Dynamic Contracting in Macroeconomics Lecture 4 Winter School 2019 Delhi School of Economics

Dynamic Contracting in Macroeconomics Lecture 4

Winter School 2019 Delhi School of Economics

V.V. Chari

University of Minnesota Federal Reserve Bank of Minneapolis

December 2019

Illustrative Paper

I Will use Dovis (2017) “Efficient Sovereign Default” as an example.

I Efficient allocation with I lack of commitment I private info

I How to implement efficient allocation with non-contingent debt + default on path

I Implications for interest rates

I Policy lessons differ from quantitative incomplete markets models

I maturity management

Sovereign Defaults in the Data

I Sovereign defaults (suspension of payments) are recurrent but infrequent events

I Associated with:

I Severe output and consumption losses I Large fall in imports of intermediate goods

I Maturity of debt shortens as default is more likely

Dynamics Around Default Episodes

-3 -2 -1 0 1 2 3

-15 -10 -5 0 5 10

15 GDP

% deviation from trend

Year after Default

-3 -2 -1 0 1 2 3

-10 -5 0 5

10 Consumption

% deviation from trend

Year after Default

-3 -2 -1 0 1 2 3

-40 -30 -20 -10 0 10 20 30

40 Intermediate Imports

Year after Default

% deviation from trend Mean

Mean +/- std

Sample of Defaults Episodes from Mendoza and Yue (2012) Source: WDI, UN Comtrade and Feenstra

Conventional Approach and Views

Incomplete market approach to sovereign debt:

I Sovereign borrower can issue only non-contingent debt I Sovereign borrower cannot commit to fully repay its debt

Typically:

I Exogenous maturity composition of debt I Exogenous cost of default

I Markov equilibrium

I Pervasive inefficiencies

I Defaults due to incomplete contracts

I Excessive reliance on short-term debt causes crises

I Roll-over risk: Cole and Kehoe (2000), Rodrik and Velasco (1999)

Dovis’ Approach

As in conventional approach:

I Sovereign borrower can issue only non-contingent debt I Sovereign borrower cannot commit to fully repay its debt Extend by allowing for:

I Endogenous maturity composition of debt

I Endogenous cost of default (Production economy) I Best equilibrium

Dovis’ View of Debt Crises

Best equilibrium outcome is constrained efficient I Solution to an optimal contracting problem with:

I Lack of commitment by sovereign borrower I Private information

I Non-contingent defaultable debt of multiple maturities sufficient to implement efficient outcome

I High reliance on short-term when default is likely part of the efficient arrangement

I Symptom,notcause

Features of the Best Outcome

Recurrent but infrequent defaults associated with:

I Output and consumption losses

I Fall in imports of intermediate goods

I Maturity of debt shortens as indebtedness increases before default

Contribution to the Literature

Incomplete market literature on sovereign default:

I Eaton and Gersovitz (1981), Aguiar and Gopinath (2006), Arellano (2008), Benjamin and Wright (2009), Chatterjee and Eyigungor (2012), Mendoza and Yue (2012), Arellano and Ramanarayanan (2012)

I Cole and Kehoe (2000), Conesa and Kehoe (2012) Extend by allowing for:

I Endogenous maturity composition of debt

I Endogenous cost of default (Production economy) I Best equilibrium

Develop efficiency benchmark useful for policy analysis

Contribution to the Literature, cont.

Optimal dynamic contracting literature:

I Private Information: Green (1987), Thomas and Worrall (1990), Atkeson and Lucas (1992)

I Lack of Commitment: Thomas and Worrall (1994),

Kocherlakota (1996), Kehoe and Perri (2002), Alburquerque and Hopenhayn (2004), Aguiar, Amador and Gopinath (2009) I Atkeson (1991) and Ales, Maziero, and Yared (2012)

I Quadrini (2004), Clementi and Hopenhayn (2006), DeMarzo and Fishman (2007), DeMarzo and Sannikov (2006), Hopenhayn and Werning (2008)

Implementation: Relate efficient outcome to data on default, bond prices, maturity composition of debt

Outline

I Environment

I Characterization of Efficient Allocation I Optimality of ex-post inefficiencies

I Implementation with Non-Contingent Defaultable Bonds (See Dovis’ paper)

I Relate to the evidence (See Dovis’ paper)

Baseline environment

I t =0,1, ...,∞ I 2 types of agents:

I Foreign lenders

I Domestic agents (government) I 3 types of good:

