Optimal Financial Regulation and the Concentration of Aggregate Risk Sebastian Di Tella
Optimal Financial Regulation and the Concentration of Aggregate Risk
Sebastian Di Tella
∗
Stanford GSB
July 2015
Abstract
The concentration of aggregate risk on the balance sheets of financial intermediaries can lead to financial crises. If agents can share aggregate risk in complete markets, however, this concentration of aggregate risk might be an efficient feature of a market economy. This paper studies the optimal financial regulation policy in a standard macroeconomic environment where financial frictions are derived from a moral hazard problem. I find that even if agents are able to write complete long-term contracts with full commitment, the competitive equilibrium may feature overinvestment and an inefficient allocation of aggregate risk. Instead of committing to a set of policy instruments, I characterize the best allocation that can be achieved by a social planner who faces the same contractual problems as the market, and then show how it can be implemented as a competitive equilibrium. I provide a simple sufficient statistic for the optimal policy.
1 Introduction
The concentration of aggregate risk on the balance sheets of financial intermediaries can lead to financial crises. Macroeconomic shocks will be amplified and propagated throughout the economy when they face large losses, depressing asset prices and investment, and creating a balance sheet recession. If agents can share aggregate risk in complete markets, however, this concentration of aggregate risk might be an efficient feature of a market economy. This paper studies the optimal financial regulation policy, with a focus on the allocation of aggregate risk.
I use a standard macroeconomic model of financial crises, similar to
He and Krishnamurthy ( 2011 ), or
Di Tella ( 2013 ). Specialized agents, whom I call experts,
∗
I’d like to thank Andy Skrzypacz, Peter DeMarzo, Pablo Kurlat, Yuliy Sannikov, Bob Hall, Martin Schneider,
Monika Piazzesi, V.V. Chari, Chad Jones, Chris Tonetti, Florian Scheuer, Eric Madsen, Alex Bloedel, Javier Bianchi,
Fernando Alvarez, Takuo Sugaya, Victoria Vanasco, Mike Harrison, Peter Kondor, and seminar participants at Stanford, Chicago, Minnesota, NYU, EIEF, FRB of Philadelphia, TSE, SITE, UTDT, RIDGE, NBER Summer Institute, and Columbia. e-mail: sditella@stanford.edu.
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trade and manage capital on behalf of households. We can think of experts as financial intermediaries. Experts, however, face a moral hazard problem: they are able to secretly divert returns, so they must keep some exposure to their idiosyncratic risk to provide incentives. As a result, when experts face large financial losses they become less willing to hold capital, creating a financial amplification channel. Importantly, I allow agents to write complete, long-term contracts with full commitment.
Turning to financial regulation, instead of committing to a set of policy instruments, I characterize the best allocation that can be achieved by a social planner who faces the same contractual problem as the market, and then show how it can be implemented as a competitive equilibrium.
To this end, I use a model of hidden trade as in
The social planner can control the allocation of capital and consumption, but cannot prevent agents from trading capital.
Even in the absence of aggregate shocks, this simple environment features an externality because the private benefit of diverting returns depends on the value of assets under management. Experts don’t internalize that by demanding capital and bidding up its price, they worsen the moral hazard problem for everyone else. Experts’ leverage is too high in equilibrium, and as a result they are exposed to too much idiosyncratic risk, for which they must be compensated. The social planner prefers to reduce the price of capital, at the cost of less investment, in order to improve idiosyncratic risk sharing and reduce the cost of providing incentives to experts. He can implement the optimal allocation as a competitive equilibrium with a tax on capital holdings that internalizes the externality.
He uses the proceeds to create a safe government security (without idiosyncratic risk) that agents can hold on their portfolio. Risky capital is then a smaller fraction of aggregate wealth and experts’ equilibrium leverage is lower.
Notice that the externality is unrelated to aggregate risk sharing, and since moral hazard does not limit agent’s ability to share aggregate risk, it is natural to expect that the competitive equilibrium will allocate aggregate risk efficiently. However, I find that the allocation of aggregate risk will also be inefficient if, and only if, aggregate shocks are correlated with the size of the externality. To see why, consider private and social incentives for sharing aggregate risk. When experts can get high returns from the capital they manage, the private cost of delivering utility to experts is low (in terms of forgone utility by households). In the empirically relevant case with risk aversion greater than one, privately optimal contracts give experts more utility and less net worth, relative to households, after shocks that reduce the cost of experts’ utility. Intuitively, experts are willing to take large financial losses after these aggregate shocks because they expect high returns looking forward. This is an privately efficient property of optimal contracts that can create a financial amplification channel, as in
The social planner has the same incentives to hedge the social cost of experts’ utility. Because of the moral hazard externality there is a wedge between the social and private cost of experts’ utility, but if the wedge is not affected by aggregate shocks, aggregate risk sharing will still be efficient. The
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social cost of experts’ utility is higher than the private cost, but equally so across aggregate states.
In contrast, financial losses will be too concentrated on experts’ balance sheets after aggregate shocks that increase the wedge. Indeed, after taking into account the externality, the social planner realizes that these aggregate shocks increase the social cost of experts’ utility relative to the private cost, so he prefers to distribute financial losses more evenly or even concentrate them on households. For the same reason, in the competitive equilibrium utility losses will be too concentrated on households after shocks that increase the wedge.
The optimal financial regulation policy admits a simple sufficient statistic representation. The value of the safe government security (the present value of taxes), relative to the market value of capital, captures the size of the externality and satisfies at any point in time
T p
= γ × leverage ratio × ( uninsurable id. risk )
2
× p,g
Since the externality involves an excessive exposure to idiosyncratic risk, the optimal policy is proportional to the relative risk aversion γ , the leverage ratio, and the portion of capital’s idiosyncratic risk that cannot be offloaded onto the market due to the moral hazard problem. In addition, p,g is the semi-elasticity of the price of capital with respect to the growth rate of the economy. To see why this matters, recall that the planner can improve idiosyncratic risk sharing by reducing the price of capital at the cost of lower growth. We can obtain an even simpler formula T /p = α , where
α is experts’ equilibrium risk adjusted excess return. To get an idea of the economic size of the externality, these formulas yield a T /p ratio between 4 .
5% and 12% . We can use this expression to understand how aggregate shocks affect the size of the externality, and how optimal policy should respond.
I solve the model numerically for two settings that illustrate these general results. The first one is useful as a theoretical benchmark. If the economy is hit only by Brownian TFP shocks, the social planner implements lower asset prices and investment than the unregulated competitive equilibrium, in order to reduce the cost of providing incentives. In the competitive equilibrium, experts’ leverage is too high and they are being paid too much to compensate them for the large idiosyncratic risk they must take. Aggregate risk sharing, however, is efficient, because the wedge is uncorrelated with
TFP shocks. In fact, in this case neither the competitive equilibrium, nor the optimal allocation, feature any concentration of aggregate risk.
In the second setting, the economy is hit by uncertainty shocks that raise idiosyncratic risk.
There is a large literature exploring the role of countercyclical risk in both finance and business
In this case we obtain an excessive concentration of aggregate risk on experts’ balance sheets, which drives a financial amplification channel. When idiosyncratic risk is high capital must pay a high excess return to convince experts to hold it. As a result, the price of capital and investment falls after uncertainty shocks, and the resulting financial losses are disproportionally concentrated
1 See for example
Bansal and Yaron ( 2004 ), Bansal et al.
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on experts - with high excess returns the cost of experts’ utility is lower after uncertainty shocks.
For the same reason utility losses are concentrated on households. This concentration of aggregate risk is inefficient, however, because uncertainty shocks drive up the wedge between the social and private cost of experts’ utility. Indeed, the externality overexposes experts to idiosyncratic risk, and its cost is higher when there is more idiosyncratic risk. The social planner may allow experts’ equilibrium leverage to go up after uncertainty shocks, but less so than in the unregulated competitive equilibrium. In summary, in the unregulated competitive equilibrium asset prices and investment are too high and experts are too highly leveraged, financial losses are excessively concentrated on experts’ balance sheets, and utility losses are excessively concentrated on households.
Source of inefficiency.
I use an environment similar to
Brunnermeier and Sannikov ( 2012 ),
He and Krishnamurthy ( 2011 ), where financial frictions are created by a moral
hazard problem that forces experts to keep some “skin in the game”.
The main contribution of this paper is the characterization of the optimal financial regulation policy in this setting. I show that this class of models features an externality that leads to overinvestment and an inefficient allocation of aggregate risk. The essential assumptions for this result are that 1) there is moral hazard, and
2) the private benefit of the hidden action depends on the value of assets under management.
This is very reasonable in the case of financial intermediaries (and also in other environments), and generates commonly observed “skin the game” compensation schemes that expose agents to part of the risk in the capital they manage. In order to capture the same contractual problem as in the private competitive equilibrium, I study the planner’s problem using a model of hidden trade, as in
Kehoe and Levine ( 1993 ). The externality arises because the
planner realizes that by distorting the terms of trade in the hidden market, he can relax the moral hazard problem. If the private benefit of diverting funds was specified directly in utility terms (like shirking), the competitive equilibrium would be efficient. Note that the distinction between spot and forward markets plays no role here, everything can be decided at time zero. In Arrow-Debreu terms, the incentive compatibility constraint from the moral hazard problem defines the admissible consumption set for experts. Since a price appears there, the First Welfare Theorem doesn’t apply.
allocation of aggregate risk in his setting is the result of incomplete markets.
More productive agents are unable to obtain enough insurance against downturns, so the marginal rates of substitution don’t equalize. Raising the equilibrium price of assets that more productive agents hold is a way of transferring resources to them in that state. Other papers, such as
Bianchi ( 2011 ), study optimal
regulation in an economy with borrowing constraints and incomplete markets for aggregate risk. In contrast,
Rampini and Viswanathan ( 2010 ) also work with a borrowing constraint that depends on
2 More broadly, this paper fits into the literature on financial amplification channel going back to
Korinek ( 2012 ) features a similar externality. In a different environment, see also Hart ( 1975 ),
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prices, but allow complete financial markets. Aggregate risk might be concentrated in their setting because credit constrained firms may decide to forgo insurance in order to obtain more funds to invest up front. Since financial markets are complete, however, this concentration of aggregate risk is efficient. In contrast, here agents are completely free to share aggregate risk as they see fit, private marginal rates of substitution equalize, and yet we can get an excessive concentration of aggregate risk if the wedge created by the externality is correlated with aggregate shocks.
One of the methodological contributions, relative to the setting in
( 2012 ), Di Tella ( 2013 ), and
He and Krishnamurthy ( 2011 ) is that I allow agents to write complete,
long-term contracts with full commitment. Long-term contracts are not only interesting in their own right, they are also a cleaner contractual environment that makes the comparison with the social planner’s allocation, and the source of the externality, clearer. It is therefore a more natural starting point for the study of optimal financial regulation. In addition, the long-term contract can be implemented in a simple way as a portfolio problem with a “skin in the game” constraint that depends on the aggregate state of the economy. The contractual setting is related to the partial equilibrium settings in
DeMarzo and Sannikov ( 2006 ),
( 2007 ). In particular, I use the same contractual setting as in
Sannikov ( 2014 ) who characterize optimal contracts in a stationary environment where agents have
access to hidden savings. Here I rule out hidden savings, but the impact of hidden savings and limited commitment on the optimal financial regulation policy seem like fruitful avenues for future research.
Layout.
The rest of the paper is organized as follows. In Section 2 I present the model and define the competitive equilibrium. In Section 3 I provide a recursive characterization of the competitive equilibrium. In Section 4 I introduce the planner’s problem and characterize it recursively. I also show how to implement the optimal allocation as a competitive equilibrium, and derive a sufficient statistic for optimal policy. In Section 5 I apply the general results to two settings: TFP shocks and uncertainty shocks. Section 6 concludes.
2 The model
I use a simple macroeconomic model of financial crises where specialized agents, called experts, trade and manage capital on behalf of households. Financial frictions are derived from a moral hazard problem, but I allow agents to write complete, long-term contracts with full commitment.
Technology.
The economy is populated by two types of agents: “experts” i ∈
I
= [0 , 1] and
“households” j ∈
J
= (1 , 2] , identical in every respect except that experts can trade and manage capital on behalf of households. We can think of experts as financial intermediaries. There are two goods, consumption and capital. Denote by k t the aggregate “efficiency units” of capital in the economy, and by k i,t the individual holdings on expert i , where t ∈ [0 , ∞ ) is time. Each efficiency
5
unit of capital produces a flow of consumption goods y i,t
= ak i,t
Capital can be costlessly reallocated between experts, so k i,t will be a choice variable. However, capital is exposed to both aggregate and expert-specific idiosyncratic risk. If an expert holds k i,t
units of capital, he gets a gain in his capital stock
d ∆ i,t
= σ t k i,t dZ t
+ ν t k i,t dW i,t where Z is an aggregate d -dimensional brownian motion, and W i is an idiosyncratic brownian motion for each expert i , in a filtered probability space (Ω , P, F ,
F
) . Here Z represents an aggregate TFP shock, and W i the idiosyncratic outcome of expert i
In addition, there’s a competitive investment sector that uses capital and consumption goods to produce new capital. Producing a flow of new capital k t g t requires the service of k t units of capital and ι t
( g t
) k t consumption goods, with ι
0 t
≥ 0 , ι
00 t
> 0 . As a result of investment and shocks, the aggregate capital stock k follows the law of motion dk t k t
= g t dt + σ t dZ t where the idiosyncratic shocks W i have been aggregated away.
We can let several features of the environment (such as σ t
, ν t
, ι t
( g ) , or φ t introduced below) depend on the history of aggregate shocks (in one of the examples, it will be idiosyncratic volatility
ν t
). To this end, I introduce an exogenous aggregate state of the economy Y t
∈
R d following a diffusion dY t
= µ
Y
( Y t
) dt + σ
Y
( Y t
) dZ t
We can later specify how this aggregate state affects the economy, e.g.
ν t
= ν ( Y t
) for uncertainty shocks, or ι t
( g ) = ι ( g ; Y t
) for shocks to investment technology. Notice that Y is driven by the same
Z we called a TFP shock above, but this is wlog because Z (and Y ) can be multidimensional ( d > 1 ) so we can think of different types of aggregate shocks.
Preferences.
Both experts and households have Epstein-Zin preferences with the same discount factor ρ , risk aversion γ , and elasticity of intertemporal substitution ψ :
ˆ
∞
U t
=
E t f ( c u
, U u
) du t
(1)
4
In other words, while k i,t is a choice variable, the cumulative change in the capital stock for which expert i is responsible up to time t is t
.
5
0 d ∆ i,t
For example, if we give $1 to two intermediaries, they will obtain different returns depending on their precise trading strategies, or which projects they choose to finance.
6
where the EZ aggregator takes the form
f ( c, U ) =
1
c
1 − 1 /ψ
1 − 1 /ψ
[(1 − γ ) U ]
γ − 1 /ψ
1 − γ
− ρ (1 − γ ) U
I will focus on the case where relative risk aversion is larger than log: γ > 1 , and elasticity of intertemporal substitution is larger than two: ψ > 2
In order to obtain a stationary distribution, for the numerical solutions I will introduce retirement among experts, which will arrive with Poisson intensity θ . For most of the paper, however, we can focus on θ = 0 .
Markets.
