us show that maxf ′ A(c, v(c), f ′) > 0 everywhere, i.e. v(c) 6= vs(c). At anypoint c ∈ (ca, ch), when v′s(c) < 0 then the principal can get a better value than vs(c) byswitching to the optimal stationary contract slightly above c. At any point c ∈ (0, ca),

77

when v′s(c) < 0, if it were the case that v(c) = vs(c), then the principal could achieve v(c)

at c − ε, so v(c − ε) ≤ v(c) < vs(c − ε), which we know cannot be. When c ∈ (ca, ch) andv′s(c) > 0, we can conclude that v(c) < vs(c) by the following argument. Any C2 functionwhich satisfies f(c) = vs(c), f

′(c) = v′s(c) > 0, including those that go above vs in theneighborhood of c, is locally a strict supersolution. Hence, vs(c)− ε is locally attainable forsufficiently small ε.

The following lemma is helpful to deal with the remaining cases (recall cp is defined in(O.33) as the myopic stationary contract).

Lemma. When A(c, f) ≤ 0 and f < cγ , then A3(c, f, 0) > 0. Hence, for any c ∈ (cp, ch)

and any c ∈ (0, ca), at (c, vs(c)), f′L = f ′R > 0.

Proof. We have

A3(c, f, 0) =

(c1−γ − (cu)1−γ

1− γ+

(σx)2

2

).

Since A(c, f) ≤ 0, it means that

c− σxcγ αφσ− rf + f

(ρ− c1−γ

1− γ+γ

2(σx)2 − γ

2

(πγ

)2) ≤ 0

fγ

2(σx)2 ≥ c+ f

γ(cu)1−γ − c1−γ

1− γ, σx = cγ

α

φσ

1

γf.

Towards a contradiction, suppose A3(c, f, 0) ≤ 0. Then

(cu)1−γ − c1−γ

1− γ≥ (σx)2

2⇒

γf(cu)1−γ − c1−γ

1− γ≥ f γ

2(σx)2 ≥ c+ f

γ(cu)1−γ − c1−γ

1− γ⇒ f ≥ cγ ,

a contradiction.Now, we have A(c, vs(c)) < 0 and vs(c) < cγ for any c ∈ (0, ca) ∪ (cp, ch), so at those

points we have f ′L = f ′R > 0.

It follows from the lemma that starting from the minimum of vs on (ca, ch) to theleft, we can solve the equation (O.35) with slope f ′L > 0 (locally) with nonnegative drift.This solution is attainable,24 hence v must be at the level of this solution or below, but0 < f ′L ≤ v′− ≤ v′+ ≤ v′s(c) = 0, which contradicts this. So v is below vs everywhere onc ∈ (ca, ch), including at the minimum of vs.

Now, let us rule out the possibility that f = v(c) = vs(c) on the increasing portion ofvs (including the local maximum) in the range (0, ca). Then A3(c, f, f ′L = f ′R) = 0. Hence,

24This solution corresponds to a deterministic contract, in which ct converges slowly to the minimum ofvs, as the drift gets closer and closer to 0.

78

for

σx =

√2

(cu)1−γ − c1−γ

1− γ− ε,

The value ofc− σxcγ α

φσ+ f

(ρ− c1−γ

1− γ+γ

2(σx)2 − 1

2

π2

γ

)− rf︸ ︷︷ ︸

>0, O(ε)

+f ′c

(c1−γ − (cu)1−γ

1− γ+

(σx)2

2

)︸ ︷︷ ︸

<0, O(ε)

< 0,

on the order of ε2. Hence, we can satisfy this equation by setting f ′ slightly lower than f ′L.For that choice of σx, we get a deterministic contract with positive drift near c, and thecurve that corresponds to this contract is an upper bound on v. Likewise, by choosing

σx =

√2

(cu)1−γ − c1−γ

1− γ+ ε,

we get a contract with slope slightly higher than f ′R. These contracts allow us to achievevalues below vs (which is impossible), unless f ′L = f ′R = v′s(c). In the latter case, lettingε → 0, we find that v′ = f ′L = f ′R. Now, any C2 function which satisfies f(c) = vs(c),

f ′(c) = v′s(c) > 0, including those that go above v in the neighborhood of c, is locally a strictsupersolution. Hence, vs(c)− ε is locally attainable for sufficiently small ε, a contradiction.We conclude that maxf ′ A(c, v(c), f ′) > 0 for all c ∈ (0, ch).

