Adverse Selection, Risk Sharing and Business Cycles V. V. Chari - - PowerPoint PPT Presentation

Discussion of Veracierto’s

Adverse Selection, Risk Sharing and Business Cycles

• V. V. Chari & Keyvan Eslami

University of Minnesota & Federal Reserve Bank of Minneapolis

August 2016

Point of Paper

◮ Theoretical exercise ◮ Qestion: Are fluctuations of aggregates in private information

economies different from those in same economy with public information?

◮ Answer: No ◮ Deeper Qestion: Should macroeconomists worry about abstracting

from private information frictions?

◮ Answer: No, as long as we focus on efficient outcomes.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Our Discussion

◮ Explain part of logic ◮ Propose different way of writing paper ◮ Briefly discuss computational method ◮ Discuss decentralization ◮ Taking this kind of model to data

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Static Model with Private Information

◮ Two types of agents of equal measure ◮ Utility function:

U(c, h, s) = log c + s log(1 − h)

◮ s = sL for low types ◮ s = sH for high types ◮ Technology:

cL + cH = z(hL + hH)

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Planner’s Problem

◮ Planner solves

max

• i

log ci + si log(1 − hi) s.t. cL + cH = z(hL + hH) log ci + si log(1 − hi) ≥ log cj + si log(1 − hj), ∀i, j

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Solution to Planner’s Problem

◮ Proposition: For all z,

ci(z) = zˆ ci, and ˆ ci and hi are independent of z.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Proof

◮ Let ci(z) = zc∗ i (z). Then, planner’s problem becomes

max

• i

log ci + si log(1 − hi) s.t. z(c∗

L(z) + c∗ H(z)) = z(hL + hH)

log z + log c∗

i (z) + si log(1 − hi)

≥ log z + log c∗

j (z) + si log(1 − hj),

∀i, j

◮ Note that z disappears. ◮ So, c∗ i (z) is independent of z.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Embedding Static Framework in Growth Model

◮ Suppose technology is now

Y = K α(zH)(1−α).

◮ Consider steady-states with different values of z. ◮ Key property of steady-state: K/zH is independent of z. ◮ Across steady-states, technology looks like

Y = zH

◮ Suggests static intuition applies to steady-states of growth model. ◮ Result is too strong, because disutility of H is irrelevant.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Veracierto’s Model

◮ Each period of growth model is same as static model. ◮ Disutility shocks are iid over time. ◮ Aggregate technology shocks are AR1. ◮ Exponential death with replacement by young agents.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Veracierto’s Model

◮ uis: consumption utility for i ∈ {y, o} and s ∈ {L, H} ◮ nis: leisure utility for i ∈ {y, o} and s ∈ {L, H} ◮ wis(z′): continuation utility for i ∈ {y, o} and s ∈ {L, H} ◮ v: promised utility of the old ◮ resource constraints:

Es,µ

• euos(v)

+ (1 − σ)Es (euys) + I ≤ ezK γH1−γ, H ≤ (1 − σ)Es (1 − enys) + Es,µ

• 1 − enos(v)

, and K ′ = (1 − δ)K + I.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Recursive Planner’s Problem

◮ Planner solves

V (z, K, µ) = max Es,z′

• uys + snys + βσwys(z′) +

θ 1 − σ V(z′, K ′, µ′)

• ,

subject to the incentive constraints, uys + snys + βσE′

z

• wys(z′)
• ≥ uyˆ

s + snyˆ s + βσE′ z

• wyˆ

s(z′)

• uos(v) + snos(v) + βσE′

z (wos(v, z′)) ≥ uoˆ s(v) + snoˆ s(v) + βσE′ z (woˆ s(v, z′)) ,

promise-keeping constraint, v = Es,z′ (uos(v) + snos(v) + βσwos(v, z′)) , resource constraints, and the law of motion of µ.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Defining States and Stationary Equilibrium

◮ Aggregate state of economy: X = [K, z, µ] ◮ X is the support of X. ◮ Individual state: x = [s, v, X] ◮ Planner’s problem yields transition function Q : Xt → Xt+1. ◮ For every state today, Q gives probability distribution over states

tomorrow.