Experts can trade capital continuously at a competitive price p > 0 with law of motion dp t p t
= µ p,t dt + σ p,t dZ t
The total value of the capital stock in the economy is pk . In addition, investment firms will pay a rent q for the use of capital. There is also a complete financial market with risk-free rate r and price π for aggregate risk Z . Idiosyncratic risks W i are tradable but have zero price in the financial market since they can be aggregated away. Let Q be the equivalent martingale measure associated with r and π . Prices p , q , r , π depend on the history of aggregate shocks Z and are determined in equilibrium.
Taxes.
I will later show that the planner’s optimal allocation can be implemented with a tax on capital holdings, so it is useful to introduce it at this point. The government taxes investment in capital with an arbitrary tax τ k that may depend on the history of aggregate shocks Z . An expert who holds capital worth p t k i,t must pay a tax flow τ k t p t k i,t
. As a result the government raises a total flow of taxes τ t k p t k t
. To balance the budget the government will securitize these cash flows into a safe government security (without idiosyncratic risk) that agents will hold as part of their wealth
(because of complete markets, this is equivalent to distributing it as lump-sum transfers). The value of the safe government security per unit of capital is T
T t
=
1 k t
E
Q t
ˆ t
∞ e
−
´ u t r m dm
τ k u p u k u du (2)
Total wealth in the economy is therefore k t
( p t
+ T t
) . In the unregulated economy, we simply take
τ k
= 0 and therefore T = 0 . It should be stressed that I am not restricting the planner to this policy
6
It is natural to focus on the case with elasticity of intertemporal substitution greater than 1, especially when studying economies with stochastic volatility. The further restriction to EIS > 2 is required to ensure the existence of the competitive equilibrium. See the discussion below on hidden savings. The empirical literature on the EIS is
mixed. Several authors find an EIS less than 1 ( Hall ( 1988 ),
Vissing-Jorgensen ( 2002 )) while others an EIS of
1 .
5 , 2 ,
or even larger ( Campbell and Beeler ( 2009 ),
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instrument, but rather finding that the optimal allocation can be implemented this way.
Households’ problem.
Households are all identical and have homothetic preferences, so we may consider the problem faced by a representative household. It starts with some wealth w
0
(derived from its initial ownership of capital and government transfers) and chooses a stream of consumption c to maximize utility subject to the budget constraint st :
E
Q
V
0
= max c
U ( c )
ˆ
0
∞ e
−
´ t
0 r m dm c t dt ≤ w
0
This is equivalent to choosing c and the exposure of wealth to aggregate risk σ w to maximize utility subject to a dynamic budget constraint dw t w t
= ( r t
+ σ w,t
π t
− ˆ t
) dt + σ w,t dZ t and a solvency constraint w t
≥ 0 , where the hat on ˆ t
denotes the variable is divided by wealth.
Implicit in the second formulation is the fact that since idiosyncratic risks W i have price zero in equilibrium, it is wlog that households will never choose to be exposed to them.
Investment.
Each investment firm chooses k and g to maximize profits max g,k p t gk − ι t
( g ) k − q t k where q t is a rent for the use of capital. The FOC for g yields ι
0 t
( g ) = p t
, so all investment firms pick the same g t
. Since investment firms have a constant returns to scale technology, they must have zero profits in equilibrium. This implies q t
= p t g t
− ι t
( g t
) . An expert who holds capital k i,t will therefore receive an income flow q t k i,t
= ( p t g t
− ι t
( g t
)) k i,t from renting capital to the investment sector. Investment firms can be owned by experts or households, but since they have zero profits they can be ignored in their problems.
Experts’ contracts.
Experts can continuously trade and use capital to produce consumption goods. Each expert would like to borrow from and share risk with the market, but he faces a moral hazard problem: he can secretly divert returns for a private benefit. The contractual environment is developed in detail in the Online Appendix. In this section I drop the i subindex to avoid clutter.
Formally, the expert starts with net worth n
0
> 0 which he gives up in exchange for a contract
C = ( e, k ) that specifies his consumption stream e and the capital he will manage k . Both can depend
7
Of course, the implementation of the optimal allocation is not unique. This is one possible way to implement it, that has the advantage of being simple and highlighting the source of the externality.
8 The link is w t
=
E
Q t h t
∞ e
−
´ t u r m dm c u du i and σ w,t w t is the loading on Z of w t thus defined.
8
on the history of aggregate shocks Z and his idiosyncratic shock W . There is full commitment.
Faced with a contract C , the expert privately chooses a stealing plan s ≥ 0 , also contingent on the history of Z and W . Stealing changes the probability distribution of observed outcomes to P s
, equivalent to P . As a result, the return per dollar invested in capital p t k t
dR t
= a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− τ t k
− s t dt + ( σ t
+ σ p,t
) dZ t
+ ν t dW t s where W t s
= W t
+
´
0 t s u
ν u du is a Brownian Motion under P s
. Stealing does not affect the probability over the aggregate shocks, so Z is also a P s
-Brownian Motion. Notice that R , Z , and therefore W are all observable and contractible. The principal, however, doesn’t observe the stealing s , and so he doesn’t know the underlying probability distribution over these observed outcomes. This is the
source of the asymmetric information in this environment.
The expert only keeps a fraction φ t
∈ (0 , 1) of the stolen capital, which can also depend on the history of aggregate shocks, i.e
φ t
= φ ( Y t
) . He immediately sells it for consumption goods, which he adds to his consumption (he doesn’t have access to hidden savings). As a result, he obtains utility U s
( e + φpks ) , where the s superscript denotes the utility is computed under the probability distribution P s induced by the stealing plan s . In this environment it is always optimal to implement no stealing in equilibrium, s = 0
Therefore, we say a contract C = ( e, k ) is incentive compatible if
0 ∈ arg max s ≥ 0
U s
( e + φpks ) (3)
Let
IC be the set of incentive compatible contracts. The cost of delivering utility to the expert is
F
0
( e, k ) =
E
Q
ˆ
0
∞ e
−
´ t
0 r m dm
( e t
− p t k t
α t
) dt where α t
≡ a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− τ t k − r t
− ( σ t
+ σ p,t
) π
An incentive compatible contract is t is the risk-adjusted excess return optimal if it minimizes the cost of delivering utility to the agent:
J
0
( u
0
) = min
( e,k ) ∈
IC
F
0
( e, k ) s.t.
U ( e ) ≥ u
0
We pin down the initial utility with a break even condition. An expert with net worth n
0 can buy
9
Notice the flow of profits from holding capital is output a plus the rent q t
= p t g t
− ι t
( g t
) .
10 All these processes are fixed and adapted to the same filtration generated by Z and W . By analogy, think of a static environment where contractible output y is a fixed random variable, distributed with pdf f ( y | a ) , but the hidden action a is not observable and affects the probability distribution over y .
11
The standard argument applies: if the agent is stealing in equilibrium it’s better to just give him what he steals and have him not steal instead. See
DeMarzo and Sannikov ( 2006 ) or DeMarzo and Fishman ( 2007 ) for example.
12
In the context of investment funds, this is the “gross alpha”. Here in equilibrium experts appropriate all the excess returns, so investors only get the market rate of return (zero “net alpha”). See
Berk and Green ( 2004 ). Other financial
institutions also have an implicit “alpha”.
9
a contract with cost J
0
( u
0
) = n
0
, and get utility u
0
. At any point in time t , denote by J t the continuation cost of the contract. Notice that in equilibrium it is costly to provide continuation utility to the expert, J t
> 0 . If this were not the case, by scaling up the contract C 0
= ( κe, κk ) the principal can achieve unbounded profits.
Define the net worth of the agent as the continuation cost of his contract, n t
= J t
. There are two good reasons for this definition. First, this is the value of the only asset the expert owns so it is his net worth in an accounting sense. Even better, below I will show the optimal contract can indeed by implemented as a constrained portfolio problem with n as the expert’s net worth, with a natural dynamic budget constraint dn t n t
= ( r t
+ p t k t
α t n t
− e t n t
+ σ n,t
π t
) dt + σ n,t dZ t
φ t p t k t n t
ν t dW t where φ t is the share of idiosyncratic risk that the expert must keep on his balance sheet, which depends only on the history of aggregate shocks Z , i.e. his “skin in the game” constraint.
Competitive Equilibrium.
Take as given the initial capital stock k
0 and a tax process τ . In addition, take as given the initial distribution of wealth for experts { θ i
> 0 } i ∈
I
, such that
I
θ i di < 1
(the rest belongs to the representative household).
Definition 1.
A competitive equilibrium is a set of aggregate processes: price of capital p , value of government security T , risk-free interest rate r , price of aggregate risk π , growth rate g , and the aggregate capital stock k ; a contract C i
= ( e i
, k i
) for each expert ; and a consumption stream c for households, such that i.
The representative household’s consumption and experts’ contracts are optimal, with initial wealth n i, 0
= θ i
( p
0
+ T
0
) k
0 and w
0
= ( p
0
+ T
0
) k
0
(1 − θ i di )
I ii.
Investment is optimal ι
0 t
( g t
) = p t iii.
The value of the government security T
iv.
Market clearing ˆ
I e i,t di + c t
= ( a − ι t
( g t
)) k t
ˆ
I k i,t di = k t v.
Aggregate capital satisfies the law of motion dk t k t
= g t dt + σ t dZ t
10
Distribution of wealth and concentration of financial risk.
Let n t
=
´
I n i,t di =
´
I
J i,t di be the aggregate net worth of experts. In equilibrium the representative household and experts hold the total wealth in the economy, made up of the value of capital p t k t
, plus the value of the government security T t k t
(recall that in the unregulated economy τ k
= T = 0 ) w t
+ n t
= k t p t
+ T t
We say aggregate financial risk is concentrated on experts , if after a bad shock their net worth n t falls proportionally more than households’ wealth w t
. As I’ll show below, this will create a financial amplification channel.
3 Solving the competitive equilibrium
In this section I characterize the competitive equilibrium. Both experts and households face a dynamic problem where their optimal policies depend on endogenously stochastic investment possibility sets. The approach I take is to obtain a recursive characterization of their problems, and look for a recursive equilibrium in the exogenous state variable Y t and an endogenous state variable X t that captures the continuation utility of experts relative to the total capital stock in the economy k t
. This is the same endogenous state variable that a social planner would use, which makes comparisons between the competitive equilibrium and the social optimum clear.
The layout of this section is as follows: I first characterize a first best benchmark without moral hazard. I then provide a recursive characterization of agents’ problems and define a recursive equilibrium which can be solved as a system of PDEs.
3.1
First best without moral hazard
Consider the first best without moral hazard, and without taxes τ k
=0. Without moral hazard, optimal contracts will provide full insurance against idiosyncratic risk and there is no financial friction. Capital is then priced by arbitrage, and the distribution of wealth between experts and households doesn’t matter (except to determine consumption). Experts and households are in fact equivalent. If there are only TFP shocks ( Y is constant), the competitive equilibrium is a balanced growth path with constant asset prices p t
= p
∗ and growth g t
= g
∗
, given by p
∗
= a − ι ( g
∗
)
ρ − (1 − 1 /ψ ) g + (1 − 1 /ψ )
γ
2
σ 2
ι
0
( g
∗
) = p
∗
The interest rate and price of risk are then also constant, r t
π t
= π
∗
= γσ .
= r
∗
= ρ + g/ψ −
γ
2
σ
2
(1 + 1 /ψ ) and
This is also the case when the exogenous state Y t affects only idiosyncratic volatility ν t
= ν ( Y t
) ,
11
because with full idiosyncratic risk-sharing this volatility is irrelevant, and so are shocks to it. Instead if aggregate shocks affect other features of the environment, the equilibrium will not be a balanced growth path in general, but it will still be true that the distribution of wealth doesn’t matter.
Experts and households are equivalent, and they share aggregate risk proportionally.
3.2
Back to moral hazard
Optimal contracts.
It’s a standard result that experts’ optimal contracts are recursive in their continuation utility U i,t
. Drop the i subscript to simplify notation. We can use Lemma
in the
Online Appendix to write the law of motion of an expert’s continuation utility (see the Online
Appendix for more details) dU t
= − f ( e t
, U t
) dt + σ
U,t dZ t
+ ˜
U,t dW t
(4)
This can be interpreted as a promise keeping constraint. If the expert has been promised utility U , this must be delivered by consumption e , either today or in the future. In addition, the expert’s continuation utility can be exposed to both aggregate risk Z and idiosyncratic risk W . Exposing the expert to idiosyncratic risk W is costly, because the expert is risk averse and the market doesn’t price idiosyncratic risk (the first best has full insurance against idiosyncratic risk). However, it will be necessary to expose him to idiosyncratic risk to provide incentives. If the agent steals, he adds φpks to his consumption, but bad outcomes are more likely to be observed. To deter him from stealing, the optimal contract must give him lower continuation utility after bad outcomes are observed. Lemma
in the Online Appendix shows that, for the parameter values γ > 1 and ψ > 1 , the contract C = ( e, k ) is incentive compatible if and only if
0 ∈ arg max s ≥ 0 f ( e t s
+ φ t p t k t s, U t
) − ˜
U,t
ν t
− f ( e t
, U t
)
Taking FOC we obtain
˜
U,t
≥ ∂ e f ( e t
, U t
) φ t p t k t
ν t
= e
− 1 /ψ t
((1 − γ ) U t
)
γ − 1 /ψ
1 − γ
φ t p t k t
ν t
≥ 0 (5)
Notice we can relax the IC constraint and improve idiosyncratic risk sharing if we front-load the agent’s consumption with a high e t
. This reduces the marginal utility from consumption and therefore makes stealing less attractive. The IC constraint will be binding in the optimal contract, so we can solve for the optimal contract choosing e , k , and σ
U
( ˜
U
is given by ( 5 )). We can also verify
that if contract C = ( e, k ) is incentive compatible, so is a scaled up version of it C 0
= ( κe, κk ) . In consequence, homothetic preferences imply the value function for the cost of the optimal contract takes the form
J t
= ξ t
(1 − γ ) U t
1
1 − γ (6)
12
The process ξ captures the stochastic investment opportunities facing the expert. It depends only on the history of aggregate shocks Z and follows the law of motion dξ t
ξ t
= µ
ξ,t dt + σ
ξ,t dZ t which must be determined in equilibrium.
The HJB equation associated with the optimal contract is
(7) r t
J t dt = max e,k,σ
U e − p t kα t
+
E
Q t
[ dJ t
]
We can use Ito’s lemma together with ( 4
) and ( 7 ) to expand the last term. Since expectations are
taken under the equivalent martingale measure Q , it is useful to write Z t
Z
Q
= Z
Q t
−
0 t
π u du , where is a Brownian motion under Q . Finally, let us normalize the controls e t
= ˆ t
((1 − γ ) U t
)
1
1 − γ , k t k t
((1 − γ ) U t
)
1
1 − γ , and σ
U,t
σ
U,t
(1 − γ ) U t
. Thanks to homothetic preferences, the optimal
1 contract is linear in the monotone transformation of continuation utility ((1 − γ ) U t
) 1 − γ . All experts get the same ˆ ,
ˆ
, and ˆ
U
. The HJB BSDE then takes the following form
r t
ξ t
= min
ˆ
ˆ
U
ˆ − p t
ˆ t
+ ξ t
1
1 − 1
ψ
ρ − ˆ
1 − 1 /ψ t
−
+ µ
ξ,t
− σ
ξ,t
π t
+
1
2
γ ˆ
2
U
+
1
2
γ ˆ
− 1 /ψ
φ t p t
ˆ t
ˆ
U
2
π t
+ σ
ξ,t
ˆ
U
(8)
It is easy to see from the HJB that the optimal contract can be implemented as a portfolio problem where the expert’s net worth n t
= J t
= ξ t
((1 − γ ) U t
)
1
1 − γ follows the law of motion dn t n t
= ( r t
+ p t k t
α t
− n t e t n t
+ σ n,t
π t
) dt + σ n,t dZ t
+ ˜ t p t k t
ν t dW t n t
(9)
The expert must keep a fraction of idiosyncratic risk on his balance sheet, which depends only on the history of aggregate shocks, given by
˜ t
= ξ t e
− 1 /ψ t
φ t
= ξ
1 − 1 /ψ t
( e t n t
)
− 1 /ψ
φ t
This is his “skin in the game” constraint. He can offload the rest on the market.
Households’ problem.
Households have a value function
V t
( w t
) =
( ζ t w t
)
1 − γ
1 − γ
13
Existence of the optimal contract requires ψ > 2 . This is related to the assumption of no hidden savings, as will be explained below. Once we assume this, the objective function in the HJB is convex, and the FOC are sufficient for optimality.
13
where ζ captures their endogenously stochastic investment possibilities. It depends only on the history of aggregate shocks Z with law of motion dζ t
ζ t
= µ
ζ,t dt + σ
ζ,t dZ t which we must find in equilibrium. After some algebra, the associated HJB equation for ζ is
(10)
ρ
1 − 1 /ψ
= max
ˆ w c 1 − 1 /ψ
1 − 1 /ψ
ζ t
1 /ψ − 1
+ r t
+ σ w
π t
− ˆ + µ
ζ,t
−
γ
2
σ
2
ζ,t
−
γ
2
σ
2 w
+ (1 − γ ) σ
ζ,t
σ w
(11)
Value of the government security.
Since T only depends on the history of aggregate shocks
Z , we can write dT t
T t
= µ
T ,t dt + σ
T ,t dZ t
In equilibrium T must satisfy a no-arbitrage pricing equation
(12) p t
τ t k
T t
+ µ
T ,t
+ g t
+ σ t
σ
0
T ,t
− r t
= ( σ
T ,t
+ σ t
) π t
(13)
Recursive equilibrium.
Experts’ continuation utilities { U i,t equilibrium. In fact, contracts are linear in x i,t
= ((1 − γ ) U i,t
)
} i ∈ [0 , 1]
1
1 − γ are a state variable for the
, i.e.
e i,t
= ˆ t x i,t
, k i,t k t x i,t
.
This suggests using
X t
=
´
I x i,t di k t as an endogenous aggregate state variable, with law of motion dX t
X t
= µ
X,t dt + σ
X,t dZ t
(14)
The increasing transformation of the expert’s continuation utility x i,t
= ((1 − γ ) U i,t
)
1
1 − γ can also be interpreted as his continuation utility measured in consumption units (up to a constant). The endogenous state variable X therefore captures the aggregate continuation utility of experts, normalized by k . We look for an equilibrium with state variables X t and Y t
. where, e.g.
ξ t
= ξ ( X t
, Y t
) for some function ξ twice continuously differentiable. We can then use Ito’s lemma to obtain expressions for e.g.
µ
ξ
=
ξ
X
µ
X
X +
ξ
ξ
Y
ξ
µ
Y
+
1
2
ξ
XX
ξ
( σ
X
X )
2
+
ξ
Y Y
ξ
σ
2
Y
+ 2
ξ
XY
ξ
σ
X
Xσ
Y
σ
ξ
=
ξ
X
ξ
σ
X
X +
ξ
Y
ξ
σ
Y where the functions are evaluated at ( X, Y ) (and analogous expressions for µ
ζ
, σ
ζ
, µ p
, σ p
, µ
T
, and
σ
T
). Taking the law of motion for X
as given, this transforms the HJB equations for experts ( 8 ),
of the environment, such as ν t
, σ t
, ι t
( g ) , or φ t are already functions of the exogenous state variable
14
Y t
. In addition, for this to work it must be the case that taxes also share the same state variables, i.e.
τ t k
= τ k
( X t
, Y t
) . This will be the case both for the unregulated competitive equilibrium and for the implementation of the optimal allocation.
We can now provide a definition of a recursive equilibrium.
Definition 2.
A recursive equilibrium is a set of functions of ( X, Y ) for prices p , T , r , and π and growth g ; value functions ξ , ζ , and policy functions ˆ ,
ˆ
, ˆ
U for experts and ˆ , σ w for households; and law of motion for the endogenous state variable X , µ
X and σ
X such that: i.
The value functions ξ and ζ and the associated policy functions solve experts’ and households’
HJB equations, taking prices r , π , p and the law of motion of X as given ii.
Investment is optimal ι
0
( g ) = p iii.
The value of the government security T solves its pricing equation iv.
Market clearing
ˆ ( p + T − ξX eX = a − ι ( g ) [consumption goods]
ˆ
=
1
X
[capital] v.
The law of motion for X is derived from the policy functions
µ
X
=
ρ
1 − 1 /ψ e
1 − 1 /ψ
−
1 − 1 /ψ
+
γ
2
2
U
+
γ
2 e
− 1 /ψ
φ pν
X
)
2
− g − σ ˆ
U
+ σ
2
σ
X U
− σ
To understand the market clearing conditions, notice that the aggregate net worth of experts is n = ξXk , so household wealth is w = ( p + T − ξX ) k in equilibrium. We use this for the market clearing condition for consumption goods (and divide throughout by k ). Market clearing for capital goods requires total demand for capital
ˆ be equal to total supply k . Finally, the law of motion of the endogenous state variable X is derived using Ito’s lemma.
Demand for capital and investment.
Taking FOC in experts’ HJB, and using the equilibrium condition
ˆ
= X
− 1 we obtain a pricing equation for capital
| a − p
ι ( g )
+ g + µ p
+ σσ p
0
{z
− τ k
− r − ( σ + σ p risk-adjusted excess return ≡ α
) π
}
= γ ( ξ ˆ
− 1 /ψ
φν )
2 p
ξX
| {z } id. risk premium
(15)
Although the market price of idiosyncratic risk W i is zero, capital must pay a premium for its exposure to this risk. Because of the moral hazard problem, experts must be exposed to their
15
idiosyncratic risk proportionally to the value of capital they manage, as in ( 5 ).
This is costly because the expert is risk averse, so he will demand a premium for holding capital. Notice how this premium would disappear if we had no moral hazard ( φ = 0 ) or no idiosyncratic risk ( ν = 0 ).
To understand this premium for idiosyncratic risk better, recall from equation ( 9 ) that the net
worth of experts has idiosyncratic volatility and
˜ n pk n
ν , where pk n is the leverage ratio for the expert,
˜
= ξ ˆ
− 1 /ψ φ is his “skin in the game” constraint (the same for all experts). The premium for idiosyncratic risk can then be written idiosyncratic risk premium = γ pk
( ˜ )
2 n
(16)
The premium is higher if capital is more risky after sharing risk with the market (higher
˜
), or experts are more highly leveraged (high pk n
).
It is through this pricing equation for capital that both idiosyncratic risk ν and the endogenous state variable X affect the equilibrium. If financial losses are concentrated on the balance sheet of experts, they will demand a higher premium to hold capital, driving down its price. Since the price of capital determines investment through the FOC ι
0
( g ) = p , any shocks that depress the price of capital will result in lower investment as well. Thus we may obtain a financial amplification channel.
Consumption and “skin in the game” constraint.
Households have a standard FOC for consumption c
− 1 /ψ
ζ
1 /ψ − 1
= 1 (17)
For experts, the FOC for ˆ yields
ξ ˆ
− 1 /ψ
− 1 + ξ
γ
ψ
( φp
ν
X
)
2 e
− 2 /ψ − 1
= 0
| {z } smoothing
| {z front-loading
}
(18)
The first term is standard: more consumption today implies lower promised utility in the future.
As a result, we trade off the cost of delivering utility today against the cost of delivering it in the future. If the cost of providing utility ξ is high, it will be attractive to give the agent consumption today.
In addition, by front-loading consumption we can reduce the marginal utility of consumption and therefore make stealing (and immediately consuming) less attractive. We can relax the idiosyncratic risk sharing constraint by distorting the intertemporal consumption profile. Recall that the exposure of experts’ net worth to idiosyncratic risk is ˜ n pk n
ν , where
˜
= ξ ˆ
− 1 /ψ
φ . If we ignored the second term in the FOC, we would equalize the marginal utility of consumption with the marginal value of a dollar, ξ ˆ
− 1 /ψ
= 1 and therefore we would get
˜
= φ . By front-loading consumption, we can reduce the fraction of capital’s idiosyncratic risk that the expert must bear,
˜
.
In fact, this tool is so powerful that if the experts’ EIS is low ( ψ ≤ 2 ) the principal would be
16
Indeed, since the expert doesn’t have access to hidden savings, the principal has a lot of power over him and can eliminate the moral hazard problem. A large EIS
( ψ > 2 ) limits the usefulness of front-loading consumption and makes the contractual problem well defined. An alternative is to introduce hidden savings. The contractual problem with hidden savings is considerably more involved, but
Di Tella and Sannikov ( 2014 ) show that it can be characterized
in a tractable way suitable to macro and financial applications.
Aggregate risk-sharing.
The FOC for ˆ
U is
π − σ
ξ
= γ ˆ
U
= ⇒ ˆ
U
=
π − σ
ξ
γ
(19)
The interpretation is straightforward: to minimize the cost, give the expert more utility when a) the value of money is lower (the term π ); and b) the cost of delivering utility is lower (the term σ
ξ
).
Households have a similar FOC for their exposure to aggregate risk σ w
:
π = γσ w
− (1 − γ ) σ
ζ
= ⇒ σ w
=
π
γ
+
1 − γ
σ
ζ
γ
(20)
Households’ hold aggregate risk because it carries a positive premium π (a myopic motive), and because it correlates with their investment possibility set, as captured by ζ (hedging motive).
To understand aggregate risk sharing, it’s useful to consider the marginal rate of transformation between experts’ and households’ utility. Let V be the representative household’s continuation utility, and consider the increasing transformation
S =
((1 − γ ) V )
1
1 − γ k with law of motion dS
S
= µ
S dt + σ
S dZ t where µ
S and σ
S are of course functions of X and Y . This monotone transformation of households’ utility can also be interpreted as their utility measured in consumption units and normalized by k , analogous to X for experts. It follows that S = ζ w k experts’ and households’ utility is
= ζ ( p + T − ξX ) . The (private) MRT between
Λ = ξζ
14
As long as capital pays a premium for idiosyncratic risk. If it doesn’t then the optimal contract has this can’t be an equilibrium.
ˆ
= 0 and
17
with law of motion d Λ
Λ
= µ
Λ dt + σ
Λ dZ t
(21)
If we increase an expert’s (transformed) utility x i by ∆ , the cost of the contract increases by ξ ∆ .
With this money, a household could obtain (transformed) utility ζξ ∆ . The MRT Λ is therefore the cost of experts’ utility, measured in forgone utility by households. It is the private cost because it takes all prices as given. Later when we study the planner’s problem we will encounter the social cost of experts’ utility.
We can use Ito’s lemma and the FOC for aggregate risk sharing ( 19 ) and ( 20 ), and the definition
of σ
X
= ˆ
U
− σ to obtain an expression for the allocation of aggregate risk in utility terms
σ
X
− σ
S
= −
1
γ
( σ
ζ
+ σ
ξ
) = −
1
γ
σ
Λ
(22)
the competitive equilibrium allocates more continuation utility to experts, relative to households, after aggregate shocks that reduce the (private) cost of experts’ utility Λ .
Of course, the cost of experts’ utility Λ is endogenous and must be determined in equilibrium.
Intuitively, if experts obtain a large excess return from managing capital, the cost of giving them utility will be low. In particular, Λ depends on the endogenous state variable X , which creates a two-way feedback loop: a) aggregate risk sharing (captured by σ
X
) affects how Λ and S respond to aggregate shocks
σ
Λ
=
Λ
X
σ
X
X +
Λ
Λ
Y
Λ
σ
Y
σ
S
=
S
X
σ
X
X +
S
S
Y
S
σ
Y
and b) these in turn provide incentives for the concentration of aggregate risk given by ( 22 ). We
can then obtain an expression for σ
X
:
σ
X
=
1 −
S
Y
S
− 1
γ
Λ
Y
Λ
S
X
S
− 1
γ
Λ
X
Λ
X
σ
Y
(23)
Concentration of financial risk.
While equation ( 22 ) determines the allocation of aggregate risk
in utility terms, we might be more interested in the allocation of financial risk. Consider the ratio of experts’ net worth to households’ wealth n w
. The exposure to aggregate risk of experts’ net worth is σ n
= σ
ξ
+ σ
X
+ σ , while for households we have σ w
= σ
S
− σ
ζ
+ σ . If aggregate financial risk was distributed proportionally, then σ n
= σ w and aggregate shocks Z wouldn’t affect the distribution of wealth. If instead aggregate risk is concentrated on experts, then σ n
> σ w
definition of Λ = ξζ , we can obtain an expression for the concentration of aggregate financial risk
σ n
− σ w
=
γ − 1
σ
Λ
γ
(24)
Here we have two opposing effects. On the one hand, there is a substitution effect : experts should have more net worth when the cost of experts’ utility is low (when the MRT Λ is low) in order to get
18
more “bang for the buck”. But there is also an income effect : experts need more net worth in order to achieve any given utility level when the cost of their utility is high (when the MRT Λ is high).
In the empirically relevant case with risk aversion greater than one, the income effect dominates and the market allocates more financial wealth to experts, relative to households, after aggregate shocks that increase the (private) cost of providing utility to them.
Essentially, experts are willing to take large financial losses if they expect large excess returns looking forward. This is essentially the same mechanism as in
Di Tella ( 2013 ), and can create the
concentration of aggregate risk that drives a financial amplification channel.
Solving the competitive equilibrium.
The full competitive equilibrium can be characterized with a system of PDEs for ξ ( X, Y ) , ζ ( X, Y ) , and p ( X, Y ) . Section
provides numerical solutions for two settings that illustrate the general results.
describes the solution procedure in detail.
4 Planner’s Problem
I now turn to the problem of optimal financial regulation. I take a mechanism design approach, and ask what is the best allocation that can be achieved by a social planner who faces the same informational frictions as the market. We can then ask how the optimal allocation can be implemented as a competitive equilibrium. The main advantage of this approach is that we are not pre-committing to a given set of policy instruments which might be inappropriate for the problem at hand. We are letting the model tell us what policy instruments we should use.
4.1
Setting
Consider a social planner who faces the same informational frictions as private agents in the market.
He can a) control households’ consumption; b) give consumption and capital to experts to manage, but they can secretly divert it; c) give capital and consumption goods to invest to the investment sector, and order them to deliver a flow of new capital. In the competitive equilibrium setup experts could sell capital at price p t and consume the proceeds right away. I capture the same contractual problem with a model of hidden trade, as in
expert with a flow of stolen capital k i,t s i,t can sell it to an investment firm at a competitive black market price ˜ t
. With less consumption goods, the investment firm will then be able to produce less new capital, but it will instead present the stolen capital to the planner, so that the hidden trade is not detected.
To formalize this, consider an investment firm that receives an order to use k t units of capital and
ι t
( g t
) k t consumption goods to deliver a flow of new capital g t k t
. It can instead choose to buy a flow of stolen capital k t s and do actual investment k t
ι t
(˜ ) in order to maximize its surplus consumption
19
(that it rebates to its owners) max s, ˜ k t
( ι t
( g t
) − ι t
(˜ ) − p t s ) st : k t
˜ + k t s = k t g t
Optimality implies p t
= ι
0 t
(˜ )
So if the planner is implementing his desired investment rate, ˜ = g t and there is no stealing in equilibrium, s = 0 , the black market price of capital must be p t
= ι
0 t
( g t
) (25) and investment firms have no surplus consumption. Notice this is precisely the equilibrium price of capital in the unregulated CE, and the CE is consistent with this environment.
