Discussion of Veracierto’s Adverse Selection, Risk Sharing and Business Cycles V. V. Chari & Keyvan Eslami U niversity 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 = s L for low types ◮ s = s H for high types ◮ Technology: c L + c H = z ( h L + h H ) Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
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 Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
Solution to Planner’s Problem ◮ Proposition: For all z , c i ( z ) = z ˆ c i , and ˆ c i and h i are independent of z . Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
Proof ◮ Let c i ( z ) = zc ∗ i ( z ) . Then, planner’s problem becomes � max log c i + s i log ( 1 − h i ) i z ( c ∗ L ( z ) + c ∗ s . t . 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, 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 ◮ 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 u os ( v ) � + ( 1 − σ ) E s ( e u ys ) + I ≤ e z K γ H 1 − γ , E s ,µ � 1 − e n os ( v ) � H ≤ ( 1 − σ ) E s ( 1 − e n ys ) + E s ,µ , K ′ = ( 1 − δ ) K + I . and Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
Recursive Planner’s Problem ◮ Planner solves � � θ V ( z , K , µ ) = max E s , z ′ u ys + sn ys + βσ w ys ( z ′ ) + 1 − σ V ( z ′ , K ′ , µ ′ ) , subject to the incentive constraints, � � � � u ys + sn ys + βσ E ′ w ys ( z ′ ) ≥ u y ˆ s + sn y ˆ s + βσ E ′ s ( z ′ ) w y ˆ z z u os ( v ) + sn os ( v ) + βσ E ′ z ( w os ( v , z ′ )) ≥ u o ˆ s ( v ) + βσ E ′ s ( v , z ′ )) , s ( v ) + sn o ˆ z ( w o ˆ promise-keeping constraint, v = E s , z ′ ( u os ( v ) + sn os ( v ) + βσ w os ( 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 : 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 ) X ¯ ◮ 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; u o ( s , v , X ) = u 1 ( s ) + bv + u F 1 ( X ) , u y ( s , X ) = u 2 ( s ) + u F 2 ( X ) , w o ( s , v , X , z ′ ) = w 1 ( s ) + v + w F 1 ( X , z ′ ) , w y ( s , X , z ′ ) = w 2 ( s ) + w F 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 λ F t be multiplier on resource constraint in an equivalent full information economy. ◮ Let q F t be the wage rate in the full information economy. ◮ Let u F � λ F � 2 , t ∝ − log , t n F λ F q F � � � � and 2 , t ∝ − log − log . t t Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
How We Think Proof Works ◮ Let u F 1 , t and w F 1 , t be defined by system of difference equations, given exogenously specified λ F t and q F t , λ F + u F λ F + u F 1 , t + 1 + bw F � � � � log 1 , t = log 1 , t + 1 , t + 1 t u F sn F � w F � 1 , t + ¯ 1 , t + βσ E t = 0 , 2 , t + 1 n F 1 , t = u F q F � � 1 , t − log . t ◮ Key variable is V t , defined as � e bv d µ t V t = Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
How We Think Proof Works ◮ Veracierto claims that, if µ t is in ergodic set, � V F � � λ F � − u F ∝ − log 1 , t , log t t where u F 1 , t obtained by solving difference equations system, so that t ∝ e − u F 1 , t V F . λ F t ◮ Given claim, possible to show that aggregates do not depend on distribution of promised utilities, µ t . Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
Example: Verifying Irrelevance for Aggregate Consumption ◮ Aggregate consumption, C , is given by � e u os ( v ) � + ( 1 − σ ) E s ( e u ys ) . C = E s ,µ ◮ Using guess, get � � e u 1 ( s ) � � e u 2 ( s ) � e u F e u F e bv d µ t + ( 1 − σ ) E s C = E s 2 , t . 1 , t � e u 1 ( s ) � � e u 2 ( s ) � e u F e u F = E s 1 , t V t + ( 1 − σ ) E s 2 , t . Chari & Eslami Veracierto: Adverse Selection, Risk Sharing and Business Cycles
Example: Verifying Irrelevance for Aggregate Consumption ◮ Aggregate consumption, � e u 1 ( s ) � e u F � e u 2 ( s ) � e u F C = E s 1 , t V t + ( 1 − σ ) E s 2 , t . ◮ Using Veracierto’s claim on V F t and guess on u F 2 , t , 1 , t e − u F + ( 1 − σ ) B 1 , t C = Ae u F , λ F λ F t t where A + ( 1 − σ ) B = 1, because of steady-state considerations. ◮ With log-utility λ F t = 1 / C F 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 on 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
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