Uncertainty aversion and heterogeneous beliefs in linear
models∗Cosmin Ilut
Duke & NBER

Pavel Krivenko
Stanford

Martin Schneider
Stanford & NBER

May 2016

Abstract
This paper proposes a simple perturbation approach for dynamic models withagents who differ in their perception of exogenous shocks. The method characterizes linear dynamics around a steady state when that steady state differs from any
individual agent’s long run expectation. It applies when agents agree to disagree, as
well as when they differ in aversion to Knightian uncertainty and hence behave as if
they hold different worst case beliefs. It thus provides a simple way to study effects ofuncertainty on behavior in linear models. Our leading example looks at precautionarysavings, asset premia and gains from insurance in a borrower-lender model with agents
who differ in uncertainty aversion.

1

Introduction

In this paper we analyze a general class of dynamic models with agents that have heterogeneous perceptions over the probability distributions of the exogenous shocks. The agentsotherwise share structural knowledge of the economy, so that they know the mapping fromshocks to endogenous variables. Thus, disagreement about probability distributions of shocks
translates into a disagreement about endogenous variables that is determined in equilibrium.
We propose a perturbation approach to show how to characterize the linear dynamics of sucha heterogeneous beliefs model.
Our approach allows for disagreement that is not only temporary but survives even in
the long run. When the disagreement about future endogenous variables matters for steadystate, the solution strategy has to jointly determine the dynamics and the steady state.
∗

We thank seminar and conference participants at Minnesota, UCL and the SED 2015 in Warsaw for
helpful comments.

1

Indeed, while heterogeneity affects the steady state, the latter in turn affects the intensitywith which these beliefs about endogenous variables matter, through their equilibrium lawsof motion. Disagreement in the long run affects the applicability of perturbation methods.
The reason is that typically in dynamic models in which future endogenous variables enter
the intertemporal conditions, we cannot determine the steady state separately from thepolicy functions.
To solve jointly for the steady states and for the elasticities, we propose a solution
strategy that uses standard linear forward-looking rational expectations solution methods.
The strategy consists of the following fixed-point approach. First, guess how the future
endogenous variables respond to the exogenous shocks over which there is disagreement.
Second, stack all individual optimality conditions, where the intertemporal ones involve agent
specific beliefs directly about exogenous shocks and indirectly through the conjectured lawsof motion of future endogenous variables. Then find a candidate steady state, which we referto as the zero-risk steady state, that solves these equilibrium conditions. Third, compute allderivatives of the optimality conditions at the steady state and find elasticities for the policy
functions. To solve for this step we can appeal to standard linear rational expectations model
solution methods. To do that, we propose adding a vector of agent-specific forward-looking
endogenous variables that capture the heterogeneity in beliefs. Finally, we verify the initial
guess on the elasticities. If the guess is correct, then agents are forming heterogeneous, butmodel-consistent, expectations of the future endogenous variables.
Our proposed approach of solving dynamic heterogeneous beliefs models applies to two
broad types of economies. One is an expected utility model in which agents agree todisagree. There disagreement is wired in as a-priori assumed differences of perceptions aboutprobability distributions. A second economy is one where the heterogeneity of beliefs arises
endogenously because agents differ in their aversion to Knightian uncertainty (ambiguity).
Ambiguity aversion is described by the recursive multiple priors preferences that capture
agents’ lack of confidence in probability assessments.
To describe how linear methods can be applied for models in which agents differ in theirperceived uncertainty about the exogenous shocks, we start from a general recursive modelwith heterogeneity in ambiguity. There optimizing agents act as if they use different worstcase beliefs to evaluate future plans. When we focus attention on models with ambiguity
about conditional means, the equilibria will look like in models of disagreement. The key
property of the ambiguity model is that agents act as if the true data generating process(DGP) is given by the worst-case beliefs that support the equilibrium allocation. This meansthat the degree of disagreement is endogenous, since it will depend on equilibrium laws ofmotion that map beliefs about exogenous shocks into beliefs about endogenous variables.
2

While at the equilibrium allocations these beliefs will look to an outside observer as dogmaticdifferences, the ambiguity model is one in which policy interventions that affect equilibrium
choices will also alter this disagreement.
The endogeneity of disagreement shows up in our proposed solution approach in two
major ways. First, even in models where agents have ex-ante exogenously specified differencesof perceptions about the probability distributions of shocks, the equilibrium effects ofthese perceptions will be in general a function of the other model parameters. A second
manifestation is in models where agents face ambiguity but otherwise have ex-ante the same
degree of confidence. If in equilibrium agents take trading positions of different signs, theex-ante homogeneity manifests as ex-post heterogeneity of worst-case beliefs. For example,
as we will analyze in one of our application, when credit is in nominal terms, the worst-casebeliefs are different even if ex-ante agents shared the same uncertainty about inflation: thelender acts as if future inflation is high while the borrower acts as if future inflation is low.
The general logic of the proposed strategy to solve for heterogeneous ambiguity inlinear models starts by finding the linear equilibrium laws of motion. This entails thefollowing. First, we conjecture the worst case mean of each agent as a linear function ofstate variables. Conditional on this worst-case belief, the model is always observationallyequivalent with an expected-utility (EU) model. If there is heterogeneity in these beliefs,
then the observationally equivalent EU model is one with belief disagreement. Second, giventhese beliefs, we solve for the loglinearized model. This step usually requires jointly finding
the steady state and the law of motion. Third, we verify the initial conjecture from a firstorder expansion of value function. Having solved for the equilibrium policy functions, thelast step is to determine the actual dynamics of model by using the true Data Generating
Process (DGP) of shocks to simulate and compute moments.1
The second part of the paper consists of using our proposed strategy to solve a modelof uncertainty sharing in which two types of agents engage in financial trading to smoothconsumption. Each agent receives an endowment stream. The assets consist of a one-period
non-contingent bond, whose trading is subject to a convex borrowing cost, and a tree, whichpays dividends and has collateral value by lowering leverage. We illustrate heterogeneityin uncertainty by assuming that one type, called type A, perceives his endowment to be
uncertain while the endowment stream for the second type, called type B, is known to beconstant. In the baseline version agent A is also more patient than agent B, a feature than
we turn off for some experiments.
1

It is useful to note that the discrepancy between true and perceived moments is sometimes important.
Indeed, the econometrician that analyzes data produced by the model will measure premia or ’wedges’, suchas ex-post excess returns on uncertain assets, between equilibrium objects determined under the two different
measures.

