n?u=RePEc:red:sed016:1523&r=fmk

**Asset** **Pricing** **with** **Endogenously** **Uninsurable** **Tail** **Risks**

Hengjie Ai and Anmol Bhandari ∗

December 7, 2015

This paper studies asset pricing implications of idiosyncratic risk in labor productivities in

a model where markets are endogenously incomplete. Well-diversified owners of firms provide

insurance to workers using long-term wage contracts but cannot commit to ventures that yield

a net present value of dividends. We show that under the optimal contract subject to limited

commitment, workers are uninsured against tail risks in idiosyncratic productivities. Risk

compensations are higher when we calibrate the model to replicate the feature that tail risk

in labor income is more pervasive in recessions relative to expansions. Besides salient features

of equity and bond markets, the model is consistent **with** other empirical facts such as the

cyclicality of factor shares and limited stock market participation.

Key words: Equity premium puzzle, dynamic contracting, tail risk, limited commitment

∗ Hengjie Ai (hengjie@umn.edu) is affiliated **with** the Carlson School of Management, University of

Minnesota, Anmol Bhandari (bhandari@umn.ed) is at the Department of Economics, University of Minnesota.

We thank Chao Ying and Jincheng Tong for assistance and comments.

1 Introduction

A growing empirical literature documents the prevalence and systematic movements of

idiosyncratic tail risks in labor incomes and firm productivities over the business cycle. 1

this paper we develop a theory of incomplete insurance markets and show that idiosyncratic

risks in labor/firm productivities can explain the high equity premium. Without impediments

to risk sharing (as in an Arrow-Debreu world), idiosyncratic fluctuations in income are

diversifiable and have no bearing on the market price of aggregate risks.

In

Our theory

relies on limited commitment as a micro-foundation for the absence of insurance markets

for tail risks.

In contrast to the existing literature that assumes exogenously incomplete

markets, consumption tail risk in our model is an outcome of optimal dynamic risk sharing

arrangements. Our main results show how this risk sharing patterns accounts for higher risk

prices along **with** several other empirical facts such as the cyclicality of factor shares and

limited stock market participation.

The setting consists of two types of agents: “capital owners” and “wage earners”. Capital

owners are well-diversified and provide insurance to wager earners against idiosyncratic

fluctuations in labor productivities using long-term wage contracts. The key friction that

distinguishes our paper from standard representative agent asset pricing models is that capital

owners cannot commit to insurance/wage contracts that yield a negative net present value

of profits from the relationship. We solve for the constrained efficient allocation subject to

(one-sided) limited commitment and study its asset pricing implications.

Our theoretical framework has several predictions. First, due to limited commitment,

tail risks in labor productivities can only be partially insured. In our model, perfect risk

sharing implies that all agents in the economy consume a constant share of the aggregate

consumption. This Pareto-efficient allocation is implemented by the well-diversified capital

owners (the principal) offering an insurance contract to workers (agents) that exchanges

a constant consumption share for risky labor income.

profitable due to the welfare gain of risk sharing.

The above arrangement is ex-ante

However, a large drop in the level of

labor income will likely imply that the full risk sharing contract has a negative net present

value (NPV) from the principal’s perspective; therefore, extreme productivity shocks cannot

be fully insured if the principal cannot commit to ex post negative profit wage contracts.

Combining **with** the fact that tail risks are more pervasive in recessions, this propagation

mechanism leads to larger tail risks in consumption and agents consequently demand higher

compensation for aggregate shocks in adverse states.

Second, our model generates countercyclical movements in the share of labor

1 For instance, (Guvenen, Ozkan, and Song (2014) use a large panel from Social Security Administration

data to focus on labor income risks, Kelly and Jiang (2014) document similar patterns for firm level tail risks.

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compensation in aggregate output. Recent studies using aggregate time series, for example

Greenwald, Lettau, and Ludvigson (2014) show that variations in labor shares account for

a large fraction of stock market fluctuation. In the benchmark **with** perfect risk sharing,

capital owners and wage earners consume a constant share of aggregate output. However,

the limited commitment constraint binds for a large number of agents in recessions. As

such continuation values fall by a much larger fraction relative to the perfect risk sharing

benchmark. With recursive utility, this lack of risk sharing in the future raises the marginal

utility of both constrained and unconstrained agents in the current period. The optimal risk

sharing scheme compensates for this fall in utilities by allocating a higher share of aggregate

output to the workers. Therefore, labor share moves negatively **with** the aggregate stock

market in our model. The countercyclical behavior of labor share translates into a procyclical

consumption share of the well- diversified capital owners further amplifying the risk prices.

Third, the risk sharing arrangement in our model has an interpretation as a theory

of endogenous financial market participation. Workers who realize positive income shocks

or small negative income shocks are unconstrained, and therefore their marginal rate of

substitution are equalized **with** that of the capital owners. However, the limited commitment

constraint binds for workers who experience large negative income shocks, and therefore

the intertemporal Euler equation typically does not hold for poor households who are hit

by tail risks in labor income. This delineation of constrained and unconstrained agents in

our optimal risk sharing scheme can be implemented **with** endogenous market segmentation,

where poor agents do not participate in financial markets for portfolio diversification.

The paper builds on the literature on asset pricing **with** incomplete markets. Alvarez

and Jermann (2000) and Alvarez and Jermann (2001) build on Kehoe and Levine (1993) and

proposed a model of asset pricing **with** limited commitment which is is equivalent to the agentside

limited commitment in our setup. Their theory implies that agents who experience large

positive income shocks have an incentive to default because they have better outside options.

As a result, positive income shocks cannot be insured while tails risks in labor income are

perfectly insured. We consider principal-side limited commitment and our purpose is build

a theory of uninsured tail risks and study its asset pricing implications.

**Asset** pricing models **with** exogenously incomplete market include Mankiw (1986) ,

Constantinides and Duffie (1996), Schmidt (2015), and Constantinides and Ghosh (2014) .

This literature shows the countercyclical labor income risk require a significant compensation.

