Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Efficient Unemployment Insurance Daron Acemoglu and Robert Shimer

Presented by Olga Croitorov November 4th, 2010

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Outline Introduction Static Model Preferences and Technology Definition, Existence and Characterization of Equilibrium Comparative Statics Worker Heterogeneity Risk-Aversion, Unemployment Insurance and Output A Dynamic Model Without Wealth Effects Preferences and Technologies Optimal Consumption Decision and Value Functions Analysis Risk Aversion, Unemployment Insurance and Output

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Outline

Introduction Static Model Risk-Aversion, Unemployment Insurance and Output A Dynamic Model Without Wealth Effects

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Motivation

Conventional wisdom of risk-sharing/output tradeoff - more unemployment insurance raises the unemployment rate and therefore reduces output. Innovation - moderate amount of unemployment insurance may increase output. WHY? A risk-averse worker reduces her risk-aversion to unemployment by accepting wages, from a productive efficiency stand point, too low. A moderate amount of UI allows to raise the reservation wage.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Mechanism of the model

In absence of UI: 1. Risk-averse workers seek lower wage jobs with less unemployment. 2. In an economy with incomplete insurance firms cater to preferences and lower wages and high vacancy risk. 3. Firms reduce capital intensity, moving the economy away from ”productive efficiency”. By adding UI workers increase their reservation wage, firms respond by raising wages and capital-labor ratio ensuring productive efficiency.

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Literature Review

• General Equilibrium search literature (no risk aversion and capital

investment) - Diamond (1982), Pissarides (1990) • Wage posting models - Peters (1991), Montgomery (1991) • Models where Unemployment Benefits improves resource allocation

(risk-neutral agents) - Diamond (1981), Acemoglu (1997) • Models with optimal UI in the presence of asymmetric information

(wage and job offers exogenous)- Morensen (1977), Hansen and Imrohoroglu (1992), Atkeson and Lucas (1995)

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Outline Introduction Static Model Preferences and Technology Definition, Existence and Characterization of Equilibrium Comparative Statics Worker Heterogeneity Risk-Aversion, Unemployment Insurance and Output A Dynamic Model Without Wealth Effects

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Preferences and Technology Workers: • continuum 1 of identical • they receive income yi in form of wi or z • they are endowed with A0 and pay taxes τ , A ≡ A0 − τ • workers have a vN-M utility u(A + yi ), which is u 0 > 0, u 00 ≤ 0

Firms: • larger continuum of firms • production technology f : (0, ∞) → (0, ∞) requires one worker and

capital k > 0. Production function satisfies f 0 (k) > 0, f 00 (k) < 0, limk→0 f (k) = 0, limk→0 f 0 (k) > 1 and exist f 0 (k) ≡ 1

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Game Structure and Timing • each firm j decide whether to make an irreversible investment

kj > 0. If yes then it’s active and post wage ωj ∈ W where W = {ωj , for all j active} • worker observes all the wage offers and decides where to apply • there maybe more competition for some jobs than others. Expected

queue length qj is ratio of workers who apply for jobs at firms offering ωj to number of firms posting this wage • worker applying for wage ωj is hired with probability µ(qj ) which is

cont differentiable and decreasing • probability that firm j hires is η(qj ) which is increasing and cont

differentiable • CRS matching with search frictions

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Definition of Equilibrium An allocation is a tuple {K , W , Q, U} where K ⊂ R+ - set of capital investment, W : R+ ⇒ R+ - set of wages for given capital investment, Q : R+ → R+ ∪ ∞ - queue length associated with each wage and U ∈ R+ - workers’ utility. W ={ω|ω ∈ W (k)for some k ∈ K }

Definition An Equilibrium is an allocation {K ∗ , W ∗ , Q ∗ , U ∗ } such that: 1. [Profit Maximization] ∀ω, k, η(Q ∗ (ω))(f (k) − ω) − k ≤ 0 with equality if k ∈ K ∗ and ω ∈ W ∗ (k) 2.[Optimal Application] ∀ω, U ∗ ≥ supω0 ∈W ∗ µ(Q ∗ (ω))u(A + ω) + (1 − µ(Q ∗ (ω)))u(A + z) and Q ∗ (ω) ≥ 0, with complementary slackness, where ∗