A plan P = ( c, g, k, { e i
, k i
} i ∈ [0 , 1]
} is a consumption stream for the representative household c and a growth rate g and aggregate capital k , which can depend on the history of aggregate shocks
Z ; and consumption and capital ( e i
, k i
) for each expert i , which can depend also on his history of idiosyncratic shocks W i
. A plan P is feasible if it satisfies the aggregate consistency conditions
ˆ c t
+ e i,t di = ( a − ι t
( g t
)) k t
(26)
I
ˆ k i,t
= k t
(27)
I and aggregate capital follows the law of motion dk t
= g t k t dt + σ t k t dZ t
(28)
Faced with a feasible plan P , each expert chooses a stealing plan s i and gets consumption ˜ i
= e i
+ φ ˜ i s i
. As in the private problem, it is optimal to implement no stealing always. A feasible plan is incentive compatible if every expert i chooses s i
= 0 :
0 ∈ arg max s ≥ 0
U s
( e i
+ φι
0
( g ) k i s ) (29)
that by controlling g he can relax the moral hazard problem. This is the source of inefficiency in this model. Let
ICP be the set of incentive compatible plans. Given initial utility levels for each expert
{ u
0 i
} i ∈ [0 , 1]
, an incentive compatible plan P is optimal if it maximizes households’ utility subject to delivering utility u
0 i to each expert: max
P∈
ICP
U ( c )
20
st : U i
( e i
) = u
0 i
4.2
A recursive formulation of the planner’s problem
Just as in the competitive equilibrium, we look for an optimal mechanism that is recursive in the continuation utility of experts { U i
} i ∈
I and the aggregate state variables. Each expert’s utility still
follows the law of motion given by ( 4 ), and the IC constraint is like ( 5 ), with
p t replaced by ˜ t
= ι
0 t
( g t
)
˜
U,i,t
≥ f
1
( e i,t
, U i,t
) φ t
ι
0 t
( g t
) k i,t
ν t
= e
− 1 /ψ i,t
((1 − γ ) U i,t
)
γ − 1 /ψ
1 − γ
φ t
ι
0 t
( g t
) k i,t
ν t
≥ 0 (30) and it will be binding in the optimal allocation.
Introduce x i,t
= ((1 − γU i,t
)
1
1 − γ as in the private contract, and write e i,t e i,t x i,t and k i,t
=
˜ i,t x i,t
, and σ
U,i,t
σ
U,i,t
(1 − γ ) U i,t
. We can verify that, just as in the private contract, the planner will choose the same ˜ i,t e t
,
˜ i,t
= ˜ t
, and ˜
U,i,t
σ
U,t for all experts. For consumers write c i,t c i,t k t
. The planner’s problem must be recursive in the same endogenous state variable as the competitive equilibrium X which captures the aggregate continuation utility of experts, and the exogenous state variable Y . Thanks to homothetic preferences and the linear technology, the planner’s value at time t then takes the following power form
( S t k t
) 1 − γ
1 − γ for some process S t which depends only on the history of aggregate shocks Z . We look for a value function S and the policy functions ˜ ,
˜
, g , and σ
U
, all functions of ( X, Y ) . We can then re-write the aggregate consistency conditions
˜ eX = a − ι ( g )
˜
= 1 and the law of motion of X is given by
µ
X
=
ρ
1 − 1 /ψ
− e
1 − 1 /ψ
1 − 1 /ψ
+
γ
2
2
U
+
γ
2 e
− 1 /ψ
φ
ι
0
( g )
X
ν )
2
− g − σ ˜
U
+ σ
2
σ
X
σ
U
− σ
Using Ito’s lemma on S ( X, Y ) we obtain
µ
S
=
S
Y
µ
Y
S
+
S
X
µ
X
X +
S
1 S
Y Y
2 S
σ
2
Y
1
+
2
S
XX
S
( σ
X
X )
2
+
S
XY
S
σ
X
Xσ
Y
σ
S
=
S
X
σ
X
X +
S
S
Y
σ
Y
S
21
(31)
(32)
(33)
(34)
(35)
(36)
The HJB equation associated to the planner’s problem is
ρ
1 − 1 /ψ
= max g, ˜ ˜
U
( a − ι ( g ) − ˜ )
1 − 1 /ψ
S
1 /ψ − 1
1 − 1 /ψ
+ µ
S
+ g −
γ
2
σ
2
S
−
γ
2
σ
2
+ (1 − γ ) σ
S
σ (37)
The (social) MRT between experts’ and households’ utility is
Λ = − S
X
> 0 with law of motion d Λ
Λ
= µ
Λ dt + σ
Λ dZ t
(38)
Recall in the competitive equilibrium, the private MRT is Λ
CE counterpart, Λ
SP
= ξζ . Analogously to its private can be interpreted as the (social) cost of providing utility to experts in terms of forgone utility for households, and will play an important role in the socially optimal allocation.
When necessary for clarity, I will use CE and SP to denote object in the competitive equilibrium and the planner’s allocation, respectively.
Consumption.
The FOC for ˜ takes the following form
Λ ˜
− 1 /ψ
+
γ
ψ
( φι
0
( g )
ν
X
)
2 e
− 2 /ψ − 1
=
S
− 1 /ψ
(39)
On the right hand side we have the cost of giving consumption to experts, in terms of the necessary reduction in households consumption. The benefit, on the left hand side, is the reduction is promised utility for experts. First, by giving them consumption today, the planner can promise lower utility to them in the future. Second, by front-loading consumption the planner can relax the idiosyncratic risk-sharing problem, which is costly because experts are risk averse. The planner is willing to distort the intertemporal allocation of consumption in order to improve idiosyncratic risk sharing, just like
the private contracts. In fact, condition ( 39 ) holds in the competitive equilibrium.
Investment and idiosyncratic risk sharing.
The FOC for investment/growth g is
S externality
− 1 /ψ
ι
0
( g ) + z }|
Λ Xγ (˜
− 1 /ψ
φι
0
( g )
ν
X
)
2
ι
00
( g )
{
ι 0 ( g )
= S + Λ X (40)
On the right hand side we have the benefit of increasing the growth rate of the economy: first, we can deliver more utility to households with more capital, and second we reduce the utility promised to experts per unit of capital (we reduce X ), so we improve the representative households’ utility per unit of capital by Λ X .
On the left hand side we have the cost of increasing growth: first, we must reduce households’
22
consumption to allocate goods to investment. But in addition, the planner realizes that a higher growth rate will increase the marginal cost of capital, and therefore will require a larger exposure to idiosyncratic risk for experts. The planner internalizes the tradeoff between growth and idiosyncratic risk sharing which private agents in the competitive equilibrium don’t. This is where we see the moral hazard externality. We can obtain a similar equation for the unregulated competitive equilibrium from ι
0
( g ) = p and S = ζ ( p − ξX ) :
S
− 1 /ψ
ι
0
( g ) = S + Λ X (41)
Because of the moral hazard externality, the unregulated competitive equilibrium without taxes is not efficient.
Proposition 1.
The unregulated competitive equilibrium is inefficient.
Proof.
Because the unregulated competitive equilibrium is inefficient, there is a wedge between the private and social cost of experts’ utility
Γ =
Λ SP
Λ CE
= 1 (42)
The wedge Γ captures the inefficiency associated with the moral hazard externality, with law of motion d Γ
= µ
Γ dt + σ
Γ dZ t
Γ
(43)
In the unregulated competitive equilibrium experts don’t internalize that by demanding capital they push up its price, increasing the private benefit of diverting returns for every other expert. This tightens their moral hazard problem and limits their idiosyncratic risk sharing. The social planner is willing to forgo some investment (and growth) in order to reduce the value of capital and improve idiosyncratic risk sharing. This allows him to deliver utility to experts with less consumption, leaving more for households. Since private contracts don’t internalize the cost this imposes on others, the social cost of delivering utility to experts Λ
SP is larger than the private cost Λ
CE
, so we focus on
Γ > 0 .
It is useful to ask why the first welfare theorem fails in this environment. In the Arrow-Debreu setting, prices appear only in the budget constraint, but not in utility functions or consumption sets.
Here, incentive compatibility defines the set of admissible plans for an expert. Because the price of capital p appears in the IC constraint, the assumptions of the first welfare theorem are not satisfied.
Aggregate risk sharing.
For experts’ exposure to aggregate risk ˜
U we have the following FOC
σ
X
− σ
S
1
= −
γ
σ
Λ
(44)
23
This FOC has a straightforward interpretation.
The planner gives more utility to experts, relative to households, after aggregate shocks that reduce the social cost of experts’ utility Λ . After some algebra, using Ito’s lemma, we can obtain an expression for σ
X
σ
X
=
S
Y
S
1 − (
S
X
S
−
−
1
γ
1
γ
Λ
Y
Λ
Λ
X
Λ
) X
σ
Y
(45)
) parallel equations ( 22 ) and ( 23 ) for aggregate risk sharing in the
competitive equilibrium. The only difference is that the competitive equilibrium uses the private cost of experts’ utility Λ
CE while the planner uses the social cost Λ
SP
. Although the externality only directly relates to the tradeoff between growth and idiosyncratic risk sharing, it will also distort the allocation of aggregate risk.
Proposition 2.
The allocation of aggregate risk in the unregulated competitive equilibrium is inefficient if the wedge Γ =
Λ
SP
Λ CE is correlated with the aggregate shocks.
σ
CE
X
− σ
CE
S
− σ
SP
X
− σ
SP
S
1
= −
γ
σ
CE
Λ
− σ
SP
Λ
=
1
σ
Γ
γ
(46)
Experts gain “too much” utility, relative to households, after shocks that reduce the private cost of experts’ utility Λ
CE relative to the social cost Λ
SP
(shocks that increase the wedge Γ ).
If the wedge Γ , which reflects the moral hazard externality, is not affected by aggregate shocks, although the unregulated competitive equilibrium may be inefficient, the allocation of aggregate risk will be efficient. If instead aggregate shocks affect the wedge Γ , the unregulated competitive equilibrium will overexpose experts’ continuation utility to aggregate shocks that increase the wedge
Γ . Notice that because X is a state variable for both the planner’s allocation and the competitive equilibrium, inefficient aggregate risk sharing may result in an inefficient response of the economy to aggregate shocks.
Although Proposition
is useful to understand the efficiency of the equilibrium allocation, sometimes we may be more interested in the allocation of financial risk (i.e. how financial gains or losses are shared by experts and households). Proposition
below deals with this.
Remark.
The total loss or gain of utility for experts and households in response to an aggregate shock will in general be different in the unregulated competitive equilibrium and the optimal allocation.
Equation ( 46 ) refers to how this loss or gain is allocated between experts and households.
Solving the planner’s problem.
The planner’s problem boils down to solving a PDE for
S ( X, Y ) . Section
provides numerical solutions for two settings that illustrate the general results.
describes the procedure in more detail.
24
4.3
Implementation of the planner’s allocation
The planner’s allocation can be implemented as a competitive equilibrium with a tax on capital holdings τ k
( X, Y ) . If an expert holds p t k i,t in capital, he must pay a tax flow τ t k p t k t to the government. To balance the budget, the government securitizes the proceeds into a government security without idiosyncratic risk that agents can hold on their portfolios, with market value T t k t given by
fers). Remarkably, although the competitive equilibrium may feature an excessive concentration of aggregate risk on experts, it is not necessary to directly regulate the allocation of aggregate risk.
Aggregate risk sharing will be inefficient in the unregulated competitive equilibrium if the wedge Γ between the social and private cost of experts’ utility is correlated with aggregate shocks. But once taxes on capital internalize the moral hazard externality and align the private and social MRT between experts’ and households’ utility, Λ CE = Λ SP , aggregate risk sharing becomes optimal without the need for further regulation. The same reasoning applies to consumption and investment.
Consider an optimal plan P and the associated value function S and policy functions ˜ , g , and the law of motion of the endogenous state, µ
X and σ
X
, all functions of ( X, Y ) . We can build a recursive equilibrium using the same law of motion µ
X
ˆ = ˜ , and ˆ = and σ
X
, as well as the policy functions g , c p + T − ξX
. From the FOC for growth we get p = ι
0
( g ) (47)
And we want to build the competitive equilibrium so that the private cost of experts’ utility is
ξζ = − S
X
We build ξ and ζ as follows. Households’ wealth per unit of capital is p + T − ξX > 0 , and their utility S > 0 , so we set
S = ζ ( p + T − ξX )
In addition, from the FOC for ˆ we obtain p + T − ξX
− 1 /ψ
= ζ
1 − 1 /ψ
The FOC for ˆ will then be automatically satisfied. Putting these equations together we obtain
ζ =
S
− 1 /ψ
> 0 , ξ = −
S
X
ζ
> 0 , T =
S
ζ
− p + ξX (48)
We can use the representative household’s HJB equation to pin down r , and from his FOC for σ w we pin down π (experts’ FOC for ˆ
U
will be automatically satisfied).
The optimal tax on capital
15 This will only work if S
X
< 0 , i.e. if it is costly for households to provide utility to experts.
16
Using the envelope theorem for the planner’s HJB, we can show that experts’ HJB is satisfied, and therefore also
25
τ k that implements the optimal allocation as a competitive equilibrium is then obtained from the
FOC for capital.
a − ι ( g ) p
+ µ p
+ g + σ
0
σ p
− ( r + τ k
) = π ( σ + σ p
) + γξ (ˆ
− 1 /ψ
φν )
2 p
X
shows the procedure in detail and proves the following result:
(49)
Proposition 3.
Given the planner’s optimal allocation with value function S satisfying S
X
< 0 , and associated policy functions ˜ , g and ˆ
U
, we can implement it as a recursive competitive equilibrium with a tax on capital τ k
Proof.
The planner is in fact using the tax τ k to force agents to internalize the externality, and using the proceeds to create a safe government asset (without idiosyncratic risk) with value T k . Risky capital is now just a fraction of aggregate wealth ( p + T ) k . As a result, experts’ don’t need to lever
risk premium on capital is proportional to the equilibrium leverage of experts pk n
).
It is worth pointing out that this is not be the only way in which we can implement the optimal allocation, but it has the benefit that it highlights the source of the inefficiency and deals with it directly. What is needed is an instrument that will reduce the equilibrium price of capital, in order to reduce the private benefit of stealing and therefore improve idiosyncratic risk sharing (at the cost
Concentration of financial risk in the regulated competitive equilibrium.
We can use
expression ( 24 ) to obtain an expression for the concentration of aggregate financial risk
σ n
− σ w
=
γ − 1
σ
Λ
γ
(50)
This has a similar interpretation in terms of income and substitution effects as equation ( 24 ) in
the competitive equilibrium.
In the empirically relevant case with risk aversion greater than one, the income effect dominates and the planner allocates more financial wealth to experts, relative to households, after aggregate shocks that increase the (social) cost of providing utility to them.
We can then obtain the following equation comparing the concentration of financial risk in the unregulated competitive equilibrium and the implementation of the optimal allocation
Proposition 4.
The allocation of aggregate financial risk in the unregulated competitive equilibrium is inefficient if the wedge Γ =
Λ
SP
Λ CE is correlated with the aggregate shock: the pricing equation for taxes. See
17
The tax on capital holdings can be thought of as a disincentive to leverage.