3

We analyze two versions of modeling uncertainty sharing, which we compare to the casewhere uncertainty does not affect decision rules. In the expected utility version, uncertainty
refers only to risk, where all agents agree on the same probability distribution of shocks asgiven by the true DGP. For this ’risky’ model, we need to use nonlinear solution for thepolicy functions to analyze the general equilibrium effects of heterogeneous uncertainty. In
the second version, the uncertain endowment stream is also ambiguous. In particular, thetype A agent entertains a set of probability distributions about the conditional mean of
his endowment, while type B knows the true DGP. For this ’ambiguity’ model, we use our
proposed solution strategy to show that we can study equilibrium effects of the heterogeneous
uncertainty using linearization methods. In particular, we guess and verify what is the worstcase belief of agent A. Under this belief, which differs from the true DGP, the model looks
like one with belief disagreement.
We develop several exercises to help understand the economics of the model. First, we
confirm that the qualitative equilibrium effects of uncertainty are similar, when modeled
either as risk or ambiguity. Moreover, we find that for an appropriately chosen width of theset of beliefs of agent A the steady state of the two uncertainty versions is the same and thatpolicy rules behave very similarly. In particular, when compared to the deterministic case,
uncertainty generates in the ergodic steady state gains from trade that come in addition to
the difference in patience. This is reflected in the higher equilibrium steady state leverageand debt. The precautionary saving motive of agent A, coming either in the form of risk
or worst-case belief, incentivizes agent A to lend more. This not only raises the quantity ofdebt but also the bond price so that uncertainty aversion leads to a lower interest rate. Fromthe perspective of the borrower, the higher collateral value of the tree means that agent B
is willing to pay a higher price for the tree.
Second, we focus on the ambiguity model to study time-variation in uncertainty. An
advantage of the linearity behind the solution method is that we can expand easily the state
space. We study shocks to the ambiguity, or, in the language of heterogeneous beliefs, shocks
to the degree of disagreement between the two agents. We analyze a one-time increase inagent A’s ambiguity about his future income. This results in a strong desire to save fromagent A which leads to a fall in his consumption, a reduction in the interest rate and anincrease in debt. At the same time, the tree price increases as its collateral value is higher.
The transitional dynamics back to steady state differ whether the shock leads to an initial
negative real rate.
Third, we modify our baseline model so that there is no heterogeneity in patience. In thiscase, in the deterministic steady state there are no gains from trade. This means that there
is zero debt and, importantly, that the asset and consumption allocations are indeterminate.
4Standard perturbation methods that rely on approximations around the deterministic steadystate cannot handle such a case and need to impose some prior differences between agentsto make these allocations determinate. The purpose of this exercise is to highlight that
our strategy does not use the deterministic steady state and instead involves solving jointly
for the equilibrium elasticities and the zero-risk steady state. In the latter there will bedifferences between agents because of the heterogeneity in uncertainty.
The key property that emerges is that ambiguity generates gains from trade. The lessconfident agent, of type A, is concerned about future income and wants to save. Indeed, weshow analytically how heterogeneity in ambiguity works like differences between subjective
discount factors. Importantly, the resulting difference is endogenous. Changes in theenvironment will lead to agent A behaving as if his subjective discount factor has changed.
Part of the a-priori assumed heterogeneity in beliefs is the selection of the identity of
the agent whose income is uncertain. However, the proposed approach can be extended
to economies in which agents self-select in equilibrium into types whose beliefs will ex-postdiffer. To illustrate this, the fourth experiment we consider is a version of the baseline modelin which we introduce nominal credit and ambiguous inflation. Ambiguity is modeled as aset of conditional means for inflation. In equilibrium, the lender is concerned that the mean
for inflation is high, because this lowers the future continuation utility by eroding the realvalue of the nominal bond. Thus, agent A, the lender, acts as if the probability distribution
for future inflation is the one with the highest mean. At the same time, agent B, who inequilibrium becomes the borrower, is concerned that future inflation is low since this raises
the real value of the repayments the agent has to make, which lowers his continuation value.
We find that the endogenous disagreement generated by inflation uncertainty lowers
significantly the gains from trade, since both agents now perceive lower real returns to theirtrading strategies. The uncertainty premium lowers the price of nominal bonds and it leads
to less debt. The price of the tree also decreases due to the lower value of collateral.
The paper is organized as follows. Section 2 proposes a solution approach for a generalclass of recursive models with heterogeneous perceptions of exogenous shocks. There we firstcharacterize models with heterogeneous ambiguity and then describe a representation of arecursive model with heterogeneous beliefs. Section 3 introduces an application consistingof a model of uncertainty sharing, whose economic intuition is discussed in section 4 basedon a numerical example.

5

2

Solving heterogeneous beliefs models by linearization

In this section we first introduce a general class of recursive models with heterogeneity inambiguity. We then move towards a representation of a recursive model with heterogeneous
beliefs by noting the observational equivalence of the ambiguity model with an expectedutility (EU) model, given the worst-case beliefs. Once we focus on ambiguity about
conditional means, we develop an approach to solve for this EU model with disagreement
using log-linearization. The approach usually requires finding jointly the steady state and theequilibrium law of motion. We show that there is an easy implementation that uses standard
solution methods, implementable in packages such as Dynare, for an as if linear forwardlooking rational expectations representation. In the heterogeneous ambiguity model we needto verify the conjectured worst-case belief by analyzing the equilibrium value functions. This
step is not required for EU models, which can be solved using our solution approach, butwhich impose dogmatic disagreement. Once the linear policy functions are determined, theequilibrium dynamics of the model are found by simulating under the true DGP.