Now, at any c, the slope v′+ cannot be steeper than f ′R, or else we can improve upon thecost function v to the right of c through a deterministic solution with slope f ′R that passesthrough (c, v(c)). Likewise, the slope v′− cannot be less than f ′L and cannot be negative.

We already showed that v′− ≤ v′+. The inequality cannot be strict, or else the equation(O.34) is solvable as a second-order ODE at c with f(c) = v(c) + ε and slope f ′(c) =

(v′− + v′+)/2 ∈ (f ′L, f′R), for sufficiently small ε. Because this is a subsolution, this implies

that cost v(c) at c is unattainable. To sum up, the derivative v′ exists and must be in theinterval [f ′L, f

′R] and nonnegative.

Now, let us show that v′ ∈ (f ′L, f′R) whenever v′ > 0. Otherwise, any C2 test function

f with f = v, f ′ = v′ and arbitrarily large f ′′ is locally a strict supersolution. Supposev′ = f ′R, then the solution of (O.35) with this initial condition to the right of c is weaklyabove v and has finite second derivative. The test function f goes strictly above the solutionof (O.35) to the right of c, assuming f ′′ is large enough. We can rotate the test functionclockwise slightly, it remains a supersolution that goes below v and then above to the rightof c. When it goes below, those points are attainable, hence, we can improve upon the costfunction v, a contradiction.

79

Since v′ ∈ (f ′L, f′R), if v′ > 0, we can solve (O.34) locally with initial conditions (c, v, v′).

If the solution f does not coincide with v locally, if it goes above, then we can rotate itslightly to find points below v that are attainable. If it goes below, then likewise we canrotate it slightly to find points above v that are unattainable. Hence, the tangent solutionof (O.34) must coincide with v locally.

To sum up, whenever v′ > 0, the cost function v satisfies the HJB equation (O.34) as asecond-order ODE.

Now, whenever also A(c, v(c)) < 0 we know that v(c) < cγ , and we can rule out thepossibility that v′ = 0 because otherwise f ′L > 0 and we can improve upon v using thesolution of the deterministic equation (O.35) with slope f ′L > 0 at (c, v) (and positivedrift). To see that A(c, v(c)) < 0 implies v(c) < cγ , use Lemma O.15, and notice thatv(c) ≥ cγ can only occur for c ≤ cp because v(c) < vs(c) < cγ for c > cp. For c ≤ cp, weknow A(c− δ, cγ) ≥ 0 for any δ ∈ [0, c]. So A(c, v(c)) < 0 implies v(c) < cγ .

We also know that if v′(c) > 0 for some c ∈ (0, ch), then v′(c′) > 0 and the HJB holds forall c′ ∈ (c, ch). To see why, if v′(c) > 0 then v is C2 and the HJB holds in a neighborhoodof c. We must always have:

v′c(1 + γ) + v′′c2 ≥ 0.

Otherwise, we can set σx = 0 and σc arbitrarily large, satisfying the hidden investment con-straint and getting an arbitrarily negative value on the left-hand side of the HJB. Rearrangeto get:

v′′ ≥ −v′c−1(1 + γ).

Use Gronwall’s inequality to get

v′(c′) ≥ v′(c)× e∫ c′c −x

−1(1+γ)dx > 0 ∀c′ ∈ (c, ch),

and therefore v satisfies the HJB in (c, ch).It follows that v′ could be zero only in (0, cl], where A(c, v(c)) ≥ 0 and the HJB therefore

holds only as an inequality, but the derivative v′ must become strictly positive beforeA(c, v(c)) < 0, so cl = infc : v′(c) > 0 ∈ (0, ch). Once v′(c) > 0 it remains strictlypositive and the HJB holds for all c ∈ (cl, ch). Since we know there are no kinks, we havethe smooth pasting condition v′(cl) = 0 at cl.