U ∗ = u(A + z) if W is empty

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Existence and Characterization Proposition There always exists an equilibrium. If {K , W , Q, U} is an equilibrium, then any k ∗ ∈ K , ω ∗ ∈ W (k ∗ ), and q ∗ = Q(ω ∗ ) solve U = maxk,ω,q µ(q)u(A + ω) + (1 − µ(q))u(A + z) subject to η(q)(f (k) − ω) − k = 0

(ZPC)

and ω ≥ z. Conversely, if some {k ∗ , ω ∗ , q ∗ } solves the above program, then there exists an equilibrium {K , W , Q, U} such that k ∗ ∈ K , ω ∗ ∈ W (k ∗ ), and q ∗ = Q(ω ∗ ).

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Existence and Characterization Let define f 0 (k) ≡ 1 and z ≡ f (k) − k. Then, whenever z > z, the constraint set is empty, and there are no active firms in equilibrium. Contrary, if z < z, then K , W 6= Ø, and any ω ∗ ≥ z. Using Proposition , precisely optimal choice condition η(q ∗ )f 0 (k ∗ ) = 1 and zero profit condition, firm’s investment decision is given by ω ∗ = f (k ∗ ) − k ∗ f 0 (k ∗ ). Some useful results: • since f is concave

∂ω ∗ ∂k ∗

>0

• from optimal capital choice and equilibrium wage eq, firm’s ZPC is

an upward sloping curve in {q, ω} space • similarly, workers’ indifference curve is upward sloping in {q, ω} space

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Existence and Characterization

Figure: An equilibrium maximizes workers’ utility to firms earning zero profits. Higher wages and shorter queues raise utility and lower profits.

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Comparative Statics Proposition 1. Let {ki , ωi , qi } be an equilibrium when the utility function is ui . If u1 is strictly concave transformation of u2 , then k1 < k2 , ω1 < ω2 , and q1 < q2 . 2. Let {ki , ωi , qi } be an equilibrium when the initial asset level is Ai . If A1 < A2 and the utility function is DARA, then k1 < k2 , ω1 < ω2 , and q1 < q2 . With IARA, the inequalities are reversed. With CARA, asset levels do not affect the set of equilibria. 3. Let {ki , ωi , qi } be an equilibrium when the unemployment benefit is zi . If z1 < z2 then k1 < k2 , ω1 < ω2 , and q1 < q2 .

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Comparative Statics

Figure: The left panel shows that more risk-aversion makes indifference curves steeper, and lowers the equilibrium wage and queue length. The right panel shows that higher UI makes indifference curves flatter, having the opposite effect.

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Worker Heterogeneity Let’s assume that there are s = 1, 2, . . . , S types of workers, where type s has utility function us , after-tax asset level As , and unemployment benefit zs . Let U now be a vector in RS , and assume that zs < z for all s.

Proposition There always exists and equilibrium. If {K , W , Q, U} is an equilibrium, then any ks∗ ∈ K , ωs∗ ∈ W (ks∗ ) and qs∗ ∈ Q(ωs∗ ), solve Us = maxk,ω,q µ(q)us (As ω) + (1 − µ(q))u(As + zs ) subject to η(q)(f (k) − ω) − k = 0 for some s = 1, 2, . . . , S. If {ks∗ , ωs∗ , qs∗ } solves the above program for some s, then there exists an equilibrium {K , W , Q, U} such that ks∗ ∈ K , ωs∗ ∈ W (ks∗ ) and qs∗ ∈ Q(ωs∗ )

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Worker Heterogeneity - Comments

Equilibrium is attended for each type of the worker. That is markets endogenously segments into S different submarkets. This analysis predicts that a group of workers with higher UI will apply for higher wage jobs with longer queues and longer unemployment spells. Note: Previous result crucially depends on the assumption that worker observe all posted wages, hence can ignore the presence of other submarkets.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Outline

Introduction Static Model Risk-Aversion, Unemployment Insurance and Output A Dynamic Model Without Wealth Effects

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Risk-Aversion, Unemployment Insurance and Output Assume all workers are homogenous, and all firms invest k, and pay wage ω. Net output will be given by: Y (q, k) ≡ µ(q)f (k) − k/q

Definition {k e , ω e , q e } is output-maximazing if Y (k e , q e ) = maxk,q Y (k, q). UI z e and tax τ e are output-maximazing if any associated equilibrium {kz e ,τ e , ωz e ,τ e , qz e ,τ e } is output-maximazing and has a balanced government budget, τ e = (1 − µ(qz e ,τ e ))z e .

Proposition If agents are risk neutral, the unique output-maximazing level of UI is z e = τ e = 0.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Risk-Aversion, Unemployment Insurance and Output

Proposition If agents are risk-averse and z e = τ e = 0, output is below its maximum Changes in preferences do not affect the output-maximizing allocation. Since z e = τ e = 0 maximize with risk neutral agents it yields lower level of output when workers are risk-averse.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Optimal Unemployment Insurance There are two ways to solve this problem − via Social Planner and via Decentralized Economy. Social Planner would maximize total output and divide equally among all workers, providing full insurance. (Trivial result) Equilibrium in Decentralized Economy is given by following results.

Definition Let {k z,τ , ω z,τ , q z,τ } be an equilibrium with UI and taxes (z, τ ). Then, the policy (z 0 , τ 0 ) is optimal if it maximizes µ(qz,τ )u(A − τ + ωz,τ ) + (1 − µ(qz,τ ))u(A − τ + z) subject to τ = (1 − µ(qz,τ ))z

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Risk-Aversion, Unemployment Insurance and Output

Figure: Risk aversion makes indifference curves steeper, while UI makes them flatter. The curve for u concave and z = z e lies everywhere above the u linear curve

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Outline Introduction Static Model Risk-Aversion, Unemployment Insurance and Output A Dynamic Model Without Wealth Effects Preferences and Technologies Optimal Consumption Decision and Value Functions Analysis Risk Aversion, Unemployment Insurance and Output

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Motivation

Question: Do the previous findings change exploiting the fact that forward-looking agents may save in the absence of insurance market? Conclusion: Previous results generalize to this environment

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Preferences and Technologies Framework: • CARA utility function • Search and production technology are generalization of static model: 1. Firms • can be inactive, vacant or have a filled job R • inactive firms buy capital at unit cost R−1 • maximize expected value of profit

2. Workers • can be employed or unemployed, which can participate or not in the market economy • decide which wage, if any, to seek • worker i makes consumption and job search decision maximizing ∞ X t=0

βt

1 − e −θcit θ

• subject to dynamic budget constraint Ai,t+1 = R(yi,t + Ait − τt − cit ) • TVC and consumption smoothing over time holds (Rβ = 1 ) • nonparticipant workers’ home production xi . Distribution of x is the same within each generation

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Preferences and Technologies

Framework Continued: • Matching technology is the same as above • Unfilled jobs and unemployed workers search again next period • Productive relationship between worker and firm never ends • Maintenance of steady state population of unemployed workers by

assuming the labor force Lt grows at rate δ. New workers at the beginning of the period are unemployed with initial assets Ait = A0

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Value Functions Focus on steady state equilibria in which wages, capital stock, queue length and unemployment rate are constants. • An employed worker who starts a period with assets At , pays taxes

τ , and earns wages ω in every period, has optimal consumption cte = ω + (1 − β)At − τ , so At+1 = At . His lifetime utility is E (At , ω) =

1 − e −θ(ω+(1−β)At −τ ) θ(1 − β)

• A nonparticipating worker, who starts period t with assets At ,

pays taxes τ , earns income x, and receives unemployment benefit z in every period, has optimal consumption ctn = x + z + (1 − β)At − τ . His lifetime utility is N(At , xi + z) =

1 − e −θ(xi +(1−β)At +z−τ ) θ(1 − β)

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Value Functions • An unemployed individual who starts with assets At , pays taxes τ

in every period, applies for a job offering wage ω and queue length q and earns benefit z in every period that he fails to get a job, has optimal consumption ctu = βψ + (1 − β)At + z − τ , where ψ ∈ (0, ω − z) is implicitly defined by 0 = µ(q)

1 − e θ(1−β)ψ 1 − e −θ(ω−z−ψ) + (1 − µ(q)) , θ θ

so At+1 = At − ψ. His lifetime expected utility is J(At , ω, q) =

1 − e −θ(ψ+(1−β)At +z−τ ) θ(1 − β)

The latter is derived from: J(At , ω, q) = µ(q)E (At , ω)+ (1 − µ(q))( 1−e

−θ(βψ+(1−β)At +z−τ )

θ

+ βJ(At − ψ, ω, q)).

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Characterization of Value Functions Optimal consumption decision follows from consumption smoothing (Rβ = 1). Hence: • employed and nonparticipating workers consume their current net

income and interest rate from savings • unemployed workers consume current income and dissave a

constant amount ψ of assets each period, resulting in decreasing asset and consumption level while unemployed. Therefore each additional period of being unemployed is a bad shock • unemployed workers have the same preferences over wage-queue

combination, regardless of their asset level. Hence each time they apply for the same wage in every period.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Analysis Let P be the set of market participants and U(·) denote the Bellman value of an unemployed participant.

Definition A steady state equilibrium is an allocation {K , W , Q, U, E , J, N, P} s.t.: 1. [Profit Maximization] ∀ω, k η(Q(ω))(f (k) − ω) + (1 − η(Q(ω)))βk − k ≤ 0 with equality if k ∈ K and ω ∈ W (k). 2. [Optimal Application] ∀ω, U(·) ≥ J(·, ω, Q(ω)) and Q(ω) ≥ 0 with complementary slackness; U(·) = supω0 ∈W J(·, ω 0 , Q(ω 0 )) or U(·) = N(·, z) if W = Ø. 3. [Optimal Participation] i ∈ P if and only if N(·, xi + z) ≤ U(·).

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Equilibrium

{K , W , Q, U, E , J, N, P} is an equilibrium ⇐⇒ k ∗ ∈ K , ω ∗ ∈ W (k ∗ ), and q ∗ = Q(ω ∗ ) solves: 0 = h(ψ ∗ ) ≡ max µ(q) k,ω,q

1 − e −θ(ω−z−ψ θ



)

+ (1 − µ(q))

1 − e θ(1−β)ψ θ



(k)−ω) subject to η(q)(f 1−β(1−η(q)) = k E , J and N are defined as above; U(A) = J(A, ω ∗ , q ∗ ). i ∈ P if and only if xi ≤ ψ

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Intuition behind • similar problem to static model of choosing ω and q to maximize

J(A, ω, q), subject to ZPC • the previous problem is equivalent to maximizing dissavings while

unemployed, ψ, subject to DEE, given the wage and hiring probability • asset levels do not affect the comparison of N and U, hence there is

a cut-off level of outside production x such that all workers with higher outside productivity do not participate. Therefore x = ψ (from equalizing U(·) and N(·)) • (result) For given CARA utility functions with θ1 ≥ θ2 , and

respectively z1 ≤ z2 their corresponding equilibrium points are characterized as follow: k1 < k2 , ω1 < ω2 , q1 < q2 , and ψ1 < ψ2

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Variables for Calibration Problem Output is given by expected value of market output produced by a particular worker during lifetime, net of any investment cost on her behalf Y (q, k, p) ≡ p

µ(q)f (k) − βµ(q)k − (1 − β) kq (1 − β)(1 − β(1 − µ(q)))

Definition of unemployment and equilibrium wage are given by Proposition 9 (AS) Assumption for calibration problem: • β = 0.94 • δ = 0.01 • f (k) = 10k 0.5 • η(q) = 1 − e −0.15q • a period is given by a year

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Calibration Results Without Worker Nonparticipation

Figure: Calibration results for different coefficients of absolute risk aversion, without worker nonparticipation.

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Calibration Results With Worker Nonparticipation

Figure: Calibration results for different coefficients of absolute risk aversion, with worker nonparticipation.

Conclusion

Introduction

Static Model

Risk-Aversion, Unemployment Insurance and Output

A Dynamic Model Without Wealth Effects

Conclusion

Conclusion

• A general equilibrium model of search with risk-aversion is presented. • Increase in risk-aversion lead to lower wages job, less unemployment

risk, higher vacancy risk and lower capital investment. • Insured workers seek riskier job, hence higher wages, higher

unemployment. • UI is the tool to deal with market failures(incomplete insurance and

risk aversion) • The result carry over from static model to dynamic general

equilibrium model.

Efficient Unemployment Insurance

Nov 4, 2010 - In absence of UI: 1. Risk-averse workers seek lower wage jobs with less unemployment. ... Models with optimal UI in the presence of asymmetric information ... production technology f : (0, ∞) → (0, ∞) requires one worker and.

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