2.1

Generalized recursive model

Let Zt denote a k × 1 vector of exogenous state variables, Xt an m × 1 vector of endogenous
state variables and Yt a n × 1 vector of other endogenous variables.
There are I agents, each agent i with a vector of actions, denoted by Di . These actions
are part of X or Y and include the consumption choice C i . Let B i (X, Y, Z) denote the agent
i0 s constraint set.
Given the evolution of Z, the evolution of the endogenous state variables is given by
X 0 = T (X, Y, Z)
An example of such law of motion is the capital accumulation equation, where X will
include capital. An important component is the law of motion of the vector of ambiguities,
ai ∈ X, given by some transitionai = Ai (X, Z)
As detailed below, the vector of ambiguities controls the size of the belief sets for eachagent, and in turn for each shock. In particular, each agent is characterized by belief sets

6

such that one-step ahead conditional mean of innovation to the jth shock Zj belongs to theinterval −aij , aij . In principle, the ambiguity process can be a function of the endogenous
state-variables X, as it would happen for example in models of learning.
Finally, the model economy consists of additional restrictions that determine the otherendogenous variables, Y, through some relationshipΓ (X, Y, Z) = 0
An example would be market clearing conditions in the goods or asset markets.
The recursive equilibrium consists of functions for the endogenous variables X 0 (X, Z),
Y (X, Z), value functions V i (X.Z) that satisfy three conditions.
First, agents optimize. In particular, we describe agents’ preferences by the recursive
multiple priors according to which(
V i (X, Z) =max

Di ∈B i (Y,X,Z)ui (C i ) + β

s.t. X 0 = T (X, Y, Z)

min|µij |≤Aij (X,Z)

E


µ

)

V i (X 0 , Z 0 )

(1)
(2)

Agents’ valuation of state variables X and Z is given by V i (X, Z) , which is obtained by
maximizing per period utility ui (C i ) and the continuation value over the available actions
Di from the budget set B i (Y, X, Z) .
A key element of the setup is that each agent has a set of beliefs over the one-step ahead
conditional mean, denoted by µij , of the innovation to each shock Zj . This set is given by
the restriction that |µij | ≤ Aij (X, Z), where the ambiguity vector Ai is potentially specific toeach agent.
In the recursive representation in (1), agents react to the ambiguity (Knightian uncertainty) about the conditional probability distributions of the shocks by taking optimal
actions as if the true data generating is characterized by each shock Zj having the probability
distribution with the lowest mean µij out of the set of beliefs. The fact that agents havedifferent sets thus gives rise to heterogeneity in beliefs. Finally, in order the forecast thenext period state variables, agents use the law of motion in (2).
Second, the endogenous variables are determined according to the restrictions summarized by Γ (Y, X, Z) = 0.
Third, the dynamics of this equilibrium need to be characterized under some law of
motion for the exogenous variables Z. In the two steps above we have found the decisionrules and laws of motion that arise from agents’ aversion to uncertainty, as expressed by the
optimization of the recursive multiple priors preference. Given these functions X 0 and Y,
7the realized equilibrium outcomes are obtained feeding in the true exogenous Markov stateprocess for Z.

2.2

Recursive equilibrium with general beliefs

The general system consists of three types of equations. There are n equations that determine
the other endogenous variables as an implicit function of the state variables:f (Xt , Xt−1 , Yt , Zt ) = 0(n equations)

There are m equations that capture intertemporal behavior by agents. In a modelwith heterogeneous beliefs, we need to distinguish between what belief is used for form
P
expectations. We say there mi equations involving beliefs of agent i, where i mi = m.
Eti g i (Xt , Xt−1 , Yt , Yt+1 , Zt ) = 0(mi equations)

Notice that the expectation operator uses the conditional distribution that emerges as theworst-case distribution from the minimization in the utility represented in (1). Once theworst-case belief is determined, or guessed, the model becomes observationally equivalent
with an expected utility model with heterogeneity in beliefs.
There are k equations that describe the law of motion of the exogenous variables. We
introduce the risk directly on innovations to the log state variables:
log Zt = (I − P ) log Z̄ + P log Zt−1 + εt(3)

We assume that I − P is invertible so the mean of a stationary solution to this difference
equation is log Z̄.
We assume that agents may disagree about the exogenous variables, but they sharestructural knowledge of the economy, that is, they agree on the mapping from exogenous toendogenous variables.
For a general definition of equilibrium, suppose that each agent has in mind a probability
distribution over Zt , possibly different from (3). A recursive equilibrium is given by functions
X 0 (X, Z) and Y (X, Z) such that the equations above are satisfied for all (X, Z). The

8

resulting system of functional equations isf (X 0 (X, Z) , X, Y (X, Z) , Z) = 0,

E i g i (X 0 (X, Z) , X, Y (X, Z) , Y (X 0 (X, Z) , Z 0 ) , Z) |Z = 0,
where in the second equation Z 0 is uncertain and expectations in equation i are computed
using agent i’s conditional distribution over it. These are m + n functional equations in
m + n functions.
2.2.1

Recursive equilibrium: zero risk steady state

If agents’ beliefs differ in conditional means, then they will disagree in steady state. Denotesteady state value by bars. Suppose also there are no shocks, but agents differ in conditionalbeliefs at the steady state E i Z 0 |Z̄ . The steady state is then characterized by
f X̄, X̄, Ȳ , Z̄ = 0, 
g i X̄, X̄, Ȳ , Y X̄, E i Z i |Z̄ , Z̄ = 0,
In general, the beliefs E i Z 0 |Z̄ matter both directly, in the second equation, and throughtheir effects on future endogenous variables.

2.3

Recursive equilibrium: log-linear approximation

Disagreement at the steady state affects the applicability of perturbation methods. Indeed, if
future endogenous variables Y enter into the intertemporal conditions, we cannot determine
the steady state separately from the policy function Y (X, Z) . This problem is relevant for
example in models with an intertemporal Euler equation of a long lived agent, where Y
typically includes consumption, and it is not possible to find consumption in closed form.
It does not occur for example in models with 2 period lived agents, where we have a closed
form solution for consumption as a function of X and Z.

