Uncertainty aversion & heterogeneous beliefs in linearmodels
Cosmin Ilut
Duke

Pavel Krivenko Martin Schneider
Stanford

Stanford

CIGS Tokyo, May 2016

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

1 / 22

Basic idea

Models with heterogeneous ambiguity averse agents
IIagents use di¤erent worst case beliefs to evaluate future plans
equilibria look like in models with disagreement

Multiple priors utility with ambiguity about meansII…rst order e¤ects of uncertainty
linear approximation works well) Use linear models to study e¤ects of uncertainty

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

2 / 22

Leading example
Recent history
Ilow real interest rate + stock market boom

Candidate story: higher uncertainty about middle class incomesI

type A agents perceive uncertain labor income= behave as if mean labor income will be low
I type B agents con…dent about labor income
I agents trade trees with safe payo¤, safe bonds collateralized by trees

EquilibriumI

type B agents rich: own trees & supply safe bonds to type A agents
type A agents poor: perceive tree price uncertainty, hold no trees
I asset pricing: low riskless rate & high price of trees
IE¤ects of policyIIgovernment debt can substitute for safe private debt
nominal credit: in‡ation uncertainty reduces gains from tradeIlut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

3 / 22Computation
System of stochastic di¤erence equations
II

as typically used to characterize equilibria in macro models
but: di¤erent expectation operators for each agent

Solution strategy
IIlinearize around steady state
jointly determine steady state & dynamics

Models with heterogeneous beliefs
Iagents may agree to disagree in the long run

Models with di¤erences in ambiguity aversion
Isimilar in many ways to models with di¤erences in risk aversion
average e¤ects and response to uncertainty shocks easy to characterize
I can be accurately solved by linearizationI

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

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Outline

Ambiguity aversion & precautionary savings
Two agent model with heterogenous beliefsII

computational approachnumerical example

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

5 / 22Aversion to uncertainty
Two dates
IIdate 2 states ω 2 Ωconsumption plans (c, c̃ (ω ))Preferences over consumption plansu (c ) + βu (CE (c̃ ))IIfor certain plans (c̃ (ω ) = c̄), certainty equivalent CE (c̃ ) = c̄u captures desire to smooth consumption, IES = cu 00 (c ) /u 0 (c )Certainty equivalent with risk aversionv (CE (c̃ )) = E Q [v (c̃ )]IIsubjective belief Qv captures risk aversion; CRRA =cv 00 (c ) /v 0 (c )Certainty equivalent with ambiguity aversion (multiple priors utility)v (CE (c̃ )) = min E P [v (c̃ )]P 2PI

set of subjective beliefs P captures ambiguity

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

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Precautionary savings
Choose optimal savings s s.t.
c +s = wc̃ = Rs + ỹRisky labor income ỹ = ȳ + σz̃, E Q [z̃ ] = 0, var QI(z̃ ) = 1

assume u = v & interior solution
u 0 (c ) = βRE Q u 0 (c̃ )Iresponse of savings to risk at σ = 0ds= ∆ 1 RE Q u 00 (c̃ ) z̃ = ∆ 1 Ru 000 (c̄ ) + ...
dσ
I ∆ > 0 from 2nd order condition; response positive if u 000 > 0Multiple priors with ambiguous labor income ỹ = ȳ + z̃, E z̃ 2 [ a, a]
I same FOC, but Q = worst case belief with E z̃ =a
I response of savings to ambiguity at a = 0ds= ∆ 1 RE Qu 00 (c̃ ) = ∆ 1 Ru 00 (c̄ ) + ...
daI response positive if u 00 < 0I precautionary savings will show up in linearized FOC!
Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

7 / 22

Recursive multiple priors utilityΩ = state spaceII

one element ω 2 Ω realized every period
histories ω t 2 Ωt

Preferences over consumption plans c = (ct (ω t ))Utility process solves
Ut c; ω t = u ct ω t+β

min

p 2Pt (ω t )E p Ut +1 c; ω t , ω t +1

PrimitivesIIfelicity u, discount factor β
one-step-ahead belief sets Pt (ω t ) ; may depend on history

Consider equilibrium with RMP agentsat optimal plan, minimizers p 2 Pt (ω t ) support choice) always obs equivalent EU model; possibly heterogeneous beliefs
I if worst case belief easy to …nd, just study het beliefs modelI

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

8 / 22

Outline

Ambiguity aversion & precautionary savings
Two agent model with heterogenous beliefs
II

computational approachnumerical example

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

9 / 22Model of ambiguity sharing
Two types of in…nitely lived agents; preferences
E0j∞t∑ β u ( ct )t =0Beliefs about income (endowment of goods)
type B agents always get ȳ B ; all agents know thisat date t, type A agents believe they get yt +1 = ȳ A exp ( at )AI in fact, type A agents get yt +1 = ȳ ; type B agents know thisI at = ambiguity perceived by type A agents (stochastic)
IIAssetsI

One period noncontingent debt, price qt
Trees: dividend d, price pt ; no short sales
I Leverage `t =qt bt /pt θ t ; cost to borrower k (`t ) qt btk (`) = 0 for ` 0; k (`) , k 0 (`) , k 00 (`) > 0 for ` > 0I

Date t budget constraint
ct + pt θ t + qt bt (1 + k (`t )) = yt + (pt + dt ) θ t 1 + bt 1 =: wt
Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201610 / 22

Recursive equilibrium
Market clearing
II

goods market: ctA + ctB = ȳ A + ȳ B
debt market: b A + b B = 0qt bt k (`t )State variables
type A income y A
I ambiguity aI distribution of asset holdings; here just type B debt b =I

bB = bA

Allocations & prices…nd c i (y A , a, b ), b 0 (y A , a, b ), q (y A , a, b ), p (y A , a, b )I agents disagree only about income, know equilibrium map) …nd functions from Euler equations + budget constraints
IIlut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201611 / 22

System of stochastic di¤erence equations
Equilibria s.t. only type B agents hold tree
II

type B agents: Euler equations for both assets
type A agents: Euler equation for bonds, tree too expensive

Bond pricing"
qt = βEtA
I

ctA+1ctA! γ#= βEtB"

ctB+1ctB! γ#+ qt k (`t ) + `t k 0 (`t )lender pessimism, borrower leverage cost ) lower bond price

Tree pricing

pt = βEtBI"

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

collateral bene…t increases tree priceWith budget constrs and ` = qb/p: 6 equations in c A , c B , q, p, b, `Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201612 / 22Computational challenge
Steady state = solution to system when vol of shocks is zeroIusually just remove Et and solve for deterministic steady state

Expectations in Euler equations without shocks
II

Type B expects all state variables to remain constantType A expects next period income ȳ A exp ( ā) and consumptionc A ȳ A exp ( ā) , ā, b̄II

behaves as if perpetually surprised by high income
problem: policy function c A matters for steady stateWith di¤erent expectation operators for each agent
IIagents expect di¤erent dynamics of endogenous state variables
cannot solve for steady state independently of dynamics

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201613 / 22

Solution strategy
Guess coe¢ cients of loglinear approximation
Ie.g.type A consumptionĉA = log c AcAAcAlog c̄ A = εcAyA ŷ + εa â + εb b̂

Compute steady stateIuse guessed coe¢ cients to model expectations, e.g.
c A ȳ A exp ( ā) , ā, b̄ t c̄ A exp

εcAyA āLinearize around candidate steady state
Iterate until …xed point in coe¢ cients!
RemarksII

steady state typically 6= any agent’s long run expectation
can check accuracy via Euler equation errors

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201614 / 22

Gains from trade: sharing ambiguity vs risk

Comparison model with risk
IItype B agents as before
at date t, type A agents get yt = ȳ A (1 + z̃t ), z̃ iid with mean zero

Risk sharing in equilibriumI

type B sells safe claim to type Atype A gives up goods today since CE of future income low
I A’s precautionary savings lowers interest rate
I

Ambiguity sharing in equilibriumI

type A gives up goods today since he acts as if future income low

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201615 / 22

Outline

Ambiguity aversion & precautionary savings
Two agent model with heterogenous beliefs
II

computational approachnumerical example

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201616 / 22

Numerical example
Baseline parameters
I output = 1, income & dividends: ȳ A = ȳ B = .45, d = .1
I preferences: u (c ) = log c, β = .96, ambiguity ā = 10%
I k (`) = (.075) `2 ! steady state leverage cost k (`) qb = .016
Steady statec̄ Ac̄ Bq̄

p̄

debt q̄ b̄`¯ = q̄ b̄/p̄

baseline.454.530.997

3.2

1.3.40ā = 0γ=2.428.539.960
1.012

2.44.00
1.9

0.48

Ambiguity generates gain from trade, type A "late consumer"
Ilower interest rate, higher tree price

Lower IES ! ambiguity matters more
I negative interest rate, c̄ A < ȳ AIlut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201617 / 22

Impulse response to increase in ambiguity
ambiguity

1.5

type 1 consumption

type 2 consumption

00.3

1-0.1

0.2

0.5

-0.2

0.1

0-0.3
01020

bond price

0.4001020tree price

0.4

1020

debt

0.2

0.3

0.2000.2
0-0.2

0.1

-0.2

0010

Ilut, Krivenko, Schneider ()20-0.4010

Beliefs in linear models

2001020

CIGS Tokyo, May 201618 / 22

Impulse response to ambiguity (lo discount factor)
ambiguity

1.5

type 1 consumption

1

0.5

0

0.3

00.2

-0.1

0.1

-0.2

0-0.3
01020

bond price

0.3

-0.1
010

20

tree price

0.3

010

20

debt

0.3

0.2

0.2

type 2 consumption

0.1

0.2

0.1
0.1

0.1

00-0.1010

Ilut, Krivenko, Schneider ()200010Beliefs in linear models

200

1020

CIGS Tokyo, May 201619 / 22Government debt
GovernmentIissues safe debt b g & levies lump sum taxes; budget constraintBbtg 1 = qt btg + τ At + τtIIbond market clearing btA + btB = btg
parameters: b̄ g = .6, equal taxes for both types

Steady state comparisonc̄ Ac̄ Bq̄

p̄

baseline

.454

.530

.997

government.463

.527

.994

`¯

3.2

debt q̄ b̄(private)
1.3 (1.3)

.40

3.1

1.8 (1.2).32

Fiscal policy here alternative safe asset schemeIless private debt, higher interest rate, lower collateral values

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

20 / 22

Nominal credit & uncertain in‡ation
Nominal credit
payo¤ of bond = (1 + π ) 1 ; q now nominal bond priceactual in‡ation zero
I endogenous worst cases: borrower believes π =.01, lender believes
π = .01(worst case beliefs with ambiguous in‡ation)II

Steady state comparisonc̄ Ac̄ Bq̄

p̄

debt q̄ b̄`¯ = q̄ b̄/p̄

baseline.454.530.997

3.2

1.3.40amb. in‡ation

.456

.537

.993

2.7

0.9.32In‡ation uncertainty lowers gains from tradeII

uncertainty premium lowers price of nominal bonds
less debt, lower value of collateral

Ilut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 2016

21 / 22

Conclusion
Stochastic di¤erence equations with disagreementItransition dynamics important for behavior in steady statesolve jointly for steady state & coe¢ cients of loglinear approximation
I paper provides general formulation
IMultiple priors utility with ambiguity about meansII…nd worst case beliefs for all agentssolve implied model with disagreement

Example: di¤erences in ambiguity about incomeI

precautionary saving by scared poor type A agentstrees valuable as collateral for con…dent rich type B agents
I low interest rate and stock market boom
I ambiguity shocks may persistently lower type A consumption
I government debt may substitute for safe private debt
I in‡ation uncertainty lowers gains from trade
IIlut, Krivenko, Schneider ()Beliefs in linear models

CIGS Tokyo, May 201622 / 22