I Intermediate good,m

I Domestic consumption good,y (Non-Tradable) I Tradable domestic good

I Export denoted byy^∗

Foreign Lenders

I Risk neutral, discount factorq ∈ (0,1) I Value consumption of the export good I Large endowment of the intermediate good

I Technology of the foreign lenders is such that relative price between intermediate and domestic good is one

Preferences and Technology

I Domestic agent preferences

∑

∞ t=0

∑

θ^t

β^tPr(θ^t)θtU(c(θ^t))

I θt i.i.d. over time

I Domestic resource constraint

c+y^∗ ≤f(m)

Timing Within the Period

I Foreign lenders lend mt

I θt is realized

I Real activity occurs:

I Production I Consumption

I Domestic government chooses how muchy^∗ to export ct+y_t^∗ ≤f(mt)

I Use revelation principle to let allocation be x≡ {m(θ^t−1),c(θ^t),y^∗(θ^t)}^∞_t=0

Efficient Allocation

Two Contracting Frictions:

I Private Information: Incentive Compatibility Constraint I θis privately observed by the government

I Lack of Commitment: Sustainability Constraint I Government cannot commit to level of exports

Incentive Compatibility Constraint

∑

∞ t=0

∑

θ^t

β^tPr(θ^t)θtU ct(θ^t) ≥

∑

^∞

t=0

∑

θ^t

β^tPr(θ^t)θtU ct(σ(θ^t))

where σ(θ^t)is some reporting strategy.

Sustainability Constraint

I Let v_a =utility whenm_t =0, y_t^∗=0 ∀t I Sustainability constraint is

θsU(c(θ^s)) +

∑

∞ t=s+1

∑

θ^t

β^t−s−1Pr(θ^t)θtU ct(θ^t)

≥θsU(cˆ) +βv_a ∀s,θ^s,cˆ (SC) where cˆ satisfies

ˆ

c+yˆ ≤f(m(θ^s))

Full Information Benchmark

I ^Suppose

θHu⁰(c_H) =θLu⁰(c_L) I Production efficiencyf⁰(m) =1

I Use time path of y to provide needed payments to outside lenders

I Next, efficiency with private information and sustainability

Pareto Frontier

I Pareto frontier given by solution to maxE

∑

^∞

t=0

q^t

y(θ^t)−m(θ^t)

s.t. IC, SC and

∑

∞ t=0

∑

θ^t

β^tPr(θ^t)θtU c_t(θ^t)≥v₀

as v0 varies.

I Next we will replace IC and SC by temporary IC and temporary SC and solve the problem recursively.

Developing Temporary IC

I Green, Abreu, Whittle temporary IC result

I Any allocation c(θ^t)induces continuation expected utility v(θ^t). Suppose β^tv(θ^t)→0 a.s. then

I IC is equivalent to T.I.C. given by u c(θ^t)+βv(θ^t)≥u c(θ^t−1),θˆ

+βv(θ^t−1,θˆ) ∀θ^t−1,θt,θˆ

I Checking one-shot deviation is enough

One-shot Deviation Result

I Proof Sketch

I Easy to prove that checking one-shot deviations equivalent to checking finite deviations [exercise]

I Ifβ^tv_t →0, then any infinite deviations that yield strictly higher payoff can be approximated by a finite deviation that yields strictly higher payoffs. Result then follows from step one.

Temporary Incentive Compatibility Constraint

No gains from reportingθ⁰ 6= θt:

θtU(c(θ^t)) +βv(θ^t)≥θtU(c(θ^t−1,θ⁰)) +βv(θ^t−1,θ⁰) (IC) wherev(θ^t)is the continuation utility:

v(θ^t)≡

∑

^∞

s=t+1

∑

θ^s

β^s−t−1Pr(θ^s|θ^t)θsU(c(θ^s))

Relevant deviation: θL type reportingθH

Sustainability Constraint

No gains from increasing current consumption by not exporting and then living in autarky

θtU(c(θ^t)) +βv(θ^t)≥ θtU(f(m(θ^t−1)) +βv_a (SUST) wherev_a = value of autarky

v_a ≡ ^πLθLU(f(0)) +πHθHU(f(0)) 1−β

Relevant deviation forθ_H: Binding pattern reversed relative to model with only lack of commitment

Efficient Allocation

An allocation isefficient if it maximizes lenders’ value subject to (IC), (SUST) and participation constraint for the borrower.