Lemma O.12. Define the function

A(c, v) ≡ c− rv − 1

2

(cγαφσ

)2

vγ+ v

(ρ− c1−γ

1− γ− 1

2

π2

γ

).

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

γ − 1−γ2

(πγ

)2) 11−γ .

80

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

2 , A(c; v)

is convex and has at most two roots.

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

(ρ−r(1−γ)

γ − 1−γ2 (πγ )2

)> 0. For γ > 1,

limc→0A(c; v) =∞.For γ ≥ 1/2, to show that A(c; v) has at most one root in [0, cu] for any v ∈ (0, vh),

we will show that A′c(c; v) = 0 =⇒ A(c; v) > 0 for all c < cu. Compute the derivative(dropping the arguments to avoid clutter)

A′c = 1− vc−γ − c2γ−1

(α

φσ

)2 1

v.

So

A′c = 0 =⇒ c− vc1−γ = c2γ

(α

φσ

)2 1

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+ v

ρ− 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 , (O.36)

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

minimized at ch. So let’s plug in cu into B(c, v):

2γB(cu, v) = (2γ − 1) cu +v

1− γ((ρ− r(1− γ))2γ + (1− 3γ)(cu)1−γ)− vπ2

= (2γ − 1) cu +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) cu + v

((ρ− r(1− γ))

γ

)(1− 2γ)− 1

2(1− γ)

(πγ

)2v

(1− 3γ

1− γ+

2γ2

1− γ

)= (2γ − 1) cu + v

((ρ− r(1− γ))

γ

)(1− 2γ)− 1

2(1− γ)

(πγ

)2v(1− 2γ)

81

(2γ − 1)

(cu − v

(ρ− r(1− γ)

γ− 1

2(1− γ)

(πγ

)2)) ≥ 0,

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

plugging in (O.36) into B:

B(c, v) ≥ 2γ − 1

2γcm −

2γ − 1

2γ

cm1− γ

+ vρ− r(1− γ)

1− γ− v

2

π2

γ

=1− 2γ

2

cm1− γ

+ vρ− 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

),

and dividing throughout by 2γ − 1 > 0

= −1

2

cm1− γ

+cγm

3γ − 1

(ρ− r(1− γ)

1− γ− 1

2

π2

γ

),

and multiplying by c−γm > 0 and using c1−γm1−γ <

(cu)1−γ

1−γ :

> −1

2

(cu)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, cu].For the case with γ < 1

2 , the second derivative of A is

A′′c = γvc−γ−1 − (2γ − 1)c2γ−2

(α

φσ

)2 1

v> 0.

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

Lemma O.13. Assume there are some constants λ1, λ2, λ3, and a constant λ4 > 0 suchthat for any feasible strategy (c, a, z, z) there is a non-negative process N with

dNt ≤ ((λ1t + λ2tσNt + λ3tσ

Nt )Nt − λ4ct)dt+ σNt NtdZ

at + σNt NtdZt,

for some locally bounded processes σN and σN , which can depend on the strategy. Then for

82

a given T > 0, there is a constant λ5 > 0 such that for any feasible strategy (c, a, z, z)

Ea[∫ T

0e−ρt

c1−γt

1− γdt

]≤ λ5

N1−γ0

1− γ.

Proof. First define nt as the solution to the SDE

dnt =((λ1 + λ2σ

Nt + 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,

andnt =

∫ t

0ζscsds+ ζtnt.

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

Ea[∫ τm∧T

0ζscsds

]≤ Ea

[∫ τm∧T

0ζscsds+ 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[∫ T

0ζscsds

]≤ n0.