26
( σ n
− σ w
)
CE
− ( σ n
− σ w
)
SP
=
γ − 1
γ
σ
CE
Λ
− σ
SP
Λ
= −
γ − 1
γ
σ
Γ
(51)
When the income effect dominates ( γ > 1 ), experts lose too much net worth, relative to households, after shocks that reduce the private cost of experts’ utility Λ
CE relative to the social cost Λ
SP
(shocks that increase the wedge Γ ).
Remark.
As in the case of utility in equation ( 46 ), the total loss or gain of aggregate wealth for experts
and households in response to an aggregate shock will in general be different in the unregulated
allocated between experts and households.
) and ( 4 ) together to get the following picture. Imagine an adverse
shock hits the economy, producing both financial and utility losses. If the shock reduces the private cost of experts’ utility Λ CE
, utility losses will be concentrated on households, but financial losses will be concentrated on the balance sheets of experts, which can create a financial amplification channel. If the private cost Λ
CE falls proportionally more than the social cost Λ
SP
(so the wedge
Γ = Λ
SP
/ Λ
CE becomes larger), then financial losses will be too concentrated on experts’ balance sheets, and utility losses too concentrated on households.
4.4
A sufficient statistic for optimal policy
Without taxes (but keeping the optimal allocation) the market value of capital would be p + T .
This is in fact the shadow value of capital for the social planner, in terms of consumption goods:
− ψ
∂ k
Sk = ζ ( p + T ) , where ζ = captures the marginal value of an extra unit of consumption
S for households. It would seem that investment should be pinned down by ι
0
( g ) = p + T . That is, devote consumption goods to create new capital up to the point where the cost in consumption goods equals the shadow value, also in consumption goods. However, this is the shadow value of a unit of capital that falls from the sky. The planner knows that in order to create more capital using his investment technology, he must distort idiosyncratic risk sharing, so in the implementation of the optimal allocation we have ι
0
( g ) = p = shadow value − T . The ratio T /p therefore measures the size of the externality.
We can use the FOC for g
in equation ( 40 ) to obtain a sufficient statistic for the optimal tax.
Proposition 5.
In the regulated competitive equilibrium, the present value of the tax on capital holdings, relative to the market value of capital, satisfies the expression
T p
= γ pk
( ˜ )
2 n
ι
00
( g )
ι 0 ( g )
(52)
risk sharing, that’s why relative risk aversion γ is there. Next we have the leverage ratio for experts, pk/n , since the more leveraged they are, the bigger their exposure to idiosyncratic risk. Next we
27
have the fraction of capital’s idiosyncratic risk that can’t be insured away due to the moral hazard problem, ( ˜ )
2
. Finally, the semi-elasticity of the price of capital with respect to the growth rate
ι
00
( g )
ι ( g )
= p,g
. To understand why this matters, recall that the social planner relaxes the idiosyncratic risk sharing problem by distorting the optimal growth rate - this reduces the black market price of capital ˜ = ι
0
( g ) and therefore reduces the private benefit of diverting returns. If the function ι ( g ) was linear, however, the planner wouldn’t be able to affect the price of capital, so there is no point in distorting the growth rate. This is the same intuition as in
Kehoe and Levine ( 1993 ). We can
T /p = α p,g
. This makes sense, since in equilibrium α measures the marginal cost for an expert of managing an extra dollar of capital, from the required exposure to idiosyncratic risk.
) to get an idea of the size of the optimal tax.
For example, following
He and Krishnamurthy ( 2014 ), use a “skin in the game” constraint
˜
= 20% , leverage ratio pk/n = 3 , and an elasticity = 3 . For the idiosyncratic risk of capital use ν = 25%
For a RRA γ = 2 we get a T /p ratio of 4 .
5% , while for a γ = 5 we get a T /p ratio of 11 .
25% . Alternatively, using T /p = α p,g and a value of α between 2% and 4% we get an optimal T /p ratio between 6% and 12% .
The optimal policy is state dependent. Taking γ and p,g fixed, the value of taxes relative to capital goes up after shocks that increase experts’ leverage pk/n or the uninsurable idiosyncratic risk
˜
. Since this ratio measures the size of the externality, this can help us understand how different aggregate shocks may affect the wedge, and therefore the optimality of aggregate risk sharing in the unregulated competitive equilibrium. We can also evaluate how optimal policy should respond to
aggregate shocks. Notice that in order to use formula ( 52 ) we don’t need to specify what are the
aggregate shocks hitting the economy.
5 Two numerical examples: TFP and uncertainty shocks
In this section I provide numerical solutions for the competitive equilibrium and the social planner’s allocation for two different setups that illustrate the general results. First I consider an economy driven purely by TFP shocks. In this case, the unregulated competitive equilibrium features overinvestment, but the allocation of aggregate risk is efficient since the wedge is not correlated with aggregate shocks. Experts and households share aggregate risk proportionally, in both utility and financial terms, and there is no financial amplification channel. As a result, this case holds less interest as a positive theory of financial crises, but is very useful as a benchmark.
I then consider an economy hit by uncertainty shocks that increase idiosyncratic risk ν t
, as in
Di Tella ( 2013 ). As in in the TFP case, the unregulated competitive equilibrium is inefficient
because of the moral hazard externality. In contrast with the first example, after an uncertainty shock
18
Of course, introducing taxes may affect equilibrium objects such as leverage ratios, but we can use this formula to evaluate whether the current policy is in fact optimal and to determine properties of the optimal policy. In addition, notice that any equilibrium leverage is consistent with the optimal allocation in some region of the state space ( X, Y ) .
19 See
28
20
15
10
5
30
25
S
5 10 15 20 25 30
X pk n
10
8
6
4
2
0 5 10 15 20 25 30
X
0.020
0.015
0.010
0.005
g
5 10 15 20 25 30
X
0.8
0.6
0.4
0.2
0.0
L
5 10 15 20 25
X
0.6
0.5
0.4
0.3
0.2
0.1
0.0
5 10 15 20 25
T p
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0 5 10 15 20 25
X
X
Figure 1: Households’ utility S , growth g , and experts’ idiosyncratic risk ˜ leverage pk n
, MRT Λ , and
T p x
(above), and experts’ ratio (below), as functions of X . Solid line is the CE, dashed line is the
SP.
financial losses are concentrated on the balance sheet of experts, creating a financial amplification channel. This concentration of aggregate risk is inefficiently large, because the wedge created by the moral hazard externality is correlated with uncertainty shocks. Utility losses, on the other hand, are too concentrated on households.
5.1
Brownian TFP Shocks
In this section I solve for a recursive equilibrium in an economy driven only by TFP shocks. There are no shocks Y to the economic environment ( µ
Y
= σ
Y
= 0 ) which is fixed: ν t
= ν , σ t
= σ ,
ι t
( g ) = ι ( g ) , and φ t
= φ . In order to obtain a stationary distribution for X , I introduce retirement.
Experts retire with independent Poisson arrival rate θ . When they retire they become households and cannot manage capital any longer. The contractual environment with retirement is explained in detail in the Online Appendix, while
and
show how to incorporate retirement into the competitive equilibrium and the planner’s problem.
Parameter values: I use the following parametrization for this numerical example.
Preferences :
γ = 5 , ψ = 3 , ρ = 0 .
1 , θ = 0 .
1 ; technology : a = 1 , σ = 0 .
0125 , ν = 0 .
25 , ι ( g ) = 200 g
2
; moral hazard :
φ = 0 .
2 .
Competitive equilibrium vs.
social planner.
Figure 1 shows the unregulated competitive equilibrium and the social planner’s allocation.
The social planner can deliver more utility to households for any level of utility for experts. He achieves this by improving idiosyncratic risk sharing, which reduces the cost of providing incentives to experts, but comes at the cost of lower price of capital, and therefore lower investment and growth. As a result, experts are exposed to too
29
much idiosyncratic risk, and must be paid too much to compensate them, from a social perspective
(but contracts are privately optimal). The planner can implement his allocation as a competitive equilibrium with a tax on capital to reduce its equilibrium price, internalizing the externality. He uses the proceeds to create a safe government asset (without idiosyncratic risk), which agents can hold, reducing the weight of risky capital on their portfolios. As a result, experts’ equilibrium leverage ratio goes down.
In both the unregulated competitive equilibrium and the planner’s allocation, growth and investment are lower when experts’ continuation utility X is lower. This is because experts must be exposed to a large amount of idiosyncratic risk, relative to their continuation utility (in terms
ψ > 1
its equilibrium price falls, along with investment and growth.
The size of the moral hazard externality is larger as well when X is low. When experts’ exposure to idiosyncratic risk is large, so is its marginal cost. The planner is therefore willing to give up more growth, relative to the unregulated competitive equilibrium, to improve idiosyncratic risk sharing. This is reflected in a high value of the safe government asset in this region, as captured by the tax ratio T /p in expression
In both the unregulated competitive equilibrium and the planner’s allocation aggregate risk is shared proportionally between experts and households, both in utility and financial terms. There is therefore no financial amplification channel, and TFP shocks don’t affect investment, growth, or prices (only the level of effective capital k
σ
Y
= 0 , we have
σ
X
= 0 in both the competitive equilibrium and the planner’s allocation. As a result, aggregate TFP shocks don’t affect either the private or social cost of experts’ utility, σ
CE
Λ
σ
X
= σ
S
= σ
SP
Λ
= 0
in both the unregulated competitive equilibrium and the planner’s
allocation. Likewise, equations ( 24
) and ( 50 ) show aggregate financial risk is shared proportionally
between experts and households in both the unregulated and regulated competitive equilibrium.
Indeed, although the unregulated competitive equilibrium is inefficient, the allocation of aggregate risk is efficient because aggregate TFP shocks are not correlated with the wedge Γ created by the moral hazard externality.
5.2
Uncertainty Shocks
Now consider an economy hit by uncertainty shocks. The idiosyncratic risk of capital, ν t
= Y t
, now
follows an autoregressive stochastic process
dν t
|{z} dY t
= β (¯ − ν t
) dt +
√
ν t
σ
ν dZ t
| {z }
µ
Y
( Y t
)
| {z }
σ
Y
( Y t
)
20
With EIS ψ < 1 , the interest rate would adjust instead, and the price of capital would actually be higher.
21 This is a CIR process. If 2 β ¯ ≥ σ 2
ν
, then ν t
> 0 always, and it has a long-run distribution with mean ¯ .
30
By convention, we will take σ
ν
< 0 , so that we may think of Z as a “good” shock that drives idiosyncratic risk ν t down. The aggregate risk of capital is still fixed σ . For simplicity we consider only one aggregate shock ( Z is unidimensional). If σ > 0 the good shock Z also improves TFP, but there is no loss in intuition from setting σ = 0 .
Parameter values: I use the same parametrization as for the TFP shocks case and add a stochastic process for ν t
.
Preferences : γ = 5 , ψ = 3 , ρ = 0 .
1 , θ = 0 .
1 ; technology : a = 1 , σ = 0 .
0125 ,
ι ( g ) = 200 g
2
; moral hazard : φ = 0 .
2 ; uncertainty shock : β = 0 .
69 , ¯ = 0 .
25 , σ
ν
= − 0 .
17 .
Competitive Equilibrium vs. social Planner.
Figures 2, 3, 4, and 5 show the unregulated competitive equilibrium and the social planner’s allocation. As in the TFP shocks case, the social planner can deliver more utility to households S for any level of utility for experts X . He achieves this by improving idiosyncratic risk sharing, which allows him to deliver utility to experts at a lower cost, but comes at the cost of lower price of capital and therefore lower investment and growth.
The optimal allocation can be implemented with a tax on capital that internalizes the moral hazard externality, and using the proceeds to create a safe government security (no idiosyncratic risk) that agents can hold on their portfolio. Since capital is now only a fraction of aggregate wealth, experts don’t need such a high leverage ratio to hold the capital stock.
In both the unregulated competitive equilibrium and the planner’s allocation, investment and growth are lower when experts’ continuation utility X is low, and when idiosyncratic risk ν is high.
It is costly to provide incentives to experts when capital is very risky relative to their continuation
utility. From the pricing equation for capital ( 15 ), we see that with higher idiosyncratic risk
ν and lower X
experts will demand a higher premium to hold capital (in terms of expression ( 16 ) both
ν and experts’ leverage ratio is higher). With EIS ψ > 1 the price of capital must then be lower, as well as investment and growth. For the same reason, the externality is also larger when X is low and ν high (the wedge Γ is larger). The planner is then willing to reduce growth to a larger extent, relative to the unregulated competitive equilibrium, in order to improve idiosyncratic risk sharing.
This is also reflected in a higher T /p ratio.
We can check that σ
X
> 0 in the unregulated competitive equilibrium. After a bad uncertainty shock, idiosyncratic risk ν goes up and experts’ utility X goes down. This is not surprising. Since the economic environment is simply worse both households and experts will lose utility.
In contrast to the TFP shocks case, aggregate risk is not shared proportionally here. In the unregulated competitive equilibrium, after a bad uncertainty shock, financial losses will be concentrated on the balance sheets of experts, σ n
> σ w
. As a result, experts’ equilibrium leverage pk n must
go up for them to hold the capital stock, and we can see from ( 16 ) that this further increases the
idiosyncratic risk premium on capital. As a result, the endogenous concentration of financial losses on the balance sheets of experts amplifies the effect of higher idiosyncratic risk, further depressing asset prices and investment through a financial amplification channel.
31
30
25
20
15
10
5
S
5 10 15 20 25 30
X
27
26
25
24
23
22
S
0.1
0.2
0.3
0.4
0.5
v
0.020
0.015
0.010
0.005
g
0.020
0.015
0.010
0.005
g
5 10 15 20 25 30
X
0.6
0.5
0.4
0.3
0.2
0.1
0.0
5 10 15 20 25
X
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
v
0.1
0.2
0.3
0.4
0.5
v
Figure 2: Households’ utility S , growth g , and experts’ idiosyncratic risk ˜ x
, as functions of X for a fixed ν = 0 .
25 (above), and as functions of ν for a fixed X = 4 .
86 (below). Solid line is the CE, dashed line is the SP.
Σ n
0.03
-Σ w
0.02
0.01
0.00
0.01
5 10 15 20 25
X
Σ n
0.03
-Σ w
0.02
0.01
0.00
0.01
0.1
0.2
0.3
0.4
0.5
v
L
0.8
0.6
0.4
0.2
0.0
L
0.8
0.6
0.4
0.2
0.0
5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
X v
1.5
G
1.4
1.3
1.2
1.1
1.0
0 5 10 15 20 25
X
1.7
1.6
1.5
1.4
G
1.3
1.2
1.1
1.0
0.0
0.1
0.2
0.3
0.4
0.5
v
Figure 3: Concentration of financial risk σ n
− σ w
, MRT Λ , and wedge Γ , as functions of X for a fixed ν = 0 .
25 (above), and as functions of ν for a fixed X = 4 .
86 (below). Solid line is the CE, dashed line is the SP.
32
pk n
10
8
6
4
2
0 5 10 15 20 25 30
X pk n
10
8
6
4
2
0.0
0.1
0.2
0.3
0.4
0.5
v
Σ
X
0.008
0.006
0.004
0.002
0.000
X
5 10 15 20 25
0.030
Σ
X
0.025
0.020
0.015
0.010
0.005
0.000
0.1
0.2
0.3
0.4
0.5
v
Σ
X
0.000
-Σ
S
0.002
5 10 15 20 25
0.004
0.006
0.008
0.010
X
Σ
X
-Σ
S
0.000
0.002
0.004
0.006
0.008
0.010
0.1
0.2
0.3
0.4
0.5
v
Figure 4: Experts’ leverage ratio pk/n , volatility of X , σ
X
, and concentration of utility risk σ
X
− σ
S
, as functions of X for a fixed ν = 0 .
25 (above), and as functions of ν for a fixed X = 4 .
86 (below).
Solid line is the CE, dashed line is the SP.
T p
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0 5 10 15 20 25
X
T
10
8
6 p
4
2
0
0.0 0.1 0.2 0.3 0.4 0.5
v vol H pk + Tk L
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
5 10 15 20 25
X vol H pk + Tk L
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.1
0.2
0.3
0.4
0.5
v
0.5
0.4
0.3
0.2
0.1
0.0
vol H pk L
0.5
0.4
0.3
0.2
0.1
0.0
5 10 15 20 25 vol H pk L
0.1
0.2
0.3
0.4
0.5
X v
Figure 5:
T p ratio, volatility of total wealth, and volatility of value of capital, as functions of X for a fixed ν = 0 .
25 (above), and as functions of ν for a fixed X = 4 .
86 (below). Solid line is the CE, dashed line is the SP.
33
To understand why financial losses are concentrated on experts’ balance sheets, we need to look at the private cost of experts’ utility Λ
CE
. After an uncertainty shock, the premium on idiosyncratic risk goes up. Since experts are able to invest in capital and receive this premium, while households’ cannot, it is privately less costly now to deliver utility to experts, relative to households. For this reason Λ
CE falls after an uncertainty shock (both because ν goes up and because X goes down) so σ CE
Λ
> 0 . With γ > 1 the income effect dominates: experts are willing to take large financial losses after an uncertainty shock because they expect very high returns looking forward, relative
to households. Equation ( 24 ) reflects this and shows that
σ n
> σ w
. Utility losses, however, are
concentrated on households, as equation ( 22 ) shows.
This concentration of aggregate risk is inefficient.
In the regulated competitive equilibrium aggregate risk is shared close to proportionally. The reason for this inefficiency is that, in contrast to TFP shocks, the wedge Γ is correlated with uncertainty shocks. After an uncertainty shock, idiosyncratic risk ν is high and experts’ utility X is low. As explained above, this means the moral hazard externality is large, so we have σ
Γ
< 0
. As a result, equation ( 51 ) tells us that financial losses
are too concentrated on experts’ balance sheets after an uncertainty shock. The planner would like to share financial losses more evenly, or even concentrate them on households.
The planner implements the optimal allocation with a state contingent tax on capital, and uses the proceeds to create a safe government asset with value T per unit of capital. After an uncertainty shock, the T /p ratio goes up, both because ν is higher and X is lower. As a result, the price of capital, and investment, falls more in the regulated competitive equilibrium than in the unregulated one. This may seem counterintuitive, but it just reflects the fact that the externality
is larger after uncertainty shocks, and is captured by equation ( 52 ). This state contingent tax
also eliminates incentives for excessive concentration of aggregate risk, by making experts expect lower excess returns after uncertainty shocks, compared to the unregulated competitive equilibrium.
Notice also that the countercyclical government security stabilizes total wealth n + w = pk + T k .
As a result, the volatility of aggregate wealth is actually lower under regulation. Finally, with less concentration of financial risk on experts, and a higher T /p ratio after uncertainty shocks, experts’ equilibrium leverage ratio pk n
=
1
1+ T /p n + w n is not only lower, but also increases less after uncertainty shocks than in the unregulated competitive equilibrium.
6 Conclusion
This paper studies the optimal financial regulation policy in a setting where financial frictions are derived from a moral hazard problem. Even if agents can write complete long-term contracts, the competitive environment is inefficient. Agents don’t internalize that by competing for capital and bidding up its price they create a moral hazard problem for everyone else. A social planner who faces the same environment prefers to reduce the price of capital, at the cost of lower investment and growth, to improve idiosyncratic risk sharing. In addition, although the externality is unrelated
34
to aggregate risk sharing, the allocation of aggregate risk will also be inefficient if aggregate shocks are correlated with the externality. The optimal allocation can be implemented as a competitive equilibrium with a tax on capital holdings, used to finance a safe government security (without idiosyncratic risk) that agents can hold on their portfolio. Since risky capital is then a smaller fraction of aggregate wealth, experts’ equilibrium leverage is lower. I derive a simple sufficient statistic for the optimal value of taxes.
I then study two specific settings that illustrate the general results. In the first, the economy is hit only by TFP shocks. Asset prices, investment, and growth are too high in the unregulated competitive equilibrium, but the allocation of aggregate risk is efficient: experts and households share it proportionally, both in utility and financial terms, and there is no financial amplification channel. In contrast, in the second setting the economy is hit by uncertainty shocks that raise idiosyncratic risk. Now in addition the competitive equilibrium concentrates aggregate financial risk on experts, creating a financial amplification channel, while concentrating aggregate utility risk on households. Furthermore, this allocation of aggregate risk is inefficient, because uncertainty shocks are correlated with the wedge between the private and social cost of experts’ utility. After a bad uncertainty shock, financial loses are too concentrated on experts, and utility loses too concentrated on households.
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Appendix A
For the numerical solution we want to introduce retirement for experts with independent Poisson arrival θ , in order to obtain a non-degenerate stationary distribution (we can get the results in the main text by setting θ = 0 ). When an expert retires at time τ , he cannot manage capital any longer, so the cost of delivering utility to him is ξ
τ
= ζ
− 1
τ
. Since retirement is observable and contractible,
37
his utility can jump down by an amount λ
τ
= (1 − γ ) U
τ
−
λ
τ
. This requires modifying the HJB equation for experts and the law of motion of X . The Online Appendix develops the contractual environment with retirement in detail and provides a verification theorem for the following HJB.
r t
ξ t
= min
ˆ
ˆ
U
, λ
ˆ − p t
ˆ
( a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− ( r t
+ τ t k
) − ( σ t
+ σ p,t
) π t
)
+ ξ t
1
1 − 1 /ψ n
ρ − ˆ
1 − 1 /ψ o
+ θ
ˆ − ˆ
U
π t
+ µ
ξ,t
− σ
ξ,t
π t
+
1
2
γ ˆ
2
U
+
1
2
γ (ˆ
− 1 /ψ t
φ t p t kν t
)
2
+ σ
ξ,t U
+ θ 1 − ˆ
(1 − γ )
1
1 − γ
1
ζ t
ξ t
− 1
The FOC for ˆ ,
ˆ
, and σ
U are unchanged. For
ˆ we get
1 = 1 −
ˆ
(1 − γ )
γ
1 − γ
1
ζ t
ξ t
= ⇒
ˆ
=
1 − Λ
1 − γ
γ
1 − γ
The economic intuition is related to the hedging of aggregate risk. It is more costly to deliver utility to the expert after he has retired, because he can’t manage capital any longer. The principal is then happy to give him more utility while he doesn’t retire and it’s less costly to deliver utility to him, and in exchange give him less utility after retirement, when delivering utility is more costly. The law of motion for X must also be modified
µ
X
=
ρ
1 − 1 /ψ e
1 − 1 /ψ
−
1 − 1 /ψ
+
γ
2
2
U
+
γ
2 e
− 1 /ψ
φpν
X
)
2
− g − σ ˆ
U
+ σ
2
+ θ
ˆ
− 1
Because each expert’s utility will jump down on retirement, it must drift up while they don’t retire to compensate them. As a result, X gains a positive drift θ
ˆ
. On the other hand, when experts retire their continuation utility is not counted in X any longer, since they are now households, so we get the term − θ .
The strategy to solve the competitive equilibrium is to use Ito’s lemma to transform the problem into a system of PDEs for p , ξ , and ζ .
Suppose we are given these functions.
We can build
S = ζ ( p − ξX ) and Λ = ξζ , and use Ito’s lemma to compute the drift and volatility of all these objects, in terms of µ
X and σ
X
, which we still don’t know, and µ
Y and σ
Y
, which are exogenously
given. We use equation ( 23 ) to obtain
σ
X
. Then use the definition of σ
X to write ˆ
U
= σ
X
+ σ , and use the FOC for ˆ
U
π = γ σ
U
+ σ
ξ
. Then using the FOC for σ w
, equation
σ w
=
π
γ
+
1 − γ
γ
σ
ζ
. Now we need to compute the drifts. First, use the FOCs to obtain ˆ and ˆ , and we can use the definition of µ
X to compute it. Now use households’ HJB to compute r .
We now have experts’ HJB (with
ˆ
= X
− 1 from market clearing for capital), the FOC for capital, and the market clearing condition for consumption goods (this is an algebraic constraint), and we
38
need to find ξ , ζ and p .
The system of equations can be solved by adding a fictitious finite time horizon T , with some terminal values for these functions. A time derivative must be added to the computation of all drifts, and we can then solve backwards in time. In this respect we have a system of first order ODEs with respect to time, which can be solved with a standard integrator, such as Runge-Kutta 4 for example.
If the time derivatives vanish as we solve backwards, we have a solution to the system of PDEs we were interested in (infinite horizon). Terminal conditions are not important as long as the time derivatives vanish in the limit. Since the market clearing condition for consumption is an algebraic constraint, it is easier to differentiate it with respect to time to obtain a differential equation. We just need to make sure that terminal conditions are consistent with market clearing for consumption goods, and the algorithm will preserve this as we solve backwards. We can also verify ex-post that this condition is satisfied by the solution.
There are two complications. The first is that the FOC for ˆ cannot be solved analytically, and solving it numerically at each step would make the algorithm much slower. What we can do is add ˆ as a function to be solved for, and differentiate the FOC for e with respect to time, like we did for market clearing for consumption. We get an extra unknown but also an extra differential equation, and terminal conditions must be chosen so that the FOC for ˆ is satisfied. This can also be verified ex-post (the benefit is we only solve the FOC for ˆ once at the beginning). The second complication is that the domain of the system ( X, Y ) ∈ D ⊂
R
2
+ we know that X ∈ (0 , is unknown. Basically, for a given Y
¯
( Y )) , but we don’t know what is the maximum utility that can be delivered to experts for each exogenous state Y . To deal with this we can do a change of variables, such as
˜
=
X
X + ζ ( p − ξX )
∈ (0 , 1) , and solve the resulting system.
Appendix B
We want to introduce retirement for experts with independent Poisson arrival rate θ in order to obtain a stationary distribution (we can get the results in the main text by setting θ = 0 . ). This requires modifying the planner’s HJB equation and the law of motion of X . First, X now captures the continuation utility of currently remaining experts. Likewise, S is the continuation utility of current households, including previously retired experts. The HJB becomes
ρ
1 −
1
ψ
= max g, ˜ ˜
U
, λ
( a − ι ( g ) − ˜ )
1 −
1
ψ
1 −
1
ψ
S
1
ψ
− 1
+ µ
S
− θ
X
S
1 −
˜ t
(1 − γ )
1
1 − γ
+ g −
γ
2
σ
2
S
−
γ
2
σ
2
+(1 − γ ) σ
S
σ
(53)
The θ term captures the fact that current households only get a part of future continuation utility of households in the future, because future households include current experts that will retire in the meantime. Because retirement for each expert is observable and contractible, their continuation utility can jump down on retirement. The
˜ term captures this, as in
39
Likewise, µ
X is the same as in the CE with retirement (see
µ
X
=
ρ
1 −
1
ψ
− e
1 −
1
ψ
1 −
1
ψ
+
γ
2
2
U
+
γ
2 e
−
ψ
1
φ
ι
0
( g )
X
ν )
2
− g − σ ˜
U
+ σ
2
+ θ
˜
− 1
Because each expert’s utility will jump down on retirement, it must drift up while they don’t retire to compensate them. As a result, X gains a positive drift θ
ˆ
. On the other hand, when experts retire their continuation utility is not counted in X any longer, since they are now households, so we get the term − θ .
Now we have an extra FOC for
˜
:
S
0
X
S
X +
X
S
1 −
˜
(1 − γ )
1
1 − γ
− 1
= 0
= ⇒
˜
=
1 − Λ
1 − γ
γ
1 − γ
All the other FOC are unchanged.
The planner’s HJB is a PDE for S ( X, Y ) . As in the competitive equilibrium, we can solve it by adding a fictitious finite horizon T . This requires us to add a time derivative when computing µ
S
.
We can then solve backward for arbitrary terminal conditions. If the time derivative vanishes as we solve back, we found the original PDE.
Just like in the competitive equilibrium case, we need to deal with two complications. The first is that now the FOC for both ˜ and g are difficult to solve analytically, so we add both as functions of
( X, Y ) and differentiate the FOCs with respect to time to obtain two more PDEs. We just need to ensure that terminal conditions satisfy the FOCs (the benefit, as before, is that we only solve them numerically once). We can check at the end that the FOC are satisfied. The second problem is that as before we don’t know the domain, so we need to do a change of variables as in the competitive equilibrium, such as
˜
=
X
X + S
∈ (0 , 1) , and solve the resulting system.
Proof of Proposition
Using the FOCs for consumption we obtain in the CE:
Λ ˆ
− 1 /ψ
+ γψ ( φι
0
ν
( g )
X
)
2 e
− 2 /ψ − 1
=
S
− 1 /ψ where
Λ
CE
˜ = c k
. This is the same as equation ( 39 ) in SP. It follows that if the CE is efficient, then
= Λ
SP
, since all other objects in the equation must coincide, including S .
Now use the FOC for g , ι
0
( g ) = p , and for consumption to obtain in the CE
S
− 1 /ψ
ι
0
( g ) = S + Λ X
40
while in the SP we have
S externality
− 1 /ψ
ι
0
( g ) + z }|
Λ Xγ (˜
− 1 /ψ
φι
0
( g )
ν
X
)
2
ι
00
( g )
{
ι 0 ( g )
= S + Λ X
This shows the CE is not constrained efficient.
Proof of Proposition
Straightforward from equilibrium and optimal allocation conditions.
Implementation of the planner’s allocation (and proof of Proposition
Using the construction described in Section
, we only need to check that experts’ and the representative
household’s HJB equations are satisfied, and the pricing equation for taxes holds. Start with households. By construction, their FOC for consumption is satisfied. Using w = ( p + T − ξX ) k > 0 we derive an expression for σ w and then set π = γσ w
− (1 − γ ) σ
ζ
. We then set r so that their HJB equation is satisfied. We are in effect choosing r and π so that ( a − ι ( g ) − ˆ ) k ck is the optimal choice of consumption for the household, and their wealth ( p + T − ξX ) k . Notice that the FOC for experts’ ˆ , σ
U
, and
ˆ will be satisfied automatically. To see this, first use the planner’s FOC for ˜
Λ ˜
−
1
ψ + γ
1
ψ
( φι
0
( g )
ν
X
)
2 e
− 2
1
ψ
− 1
=
S
−
1
ψ and using
S
−
1
ψ = ζ and Λ = ξζ we get experts’ FOC for ˆ in the private contract (recall ˜ = ˆ )
ξ ˆ
−
ψ
1
+ γ
1
ψ e
−
1
ψ φι
0
( g )
ν
X
)
2
= 1
From planner’s optimal aggregate risk sharing we get
σ
X
= σ
S
−
1
γ
σ
Λ
= σ
ζ
+ σ w
− σ −
1
γ
( σ
ζ
+ σ
ξ
) and using π = γσ w
− (1 − γ ) σ
ζ and σ
X
= ˆ
U
− σ we get experts’ FOC for σ
U in the private contract:
ˆ
U
=
π − σ
ξ
γ
For
ˆ we get the FOC in the planner problem
˜
=
1 − Λ
1 − γ
γ
1 − γ which coincides with the FOC in the CE since Λ = ξζ .
Now we want to prove that ξ = −
S
X
ζ satisfies experts’ HJB equation. For this we will use the
planner’s HJB equation ( 53 ). Multiply by
S on both sides, take the derivative with respect to X
41
using the envelop theorem, and divide throughout by S
X to obtain
ρ
1 − 1
ψ
=
+ µ
X
( a − ι ( g ) − ˜ )
1 −
1
ψ
1 − 1
ψ
1
ψ
S
1
ψ
− 1
−
( a − ι ( g ) − ˜ )
S
X
−
1
ψ S
1
ψ
ˆ + ( g −
γ
2
σ
2
)
− γ (˜
−
1
ψ
θ
−
S
X
φ
ι
0
( g )
X
ν )
2
+
1 − λ t
(1 − γ )
1
2
S
XY Y
S
X
σ
2
Y
+
+
S
S
XY
X
σ
X
σ
Y
+ (1 − γ ) σ
1
2
1
1 − γ
+
S
XXX
S
X
S
XY
S
X
( σ
S
XX
S
X
σ
X
X +
X
µ
S
X )
2
S
Y
XY
X
+
σ
Y
S
+
XX
S
X
S
µ
XX
S
X
X
σ
2
X
X +
+ (1
X
− γ ) σσ
S
XXY
X
S
X
σ
X
Xσ
Y
2
− γ
S
X
S
σ
X
X +
S
Y
S
σ
Y
S
XX
S
X
σ
X
X +
S
XY
S
X
σ
Y
+ σ
X
+
γ
2
S
X
S
σ
X
X +
S
Y
S
σ
Y
(54)
Now use − ξζ = S
X to obtain
S
S
XX
X
=
ξ
X
ξ
+
ζ
X
,
ζ
S
XY
S
X
=
ξ
Y
ξ
+
ζ
Y
ζ
,
S
XXX
S
X
=
ζ
XX
ζ
+ 2
ξ
X
ξ
ζ
X
ζ
+
ξ
XX
,
ξ
S
XY Y
S
X
=
ζ
Y Y
ζ
+ 2
ζ
Y
ζ
ξ
Y
ξ
+
ξ
Y Y
,
ξ
S
XXY
S
X
=
ζ
XY
ζ
+
ξ
Y
ξ
ζ
X
ζ
+
ξ
X
ξ
ζ
Y
ζ
+
ξ
XY
ξ
Now plug this into ( 54 ), use the definition of
µ
X we already know holds), and simplify to obtain and the FOC for ˆ in the private contract (which
0 =
1
ψ
1 −
1
ψ c
1 −
1
ψ ζ
1
ψ
− 1
+ µ
ξ
+ µ
ζ
−
1
ψ e
1 −
1
ψ
1 −
1
ψ
+
γ
2
σ
2
X
− γ (
1
2
1
− ψ )(˜
−
ψ φ
ι
0
( g )
ν )
2
X
+ σ
ξ
σ
ζ
+ σ
X
( σ
ξ
+ σ
ζ
) + (1 − γ ) σ ( σ
ξ
+ σ
ζ
)
− γσ
S
( σ
ξ
+ σ
ζ
+ σ
X
) +
γ
2
σ
2
S
+ θ
˜ − 1 + Λ
1 − γ
γ where from the FOC for
˜ the last term is just θ
˜
. Now from household’s HJB and using σ w
=
σ
S
− σ
ζ
+ σ we get
ρ
1 − 1
ψ
=
1
ψ 1 c
1 −
1
ψ
− 1
ψ
ζ
1
ψ
− 1
+ r +
γ
2
( σ
S
− σ
ζ
+ σ )
2
+ µ
ζ
−
γ
2
σ
2
ζ which we plug into our expression. After some algebra using σ
S
= σ
X
+
1
γ
( σ
ξ
+ σ
ζ
) and ˆ
U as well as the FOC for ˆ and σ
U
, we get
= σ
X
+ σ ,
γ (˜
−
1
ψ φ
ι
0
( g )
ν )
2
X
− ˆ
42
+ ξ
( e
1 −
1
ψ r +
1 − 1
ψ
ρ
−
1 − 1
ψ
−
γ
2 e
−
1
ψ φ
ι
0
( g )
ν )
2
X
− µ
ξ
+ ˆ
U
π + σ
ξ
π −
γ
2
2
U
− σ
ξ
ˆ
U
− θ
Because τ k is chosen so that the pricing equation for capital holds we get
)
= 0
γ (˜
−
1
ψ φ
ι
0
( g )
X
ν )
2
= p
X a − ι ( g ) p
+ µ p
+ g + σ
0
σ p
− ( r + τ k
) − π ( σ + σ p
) and plugging this in, we obtain experts’ HJB.
Finally, we just need to check that the pricing equation for taxes is satisfied. First, use the planner’s HJB and households’ HJB to obtain a version of the dynamic budget constraint of the household (write S = ζ ( p + T − ξX ) , w = ( p + T − ξX ) k , and use the FOC for
˜
) pµ p
+ T µ
T
− ξX ( µ
ξ
+ µ
X
+ σ
ξ
σ
X
)+ g ( p + T − ξX )+ σ ( pσ p
+ T σ
T
− ξX ( σ
ξ
+ σ
X
)) − θξX 1 − ˜
(1 − γ )
= r ( p + T − ξX ) + π ( pσ p
+ T σ
T
− ξX ( σ
ξ
+ σ
X
) + σ ( p + T − ξX )) − ˆ ( p + T − ξX )
Multiply experts’ HJB by X to obtain a − ι ( g ) − ˆ + p ( g + µ p
+ σσ p
− ( r + τ k
) − ( σ + σ p
) π )
+ ξX
(
1 r +
1 −
1
ψ e
1 −
1
ψ
− ρ ) −
γ
2
2
U
−
γ
2 e
−
1
ψ φ
ι ( g )
X v )
2
U
( π − σ
ξ
) − µ
ξ
+ σ
ξ
π − θ
˜
)
= 0
Combining these two expressions, and using the definition of µ
X
, σ
X
, and ˆ = a − ι ( g ) − ˆ p + T − ξX pricing equation for taxes. This completes the proof of Proposition
we get the
Proof of Proposition
Straightforward from equilibrium and optimal allocation conditions.
Proof of Proposition
Straightforward from equilibrium and optimal allocation conditions.
43
Online Appendix: Optimal Contracts
This Appendix develops the contractual environment in detail. It will be useful for the numerical solutions to include an exogenous retirement time τ i which arrives with independent Poisson intensity
θ (this will yield a stationary distribution). After retirement the agent cannot manage capital any longer, and the contract just delivers a terminal utility by giving the agent consumption. The results in the main paper can be obtained by letting θ → 0 .
Setting
Let (Ω , P, F ) be a complete probability space. Throughout this appendix, all stochastic processes are adapted to the filtration
F i generated by the aggregate brownian motion Z , the idiosyncratic brownian motion W i
, and an idiosyncratic Poisson process N i with arrival rate θ associated with retirement of expert i . Prices and coefficients depends only on the history of aggregate shocks, e.g.
r , π , σ , ν , φ , p , g , ι ( g ) , τ k
, ζ . In what follows I will drop the reference i to the expert in order to simplify notation.
There is a complete financial market, with risk free interest rate r and price of aggregate risk π
(and associated equivalent martingale measure Q ). The agent can continuously trade capital k at price p , and obtains a return per dollar invested in capital dR t
= a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− τ t k dt + ( σ t
+ σ p,t
) dZ t
+ ν t dW t
The agent starts with net worth n
0 contract specifies consumption e = { and signs a contract C = e,
¯ with full commitment. The e t
≥ 0; t ≤ τ } , a terminal utility
¯
= { ¯ t
; t ≤ τ } , and capital under management k = { k t
≥ 0; t ≤ τ } . After retirement the agent cannot manage capital any longer, and the principal delivers utility
¯
τ
.
After signing the contract the expert can choose a hidden action s = { s t
; t ≤ τ } which we interpret as a stealing process. Stealing changes the distribution of observed outcomes from P to
P s so that the return can be written dR t
= a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− τ t k
− s t dt + ( σ t
+ σ p,t
) dZ t
+ ν t dW t s where W t s
= W t
+
´
0 t s u
ν u du is a brownian motion under P s
. For each dollar stolen, the expert keeps a fraction φ t
∈ (0 , 1) which he adds to his consumption: ˜ = e + φpks (the expert doesn’t have access to hidden savings). As a result he gets utility U s
(˜ ) = U s
0
, where the utility process U s
= { U t s
; t ≤ τ } is given by
U s t
=
E s t
ˆ t
τ f (˜ t
, U t s
) ds + ¯
τ
(55)
In this environment it is always optimal to implement no stealing s = 0 , for the same reasons
44
as in
The principals’ objective is to minimize the cost of delivering utility u
0 to the expert F ( C ) = F
0 where the continuation cost of the contract
F = { F t
; t ≤ τ } is
F t
=
E
Q t
ˆ t
τ e
−
´ u t r m dm
( e u
− p u k u
α u
) du + e
−
´
τ t r m dm
F
τ τ
) where utility
α t
≡ a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− r t
− τ k t
− ( σ t
+ σ p,t
) π t
. Here
¯ t
U ) is the cost of delivering
¯ to the expert who has retired and cannot manage capital any longer. The expert has in fact become a household, so the cost of delivering utility takes the form
F t
( ¯ ) = ζ
− 1 t
(1 − γ ) ¯
1
1 − γ and we assume ζ
− 1 t
> 0 is bounded.
We say a contract C = ( e,
¯
) is admissible if 1) there is a solution U
0
E h
ˆ
0 t f ( e u
, U
0 u
) du
2
+ ( U t
0
)
2 i
< ∞ for all t , and 2)
E
Q
ˆ
0
τ e
−
´ t
0 r m dm
| e t
− p t k t
α t
| dt + sup u ≤ τ e
−
´ u
0 r m dm ¯ u
( U u
) < ∞
(56)
(57)
Given a contract C , we say that a stealing process s is valid if 1) there is a U s
2) s
ν
≥ 0 is bounded and there is a constant T ∈
R + such that s t
= 0 ∀ t ≥ T
Let
S
( C ) be the set of valid stealing plans given contract C . We say an admissible contract C is incentive compatible
0 ∈ arg max s ∈
S
( C )
U s
( e + φpks )
Let
IC be the set of incentive compatible contracts. For an initial utility u
0 for the agent, an incentive compatible contract is optimal if it minimizes the cost of delivering initial utility u
0 to the agent, that is
J ( u
0
) = min
C∈
IC
F ( C ) st : U
0
( C ) ≥ u
0
By changing u
0 we can trace the Pareto frontier for this problem. In particular, at time t = 0 the expert has net worth n
0 which he gives to the principal in exchange for the contract. We set u
0 so
22 If a contract implements stealing s in equilibrium, then the principal can do better by giving e
0 agent as legitimate consumption, and implementing no stealing s
0
= 0 .
23 Note the expert can choose T as large as desired, as well as the bound on s t
ν t
= e + φpks to the
. These regularity conditions can be relaxed with some work, but are economically innocuous.
24 Notice
A
( C ) = ∅ because 0 ∈
S
( C ) for an admissible contract.
45
that the principal breaks even n
0
+ J ( u
0
) = 0
Let J t
= F ( C ∗
) be the continuation cost of an optimal contract C ∗
. This is how much it would cost the agent to “buy into” the contract at that time, and therefore this is the net worth of the expert at time t .
Recursive formulation
We can use the continuation utility of the expert as a state variable in order to provide incentives for not stealing. First we obtain a law of motion for continuation utility.
Lemma 1.
For any admissible contract C = ( c,
¯
) , the expert’s continuation utility U 0 satisfies dU t
0
= − f ( e t
, U t
0
) dt + σ
U,t dZ t
σ
U,t dW t
− λ t
( dN t
− θdt ) (58) for some σ
U and ˜
U in L 2
, and λ t
= U t
0
−
− ¯ t
Proof.
Consider the process
Y t
=
E t
ˆ
0
τ f ( e u
, U
0 u
) du + ¯
τ
=
ˆ
0 t f ( e u
, U
0 u
) du + U t
0 on { t ≤ τ }. Since Y is an
F
-adapted P -martingale, and
F is generated by Z , W and N , we can apply a martingale representation theorem to obtain dY t
= f ( e t
, U t
0
) dt + dU t
0
= σ
U,t dZ t
+ ˜
U,t dW t
− λ t
( dN t
− θdt )
we get
E
Y t
2
< ∞
for all t
U
τ
U
τ
, so therefore it must be that
σ
U
λ t
= U 0 t − and ˜
U are in L 2
− U t
, and from admissibility of C
Notice that because retirement is contractible, the agent’s utility can in principle “jump” when the agent retires. However, if U t
0 jumps down when the agent retires, for example, then while he
doesn’t retire it must drift up to compensate the agent. To obtain equation ( 4 ) in the main text,
we just drop the jump term.
Faced with contract C , the expert can consider a valid stealing process s ∈
S
( C ) , getting consumption ˜ = e + φpks under probability P s
. The following lemma gives necessary and sufficient conditions for an admissible contract to be incentive compatible, for the parameter configuration that is of interest to us.
25
In this context, L 2 is the set of
F
-adapted processes x such that
E h
´
0 t x 2 u du i
< ∞ for any t .
26
Notice that for t ≥ τ , Y t is constant, so σ
U,t
, ˜
U,t
, and λ t are zero, but this is not relevant for our purposes.
46
Lemma 2.
If EIS ψ > 1 and the risk aversion γ > 1 , an admissible contract C = ( e,
¯
) is incentive compatible if and only if
0 ∈ arg max s ≥ 0 f ( e t
+ φ t p t k t s, U
0 t
) − ˜
U,t s t
ν t
− f ( e t
, U t
0
) (59)
Remark.
The result of this lemma can be extended to other combinations of ψ and γ .
Proof.
Suppose the agent picks a valid stealing plan s . His utility is U s
, defined by
U t s
=
E s t
ˆ t
τ f ( e u
+ φ t p u k u s u
, U s u
) du + ¯
τ
We would like to compare this with the utility from good behavior U
0
. To do this, it’s useful to first express U
0 as an expectation under P s
dW t
= dW t s − s t
ν t dt , we obtain dU
0 t
= − f ( e t
, U
0 t
) − s t
ν t dt + σ
U,t dZ t
+ ˜
U,t dW t s
− λ t
( dN t
− θdt )
Now we can integrate, bearing in mind that since s
ν is bounded σ
U well. We get: and ˜
U are both in L 2
( P s
) as
U t
0
=
E s t
ˆ
T ∧ τ f ( e u
, U
0 u
) + s u
ν u
σ
U,u du + U
0
T ∧ τ t where s t
= 0 for all t ≥ T . Notice U s
T ∧ τ
ˆ
T ∧ τ
U t s
− U t
0
=
E s t t
= U
0
T ∧ τ
.
Now we can obtain f ( e u
+ φ t p u k u s u
, U s u
) − f ( e u
, U
0 u
) − s u
ν u
˜
U,u du (60) on { t ≤ T ∧ τ } .