2.3.1

Loglinear approximation

We shoot for approximate solutions that are loglinear expansions around the steady state.
Denote log deviation by lower case letters with hats, that is, X = X̄ex̂ and so on. We

9conjecture solutions
Xt = X̄ exp (εxx x̂ + εxz ẑ)Yt = Ȳ exp (εyx x̂ + εyz ẑ)Here x̂ corresponds to the lagged deviation of X from steady state and ẑ corresponds to thecurrent deviation of Z from steady state. The undetermined coefficients are therefore m + n
constants, and (m + n) (m + k) elasticities.
If all we want is loglinear approximate solutions, we do not have to model explicitly allfeatures of beliefs: all that matters is differences in conditional means. We assume
Eti log Zt+1 = (I − P ) log Z̄ + P log Zt + Ai log Zt
This assumes that conditional means can be written as a linear function of exogenous statevariables. The matrix Ai determines the belief adjustment relative to the true law of motion(3). In the exogenous ambiguity case, an ambiguity state variable would be part of Zt andwe can then use different Ai matrices to pick different worst cases.
In a rational expectations setup, we can solve the constants separately from the elasticities. This is not possible here since agents disagree in steady state about what happens
to the state variables next period. However, we can still loglinearize to obtain as manyequations as coefficients we need to determine.
Loglinearizing the first set of equations, we have (writing the jth partial derivative as fj
and dropping arguments):
0 = f X̄, X̄, Ȳ , Z̄ + X̄f1 (εxx x̂ + εxz ẑ) + X̄f2 x̂ + Ȳ f3 (εyx x̂ + εyz ẑ) + Z̄f4 ẑ.
This delivers n equations for constants, and n × (m + k) on elasticities in the usual way.
Loglinearizing the second set of equations, for each i, we have
0 = g i X̄, X̄, Ȳ , Ȳ exp(εyz Ai log Z̄ , Z̄) + X̄g1 (εxx x̂ + εxz ẑ) + X̄g2 x̂ + Ȳ g3 (εyx x̂ + εyz ẑ) + Ȳ g4 εyx (εxx x̂ + εxz ẑ) + εyz P + Ai ẑ + Z̄g5 ẑ

This delivers m equations from the constant terms, and m × (m + k) equations frommatching coefficients on the state variables. Note that all the derivatives fj and gj also
depend on the constants.

10

2.3.2

Proposed solution strategy: implementation

To solve jointly for the constants and the elasticities, we propose a solution strategy thatuses standard linear forward-looking rational expectations solution methods. The strategy
consists of the following steps:
1. Guess the elasticities of the other endogenous variables,Y, with respect to the endogenous and exogenous state variables, εyx and εyz , respectively. This step is useful so thatthe agents have a guess on the law of motion of future endogenous variables, which
they need to forecast.
2. Solve for candidate steady state from constant terms
f X̄, X̄, Ȳ , Z̄ = 0
g i (X̄, X̄, Ȳ , Ȳ exp(εyz Ai log Z̄), Z̄) = 0
Given the guessed elasticities, this is a nonlinear equation system of dimension m + n.
3. Compute all derivatives at steady state and find elasticities
This becomes finding a linear system of dimension(m + n) (m + k) .
To solve for this step we can appeal to standard linear rational expectations model
solution methods. To do that, we propose adding a vector of forward-looking endogei
nous variables Ỹt+1, with a size equal to the number of Y variables that enter as leadsin the expectation equations g i (.).
In particular, we modify the model to read
0 = f (Xt , Xt−1 , Yt , Zt )hiii0 = Et g (Xt , Xt−1 , Yt , Ỹt+1 , Z̄)(4)log Zt = (I − P ) log Z̄ + P log Zt−1 + εt

(6)

(5)

where the new defined variable follows
Ỹti = Ȳ exp(εyx log Xt−1 − log X̄ + εyz log Zt − log Z̄ + Ai log Zt−1 )(7)

The variable Ỹti thus controls for the forward looking variable Yt . but from theperspective of each agent i. The heterogeneity of beliefs shows up in introducing agentspecific Ỹti variables. The law of motion for this variable follows from the guess in11

step 1: it responds to the endogenous and exogenous state variables according toelasticities εyx and εyz . Importantly, the agent specific matrix Ai , which determines
the belief adjustment about the exogenous state Z, relative to the true law of motion,affects the belief about the endogenous variable through the elasticity εyz .
4. Finally, for the solution of the economy represented by the equations in (4)-(7) to be
the equilibrium in which we have imposed structural knowledge of the economy, weneed to check the consistency of these beliefs. This amounts to check if the guess ofthe elasticities εyx and εyz is correct. If the guess is correct, then agents are forming
model-consistent expectations of the future endogenous variables.
Having reached consistency of beliefs about the structure of the economy, the dynamics
and the steady state of the rational expectations model described by the equilibrium
conditions in (4)-(7) produces the dynamics and the zero-risk steady state of our economy
with heterogeneous beliefs. Indeed, this system contains: (i) the original static equations
f (.) in (4), (ii) the forward looking equations Et g i (.) in (5), where expectations about
future endogenous variables are agent specific. These functions contain the agent specificviews about future variables, reflected in the guessed laws of motion (7); (iii) dynamics are
determined by the true law of motion for the exogenous state, as in (6). At the fixed point
for the elasticities (εyx , εyz ) agent i’s expectation of next period’s endogenous variables, as
described by equation (7), is consistent with the linearization of the overall system at thezero risk steady state.

3

A model of partial insurance

We present here a model of uncertainty sharing. Agents face uncertain endowment streams
and engage in financial trading to optimally smooth consumption.

3.1

The model

Agents and beliefs
There are two types, denoted by A and B, of infinite lived agents. Each agent maximizes
"∞
X
j

E0

c1−γ
βjt t
1−γ
t=0
12

#
(8)

where γ is the CRRA coefficient and βj is the potentially agent specific subjective discountfactor. Each agent will receive a stream of endowment of goods. The beliefs of each agent
about these streams are as follows. Type B agents always get ȳ B and all agents know this.
At date t, type A agents believe they get yt+1 = ȳ A exp (−at ) . The confidence of agent A is
potentially time varying, in the sense that at is stochastic. However, this belief simply reflects
the concern of agent A, who is not confident about his own income. In fact, according to
the true data generating process type A agents get yt+1 = ȳ A . Type B agents are confident:
they know the true DGP and thus know not only that their income ytB = ȳ B but also thatagent A endowment is constant at ytA = ȳ A .
The model is casted as one of heterogeneous beliefs, in which for simplicity we haveassumed that agent B knows the true DGP and agent A is more pessimistic about his
income. A way to interpret agent A0 s belief is an as if pessimism that results from theambiguity he perceived about his income. In that case, the set of beliefs about the onestep ahead conditional mean of his log-income is given by [−at , at ] and the worst-case belief
is simply the lower bound, given by −at . This leads to agent A maximizing the present
discounted value in (8) under the worst-case belief that yt+1 = ȳ A exp (−at ) .
AssetsThere are two assets in this economy. One is a one period noncontingent debt, traded atprice qt . The second is a tree, which pays each period a constant dividend d,is sold at pricept and there are no short sales allowed.
There is a financial friction in this economy: in order for a borrower that owns θt of thetree to be able to support a leverage `t , defined as −qt bt /pt θt , there is an additional cost to
the borrower given by k (`t ) qt bt . The property of the borrowing cost k (`) are that: k (`) = 0
for ` ≤ 0; k (`) , k 0 (`) , k 00 (`) > 0 for ` > 0.
The date t budget constraint of each agent is given by
ct + pt θt + qt bt (1 + k (`t )) = yt + (pt + d) θt−1 + bt−1

(9)

The left hand side reflects expenditures on consumption, tree shares and debt. If leverage
is positive, this entails the additional cost described above. The right hand side gives theavailable resources: the realized endowment of goods plus the dividend and market value ofthe owned shares, as well as the default-free bonds that mature this period.
Optimality conditions

13

The optimality of bond holdings leads to the following Euler equations for bond pricing
"
"
−γ #
−γ #
A
B
c
c
t+1
t+1
qt = βA EtA= βB EtB+ qt (k (`t ) + `t k 0 (`t ))
B
cA
c
t
t

(10)

The first equality reflects the choice of agent A. This agent in equilibrium is the lender,
i.e. bAt > 0. This choice arises even in the case where the discount factors are the same
βA = βB . The reason is that the pessimism of agent A works like a desire for precautionarysavings, which is larger than for agent B and thus drives up the price at which agent A is
willing to hold bonds. In turn, as shown by the second equality, the borrower will hold thedebt in equilibrium because of the leverage cost. Indeed, if not for this cost, the confident
agent B will want to sell as much of the bonds as possible at the seemingly high price qt
obtained from the strong desire of agent A to save. The marginal cost of borrowing, givenby the last two terms in equation (10) lower the perceived bond price for the borrower until
equilibrium is achieved and both agents are happy to satisfy the bond-market clearing at aninterior solution. The marginal cost consists of the higher cost of an additional unit of debt,
for a given leverage `t plus the higher cost arising from the marginal increase in leverage, fora given size of debt.
We are looking for equilibria in which only type B agents hold the tree. This meansthat for the pricing of the tree we have to characterize just one Euler equation, given by the
optimality of agent B:
pt = βEtB

"

cB
t+1
cB
t

−γ

#
(pt+1 + dt+1 ) + `2t k 0 (`t ) pt

(11)

The tree is priced under agent B 0 s belief and the value of the tree is higher due to thecollateral benefit, given by the last component of equation (11). We must therefore checkthat the type A does not want trees. The reason that this may be the case is that thetree looks too expensive from the perspective of agent A, who is more pessimistic about the
future outcome of the economy.

3.2

Recursive equilibrium

The market clearing conditions for this economy are as follows. In the goods market
B
A
B
cA
t + ct = ȳt + ȳ − qt bt k (`t )14

where we assume that the borrower cost represents wasted resources. The debt market statesthat bonds are in zero net supply:BbA
t + bt = 0
For the market for trees, we use the condition that only agent B owns trees.
The state variables are therefore the type A endowment ytA , the stochastic ambiguity aand the distribution of asset holdings. From the debt market condition, this simply boils
down to tracking type B debt bt = −bBt .
The recursive equilibrium consists of finding the consumption allocations of each agent




i
c y A , a, b , the debt allocation b0 y A , a, b and the prices q y A , a, b and p y A , a, b .Noticethat agents disagree only about income but otherwise we assume that they know theequilibrium mapping given by the functions above. This means we are left with finding findfunctions from the Euler equations in (10) and (11) together with the budget constraints in(9).
We are looking to solve for the equilibrium using a log-linear approximation. This
amounts to finding elasticities for the unknown functions. For example:
A
A
cA A
cA
cA
ĉA
t = log ct − log c̄ = εyA ŷt + εa ât + εb b̂twhere hats denotes log-deviations from the steady state.

3.3

Steady state

We first describe how to characterize the steady state of this economy. We are looking forallocations and prices that are constant.

3.3.1

Type B agents, leverage and collateral value

The confident agents B expect some constant consumption c̄B . To describe the steady state
prices, we find it convenient to write the asset pricing conditions with returns.
From the type B Euler equation for bonds we have that the steady state return on bondis given by¯ + `k¯ 0 (`)¯ /βBrbond = q̄ −1 − 1 = δB − k(`)
where δB = βB−1 − 1, which shows that a higher leverage lowers the interest rate. The
borrower is compensated in equilibrium for the higher cost of borrowing by paying a lower
interest rate on debt.
15

From the type B Euler equation for tree holdings we have that the return on holding
trees isdrtree = = δB − `¯2 k 0 `¯ /βBp̄which also shows that a higher leverage lowers the tree return. This happens because ahigher leverage implies that the tree collateral benefit is larger which raises the value of thetree price, and thus lowers its return.
Using the two conditions for returns, we can compute the excess return as
rtree − rbond = k `¯ + k 0 `¯ ` 1 − `¯ /βB
which, for low leverage, results in a positive premium on tree.

3.3.2

Type A agents and precautionary savings

Agent A is in turn ambiguous about his income and thus expects his consumption to dropfrom the steady state c̄A to a lower value, given byc̃A = cA ȳ A exp (−ā) , ā, b̄ ≈ c̄A exp −εcA
yA ā

(12)

This agent behaves as if he is perpetually surprised by high income. He expects consumption to fall in proportion to εcAyA , the elasticity response of his equilibrium consumptionfunction to the ambiguous income. The latter is expected to be lower by ā and thus future
consumption is lower, in logs, by εcA
yA ā.
Agent A holds bonds due to his precautionary motives. The Euler equation results in
rbond = q̄ −1 − 1 ≈ δA − γεcAyA āwhere δA = βA−1 − 1. More pessimism, in the form of a larger ā, lowers the interest rate, as
the agent is willing to pay a larger bond price to achieve more stability in his consumption
profile. This behavior works like a higher discount factor, but it is endogenous, in the sensethat the equilibrium response εcA
yA matters for this effect. In addition, the curvature in utility
matters. A lower intertemporal elasticity of substitution, given here by γ −1 , means that theagent is less willing to accept an intertemporally volatile consumption profile and thus iswilling to accept an even lower interest rate to smooth his consumption.
The steady state conditions show clearly that we need to solve jointly for the steadystates and the elasticities εyz . In particular, here agent A has to have a belief about how his
16consumption will change next period as he expects a lower income. This will determine hissaving decisions, which affects the steady state consumption and pricing decisions. In turnthe elasticities are computed around the steady state values of these endogenous decisions.

3.3.3

Tree market participation

It is useful now to revisit the decision of agent A to hold shares in the tree. Type A expects
the tree price to drop from its steady state value p̄ to a valuep̃ = p ȳ A exp (−ā) , ā, b̄ ≈ p̄ exp −εpyA ā

(13)

As long as the elasticity of the tree price with respect to agent A income is positive, i.e.
εpyA > 0, then agent A is also pessimistic about the (endogenous) price.
For agent A not to hold the tree we mush have his perceived tree return be lower thanthe bond returnd− εpyA ā < rbondp̄
Notice that this can true, as it turns out to be in the numerical example presented below,
even if rbond < rtree = d/p̄. For this the price impact of y A , through the elasticity εpyA > 0, is
important.

3.4

Implementing an as if RE dynamic model

We follow the solution strategy proposed in section (2.3.2). The log-linearized equilibrium
consists of the following equations.
Budget constraints: For the lender, type A :A
cAt + qt bt = yt + bt−1

and for the borrower, type B :B
cBt − qt bt (1 + k (`t )) = y + d − bt−1

where we have imposed the market clearing condition in bonds and tree shares.

17

Euler equations. For the tree holdings:
Pt = `2t κ0 (`t ) Pt + βB Et

"

cB
t+1
cB
t

−γ

#
(Pt+1 + Dt+1 )and for the bond holdings the two Euler equations

Qt = βA Et

ew
cA,N
t+1
cA
t

!−γ

 B −γc+ Qt (κ (`t ) + `t κ0 (`t ))= βB Et t+1
cBtwhere we introduce the additional forward looking variable for the perspective of agent A
about how his future consumption will evolve:
ew
cA
cA
log cA,N= log cA + εcAtb (bt−1 − b) + εyA (yA,t − y − at−1 ) + εa (at − a)The solution of this system if the fixed point of jointly solving for constants and elasticities.
The steady state of this as if rational expectations model gives us the zero risk steady state,ewwhere the conjectured policy function for needs to cA,Nneeds to be consistent with thetAelasticities of the actual log ct policy function.

4

Numerical example

We now take the model of partial insurance described above and perform a range ofsimulation exercises. The parametrization for the benchmark model is as follows. Total
output is normalized to one, with each agent receiving equal endowments of ȳ A = ȳ B = 0.45
and the tree dividends amounting to d = 0.1. We set the common CRRA coefficient toγ = 2 and the two discount factors as βA = .98 and βB = .963. The borrowing cost is
k (`) = .075`2 . We choose these latter three parameters to target the following three moments
for the deterministic version of the model: the tree price, the price of bond and the leverage,
whose values are reported in the first row of Table (1).

4.1

Steady state effects of uncertainty

We find it useful to compare two versions of how uncertainty about agent A0 s income isincorporated into our model of partial insurance. One, which we call ”risk model”, is astandard model of uninsured income risk. There agent A’s income is hit by iid Gaussian18

shocks with a standard deviation σ = 0.1. To capture the effects of risk on decision rules wewill solve that model nonlinearly using interpolation methods.
In the second version, which we call the ”ambiguity model”, agent A’s income is ambiguous. Steady state ambiguity is normalized as a multiple of the standard deviation, i.e.
ā = ησ. We solve this model linearly, using the solution strategy that we described above. We
choose the parameter η to get a similar zero-risk steady state of debt as the ergodic steadystate obtained for the risk model. We find that a value of η = 0.12 produces this match.
In the process, the other variables also are very similar between the risk and the ambiguity
steady states, as reported in the second and third rows of Table (1). The interpretation ofthis parameter is that uncertainty in the form of risk generates similar effects on decisionrules through nonlinearities as it does through ambiguity with a set of beliefs on the one-stepahead conditional mean ranging from [−0.12σ, 0.12σ] . This parameter is much smaller thatthe upper bound of η = 2 that Ilut and Schneider (2014) argues is theoretically reasonablefrom the perspective of the statistical fit of the worst-case forecast.

Steady state
Deterministic
Risk
Ambiguity

Table 1: Steady states
c̄A
c̄B
q̄
p̄
0.48 0.518 .98 3
0.47 0.517 .987 3.36
0.47 0.517 .987 3.33q̄ b̄`¯ = q̄ b̄/p̄
1.2
.40
1.59
.48
1.59
.48

Table (1) shows that uncertainty, either in the form of risk or ambiguity, generates inthe ergodic steady state gains from trade that come in addition to the difference in patience.
This is reflected in the higher equilibrium steady state leverage and debt. The precautionarysaving motive of agent A, coming either in the form of risk or worst-case belief, incentivizes
agent A to lend more. This not only raises the quantity of debt but also the bond price sothat uncertainty aversion leads to a lower interest rate. From the perspective of the borrower,
the higher collateral value of the tree means that agent B is willing to pay a higher price forthe tree.

4.2

Policy functions

Figure (1) plots the equilibrium decision rules as function of debt in the case of thedeterministic, risk and ambiguity model. The green vertical dotted line represents the lower
steady state value of debt in the deterministic model compared to the one in the risky and,by construction, the ambiguity model, shown as the vertical red line. There several points
of interest.
19

Debt

Bond Price

Tree Price
3.4

1.7

0.988

3.3

1.6
0.986

3.2

1.5
0.984

1.4
1.3

0.982

1.2

0.98

Deterministic
Risk
Ambiguity

3.1
3
2.9

1.2

1.4

1.6

1.2

1.4

1.6

B

B

Lender Consumption

Borrower Consumption

1.4

1.6

B
Leverage
0.6

0.56

0.5

1.2

0.55

0.49

0.54
0.5

0.48
0.52

0.47
0.46

0.45

0.5

0.4

0.48

0.35

0.45
0.44
1.2

1.4

1.6

1.2

1.4

B

B

1.6

1.2

1.4

1.6

B

Figure 1: Equilibrium policy rules, as functions of debt, for the deterministic, risky andambiguous models.