Our paper provides a micro-foundation for the incomplete market exogenously assumed in the

above papers. Our paper is also related to the asset pricing literature on the time variation

in idiosyncratic volatility and tail risks, for example, Kelly and Jiang (2014), Herskovic,

Kelly, Lustig, and Nieuwerburgh (2015) and the asset pricing literature that emphasizes the

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importance of the cyclical variation in labor shares, for example, Greenwald, Lettau, and

Ludvigson (2014) and Ludvigson, Lettau, and Sai (2014).

Our computational method builds on the work of Krusell and Smith (1998). Using

techniques standard in the dynamic contracting literature such as Thomas and Worrall

(1988), Atkeson and Lucas Robert E (1992), we represent our equilibrium allocations

recursively **with** the help of distribution of promised values as a state variable. However,

in contrast to those papers our environment has aggregate shocks and the distribution of

promised values responds to such shocks even in the ergodic steady state. As in Krusell and

Smith, we approximate the forecasting problem of long lived agents by assuming that agents

use few relevant moments of the distribution of promised values to guess future state prices.

The paper is organized as follows. Section 2 describes the environment - we start

preferences and technology and layout the contracting problem and define a competitive

equilibrium. In section 3, we use a simple two period example (and assume complete markets

from t = 3 onwards) to explain all of our main results. Lastly, in section 4 we extend the

analysis to the infinite horizon economy.

2 The Model

2.1 Model Setup

We consider a discrete time, infinite horizon economy, t = 0, 1, 2 · · · . There are two groups

of agents, a unit measure of capital owners and a unit measure of workers. Preferences are

homogeneous across both groups of agents and represented by the Kreps-Porteus form **with**

risk aversion γ and intertemporal elasticity of substitution (IES) ψ. Production takes place

in a firm (say j) that hires workers using long term wage contracts to operate a technology

y j,t = Y t s j,t ,

where s j,t is a match-specific productivity shock and Y t is common across all firms.

For now we assume workers work in the same firm unless they are exogenously separated at

a rate κ ∈ (0, 1). There is a competitive market between the capital owners for the ownership

rights to the firms; however, firms **with** negative net present value can be shut down by the

owners and obligations on the wage contracts are rescinded **with**out any legal recourse. This

one-sided lack of commitment is the only restriction on the set of wage contracts that are

offered. We adopt a simple convention, whereby new wage contracts are set such that the

present value of the match is shared in fixed proportions, α and (1 − α), between the capital

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owner and the workers respectively. 2

The technological possibilities evolve stochastically **with** the growth rate of the aggregate

productivity shock following

Y t+1

Y t

= e g t+1

,

where g t is a finite state Markov process **with** transition matrix Π. A typical element in Π

is denote π (g ′ | g). We assume s j,t+s = e ∑ s

k=1 ε(k) , where ε is a random shock i.i.d. across

firms. We assume E [e ε ] = 1 so that s j,t+s can also be interpreted as the firm’s productivity

relative the economy-wide average productivity. This is a convenient normalization because

Y t = E [y t ] is the total output of the economy by the Law of Large Numbers. Importantly,

we assume that the distribution of ε depends on the aggregate state of the economy, g and

denote the conditional density of ε given g as f g (ε). As we show later, this specification

allows us to capture the idea that there is more tail risks in labor income and consumption

in recessions than booms.

2.2 Dynamic contracting

We describe the equilibrium in our economy by starting directly from the recursive

formulation of the dynamic contracting problem. In Appendix A of the paper, we show that

our notion of equilibrium implement the constrained efficient allocation subject appropriate

limited commitment constraints.

The equilibrium we construct has two state variables, (Φ, g), where Φ denote the

distribution of agent types and g is the Markov state of aggregate productivity. We use

the notation Λ (g ′ | Φ, g) to denote the stochastic discount factor (SDF) in state (Φ, g) that

measures the price of one unit of state-g ′ consumption good in state (Φ, g) consumption

numeraire.

Upon arrival, a worker is offered a wage/insurance contract by the well-diversified

principal. The profits, i.e., output net of wage payments when current output is y and

2 The parameter α can also be interpreted as the bargaining weight of capital owners in a Nash bargaining

setup.

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promised utility U is calculated as

{

P (y, U| Φ, g) = max (y − C) + ∑ ∫

π (g ′ | g) Λ (g ′ | Φ, g)

C,{U ′ (g ′ ,ε ′ )}

g ′

⎡

( ∑

∫

subject to : U = ⎣(1 − β) C 1− 1 ψ + β π (g ′ | g)

g ′

}

(

)

P ye g′ +ε ′ , U ′ (g ′ , ε ′ ) ∣ Φ ′ , g ′ f g ′ (ε) dε

U (g ′ , ε) 1−γ f (ε| g ′ ) dε

(

)

P ye g′ +ε ′ , U (g ′ , ε) ∣ Φ ′ , g ′ ≥ 0, for all (g ′ , ε) .

) 1−1/ψ

1−γ

⎤

⎦

1

1−1/ψ

Under our assumptions, the growth rate of firm-level output is y′

= +ε ′

y

eg′ . The constraint

P ( ye g′ +ε ′ , U (g ′ , ε) ∣ Φ ′ , g ′) ≥ 0 in the above optimal contracting problem reflects the limited

commitment on the principal side: the well-diversified capital owners cannot commit to

contracts the yields negative present value of profit. Because (y, U) are the state variables of

the optimal contracting problem for each agent, the equilibrium in general depends on the

joint distribution of income and promised utility, which we denote as Φ.