◮ Stationary equilibrium has distribution G over X such that

∀A ⊂ X , G(A) =

• X

Q(y, A)G(dy)

◮ Let

¯ X be the ergodic set: smallest set with the property that if we start from a point in it, we never leave the set.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Main Theorem

◮ In a stationary equilibrium, outcomes have the following properties;

uo(s, v, X) = u1(s) + bv + uF

1 (X),

uy(s, X) = u2(s) + uF

2 (X),

wo(s, v, X, z′) = w1(s) + v + wF

1 (X, z′),

wy(s, X, z′) = w2(s) + wF

2 (X, z′),

and similarly so for the utilities of leisure, with same value for b.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

General Idea of Proof

◮ Guess and verify ◮ First order conditions are necessary and sufficient. ◮ So, guess and verify works. ◮ Current proof leaves unclear where stationary equilibrium is used.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

How We Think Proof Works

◮ Let λF t be multiplier on resource constraint in an equivalent full

information economy.

◮ Let qF t be the wage rate in the full information economy. ◮ Let

uF

2,t ∝ − log

• λF

t

• ,

and nF

2,t ∝ − log

• λF

t

• − log
• qF

t

• .

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

How We Think Proof Works

◮ Let uF 1,t and wF 1,t be defined by system of difference equations, given

exogenously specified λF

t and qF t ,

log

• λF

t

• + uF

1,t = log

• λF

t+1

• + uF

1,t+1 + bwF 1,t+1,

uF

1,t + ¯

snF

1,t + βσEt

• wF

2,t+1

• = 0,

nF

1,t = uF 1,t − log

• qF

t

• .

◮ Key variable is Vt, defined as

Vt =

• ebvdµt

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

How We Think Proof Works

◮ Veracierto claims that, if µt is in ergodic set,

log

• V F

t

• ∝ − log
• λF

t

• − uF

1,t,

where uF

1,t obtained by solving difference equations system, so that

V F

t ∝ e−uF

1,t

λF

t

.

◮ Given claim, possible to show that aggregates do not depend on

distribution of promised utilities, µt.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Example: Verifying Irrelevance for Aggregate Consumption

◮ Aggregate consumption, C, is given by

C = Es,µ

• euos(v)

+ (1 − σ)Es (euys) .

◮ Using guess, get

C = Es

• eu1(s)

euF

1,t

• ebvdµt + (1 − σ)Es
• eu2(s)

euF

2,t.

= Es

• eu1(s)

euF

1,tVt + (1 − σ)Es

• eu2(s)

euF

2,t. Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Example: Verifying Irrelevance for Aggregate Consumption

◮ Aggregate consumption,

C = Es

• eu1(s)

euF

1,tVt + (1 − σ)Es

• eu2(s)

euF

2,t.

◮ Using Veracierto’s claim on V F t and guess on uF 2,t,

C = AeuF

1,t e−uF 1,t

λF

t

+ (1 − σ) B λF

t

, where A + (1 − σ)B = 1, because of steady-state considerations.

◮ With log-utility λF t = 1/CF t , so aggregate consumption is the same in

both economies.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Computational Procedure in Steady-State

◮ Start with guess on λ ◮ Use FOC’s to computes decision rules ◮ Simulate long stream to get distribution on µ ◮ Check resource constraints ◮ Update λ

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Computational Procedure in Stochastic Economy

◮ Replace µ by history of past decision rules on promised utility ◮ Now states are z, K, s, and coefficients of the decision rules ◮ Linearize FOC’s ◮ Simulate long stream ◮ Use implied µ to check resource constraint ◮ Update coefficients on decision rules

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Need Decentralization

◮ Introduction suggests Marcelo will develop theory with endogenous

borrowing constraints.

◮ Must be possible to do. ◮ Decentralization in public finance replaces promised utility with limits on

borrowing and saving.

◮ Make sure inverse Euler equation satisfied, with appropriate interest rate

• n borrowing/saving.

◮ Would be helpful to do this exercise.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Taking Theory to Data

◮ Key issue: What is income? ◮ Tough to know, because much risk-sharing is done within the firm. ◮ Key implication of theory: consumption inequality pro-cyclical, hours

inequality counter-cyclical

◮ Some evidence would be nice here.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles

Conclusion

◮ We really like the paper! ◮ Cool result! ◮ Beter explanation of proof is desirable. ◮ Small data motivation would help.

Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles