Adverse Selection, Risk Sharing and Business Cycles

Transcription

1 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

2 Point of Paper Theoretical exercise Question: Are fluctuations of aggregates in private information economies different from those in same economy with public information? Answer: No Deeper Question: Should macroeconomists worry about abstracting from private information frictions? Answer: No, as long as we focus on efficient outcomes.

3 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

4 Static Model with Private Information Two types of agents of equal measure Utility function: s = s L for low types U(c, h, s) = log c + s log(1 h) s = s H for high types Technology: c L + c H = z(h L + h H )

5 Planner s Problem Planner solves max log c i + s i log(1 h i ) i s.t. c L + c H = z(h L + h H ) log c i + s i log(1 h i ) log c j + s i log(1 h j ), i, j

6 Solution to Planner s Problem Proposition: For all z, and ĉ i and h i are independent of z. c i (z) = zĉ i,

7 Proof Let c i (z) = zci (z). Then, planner s problem becomes max log c i + s i log(1 h i ) i s.t. z(c L (z) + c H(z)) = z(h L + h H ) log z + log c i (z) + s i log(1 h i ) log z + log c j (z) + s i log(1 h j ), i, j Note that z disappears. So, ci (z) is independent of z.

8 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.

9 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.

10 Veracierto s Model u is : consumption utility for i {y, o} and s {L, H} n is : leisure utility for i {y, o} and s {L, H} w is (z ): continuation utility for i {y, o} and s {L, H} v: promised utility of the old resource constraints: E s,µ (e uos(v)) + (1 σ)e s (e uys ) + I e z K γ H 1 γ, H (1 σ)e s (1 e nys ) + E s,µ (1 e nos(v)), and K = (1 δ)k + I.

11 Recursive Planner s Problem Planner solves V (z, K, µ) = max E s,z ( u ys + sn ys + βσw ys (z ) + θ ) 1 σ V(z, K, µ ), subject to the incentive constraints, u ys + sn ys + βσe z ( wys (z ) ) u yŝ + sn yŝ + βσe z ( wyŝ (z ) ) u os (v) + sn os (v) + βσe z (w os (v, z )) u oŝ (v) + sn oŝ (v) + βσe z (w oŝ (v, z )), promise-keeping constraint, v = E s,z (u os (v) + sn os (v) + βσw os (v, z )), resource constraints, and the law of motion of µ.

12 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 : X t X t+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) = 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. X

13 Main Theorem In a stationary equilibrium, outcomes have the following properties; u o (s, v, X) = u 1 (s) + bv + u1 F (X), u y (s, X) = u 2 (s) + u2 F (X), w o (s, v, X, z ) = w 1 (s) + v + w1 F (X, z ), w y (s, X, z ) = w 2 (s) + w2 F (X, z ), and similarly so for the utilities of leisure, with same value for b.

14 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.

15 How We Think Proof Works Let λ F t be multiplier on resource constraint in an equivalent full information economy. Let q F t Let be the wage rate in the full information economy. and u2,t F log ( λ F ) t, n F 2,t log ( λ F ) ( ) t log q F t.

16 How We Think Proof Works Let u1,t F and w1,t F be defined by system of difference equations, given exogenously specified λ F t and qt F, log ( λ F ) t + u F 1,t = log ( λ F ) t+1 + u F 1,t+1 + bw1,t+1, F u1,t F + sn F ( ) 1,t + βσe t w F 2,t+1 = 0, n F 1,t = u1,t F log ( qt F ). Key variable is V t, defined as V t = e bv dµ t

17 How We Think Proof Works Veracierto claims that, if µ t is in ergodic set, log ( Vt F ) ( ) log λ F t u F 1,t, where u F 1,t obtained by solving difference equations system, so that V F t F e u 1,t λ F. t Given claim, possible to show that aggregates do not depend on distribution of promised utilities, µ t.

18 Example: Verifying Irrelevance for Aggregate Consumption Aggregate consumption, C, is given by C = E s,µ (e uos(v)) + (1 σ)e s (e uys ). Using guess, get C = E s (e u1(s)) e uf 1,t e bv dµ t + (1 σ)e s (e u2(s)) e uf 2,t. ( = E s e u1(s)) e uf 1,t Vt + (1 σ)e s (e u2(s)) e uf 2,t.

19 Example: Verifying Irrelevance for Aggregate Consumption Aggregate consumption, C = E s ( e u1(s)) e uf 1,t Vt + (1 σ)e s (e u2(s)) e uf 2,t. Using Veracierto s claim on V F t and guess on u F 2,t, C = Ae uf 1,t e uf 1,t λ F t + (1 σ) B λ F, t where A + (1 σ)b = 1, because of steady-state considerations. With log-utility λ F t both economies. = 1/C F t, so aggregate consumption is the same in

20 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

21 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

22 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 on borrowing/saving. Would be helpful to do this exercise.

23 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.

24 Conclusion We really like the paper! Cool result! Better explanation of proof is desirable. Small data motivation would help.