Due to the homogeneity of the problem, it is not hard to show that the profit function

satisfies

( U

P (y, U| Φ, g) = p

y

) ∣ φ, g y,

for some function p (·| φ, g). Here, we use the fact that the dependence of p on the distribution

of agent types can be summarized as a one-dimensional measure φ, which we will show in

the next section. We therefore introduce the notation for normalized utility and normalized

consumption:

u = U y ; c = C y . (1)

The above optimal contracting problem can therefore be written as:

{

p (u| φ, g) = max (1 − c) + ∑ ∫

π (g ′ | g) Λ (g ′ | φ, g)

c,{u ′ }

g ′

s.t : u =

⎡

⎢

⎣ (1 − β) c1− 1 ψ

+ β

( ∑

g

e g′ +ε ′ p (u ′ (g ′ , ε ′ )| φ ′ , g ′ ) f g ′ (ε ′ ) dε

π (g ′ | g)

∫ [

e g′ +ε ′ u ′ (g ′ , ε ′ )] 1−γ

fg ′ (ε ′ ) dε

) 1− 1 ψ

1−γ

⎤

⎥

⎦

1

1− 1 ψ

}

p (u ′ (g ′ , ε ′ )| φ ′ , g ′ ) ≥ 0, for all (g ′ , ε ′ ) , (2)

where p (u ′ (g ′ , ε ′ )| φ ′ , g ′ ) ≥ 0, for all (g ′ , ε ′ ) is the limited commitment constraint.

5

2.3 Equilibrium

Let C (U, y) be the consumption policy for any agent **with** promised utility U and current level

of endowment y. Homogeneity implies that C (U, y) = c (u) y for any y for some function

c, where u = U as defined in (1). In general, the joint distribution of (u, y), Φ (u, y) is

y

needed as a state variable in the construction of a recursive equilibrium, because the resource

constraint,

∫ ∫

c (u) yΦ (u, y) dydu + C P = Y.

depends on Φ (u, y). The notation C P stands for the total consumption of the well-diversified

capital owners. We can write the integral in the above equation as

∫ ∫

∫ ∫

c (u) yΦ (u, y) dydu = Y c (u) y Φ (u, y) dydu

Y

∫ [∫ ]

y

= Y c (u)

Y Φ (u, y) dy du.

Define a new measure φ (u) = ∫ y

Φ (u, y) dy, then the resource constraint can be written as

Y

∫

c (u) φ (u) du = 1 − x (φ, g) , (3)

where x (φ, g) = CP denotes the principal’s share in aggregate consumption in state (φ, g).

Y

The above procedure reduces the two-dimensional distribution Φ into a one-dimensional

measure φ and greatly simplifies our numerical computation.

Because we will use φ as one of the state variables in the construction of the recursive

equilibrium, it is useful to describe the law of motion of φ. Let u ′ = u ′ (u, g ′ , ε ′ | φ, g) be

the law of motion of continuation utility implied by the optimal contracting problem in (2).

That is, conditioning on the current state (φ, g), u ′ = u ′ (u, g ′ , ε ′ | φ, g) is the the continuation

utility for an agent **with** current promised utility u, in the next period in the state where g ′

and ε ′ are realizations of aggregate and idiosyncratic shocks. Let I be the indicator function,

that is, I {u ′ ( u,g ′ ,ε|φ,g)=v} (u) is the set of u such that u ′ (u, g ′ , ε| φ, g) = v. Under these notation,

the law of motion of φ can be written as:

∫

∀ũ, φ ′ (ũ| g ′ ) =

∫

φ (u)

e ε f (ε) I {u ′ ( u,g ′ ,ε|φ,g)=ũ} (u) dεdu. (4)

For simplicity, we specify the equilibrium in terms of normalized consumption and

continuation utility as defined in (1). Formally, an equilibrium consists of the following

price and allocations:

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• A law of motion of Γ: φ (·) → φ ′ (·| g ′ ) for all g ′ .

• A SDF as a function of φ and g: {Λ (g ′ | φ, g)} g ′.

• Principal’s share x (φ, g) in total consumption as a function of state variables.

• Value functions p (u| φ, g) for each u, and the associated policy functions c (u| φ, g),

u ′ (u, g ′ , ε ′ | φ, g) such that:

1. The SDF is consistent **with** the principal’s consumption:

[ x (φ ′

Λ (g ′ (g ′ ) , g ′ ) e g′

| φ, g) = β

x (φ, g)

] −

1

[ ψ

v (φ ′ ]

(g ′ ) , g ′ ) e 1 g′ ψ −γ , (5)

n (φ, g)

where the principals utility is defined by the fixed point of the following recursive

relationship:

v (φ, g) =

⎡

⎤

[ ] ∑ [

] 1− ψ

1

⎢

⎣ (1 − β) x (φ, 1 1−γ 1−γ g)1− ψ + β π (g ′ | g) e g′ v (φ ′ (g ′ ) , g ′ ) ⎥

⎦

g ′

1

1−1/ψ

.

and the certainty equivalent, n (φ, g) is defined by:

[ ] 1

∑ [

] 1−γ 1−γ

n (φ, g) = π (g ′ | g) e g′ v (φ ′ (g ′ ) , g ′ ) .

g ′

2. Given the SDF, and the law of motion of φ, for each u, the value function and the

policy functions solves the optimal contracting problem in (2).

3. Give the policy functions, the law of motion of the summary measure φ satisfies

(4).

4. The policy functions and the summary measure satisfy the resource constraint (3).

Here we specify the SDF as a function of the principal’s consumption directly **with**out

explicitly specifying the principal’s consumption and portfolio problem for brevity. Because

the principal is well-diversified, their consumption and investment choices are standard.

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2.4 Optimality Conditions

Let λ be the Lagrangian multiplier on the limited commitment constraint in the optimal

contracting problem in (2), the first-order conditions and the envelope condition imply:

∂

∂u p (u| φ, g) = −λ = − 1

(1 − β) ( )

c −

1

, (6)

ψ

β

(

)

1 − β c (u| φ, g) 1 ψ m (u| φ, g)

γ− 1 −γ

ψ

e g′ +ε u ′ (g ′ , ε ′ ) ≥ −Λ (g ′ | φ, g) ∂

∂u p (u′ (g ′ , ε ′ )| φ ′ , g ′ ) ,

where we use the notation m (u| φ, g) for the certainty equivalent of the agent’s continuation

utility:

m (u| φ, g) =

( ∑

g

π (g ′ | g)

∫ [

e g′ +ε ′ u ′ (g ′ , ε ′ )] 1−γ

fg ′ (ε ′ ) dε

u

) 1

1−γ

(7)

. (8)

It is not hard to show that the profit function p (u| φ, g) must be strictly decreasing in

u: higher promised utility to the agent implies a lower profit to the principal. Therefore,

the limited commitment constraint, p (u ′ (g ′ , ε ′ )| φ ′ , g ′ ) ≥ 0 can be written as u ′ (g ′ , ε ′ ) ≤

ū (φ ′ (g ′ ) , g ′ ) for all g ′ , where ū (φ ′ (g ′ ) , g ′ ) is defined as p (ū (φ ′ , g ′ )| φ ′ , g ′ ) = 0. The

complementary slackness condition implies that (7) has to hold **with** equality whenever

u ′ (g ′ , ε ′ ) ≤ ū (φ ′ (g ′ ) , g ′ ).