An efficient allocation solves J(v₀) =max

x

∑

∞ t=0

∑

θ^t

q^tPr(θ^t)−m(θ^t−1) +f m(θ^t−1)−c(θ^t)

subject to (IC), (SUST) and

∑

∞ t=0

∑

θ^t

β^tPr(θ^t)θtU(c(θ^t))≥v0 (PC) Assumption s.t. constraint set is convex (⇒ no randomization)

Efficient Allocation Solves Nearly Recursive Problem

I Problem at t =0

I Recursive problem for t ≥1

I Key idea is to add promised continuation utility v as state variable

I Can then show that efficient allocation solves nearly recursive problem using dynamic programming technique from Stokey, Lucas with Prescott

Recursive Problem for t ≥ 1

Letv = borrower’s value and B =lenders’ value B(v) = max

m,{cs,v_s⁰}s=H,L

−m+

∑

s∈{L,H}

πs

f(m)−c_s+qB(v_s⁰)

subject to

θHU(c_H) +βv_H⁰ ≥θHU(c_L) +βv_L⁰ (IC) θLU(c_L) +βv_L⁰ ≥ θLU(f(m)) +βv_a (SUST) v_H⁰,v_L⁰ ≥v_a (SUST’) πH

θHU(c_H) +βv_H⁰ +πL

θLU(c_L) +βv_L⁰

=v (PKC)

Problem at t = 0

The efficient allocation solves J(v) = max

m,{cs,v_s⁰}s=H,L

−m+

∑

s∈{L,H}

πs

f(m)−cs+qB(v_s⁰)

subject to (IC), (SUST), (SUST’) and πH

θHU(cH) +βv_H⁰ +πL

θLU(cL) +βv_L⁰

≥v (PC) whereJ is the Pareto Frontier

Asymmetry betweent=0 and t ≥1 because:

I (PKC) is an equality constraint I (PC) is an inequality constraint

I IfB(v)is increasing then (PC) is slack andJ(v)>B(v)

Results

I Region with ex-post inefficiencies I Lack of commitment has critical role

I Transit to the region with ex-post inefficiencies I Private information has critical role

Region of Ex-Post Inefficiencies

0

Lenders’Value

J

Efficient Region

Borrower’s Value Region with

Ex-Post Inefficiencies

va

B

When borrower’s value is low, can make both borrower and lenders better off ex-post

B ( v ) has the Depicted Shape

∃ v˜ >v_a such that there exist

I Region with ex-post inefficiencies:

B is increasing forv ∈[va,v˜) I Efficient region

B is decreasing for v∈[v,_˜ v¯)

To showB is increasing forv ∈ [va,v˜): Step 1: Imports are zero at autarky

Step 2: From autarky, both agents can be made better off

B ( v ) has the Depicted Shape

∃ v˜ >v_a such that there exist

I Region with ex-post inefficiencies:

B is increasing forv ∈[va,v˜) I Efficient region

B is decreasing for v∈[v,_˜ v¯) To showB is increasing forv ∈ [va,v˜): Step 1: Imports are zero at autarky

Step 2: From autarky, both agents can be made better off

B ( v ) has an Upward Sloping Portion

0

Lenders’Value

J

Borrower’s Value va

B

Lenders’ value is increasing in the borrower’s value when this is close to the value of autarky

B ( v ) is Eventually Downward Sloping

0

Lenders’Value

J

Borrower’s Value

va v^∗

m^∗: f(m^∗) = 1 can be supported

B

If the borrower’s value is high enough, the statically efficient level of intermediates can be supported⇒ B is decreasing

B ( v ) has a Peak at v ˜

0

Lenders’Value

J

Efficient Region

Borrower’s Value Region with

Ex-Post Inefficiencies

v_a

˜

v v^∗

B

Lenders’ value is maximized atv˜ ∈(va,v^∗)

Economics Behind Ex post Inefficiencies

I Desirable to provide insurance, high transfers if θ =θH and low transfers if θ =θL.

I Provide incentives (guard against misreporting) by reducing future utility ifθ =θH and raising ifθ =θL.

I Reducing future utility while steel keeping m atf⁰(m) =1 gives incentive to default.

I Optimal contract reducesm below statically efficient level to provide insurance while maintaining incentives.

SHOW THAT CONTINUATION VALUE TRANSITS TO REGION WITH EX-POST INEFFICIENCIES

Any Efficient Allocation Starts on the Efficient Region

0

Lenders’Value

J

Efficient Region

Borrower’s Value Region with

Ex-Post Inefficiencies

v_a

˜ v

B

Transits to the Region with Ex-Post Inefficiencies

0

Lenders’Value

J

Efficient Region

Borrower’s Value Region with

Ex-Post Inefficiencies

va ˜v

B

Starting fromv₀, a sequence of high taste shocks pushes the economy to the region with ex-post inefficiencies

Borrower’s Value Decreases After High Taste Shock

Borrower’sValue NextPeriod v

L(v)

v_a v_a

˜ v 45^o

v H(v)

Borrower’s Value Today

After the realization of a high taste shock, the continuation value is lower than the current one: v_H⁰ (v)<v

Two Countervailing Forces

I Incentive force: Want to spread continuation values I Cheapest way to provide utility

I Makec_L large andc_H small I Spread out continuation values I Desirable to makev_L⁰ low

I Commitment force: Want to back-load borrower payoff I By back-loading, relax future sustainability constraint I Low production distortions in the future

I Want highv_L⁰

Optimality of Ex-Post Inefficiencies

If either(i) θL−θH sufficiently highor (ii) µL sufficiently low, then any efficient allocation transits to the region with ex-post

inefficiencies with strictly positive probability.

Lemma

If either (i) or (ii) then ∀ v ∈[v,˜ v¯] v_L⁰(v)<v Formally, (i) and (ii) are:

Assumption. The parametersz_L,z_H,µ_H,andγsatisfy the following conditions:

µ_Hz_L^1−γ ≥_E z^1−γ

(1) µL

z_L^1−γ E(z^1−γ)−1

! +µH

zH

z_L 1−γ

≥1 (2)

Proof.

The necessary and sufficient conditions for an interior solutions are:

uH : −λpkc =−^C

0(u_H) θ_H + ^λic

µ_H (3)

u_L : −λpkc =−^C

0(u_L) θL

− ^λic µL

θH

θL

+ ^λsust µL

(4) v_H⁰ : −λpkc = ^q

βB⁰(v_H⁰ ) + ^λic

µ_H (5)

v_L⁰ : −λpkc = ^q

βB⁰(v_L⁰)− ^λic µL

+^λsust µL

(6) m : f⁰(m)−1= λsustθLU⁰(f(m))f⁰(m) (7) and the envelope condition is

B⁰(v) =−λpkc (8)

Proof, cont.

Consider anyv in the efficient region. Combining the envelope condition (8) with (6), I can write:

B⁰(v)≤ ^β

qB⁰(v) =B⁰(v_L⁰)− ^β q

λic−λsust

µ_L

where I used the fact thatB⁰(v)≤0 for allv ∈ [v˜,v¯]and that, by Assumption 1, β/q <1. If λic >λsust, I can rewrite the above inequality as

B⁰(v)≤ ^β

qB⁰(v) =B⁰(v_L⁰)− ^β q

λic−λsust

µ_L

<B⁰(v_L⁰) By concavity ofB, it follows that v_L⁰(v)<v ∀v ∈[v,˜ v¯]. Hence, it is sufficient to show thatλic >λsust.

Proof, cont.

Consider first the case in which there is partial insurance, i.e.

C⁰(uL)/θL≤C⁰(uH)/θH. Combine (3) and (4) to get 0≥ ^C

0(uL) θ_L −^C

0(uH) θ_H = ¹

µ_L

λsust−λic

θH

θ_L

− ¹ µ(θ_L)^λic

Rearranging terms, I obtain λsust ≤λic

µL

µH

+ ^θH θL

=λic

E(θ) µHθL

≤λic

where in the last step I use the fact thatµHθL≥_E(θ), which is true under stated Assumption.

In Appendix of the paper consider case with “over-insurance”

C⁰(u_L)/θL>C⁰(u_H)/θH

I (I cannot rule it out, never shows up in simulations)

Intuition for the Sufficient Conditions in the Lemma

I θL−θH =z_L^1−γ−z_H^1−γ large: Incentive force large I Insurance motive is large

I Incentive compatibility makesv_H⁰ (v)lower thanv

I µ_L low: Small cost of not back-loading

I Low probability of reaching state in which future (SUST) is tight and production is highly distorted

Incentive force outweighs commitment force ⇒ v_L⁰(v)<v

Transits to the Region with Ex-Post Inefficiencies

Borrower’sValueNextPeriod

va

˜ va v

45⁰ v^′(v , zH)

v^′(v , zL)

Borrower’s Value Today

Starting fromv₀, a sequence of high taste shocks pushes the economy to the region with ex-post inefficiencies

Role of Frictions

I Lack of commitment

I Critical role in generating an upward sloping portion forB I Alone: Efficient region is an ergodic set

I Private information

I Critical role in generating downward drift in borrower’s value afterθL(z_L)

I Alone: B is downward sloping over its entire domain

I To get ex-post inefficiencies need both