Now we want to maximize Ea[∫ T

0 e−ρtc1−γt1−γ dt

]subject to this budget constraint. Notice

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

e−ρtc−γt = ζtµ,

where µ > 0 is the Lagrange multiplier and is chosen so that the budget constraint holdswith equality. For any c that satisfies the budget constraint we have

Ea[∫ T

0e−ρt

c1−γt

1− γdt

]≤ Ea

[∫ T

0e−ρt

(c1−γt

1− γ+ c−γt (ct − ct)

)dt

]

= Ea[∫ T

0e−ρt

c1−γt

1− γdt

]+ µEa

[∫ T

0ζt(ct − ct)dt

]≤ Ea

[∫ T

0e−ρt

c1−γt

1− γdt

].

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

[∫ T0 e−ρt

c1−γt1−γ dt

]

83

is finite. Because of homothetic preferences, we know that Ea[∫ T

0 e−ρtc1−γt1−γ dt

]= λ5

n1−γ0

1−γ =

λ5N1−γ

01−γ for some constant 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

[∫ τn+T

τne−ρ(t−τn) c

1−γt

1− γdt+ e−ρ(T−τn)U c,aτn+T

]

≤ Eaτn

[∫ τn+T

τne−ρ(t−τn) c

1−γt

1− γdt

]≤ λ5

N1−γτn

1− γ.

So at t = 0 we get

U c,a0 = Ea[∫ τn

0e−ρt

c1−γt

1− γdt+ e−ρτ

nU c,aτn

]≤ Ea

[∫ τn

0e−ρt

c1−γt

1− γdt+ e−ρτ

nλ5N1−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 O.14. Let cl ∈ (0, ch) and vl ≤ vp. If σc = σc = 0 , σx = ασγ

cγlvlφ

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

A(c, v) ≡ c− rv − 1

2

(cγαφσ

)2

vγ+ v

(ρ− c1−γ

1− γ− 1

2

π2

γ

),

then vl < cγl and

µc =r − ργ−ρ− c1−γ

l

1− γ+

1

2(σx)2 +

1

2(π

γ)2 > 0.

Proof. Looking at (O.11), 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− γ.

Since 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)

84

< 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 haveM > 0. So for A(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

, (O.37)

then we have A = 0 and µc = 0. In this case, because we have µc = 0 we therefore have thevalue of a stationary contract, i.e. v = vs(c) given by (O.32). This point corresponds tothe myopic stationary contract with (cp, vp). We know from Lemma O.15 that cp ∈ [ca, ch].By assumption, vl ≤ vp.

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

A′c(cl, vl) = 1− vl

(c−γl + c2γ−1

l

(α

φσ

)2 1

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−γ

m

1− γ− 1

2

(α

φσ

)2 1

γ− γ

2

(π

γ

)2)

= cm − rvl + vl

(ρ− c1−γ

m

1− γ− γ ρ− c

1−γp

1− γ− (ρ− r)

)

=⇒ A(cl, vl) > cm + vlγc1−γp − c1−γ

m

1− γ= cγmγ

c1−γp − c1−γ

m

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 pointwhere vs(c) touches the locus vm(c) defined by A(c; vm(c)) = 0. If (cl, vl) = (cp, vp) thenthis must be the minimum point for vs(c), so the derivative of both vs(c) and vm(c) mustbe zero. This means that A′c(cl, vl) = 0. However,

1− vl

(c−γl + c2γ−1

l

(α

φσ

)2 1

v2l

)< 0,

where the inequality follows from vl = vp = cγl (note that cl > 0 because as Lemma

85

O.12 shows A(c, vl) is strictly positive for c near 0). This cannot be a minimum of vs(c).Therefore (cl, vl) 6= (cp, vp) and µc(cl) > 0. This completes the proof.

Lemma O.15. 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 myopic stationary contract. We have the followingproperties

1) ca < cp < cr ≤ ch, for any valid hidden investment setting

2) cγ intersects vs(c) only at cp and cu =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 1

1−γin [0, cu]. Fur-

thermore, cγ ≥ vs(c) for all c ∈ [cp, cu], and cγ ≤ vs(c) for all c ∈ [ca, cp], with strictinequality 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] andA(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 ∈ (ca, ch). Clearly, cp < ch for any type of valid hiddeninvestment, because φ < 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−γ2

(πγ

)2. Notice 1− 1−γ

1+γ =

2γ1+γ and use the definition of ca,

ca =

(2γ

1 + γ

) 11−γ(ρ− r(1− γ)

γ− 1− γ

2

(π

γ

)2) 1

1−γ

,

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

c− α

φσcγ√

2

√√√√√(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)− c1−γ

1− γ= cγ

(2r − ρ− 1 + γ

1− γρ+ γ

(π

γ

)2

+c1−γ

1− γ(1 + γ)

).

86

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−γ2

(πγ

)2)− c1−γ

1− γ= −2γ

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)

1− γ+c1−γ

1− γ(1+γ)

− α

φσ

√2

√√√√√(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)− c1−γ

1− γ= −2γ

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)

1− γ+c1−γ

1− γ2γ

α

φσ

√2

√√√√√(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)− c1−γ

1− γ= 2γ

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2)− c1−γ

1− γ.

If c =

(ρ−r(1−γ)

γ − 1−γ2

(πγ

)2) 1

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

α

φσγ=√

2

√√√√√(ρ−r(1−γ)

γ − 1−γ2

(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 vs(c) is always positive above ca and diverges toinfinity as c ca. So we know that cp is the first time they intersect and therefore cγ

intersectsvs(c) from below. Since they won’t intersect again until cu, we get the otherinequality.

Back to 1), consider the locus vm(c) defined by A(c, vm(c)) = 0. Since A(c, v) minimizesover σx, it is always below vs(c). At (cp, vp) we have vm(c) = vs(c) by part 3) below, whichmeans this is a tangency point of vm and vs. We can now show that A′c(cp, vp) < 0 andA′v(cp, vp) < 0, so that vm(cp) = vs(cp) < 0 which means that the Cp is not the optimalstationary contract, since cp < ch. Write

A′c(cp, vp) = 1− vp

(c−γp + c2γ−1

p

(α

φσ

)2 1

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−γ

p

)+

1

2

(cγpαφσ

)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))

87

+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−γ2

(πγ

)2.

To find the best stationary contract, use the HJB

rvr = mincc− σxs (c)cγ

α

φσ+ vr

(ρ− c1−γ

1− γ+γ

2(σxs (c))2 +

γ

2(π/γ)2

),

with FOC for c:

1− γcγ−1 α

φσσxs (c)− vr c−γ + (vrγσ

xs (c)− cγ α

φσ)∂cσ

xs (c) = 0.

We already know that for c ≤ cp we have vs(c) ≥ cγ and σxs (c) ≥ αγφσ . We can them show

that for c ≤ cp the left hand side of the FOC is strictly negative:

lhs = 1− γcγ−1 α

φσσxs (c)− vs(c)c−γ + (vs(c)γσ

xs (c)− cγ α

φσ)∂cσ

xs (c).

Use vs(c) ≥ cγ and ∂cσxs (c) < 0 to obtain:

lhs ≤ −γcγ−1 α

φσσxs (c) + (vs(c)γσ

xs (c)− cγ α

φσ)∂cσ

xs (c)

andlhs ≤ −γcγ−1 α

φσσxs (c) + cγ(γσxs (c)− α

φσ)∂cσ

xs (c).

Finally, σxs (c) ≥ αγφσ yields lhs < 0. This means the best stationary contract must have

cr > cp. We know cr ≤ ch from the definition of cr.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− γγ

88

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 forc ∈ [cp, ch], and A(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

)2 1

cγ

∂1A(c, cγ) = 1− 1− cγ−1

(α

φσ

)2

= −cγ−1

(α

φσ

)2

< 0

for all c ∈ (0, ch].

Lemma O.16. Suppose µc and σc are derived from first-order conditions from a solution tothe HJB equation (O.20) with the properties in Theorem O.2. Without hidden investment,H = 0, the drift and volatility of c near ch are approximately,

µcc ≈ (4γ − 6(1 + γ)2)c−γh ε

σcc ≈ −√

22(1 + γ)c−γ/2h ε3/2,

where ε = ch − c. With hidden investment, H = R+, we have

µcc ≈ (η − 2)1

2

(α

σγ

)2( γ

1− η

)2

ε < 0

σcc ≈ −(α

σγ

)γ

1− ηε,

with η ∈ (0, 1).

Proof. WITHOUT HIDDEN INVESTMENT. First we derive the drift of v′ using the HJBequation (21). Differentiating with respect to c and using the envelope theorem, we obtain

rv′ = 1− γσxcγ−1 α

φσ+ v′

(ρ− c1−γ

1− γ+γ

2(σx)2 − γ

2

(π

γ

)2)− vc−γ

+v′′c

(c1−γ − c1−γ

h

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2

)+v′′′

2c2(σc)2

89

+v′c1−γ + v′

(c1−γ − c1−γ

h

1− γ+

(σx)2

2+ (1 + γ)σxσc +

1 + γ

2(σc)2

)+ v′′c(σc)2.

The middle line has the drift of v′ plus an extra term v′′cσxσc, which we can combine withthe term containing v′′ in the third line. We also know that v′′σcc = −v′(1 + γ)(σc + σx)

from the FOC for σc. Using this we find the drift of v′ to be

+v′′c

(c1−γ − c1−γ

h

1− γ+

(σx)2

2+ γσxσc +

1 + γ

2(σc)2

)+v′′′

2c2(σc)2

= v′(

1 + γ

2(σx + σc)2 + c1−γ

h − c1−γ)

+ γσxcγ−1 α

φσ+ vc−γ − 1.

Now we approximate the cost function near ch. Conjecture, and later verify, that v(c) =

cγh −K√ε. Then

v′ =K

2ε−1/2, v′′ =

K

4ε−3/2, v′′′ =

3K

8ε−5/2,

plus smaller order terms.Now conjecture that σc is of smaller order than σx (also verified later) and use the FOC

for σx to obtaincγ

α

φσ=K

2ε−1/2cσx =⇒ σx =

2

Kcγ−1 α

φσ

√ε,

plus smaller order terms. Now plug into the FOC for σc:

K

2ε−1/2(1 + γ)(σx + σc) +

K

4ε−3/2cσc = 0 =⇒ cσc = −2(1 + γ)σxε.

This verifies that indeed σc is of smaller order than σx.Now we plug everything into the HJB equation and collect terms of order

√ε (the

constant order terms match because the cost function we specified works at ch). The onlyterms of order

√ε are

−σxcγ αφσ

+

(c+ v

γc1−γh − c1−γ

1− γ

)+ v′c

((σx)2

2−c1−γh − c1−γ

1− γ

)= 0

−2c2γ−1h

(α

φσ

)2

−K2γc1−γh − c1−γ

h

1− γ+K

2ch

(4

2

1

K2c2γ−2h − c−γh

)= 0.

We can solve forK =

√2c1.5γ−1h

α

φσ.

90

Now we plug into our expression for σx and σc

σx =√

2c−γ/2h

√ε

σcc = −√

22(1 + γ)c−γ/2h ε3/2,

as desired.For the drift, evaluate the drift of v′ using the formula above

K

2ε−1/2

(1 + γ

22c−γh ε+ (1− γ)c−γh ε

)+ γ(√

2c−γ/2h

√ε)cγ−1 α

φσ+ (cγh −K

√ε)c−γ − 1

= γKc−γh√ε.

But we can also use Ito’s lemma to obtain the drift of v′

v′′cµc +1

2v′′′(cσc)2 =

K

4ε−3/2cµc +

1

2

3K

8ε−5/28(1 + γ)2c−γh ε3 = γKc−γh

√ε.