The above conditions together imply that for all possible realizations of (g ′ , ε ′ ),

[ e

g ′ +ε ′ c (u ′ (u, g ′ , ε ′ | φ, g)| φ ′ (g ′ ) , g ′ )

β

c (u| φ, g)

] −

1

[ ]

ψ

e g′ +ε ′ u ′ (u, g ′ , ε ′ | φ, g)

1 ψ −γ ≥ Λ (g ′ | φ, g) , (9)

m (u| φ, g)

and u ′ (u, g ′ , ε ′ | φ, g) < ū (φ ′ (g ′ ) , g ′ ) implies that ”=” must hold in (9), where m (u| φ, g) is

defined in (8). If the limited commitment constraint does not bind, then equation (9) is

simply the optimal risk-sharing condition that equalizes the marginal utility of the agent

**with** that of the principal. A binding limited commitment constraint allows the marginal

utility of the agent exceeds that of the principal, and this is the key mechanism in our model

that results in limited risk-sharing **with** respect to tails risks. Small drops in output y are

not associated **with** reductions in agent’s consumption share relative to that of the principal,

because of risk sharing. However, large declines in output send the profit of the insurance

contract to the negative region. Because the principal cannot commit to negative NPV

insurance contracts, the optimal contract requires a permanent reduction in agent’s future

consumption to respect the limited commitment constraint. More negative shocks in output

can only be accompanied by further reduction in agent’s consumption — tail risks cannot be

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fully insured.

3 Two-Period Limited Commitment

To gain a better understand the optimal risk-sharing contract and the implications of the

model on the cyclical properties of labor share, we study a model **with** two-period limited

commitment, where closed form solution or semi-closed form solutions are available, and

compare it **with** a model **with** perfect risk sharing. To capture the countercyclicality of tail

risks and to facilitate close-form solutions, we assume this section that the distribution of

ε in booms is degenerate, that is, V ar [ε| g H ] = 0, and ε follows a negative exponential

distribution (Appendix 1) in recessions.

3.1 Full commitment

We first consider the case of full commitment. Here, perfect risk sharing implies that the

SDF can be computed from the marginal rate of substitution of a representative agent who

consume the aggregate output. This is effectively a Mehra-Prescott economy **with** recursive

preference (or a Bansal and Yaron (2004) economy **with** Markov consumption growth rates).

Utility and the SDF

We denote ū (g) to be the utility of the representative agent

normalized by aggregate output, then {ū (g)} g=gH ,g L

can be calculated from the definition of

recursive utility:

ū (g H ) =

[

(1 − β) + β [ π (g H | g H ) (ū (g H ) e g H

) 1−γ + π (g L | g H ) (ū (g L ) e g L

) 1−γ] 1− 1 ψ

1−γ

] 1

1−1/ψ

,

ū (g L ) =

[

(1 − β) + β [ π (g H | g L ) (ū (g H ) e g H

) 1−γ + π (g L | g L ) (ū (g L ) e g L

) 1−γ] 1− 1 ψ

1−γ

] 1

1−1/ψ

.

where m H and m L are certainty equivalents of the representative agent:

m (g H ) = [ π (g H | g H ) (ū (g H ) e g H

) 1−γ + π (g L | g H ) (ū (g L ) e g L

) 1−γ] 1

1−γ

,

m (g L ) = [ π (g H | g L ) (ū (g H ) e g H

) 1−γ + π (g L | g L ) (ū (g L ) e g L

) 1−γ] 1

1−γ

.

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Given the utility of the representative agent, the SDF can be calculated as:

( )

Λ (g ′ | g) = βe − 1 ψ

u (g ′ ) e 1 g′ ψ −γ

g′ .

m (g)

Perfect risk sharing implies that the principal’s share, x is constant over time, and the

principal’s utility is xū (g).

The profit function p ∗ (u| g)

We can compute the normalized consumption and

profit function, which we denote as c ∗ (u| g) and p ∗ (u| g), respectively, as follows.

Perfect risk-sharing implies that all agents’ consumption must be proportional to that

of the representative agent. Because the utility function is homogenous of degree one in

consumption, consumption must be proportional to utility, we therefore have:

c ∗ (u| g) =

u

ū (g) , for g = g H, g L . (10)

That is, if the promised utility of an agent is uy t , then the ratio of his utility relative to that

uy t

of the representative agent is

ū(g)Y t

. Under the perfect risk-sharing rule, his consumption is a

u y

fixed fraction of that of the representative agent: t

ū(g) Y t

Y t+j for all j = 0, 1, 2 · · · . This means

u y

his current period consumption-to-output ratio is t

ū(g) y t

= u . ū(g)

Denote w (g) as the wealth-to-consumption ratio of the representative agent. It is not

hard to show that w (g) and ū (g) are related by:

w (g) = 1

1 − β ū (g)1− 1 ψ . (11)

Consider a worker **with** current output level y t and normalized promised utility u t . The future

s

cash flow produced by the firm is Y t+j s t+j = Y t+j s

t+j

t s t

= Y t+j s t e ∑ j

k=1 ε(k) , for j = 0, 1, · · · .

