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.