Solve for cµc

cµc =(4γ − 6(1 + γ)2

)c−γh ε2,

which completes the proof.WITH HIDDEN INVESTMENT. The IC constraints for hidden investment will be

binding near ch, so we have

σx =cγ α

φσ + v′′c2 ασγ

γ(v − v′c) + v′′c2

σc =α

γσ− σx.

In this case we use the approximation

v = vh −Kεη

v′ = Kηεη−1

v′′ = −Kη(η − 1)εη−2.

Divide the FOC for σx by v′′c on both sides (v′′ 6= 0, or we would have σx > α/(γσ) andthe IC wouldn’t be binding):

σx =

ασγ +

cγ αφσ

Kη(η−1)ε2−η

1 + γ(vs−Kεη−Kηεη−1c)Kη(η−1)c2

ε2−η.

The largest terms are of order ε because η ∈ (0, 1), so we get:

91

σx ≈ α

σγ(1 +Aε) ,

where A = γc−1h

11−η > 0, and therefore

σc ≈ − α

σγAε.

We need to make sure the HJB holds up to terms of order εη. Plug into the HJB to obtain

0 = (ch − ε)−(α

σγ

)(1 +Aε) (cγh − γc

γ−1h ε)

α

φσ

+(vh −Kεη)

γ(ρ−r(1−γ)

γ − 1−γ2 (πγ )2

)− c1−γ

h

1− γ+ c−γh ε+

γ

2

(α

σγ

)2

(1 +Aε)2

+Kηεη−1(ch − ε)

( α

γσ

)2((1 +Aε)2

2− (1 + γ)(1 +Aε)Aε+

1 + γ

2A2ε2

)

−

(ρ−r(1−γ)

γ − 1−γ2 (πγ )2

)− c1−γ

h

1− γ− c−γh ε

−Kη(η − 1)εη−2(c2h − 2chε)

(α

γσ

)2

A2ε2.

The constant terms match. Then there are terms of order εη−1:

Kηch

( α

γσ

)2 1

2−

(ρ−r(1−γ)

γ − 1−γ2 (πγ )2

)− c1−γ

h

1− γ

= Kηch

( α

γσ

)2 1

2−

1−γ2

(αγσ

)2

1− γ

= 0.

Then there are terms of order εη:

−K

γ(ρ−r(1−γ)

γ − 1−γ2 (πγ )2

)− c1−γ

h

1− γ+γ

2

(α

σγ

)2

+Kηch

((α

γσ

)2

(A− (1 + γ)A)− c−γh

)

−1

2Kη(η − 1)c2

h

(α

γσ

)2

A2.

We want this to be zero. K factors out, and there is a unique η ∈ (0, 1) that makes thisexpression zero. After some algebra we obtain

c1−γh (1− η)2 + η(1− γ

2)γ

(α

σγ

)2

− γ(α

σγ

)2

= 0. (O.38)

92

The bound α < α implies c1−γh > γ

(αγσ

)2> 0, so at η = 0 the rhs is strictly positive. At

η = 1 we have γ(ασγ

)2− (1 − γ

2 )γ(ασγ

)2= γ2

2

(ασγ

)2> 0, so the rhs is negative. And

because the vertex of the quadratic term is η = 1 there is a unique η that satisfies theexpression. So we have

σx =

(α

σγ

)(1 + γc−1

h

1

1− ηε

)σcc ≈ −

(α

σγ

)γ

1− ηε.

Now let’s find the drift of c. Using (17) and plugging in the expression for σx and σc, theconstant terms cancel, and we get terms of order ε (plus smaller terms)

µcc =

(−c1−γ

h +

(α

σγ

)2 γ

1− η(1− γ)

)ε.

Now use (O.38) to replace c1−γh and obtain

µcc =

(α

σγ

)2( γ

1− η(1− γ) + η

2− γ2

γ

(1− η)2− γ

(1− η)2

).

After some algebra we get

µcc ≈ η − 2

2

(α

σγ

)2( γ

1− η

)2

ε < 0,

as desired.

93