Because E [e ε ] = 1, it is not hard to show that the price-to-dividend ratio of this cash

flow stream is the same as that of the aggregate output. Therefore, the total value of the

agent’s output is w (g t ) y t . Under the perfect risk-sharing plan (10), the present value of the

consumption stream promised to the agent is w (g) u y ū(g) t. Therefore, the normalized profit

function under full commitment satisfies:

[

p ∗ (u| g) y = w (g) 1 − u ]

y, for g = g H , g L .

ū (g)

10

Using (11), we have

[

p ∗ (u| g) = w (g) 1 − u ]

= 1

]

[ū (g) 1− 1 ψ − ū (g)

− 1 ψ u . (12)

ū (g) 1 − β

We make several observations that are useful in contrasting **with** our model **with** limited

commitment.

First, under the perfect risk-sharing plan, profit p ∗ (u| g) can be negative.

Consider an agent **with** promised utility u 0 Y 0 in period zero. Perfect risk-sharing implies that

the agent’s utility-to-aggregate output ration will be constant over time, u 0 . The normalized

utility of the agent at time t, u t = u 0Y t

y t

= u 0

s j,t

. A large negative shock in s j,t leads a value

of normalized utility u t > ū (g), and a negative profit for the insurance contract from the

principal’s perspective. As a result, the perfect risk-sharing plan will not be feasible in the

economy **with** limited commitment.

Second, p ∗ (ū (g)| g) = 0, therefore, ū (g) is the zero-profit level of promised utility. In the

full commitment case, the representative agent consumes all output, and the profit on the

insurance contract for the representative agent is zero.

Third, perfect risk-sharing implies that profit function is linear in u. In fact,

∂

∂u p∗ g (u) =

− 1

1−β ū (g)− 1 ψ by (12), which is consistent **with** envelope condition in (6) given the functional

form of consumption policy in (10). As we will see later, limited commitment introduces

convexity in the profit function: the marginal cost of utility provision is larger at higher

levels of u because the limited commitment constraint is more likely to bind in the future.

3.2 One-period lack of commitment

In this section, we study our model **with** one-period lack of commitment. That is, we assume

that the principal cannot commitment insurance contracts **with** negative NPV in period 1,

but can fully commit afterward. This assumption implies that the profit function in period

one is the same as that in the model **with** full commitment, and we can solve the model

in closed-form. We use this setup to illustrate two main results. First, tail risks cannot be

fully insured due to limited commitment. Second, One period limited commitment implies

a procyclical labor share in period one and a lower market price of risk. We show in later

sections that to generate a countercyclical labor share and a higher market price of risk, we

need a fully dynamic model.

The principal’s marginal utilities In the one-period lack of commitment case, we

denote the utility of the agent u, and the principal’s share in aggregate consumption in period

0 to be x 0 . Also, the principal’s share in period 1 to be x H , x L , respectively for state g H

11

and g L . Because the principal can perfect commit beginning from period 1, the (normalized)

continuation utility of the principal after date-1 is x H ū H in state g H and x L ū L in state g L .

Therefore, the SDF in period 0 is given by:

Λ (g H ) = β

( )

xH e g H −

1 ( ψ

e g H

) 1

x H ū

ψ −γ ( )

H

xL e g L −

1 ( ψ

e g L

) 1

x L ū

ψ −γ L

; Λ (g L ) = β , (13)

x 0 n 0 x 0 n 0

where n 0 is the certainty equivalent computed by:

n 0 = { π (e g H

x H ū H ) 1−γ + (1 − π) (e g L

x L ū L ) 1−γ} 1

1−γ

.

Given n 0 , the principal’s utility in period 0 is simply

v 0 =

[

]

(1 − β) x 1− 1 ψ

0 + βn 1− 1 1

1−1/ψ

ψ

0 . (14)

The optimal contracting problem

Denote the profit function to be p 0 (u). With

one-period lack of commitment, the profit function starting from period 1 is p ∗ (u| g). The

profit maximization problem is only non-trivial in period 0:

{

p 0 (u| g) = max (1 − c) + ∑ ∫

π (g ′ | g) Λ (g ′ | g)

g ′

e g′ +ε ′ p ∗ (u ′ (g ′ , ε ′ )| g ′ ) f (ε ′ | g ′ ) dε ′ }

⎡

( ∑

∫ ) 1−1/ψ ⎤

[ ]

s.t : u = ⎣(1 − β) c 1− 1 1−γ

1−γ

ψ + β π (g ′ | g) e g′ +ε ′ u ′ (g ′ , ε ′ ) f (ε ′ | g ′ ) dε ′ ⎦

g ′

1

1−1/ψ

u ′ (g ′ , ε ′ ) ≤ ū (g) , for all (g ′ , ε ′ ) ,

where we denote the consumption policy of the above problem as c, and the continuation

utility policy as u ′ (g ′ , ε ′ ). We have also used the function form of p ∗ (u| g) to replace the

limited commitment constraint p ∗ (u ′ (g ′ , ε ′ )| g ′ ) ≥ 0.

We conjecture that the limited commitment constraint does not bind in state H. Again,

perfect risk sharing after the first period implies that the continuation utility of the agent

in state H must be (1 − x H ) ū H . In addition, the consumption policy in state H is simply

c ∗ (u| g H ). The first order condition (9) can therefore be written as

[ e

g H

] −

1

(1 − x H )

[ ψ

e g H

] 1

(1 − x H ) ū

ψ −γ H

β

= Λ (g H |g) . (15)

(1 − x 0 )

m 0

where m 0 is the certainty equivalent of the agent.

12

The choice of continuation utility for state L depends on the realization of ε. By (9),

there exists a ¯ε such that

• At ¯ε, the limited commitment constraint holds **with** equality, but the Lagrangian

multiplier is zero. Therefore, at ¯ε, u ′ (g L , ε ′ ) = ū L , and

[ e

g L

] +¯ε c ∗ −

1

(ū L | g L )

[ ψ

e g L+¯εū

] 1

ψ −γ L

β

= Λ (g L |g) (16)

(1 − x 0 )

m 0

• For all ε ′ ≥ ¯ε, the limited commitment constraint does not bind. Using condition (9)

together **with** the faction that c ∗ (u| g L ) is linear, we have:

u ′ (g L , ε) = e¯ε−ε′ ū L .