20

First, policy functions in the risk and ambiguity model are quite similar. While theformer was obtained through nonlinear solution methods, the latter captures most of its
qualitative properties even if it is solved only up to a first-order approximation.
Second, the shape of the decision rules are as expected. Agent A0 s consumption andlending is increasing in the amount of bonds that he has entering the period, while agent B’s
consumption is decreasing since he has to repay that debt. If the lender has lower bonds thanin steady state, the transition path he expects is one with a positive consumption growth.
This means that the bond price along this path will be lower than the steady state, andwill converge to it from below as the state of the economy is one with lower debt. Similar
logic holds for the region of debt above its steady state. This explains the upward sloping
bond price. Along this path, the borrower expects a negative consumption growth which
increases, caeteris paribus, the price he is willing to pay for the tree since this serves as a way
to smooth consumption. On the other hand, in this region the collateral value of the tree islower since debt is lower than the steady state. Thus, there are two competing equilibrium
effects. For the linear policy functions, such as the deterministic and ambiguity model, one
force is bound to dominate. In this case it is the former and the equilibrium price is higher
when debt is lower. In the nonlinear solution, the price is slightly increasing and about flataround the ergodic steady state.
Third, the qualitative effects of uncertainty in shifting the deterministic policy functions
are the same, whether it is risk or ambiguity. Sharing uncertainty leads to higher gains fromtrade, which shifts up debt, for any state value. Even if the shift is small, because debt is very
persistent, the fixed point where the debt policy returns a value equal to that of the statevalue of debt is much larger in the uncertainty case. The uncertainty-averse agent A engages
in precautionary saving, so that the consumption function is shifted down with uncertainty.
For the same value of debt this results in significantly lower consumption. However, becauseof the same precautionary reasoning, in steady state agent A also accumulates more debt so
the net effect on the steady state consumption is not obvious. For this parametrization itturns out that that the shift down in the function dominates. The desire to save also shifts
up the bond price. Together with the higher debt characterizing the new steady state thisleads to a significant rise in the bond price. In turn, agent B 0 s consumption is shifted up
since he enjoys a lower interest rate on the same debt. However, in the new steady state hewill have to pay the now lower interest rate on a larger amount of debt. The overall effecthere is such that the second effect dominates and the agent B consumption is also lower inthe ergodic steady state than in the deterministic case, as shown in Table (1). The fact thatboth agents’ consumption are lower is possible because of the larger borrowing cost that is
a waste in the resource constraint. Because of the larger gains of trade and the higher debt,
21

the collateral value of the tree shifts up with uncertainty and remains high in the steadystate. For leverage the price effect dominates the bond effect and the function is shifted
down. However, due to its strong upward shape, and intuitively because of the larger gains
from trade, the ergodic steady state of leverage is larger.

4.3

Time-varying disagreement

We now focus on the ambiguity model to study time-variation in uncertainty. We havedescribed it as one with heterogeneity in beliefs, where agent A is less confident and thus
as if more pessimistic about his income. We have developed tools that allows us to study
this heterogeneity that remains even in the steady state using linear methods. An advantageof the linearity is that we can expand easily the state space. For example, we can study
shocks to the ambiguity, or, in the language of heterogeneous beliefs, shocks to the degreeof disagreement between the two agents.
Figure (2) plots the response of the ambiguity model economy to a one-time increasein agent A’s ambiguity about his future income. Following a similar logic as with thesteady state, this results in a strong desire to save from agent A. This leads to a fall in his
consumption, a reduction in the interest rate and an increase in debt. At the same time,
the tree price increases as its collateral value is higher. The periods following this shock arecharacterized by different dynamics, depending on the interest rate effect. For the baseline
economy, the fall in the interest rate is so large that the rate becomes negative. This has
interesting effects on the dynamics. Even if uncertainty has reverted to its steady state
following the first period shock, agent A now sees a negative wealth effect since the debt
that he owns comes with a negative interest rate. Thus, starting in period two, we see typicaltransitional dynamics back to steady state when the agent has suffered a negative wealthshock: consumption is lower than steady state and he reduces debt to smooth this transition.
This is the reason why consumption of agent A not only falls in the initial period, whichcomes from the precautionary savings, but continues to converge back to steady state frombelow, and at the same time, why debt initially rises and then falls.
To verify this intuition, Figure (3) plots the impulse response to the same ambiguity
increase, but for a parametrization where βA is lower so that the steady state interest rate islarger to the extent that the fall in the interest rate after the initial uncertainty shock still
keeps the rate at a positive level. In this case, the dynamics are more standard. Consumptionof agent A initially declines strongly and then converges back, slightly from above. This
agent has accumulated more debt following the higher perceived uncertainty and, following
22

ambiguity

1.5

type 1 consumption type 2 consumption
0

0.3

1

-0.1

0.2

0.5

-0.2

0.1

0

-0.3
0

10

20

bond price

0.4

0
0

10

20

tree price

0.4

10

20

debt

0.2

0.3

0.2

0

0

0.2
0

-0.2

0.1

-0.2

0
0

10

20

-0.4
0

10

20

0

10

20

Figure 2: Increase in ambiguity
standard transitional dynamics logic, will now consume out of it to converge smoothly backto steady state. Indeed, debt is initially higher and then it slowly decreases.

4.4

Perturbation without needing deterministic steady state

Our proposed perturbation method involves solving jointly for the zero-risk steady state andthe equilibrium elasticities, which has the implication that we do not use the deterministic
steady state in the solution method. This is an important technical advantage over standard
perturbation methods which usually approximate around the deterministic steady state.
That standard approach requires that such a steady state is determinate and thus cannot
handle models where the Jacobian of the entire dynamic system is rank deficient. An example
for such a situation where one cannot approximate around the deterministic steady state isan asset allocation problem where expected returns are the same, such as in models ofinternational portfolio choice.
In our model of partial insurance we would encounter the same problem is agents have23

ambiguity

1.5

type 1 consumption type 2 consumption

1
0.5
0

0.1

0.3

0

0.2

-0.1

0.1

-0.2

0

-0.3
0

10

20

bond price

0.3

-0.1
0

10

20

tree price

0.3

10

20

debt

0.3

0.2

0.2

0

0.2

0.1
0.1

0.1

0

0

-0.1
0

10

20

0
0

10

20

0

10

Figure 3: Increase in ambiguity: low discount factor economy

24

20

the same discount factor β. In this case, in the deterministic steady state there are no gains
from trade. This means that there is zero debt and, importantly, the asset and consumption
allocations are indeterminate. Standard approaches cannot handle such a case and need to
impose some prior differences between agents to make these allocations determinate. Our
strategy does not use the deterministic steady state and instead goes directly after thezero-risk steady state. In the latter there will be differences between agents because of theassumed heterogeneity in confidence.
Consider the following numerical example where the only difference from the baselinemodel is that both agents have the same discount factor: βA = βB = .963. Table (2)indicates that while in the deterministic steady state there is no debt and allocations areindeterminate, in the zero-risk steady state there is positive debt and the allocations are
determinate.