To prove necessity, suppose ( 59 ) fails. Pick a bounded stealing strategy such that
f ( e t
+ φ t p t k t s, U t
0
) − ˜
U,t s t
ν t
− f ( e t
, U t
0
) > 0 on a set of positive measure A , and zero outside (we can pick T as large as desired). Look at the
integrand in ( 60 ), and write
f ( e u
+ φ t p u k u s u
, U s u
) − f ( e u
, U
0 u
) − ˜
U,u s u
ν u
= f ( e u
+ φ t p u k u s u
, U s u
) − f ( e u
+ φ t p u k u s u
, U
0 u
)
+ f ( e u
+ φ t p u k u s u
, U
0 u
) − f ( e u
≥ f ( e u
+ φ t p u k u s u
, U s u
) − f ( e u
, U
0 u
) − ˜
U,u s u
ν u
+ φ t p u k u s u
, U
0 u
) with strict inequality on A . Now we use an interesting fact about the EZ aggregator f ( c, U ) : if
γ > 1 and ψ > 1 , then there is a constant κ > 0 such that f ( c, y ) − f ( c, x ) ≤ κ ( y − x ) for y ≥ x ,
47
and any c
We can then write f ( e u
|
+ φ t p u k u s u
, U s u
) − f ( e u
, U
0 u
) − ˜
U,u s u
{z
H u
ν u
}
≥ κ ( U s u
− U
0 u
)
| {z
M u
} when U s u
− U
0 u
≤ 0
| {z }
M u and the inequality is strict on A . Now define the process M t
= U t s − U t
0 condition as and rewrite the previous
M t
= U t s
− U t
0
=
E s t
ˆ
T ∧ τ
H u du with H t
≥ κM t whenever M t
≤ 0 t
We can now use a generalized version of Skiadas’ Lemma
to obtain that M t
= U t s − U t
0 ≥ 0 as follows. Let τ
0
= inf { t ≥ 0 : U t
0 ≤ U s t
} and write
M t
1
{ τ
0
>t }
≥
E s t
ˆ
τ
0
∧ T ∧ τ
κM u
1
{ τ
0
>t } du + M
τ
0
∧ T ∧ τ
1
{ τ
0
>t } t
M t
1
{ τ
0
>t }
≥
E s t
ˆ
T ∧ τ t
κM u
1
{ τ
0
>u } du + M
τ
0
1
{ τ
0
>T ∧ τ }
=
E s t
ˆ t
T ∧ τ
κM u
1
{ τ
0
>u } du
Applying the stochastic Gronwall-Bellman inequality,
we get that M t
1
{ τ
0
>t }
≥ 0 for 0 ≤ t ≤ T ∧ τ .
Since M
0
1
{ τ
0
=0 }
≥ 0 , we conclude M the stopping time τ u
= inf { t ≥ u : U t
0
0
≥ 0 . We can apply a similar argument for any u (redefining
≤ U t s } ) and get M u
≥ 0 for all 0 < u < T ∧ τ . For u ≥ T ∧ τ we already know that M u ∧ τ
= 0 .
Now to make the inequality strict, if M t
= 0 a.e. on [0 , τ ] × Ω , then H t
≥ κM t
= 0 , and the inequality is strict on positive measure subset A , and therefore M
0
> 0 . If M t
> 0 for at least some ( ω, t ) with t < τ , with positive probability, we do the following. For some small > 0 , let
τ = inf { t : M t
≥ } . If we take small enough, the probability that we get to such a point before τ is positive: P s
( { τ ∧ τ < τ } ) > 0 (for any P s since they are equivalent). It must be that there is some stealing going on after this, since otherwise M
τ would be zero. Now consider the alternative stealing plan and s
0
= 0 s
0 that steals only until τ and then stops, that is s
0 t after this. By a similar argument as before, U
0
0
≤ U s
0
0
= s t for t < τ
. Utility under this plan satisfies
U s
0
τ ∧ τ
= U
0
τ ∧ τ
< U s
τ ∧ τ if τ ∧ τ < τ , and equal otherwise. Now if we compare s and s
0
, both plans induce the same probability measure until τ ∧ τ , and the same consumption stream, but the payoff at τ ∧ T is larger for s (strictly so with positive probability):
U t s
0
=
E s t
ˆ t
τ ∧ τ f ( e u
+ φ t p u k u s u
, U s
0 u
) + U
0
τ ∧ τ
27 see Proposition 3.2 in
28 The strategy is similar to Theorem A.2 in
29
See
48
U t s
=
E s t
ˆ t
τ ∧ τ f ( e u
+ φ t p u k u s u
, U s u
) + U s
τ ∧ τ
By strict monotonicity of EZ preferences with respect to terminal value, we get U s
0
This proves stealing is attractive if ( 59 ) fails.
> U s
0
0
≥ U
0
0
.
For sufficiency, suppose ( 59 ) holds, so that for any valid stealing plan we have
f ( e t
+ φ t p t k t s, U t
0
) − ˜
U,t s t
ν t
− f ( e t
, U t
0
) ≤ 0
Using the same properties of the EZ aggregator as before but with the opposite inequalities, we get
M t
= U t s
− U t
0
=
E s t
ˆ
T ∧ τ
H u du with H t
≤ κM t whenever M t
≥ 0 t
The same reasoning as before now yields M
0
= U s
0
− U
0
0 compatible.
≤ 0 , so the contract is indeed incentive
∂ e f ( e t
, U t
0
) φ t p t k t
1
U,t
ν t
which yields the “skin in the game” condition ( 5 ) in the main text
˜
U,t
= ∂ e f ( e t
, U t
0
) φ t p t k t
ν t
(61)
HJB equation
Because of homothetic preferences, the value function for the principal’s cost minimization problem takes the form J t
= ξ t x t
. Here x t
= (1 − γ ) U t
0
1
1 − γ > 0 is a monotone (and concave) transformation of the expert’s continuation utility. As a result, we can also interpret it as the expert’s continuation utility. The stochastic process ξ = { ξ t
; t ≤ τ } captures the investment opportunity set, and has a law of motion dξ t
ξ t
= µ
ξ,t dt + σ
ξ,t dZ t
+ ( ξ t
ζ t
)
− 1
− 1 dN t
(62)
It depends only on the aggregate shocks Z that affect market prices, and on whether the expert has retired. The last term ensures that ξ
τ
= ζ
− 1
τ
.
Use the following normalization: k t
= ˆ t x t
, e t
= ˆ t x t
, σ
U,t
= ˆ
U,t
(1 − γ ) U t
0
, ˜
U,t e
− t
1
ψ φp t
ˆ t
ν t
(1 −
γ ) U t
0
, and λ t
λ t
(1 − γ ) U t
0
. Then we can write dx t x t
=
1
1 − 1
ψ
σ
U,t dZ t
ρ − ˆ
1 − t
1
ψ +
1
2
γ ˆ
2
U,t
+
1
2
γ ˆ
− t
1
ψ φ t p t
ˆ t
ν t
2
+ θ λ t
!
dt e
− t
1
ψ φ t p t
ˆ t
ν t dW t
+ 1 − ˆ
(1 − γ )
1
1 − γ − 1 dN t
(63)
The principal’s cost minimization problem is now a standard optimal control problem. The associ-
49
ated HJB BSDE is r t
J t dt = min e,k, ˆ
U
,λ
( e − p t kα t
) dt +
E
Q t
[ dJ t
] (64)
e ≥ 0 and value function, and the fact that Z t k ≥
= Z
Q t
0 . Using the normalization of the controls, the form of the
−
0 t
π u du , where Z
Q is a brownian motion under Q , we can rewrite the HJB equation r t
ξ t
= min
ˆ
ˆ
U
, λ
ˆ − p t
+
1
2
γ ˆ
2
U
+ kα t
+ ξ
1
2
γ ˆ
−
1
ψ t
φ t p t
1
1 −
1
ψ
ˆ t
ρ − ˆ
1 − t
1
ψ − ˆ
U
π t
+ µ
ξ,t
− σ
2
+ σ
ξ,t
σ
U
+ θ 1 −
ˆ
(1 − γ )
ξ,t
1
1 − γ
π t
1
ζ t
ξ t
− 1 + ˆ
(65)
This should be interpreted together with ( 62 ).
The expression on the right hand side is convex if
ψ > 2 , and the FOC are sufficient. If ψ ≤ 2 then the optimal contract does not exist, unless capital pays zero excess return, as explained in the paper. Focusing on the ψ > 2 case, we get the following
FOC
1
ξ t
ˆ
− t
ψ
γ
+ ξ t
ψ
φ t p t
ˆ t
ν t
2 e
− t
2
ψ
− 1
= 1 a − ι t
( g t
) p t
+ g t
+ µ p,t
+ σ t
σ
0 p,t
− ( r t
+ τ t k
) = ( σ t
+ σ p,t
) π t
| {z } agg. risk premium
+ γξ t
| e
− t
1
ψ φ t
ν t
2 p t
ˆ t
{z id. risk premium
}
σ
U,t
=
π t
γ
−
σ
ξ,t
γ
1 − ˆ
(1 − γ )
γ
1 − γ
1
ζ t
ξ t
= 1
The FOC for ˆ has the cost of delivering consumption to the expert on the right hand side, and the benefit of a lower promised utility on the left hand side. This is the standard tradeoff we would expect. In addition, however, there is another benefit to giving consumption to the agent: it relaxes the “skin in the game constraint”. By front loading consumption, the principal can reduce the marginal private benefit of stealing and consuming, and therefore improve idiosyncratic risk sharing.
As a result, there is a tradeoff between distortions in intertemporal consumption and idiosyncratic risk sharing.
The FOC for
ˆ gives us a pricing equation for capital. As usual, capital pays an excess return because it is exposed to aggregate risk with a market price of π t
. But in addition, capital must also pay an excess return for its exposure to idiosyncratic risk, even though this risk is not priced by the financial market. The reason for this is that the principal knows that if he gives more capital to the expert, he will have to expose him to risk for incentive reasons, and this is costly because the expert is risk averse.
30
That is, a solution to ( 65 ) is a process
ξ
that satisfies ( 62 ) for some
µ
ξ and σ
ξ
, and ( ξ , µ
ξ
, σ
ξ
50
The FOC for ˆ
U has the following interpretation. Delivering utility to the expert is costly for the principal. He would therefore prefer to promise him more utility in states of the world where it is relatively cheaper. This can happen because the value of a unit of consumption in that state is lower
(captured by the π t term) or because the cost of delivering utility in that state is lower (captured by the σ
ξ,t term). A similar logic explains the FOC for
ˆ
. After retirement the expert cannot manage capital any longer, so delivering utility to him is more costly. This is captured by ζ t
− 1 result, the optimal contract has
≥ ξ t
. As a
ˆ
≥ 0 . The principal prefers to promise the expert less utility after he retires (when it is more costly to deliver utility to him) even if it means promising him more utility while he doesn’t retire.
We can plug in the FOC into the HJB equation ( 65 ). If we find a solution
ξ to the HJB equation, we can use it to build the optimal contract using the policy functions ˆ ,
ˆ
, σ
U
, and
ˆ
(so the HJB holds with equality) and the law of motion of x
, ( 63 ) with initial condition
x
0
= ((1 − γ ) u
0
)
1
1 − γ .
Let C ∗
= ( e
∗
,
∗
, k
∗
) be the candidate optimal contract thus constructed, with associated state x
∗
.
We have the following verification theorem.
Theorem 1.
Let ξ
be a strictly positive solution to the HJB equation ( 65 ) bounded above by
ζ
− 1
.
Then,
ξ
0
1) For any incentive compatible contract C that delivers at least utility u
0
((1 − γ ) u
0
)
1
1 − γ ≤ F
0
( C ) .
to the agent, we have
2) Let C ∗ be a candidate contract constructed as described above. If C ∗ is admissible and delivers utility u
0 to the agent, then it is optimal, with cost F
0
( C
∗
) = J
0
( u
0
) = ξ
0
((1 − γ ) u
0
)
1
1 − γ .
Proof.
For the first part, consider an incentive compatible contract C = ( c,
¯
) that delivers utility u
0 to the agent, and has an associated state variable x . Use the HJB equation to obtain e
−
´
0
τ n ∧ τ r u du
ξ
τ n ∧ τ x
τ n ∧ τ
≥ ξ
0 x
0
−
ˆ
0
τ n ∧ τ e
−
´ t
0 r u du
( e t
− p t k t
α t
) dt
+
ˆ
0
τ n
∧ τ
+ e
−
´
0 t r u du
ξ t x t
( σ
ξ,t
σ
U,t
) dZ t
Q
+
ˆ
0
τ n
∧ τ
ˆ
0
τ n ∧ τ e
−
´ t
0 r u du
ξ t x t
θ 1 − ˆ
(1 − γ )
1
1 − γ e
−
´ t
0 r u du
ξ t x t
ζ t
1
ξ t
− 1 ( e
− t dN t
1
ψ φp t k t
− θdt )
ν t dW t
Here we are using the localizing sequence { τ n } n ∈
N
:
τ n
= inf
(
T ≥ 0 :
ˆ
T
0 e
−
´ t
0 r u du
ξ t x t
( σ
ξ,t
σ
U,t
)
2 dt +
ˆ
0
T e
−
´ t
0 r u du
ξ t x t e
− t
1
ψ
φp t k t
ν t
2 dt ≥ n
)
The stochastic integrals are therefore martingales, so we can take expectations under Q to obtain
E
Q
0 h e
−
´
τ n ∧ τ
0 r u du
ξ
τ n ∧ τ x
τ n ∧ τ i
≥ ξ
0 x
0
−
E
Q
0
ˆ
0
τ n ∧ τ e
−
´ t
0 r u du
( e t
− p t k t
α t
) dt
51
Now we will use the dominated convergence theorem to take the limit n → ∞ . First,
ˆ
0
τ n ∧ τ e
−
´ t
0 r u du
( e t
− p t k t
α t
) dt ≤
≤
ˆ
ˆ 0
τ e
−
´ t
0 r u du
0
τ n ∧ τ e
−
´ t
0 r u du
| e t
| e t
− p t k t
α t
| dt
− p t k t
α t
| dt which is integrable because C is admissible. Second, e
−
´
τ n ∧ τ
0 r u du
ξ
τ n ∧ τ x
τ n ∧ τ
≤ sup t ≤ τ e
−
´
0 t r u du
ξ t x t
≤ sup t ≤ τ e
−
´ t
0 r u du
ζ t
− 1 x t which is also integrable because C is admissible. So letting n → ∞ , we get τ n ∧ τ → τ a.s. and therefore using ξ
τ
= ζ
− 1
τ
:
E
Q
0 h e
−
´
τ
0 r u du
ζ
− 1
τ x
τ i
≥ ξ
0 x
0
−
E
Q
0
ˆ
0
τ e
−
´ t
0 r u du
( e t
− p t k t
α t
) dt
Rearranging we get the desired result.
For the second part, let C
∗ be the candidate optimal contract with associated utility process x
∗
.
By construction, the HJB holds with equality, so the same argument shows that C ∗
F
0
( C
∗
) = ξ
0
((1 − γ ) u
0
)
1
1 − γ . We know C
∗ delivers utility u
0 in fact has a cost by assumption, and Lemma
ensures it is incentive compatible.
Finally, from ( 64 ) we can easily see that the net worth of the agent
n t
= J t
= ξ t x t follows the law of motion dn t
= ( r t n t
+ p t k t
α t
− e t
+ σ n,t n t
π t
) dt + σ n,t n t dZ t
φ t p t k t
ν t dW t
+ λ n,t n t
( dN t
− θdt ) where φ t
= ξ t e
− 1 /ψ
φ , and λ n,t
λ t
( γ − 1) . With θ = 0
we obtain ( 9 ) in the main body.
52