• For all ε ′ < ¯ε, the limited commitment constraint binds and u ′ (g L , ε ′ ) = ū L .

Under the above policy, the certainty equivalent of the agent can be calculated as

m 0 =

{

∫ [ ] } 1

1−γ

π [e g H

u ′ (g H )] 1−γ + (1 − π) e g L+ε ′ u ′ (g L , ε ′ ) f (ε ′ ) dε ′ 1−γ

.

Using the fact that u ′ (g H ) = (1 − x H ) ū H , and the density of the negative exponential

distribution, we can explicitly compute the above integral to get

m (u 0 ) =

{π [e g H

(1 − x H ) ū H ] 1−γ + (1 − π) (e g L e¯ε ū L ) 1−γ [

1 − 1 − γ

λ + 1 − γ e−λ(ˆε−¯ε) ]} 1

1−γ

.

Finally, we note that the resource constraint in state L implies

∫ ∞

−∞

e ε′ c ∗ (u ′ (g L , ε ′ )| g L ) f (ε ′ ) dε ′ = 1 − x L .

Under the above described allocation rule of continuation utility, the resource constraint

implies

Finally, we can determine agent’s utility as

u 0 =

e¯ε − 1

1 + λ e−λˆε+(1+λ)¯ε = 1 − x L . (17)

[

]

(1 − β) (1 − x 0 ) 1− 1 1−

ψ + βm 1 1

1−1/ψ

ψ

0

(18)

To summarize, given u 0 , the seven equations (13), (14), (15), (16), (17), and (18) together

13

determine the seven equation price and quantities, {Λ (g)} g=gH ,g L

, v 0 , {x j } j=0,H,L

, and ¯ε.

Note that equation (17) implies that e¯ε > 1 − x L . The following proposition summarizes the

equilibrium in the model **with** one-period limited commitment.

Proposition 1. (One-period limited commitment)

1. There exists ¯ε and Ū such that ∀ε′ ≥ ¯ε, U 1 (g L , ε ′ ) = Ū, and ∀ε′ < ¯ε, U 1 (g L , ε ′ ) = e ε′ −¯ε Ū.

In particular, lim ε ′ →−∞ U 1 (g L , ε ′ ) = 0.

2. The principal’s share in aggregate consumption is countercyclical, that is, x H < x L .

3. The volatility of the SDF in period 0 is smaller than that in the economy **with** full

commitment.

Part 1 of the above proposition establishes the lack of risk-sharing **with** respect to tail

risks. For all ε ′ ≥ ¯ε, the limited commitment constraint does not bind, and there is full risksharing

**with** respect to idiosyncratic risks in the sense that continuation utility U 1 (g L , ε ′ ) does

not depend on the level of ε ′ . However, if ε ′ < ¯ε, the limited commitment constraint binds

and the level of continuation utility of the agent is determined by the zero profit condition.

In fact, as the level of income approaches zero, so does continuation utility.

Part 2 and 3 of the proposition states that although limited commitment results in

incomplete risk sharing **with** respect to tail risks, it does give rise to a higher price of risk.

The intuition for this result can be seen from the first order condition in (16). Optimal

risk sharing implies that the marginal rate of substitution of the principal must equal to

that of the unconstrained agents. The unconstrained agents are those who realizes higher

income shocks (compared to the population average), and their marginal utility is higher

than the marginal utility of an average agent. As a result, optimal risk sharing requires a

higher consumption share for the principal. In equilibrium, the marginal rate of substitution

of the principal as well as the marginal rate of substitution of agents are both valid SDF’s.

Because the unconstrained agents are those who realize higher than average income shocks,

their marginal utility in booms is lower than that the agents in a full risk sharing economy.

As a result the volatility of SDF is lower. In the next section, we show that more periods of

limited commitment generates a countercyclical labor share.

3.3 Two-period lack of commitment

We illustrate the setup of the two-period model in Figure 1. The top panel of Figure 1 depicts

the binomial tree of aggregate growth rates. The economy starts at date 0 at node 0. Node

14

H is the state in which the realized aggregate economic growth rate is g 1 =g H and node L

is associated **with** state g 1 = g L . Similarly node HH, HL, LH and LL describe the possible

aggregate states of the world in period two. The bottom panel of the same figure describes

the dynamics of agent’s normalized utility in our model **with** two-period limited commitment.

Here we assume there is only type of agent at node 0 who start **with** initial promised utility u 0 .

At node H, because the distribution of ε is degenerate, the continuation utility of the agent

depends on the aggregate state g 1 = g H . At node L, the agent’s continuation depends on

both the aggregate state and the realization of the idiosyncratic shock ε 1 . At this node, there

is a continuum of types indexed by u 1 . At node LH, for each u 1 , the continuation utility

of the agent is u 2 = u ′ (u 1 , g H ), because the distribution of ε 2 is degenerate at this node.

At node LL, agents’ continuation utility depend again on both aggregate and idiosyncratic

shocks.

The equilibrium **with** two-period lack of commitment cannot be solved analytically. We

describe several feature of the equilibrium using a numerical solution. 3

Lack of risk sharing **with** respect to tail risks

To understand the nature of the

optimal contract, in Figures 2 and 3, we plot the profit function and the consumption policy

(both normalized by income) in the full commitment case (dashed line) and contrast it **with**

the limited commitment case (dotted line). As stated in equations (10) and (12), **with** full

commitment, these functions (the dashed lines) are linear **with** respect to u. With limited

commitment, the profit function is strictly concave **with** respect to u and always below the

full commitment case while the consumption policy is strictly convex. The profits are lower

because the dividends are discounted more **with** incomplete markets. The steeper curvature

reflects the higher marginal cost of providing insurance as the limited commitment constraint

is likely to bind for high u. Since the principal cannot deliver higher u’s, in the future, the

consumption per unit of output increases more than proportionately **with** u in the limited

commitment case.

Proposition 1 stated that **with** one period-lack of commitment, tail risks are only partially

insured. In Figure 4, we plot the level of agents’ consumption (top panel) and continuation

utility (bottom panel) at node L as a function of the realization of the idiosyncratic

productivity shock ε.

As in the case **with** one-period lack of commitment, both current

consumption, C(g L , ε) and promises for future consumption, U(g L , ε) are lower for agents

who draw low a ε.

The implications of the optimal contract in our model, i.e., how the

3 We set γ = 5, ψ = 1.5 and the subjective discount factor β = 0.95. The transition matrix for

g ∈ G = {1.05, 0.99} is symmetric **with** π[g L |g L ] = pi[g h |g h ] = 0.75. The distribution of ε in a recession

is negative exponential **with** λ = 7. The initial u 0 is set to get x 0 , the share of the capital owners equal to

0.2

15

wages/consumption of workers respond to idiosyncratic productivity shocks is consistent **with**

the empirical evidence presented in Guvenen, Ozkan, and Song (2014). Figure 5 (generated

using data from their paper) shows fall in labor earnings in recessions after ranking agents

**with** respect to their 5-year average income levels before recession. 4 In our model, the (unnormalized)

promised utility is the measure of lifetime wages for the worker under the optimal

contract. Figure 6 is the a counterpart of Figure 5 in our model. It plots the expected change

in wage earnings conditioning a recession in period 2 as a function of the agents’ life time

earnings in period 1. The horizontal axis in the percentiles of the agents’ life time earnings

income in period one at node L. The vertical axis is the average change in wages for agent

in the corresponding percentile. As before the dashed line describes the implication of a

model **with** full commitment, and the dotted line is the implication of our model **with** limited

commitment.

Countercyclical labor share

As we show in the last section of the paper, labor share

in procyclical in the model **with** one-period lack of commitment. Proposition 1 in the previous

section implies that labor share must be procyclical in the second period (that is at nodes

HH, HL, LH, and LL).

Here we show that **with** two-period limited commitment and

recursive utility, labor share is countercyclical in first period, that is, x H > x L .

The key condition to understand the intuition for countercyclical labor share is the

optimality condition for risk sharing that equates the marginal rate of substitution of the

principle to that of the unconstrained agents:

β

( )

xH e g H −

1 ( )

ψ

v H e g 1

H ψ −γ [ e

g H

] −

1

(1 − x H )

[ ψ

e g H

] 1

u

ψ −γ H

= β , (19)

x 0 n 0 1 − x 0 m 0

and

β

( )

xL e g L −

1 ( )

ψ

v L e g 1

L ψ −γ [ e

g L

] eˆε −

1

c L (û L )

[ ψ

e g L

] eˆε 1

ψ û −γ L

= β . (20)

x 0 n 0 1 − x 0 m 0

The left-hand side of equations (19) and (20) are the marginal rate of substitution of the

principal at node H and node L, respectively. With recursive utility, the marginal rate of

substitution depends both on consumption growth and continuation utilities.

The righthand-side

of (19) is the marginal rate of substitution of the agent at node H. The right hand

side of (20) is the marginal rate of substitution of the agent in state (g L , ˆε), where ˆε is the

cut-off level of idiosyncratic shock below which the limited commitment constraint binds.

We use the short-hand notation û = u ′ (u 0 , g L , ˆε) for the continuation of the agent in state

4 This includes all working U.S. male in the age group 35 to 54. We averaged the data across last four

recessions [1978-2010] adjusting for the length of the recessions and report the annualized change in wages

conditioned on a recession

16

(g L , ˆε) and c L (û L ) for the consumption of the agent in state (g L , ˆε).

Dividing (19) by (20), we obtain

( ) −

1 ( ) 1

xH ψ

v

ψ −γ ( ) −

1 ( ) 1

H 1 − xH ψ

u

ψ −γ H

= . (21)

x L v eˆε L c L (û) eˆε û L

In the model **with** one-period limited commitment, full sharing is attainable starting from

period 1. Therefore, v H = x H ū H , v L = x L ū L and u H = (1 − x H ) ū H , that is, both principal

and agents consume a fixed fraction of aggregate output from period 1 on.

In addition,

û L = ū L , because principal’s profit at the first best allocation is zero. The above equation

becomes

(

xH

x L

) −γ (ūH

ū L

) 1

ψ −γ =

( 1 − xH

eˆε

) −γ ) 1

ψ

(ūH

−γ (22)

û L

It is straightforward to show that eˆε > 1 − x L because eˆε is the average consumption of the

unconstrained agents and 1 − x L is the average consumption all agents. As a result, we must

have x H < x L for the above equality to hold.

With two-period limited commitment, û < ū L , because ū L is the maximum level of utility

that yields a non-negative profit for the principal under the perfect risk sharing plan. As

show in Figures 2 and 3, limited commitment lowers the level of promised utility that can

be supported by positive NPV plans. As a result, the lower level of û allows the equality in

(21) to hold **with** x H > x L . Intuitively, the prospect of uninsurable tail risks in the future

lowers the level of the utility of the marginal agent whose limited commitment constraint

does not bind. Because the marginal rate of substitution of the principal and the agent must

equalize, optimality in risk sharing requires that the principal compensate the agent **with**

current period consumption. This implication of the model depends crucially on the forward

looking nature of recursive utility.

In fact, in a model **with** expected utility, labor share

will always be procyclical, because uninsured tail risks in the future does not raise agent’s

marginal utility in the current period. These implications are summarized in the binomial

tree in Figure 6. Each node represents the share of the capital owners.

**Asset** prices **with** tail risk As a summary of the pricing kernel we compute the

standard deviation of (log) SDF. 5 With full commitment, this measure of volatility of the

SDF is independent of current state and equals 22.5% for all t. In the model **with** limited

risk sharing, it is higher and varies ly at t = 0 taking values 23.18% and 23.24% for states H

and L respectively.