Steady state
Deterministic
Ambiguity

Table 2: Steady states
c̄A
c̄B
q̄
p̄
q̄ b̄ `¯ = q̄ b̄/p̄.963 2.63 000.47 0.52 .972 2.76 .81.29

The key property that emerges from the model with uncertainty is that ambiguity
generates gains from trade. The less confident agent, of type A, is concerned about future
income and wants to save. This desire makes agent A behave like a saver, which otherwise
could be modeled as being the more patient agent. Indeed, heterogeneity in ambiguity works
like differences between subjective discount factors. Importantly, the resulting difference isendogenous. Changes in the environment will lead to agent A behaving as if his subjective
discount factor has changed.
We can see this analytically. In steady state, optimality conditions for agent A involve
the perceived discounted intertemporal marginal rate of substitution between consumption
tomorrow and today. Using equation (12) this is given by A −γc̃= βA exp γεcA
βA A
yA ā
c̄

(14)

By defining an adjusted discount factor
βeA ≡ βA exp γεcA
yA ā



we can reinterpret zero-risk steady state optimality conditions that only involve the pricing
kernel in (14) as a deterministic steady state where, if ā > 0, agent A has a higher discount25

factor than agent B. This adjusted discount factor is however endogenous to the rest of
the parameters because the elasticity εcA
yA matters. In addition, even if we fix parameters,in general the model is not perfectly observationally equivalent to the deterministic case of
setting exogenously the discount factor of agent A to βeA . The reason is that there may beother forward-looking conditions of agent A that involve expectations of other endogenous
variables, not just consumption growth. For example, when this agent considers investing intree shares, it forms expectations over the future tree price, as shown in equation (13). The
elasticity of this price with respect to the ambiguous shock appears as an additional effectof uncertainty in this decision, on top of the effect of the pricing kernel.

4.5

Endogenous disagreement and worst-case beliefs

The baseline model features endogenous disagreement to the extent that the structuralparameters affect the equilibrium elasticities of allocations and prices with respect to theuncertain shocks. For example, the elasticity of agent A0 s consumption with respect to his
income, εcAyA , will in general be a function of many of the structural parameters, including thesize of his endowment, the subjective discount factor, the borrowing cost and so on. In thissense, the degree of disagreement is on one hand purely exogenous, controlled by the amountof ambiguity ā, and on the other hand determined by the rest of the model parameters.
Part of the a-priori assumed heterogeneity in beliefs is the selection of the identity of
the agent whose income is uncertain. However, the approach we propose can be extended
to study situations in which agents self-select in equilibrium into types whose beliefs will
ex-post differ.
To see this, let us consider a version of the baseline model in which we introduce nominal
credit and ambiguous inflation. The payoff of the bond is (1 + π)−1 , where π is the uncertain
inflation and the price q is now the nominal bond price. Suppose that the true data generatingprocess is that inflation is zero. If there would be no uncertainty about it, or, in the linear
model, there would be full confidence in this process, then we would recover the exact sameequilibrium as in the real model.
Consider the ambiguity model. Suppose that both agents believe that the one-step ahead
conditional mean of inflation belongs to the set [−π, π], where π is a positive parameter.
Agent A is in equilibrium the lender. This desire to save follows either from a-prioridifferences in the subjective discount factor, or, as we have shown above, simply fromdifferences in the ambiguity about income. The lender takes consumption and nominalbond decisions under the worst-case belief about inflation. The future payoff of the bond26

is decreasing in future inflation, and, in turn, the value function is increasing in the wealththat the agent accumulates. It follows that the lender is concerned that the true DGP for
inflation is one in which inflation is high, because this lowers the future continuation utility,
by eroding the real value of the nominal bond. Thus, agent A acts as if future inflation isπ. At the same time, agent B, who in equilibrium becomes the borrower, is concerned that
future inflation is low, and equal to −π. Indeed, a lower future inflation raises the real valueof the repayments the agent has to make, which lowers his continuation value.
Let us consider a numerical example. We set the width of the set of beliefs about inflationsuch that π = 0.01. Table (3) reports the zero-risk steady state of this model, in which the
only difference from the baseline version is the addition of the ambiguous inflation. Inflation
uncertainty lowers significantly the gains from trade, since both agents now perceive lower
real returns to their trading strategies. The uncertainty premium lowers the price of nominal
bonds and it leads to less debt. The price of the tree also decreases due to the lower valueof collateral.
Table 3: Endogenous worst-case beliefs about inflation
Steady state
c̄A
c̄B
q̄
p̄
q̄ b̄
`¯ = q̄ b̄/p̄Baseline
0.47 0.517 .987 3.33 1.59
.48
Ambiguous inflation
0.46 0.53 .981 2.74 0.77
.28
The key property of the ambiguity model is that agents act under the worst-case belief
that supports the equilibrium allocation. Agents may ex-ante have the same degree of confidence about the probability distributions of shocks. However, because they endogenously
take trading positions of different signs in equilibrium, the ex-ante homogeneity manifests
as ex-post heterogeneity of beliefs. In our case, the lender acts as if future inflation is highwhile the borrower as if future inflation is low.
When the ex-post disagreement in beliefs is sustained by the conjectured equilibrium
choices, we have been able to use our proposed strategy of studying heterogeneity. Whileat the equilibrium allocations, these beliefs will look to an outside observer as dogmaticdifferences, the model is one in which policy interventions that affect equilibrium choices will
also alter this disagreement.

27