5 For a log-normal SDF, this is the bound on the maximum Sharpe ratio.

17

4 Infinite horizon economy

In appendix B we describe the computational algorithm for the infinite horizon economy.

5 Conclusion

We present an asset pricing model **with** endogenously incomplete market. We show that

limited commitment on the principal side implies that idiosyncratic tail risks in labor

income can only be partially insured. When combined **with** recursive preferences and a

countercyclical tail risk in labor income, our model provides a potential resolution of the

equity premium puzzle based on limited commitment of financial contracts. We also show

that our model is consistent **with** the empirical evidence on countercyclical labor share and

limited stock market participation.

18

6 Appendix

A

Details of the negative exponential distribution

We assume that the density function of the idiosyncratic shock f gL (x) takes the following

form:

f (x) =

Claim 1. The integral of f (x) is

0 x > ¯x

λe λ(x−¯x)

x ≤ ¯x

∫ y

−∞

f (x) dx = e λ(y−¯x) .

In particular, ∫ ∞

f (x) dx = 1 is a proper density. Also,

−∞

∫ y

−∞

e θx f (x) dx =

λ

λ + θ e−λ¯x+(θ+λ)y for λ + θ > 0.

In particular,

∫ ∞

−∞

e θx f (x) dx =

λ

λ + θ eθ¯x for λ + θ > 0.

With θ = 1, the requirement E [e x ] = 1 requires that ¯x = ln 1+λ . This is the parameter we

λ

assume.

B

Computational Algorithm

Here we describe an algorithm to approximate Γ(g ′ |g, φ), Λ(g ′ |g, φ), x(g, φ), the value function

p(u|g, φ) along **with** the optimal policy rules for ˜c(u|g, φ) and ũ(ε ′ , g ′ , u|g, φ). This will be

based on a Krusell-Smith type approach where the distribution φ is replaced by a first k of

moments ⃗m ∈ M ⊂ R k .We will assume that G and E have finite elements.

1. We start from guessing some functions:

• Guess ˆΓ j g,g ′ (⃗m) : M → M.

This function is the perceived law of motion

that maps the vector of moments of φ today to tomorrow for each pair g, g ′ .

Many applications of KS algorithm use a log-linear function **with** log(m ′ i) =

∑

k γ i,k(g, g ′ ) log(m k ).

• Next guess ˆΛ j g,g ′ (⃗m) : M → R that maps the vector of moments of φ to the state

prices as we transition from g to g ′ .

19

• Lastly we guess ˆp j g(u, ⃗m) : U ×M → R + that maps promised utilities and a vector

of moments to discounted profits in state g.

2. Inner loop: For a fixed ˆΓ j g,g

and ˆΛ j ′ g,g

we iterate on the Bellman equation (2) to obtain

′

ˆp j+1 . Let ĉ j+1

g (u, ⃗m) and û j+1

g,g ′ ,ε

(u, ⃗m) denote the policy rules that attain ˆp j+1 . There

′

are alternatives to this step. We can use the first order conditions.

3. Simulation: For a collection of {u 0,i } i , simulate a sequence of {g t } and a panel of

{ε i,t }. 6 Use u i,t+1 = û j+1

g t,g t+1 ,ε t+1

(u i,t , ⃗m t ) to simulate a panel data for {u i,t , c i,t } i,t for a

large T . The argument m t+1 is updated using simulated data {u i,t+1 } i for each period.

Discard the first T 0 observations.

4. Outer loop: Now we update ˆΓ and ˆΛ

• We use the time series of moments of simulated data on {u t,i } T t=T 0

to get a new

j+1

guess for ˆΓ

g,g

(·). For instance, we will split the sample into transitions from g ′ t = g

to g t = g ′ and run a regression on each such sample to estimate γ i,k (g, g ′ ).

• Next use {c i,t } T t=T 0

(and market clearing) to approximate a function ˆx j+1

g (⃗m)

• Using ˆx j+1

g (⃗m) and

(⃗m)

ˆv j+1

g

ˆΓ

j+1

g,g ′ (⃗m) solve for the value function of the diversified agent

• Use the expression (5) for the state price density **with** inputs ˆx j+1

g (⃗m) and ˆv g

j+1 (⃗m)

j+1

to update the new guess for ˆΛ

g,g

(⃗m) ′

5. Repeat **with** initial conditions

j+1 j+1 ˆΓ

g,g

, ˆΛ ′ g,g

and ˆp j+1

′ g

6 There are some efficient choices for how to choose these initial u 0 ’s.

20

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22

Figure 1. Binomial Tree

Node HH

Node H

Node 0

Node HL

Node LH

Node L

Node LL

u 1 =

u ′ (u 0 , g H )

u 2 = u ′ (u 1 , g H )

u 2 = u ′ (u 1 , g L , ε 2 )

u 0

u 1 =

u ′ (u 0 , g L , ε 1 )

u 2 = u ′ (u 1 , g H )

u 2 = u ′ (u 1 , g L , ε 2 )

Figure 1 shows the binomial tree that represents the all the nodes for the two-period limited commitment

environment.

23

Figure 2. Profit function at node L

Figure 2 plots the normalized profit policy at node L.

Figure 3. Normalized consumption policy at node L

Figure 3 plots the normalized consumption policy at node L.

24

Figure 4. Consumption and Continuation Utilities at node L

Figure 4 plots consumption (top panel) and promised utilities (bottom panel) as a function of the idiosyncratic

shock realized at node L.

25

Figure 5. Change in log wages during recession, prime-age males U.S.

Figure 5 is constructed using data from Guvenen, Ozkan, and Song (2014). We computed the annualized

drop in log wages for the last four recessions adjusting for the duration of the recession

Figure 6. Change in log wages during recession

Figure 6 plots change in expected change in log wages from node L to LL for all agents (adjusting for the

drop in aggregate output) against percentiles of promised utilities at node L under the optimal contract.

26

0.2159

0.2103

0.2

0.2175

0.2178

0.1988

0.2202

Figure 7. Evolution of capital owner’s share

Figure 7 plots how the aggregate share of the capital owner’s consumption at all nodes

27