Optimal Insurance of Search Risk

Transcription

1 Optimal Insurance of Search Risk Mikhail Golosov Yale University and NBER Pricila Maziero University of Pennsylvania Guido Menzio University of Pennsylvania and NBER November 2011

2 Introduction Search and matching frictions in the labor market are a source of inequality among ex-ante identical workers: 1. Search frictions generate employment inequality, in the sense that, at a given point in time, some workers are employed and some are unemployed. 2. Search frictions generate wage inequality, in the sense that, at a given point in time, some workers are employed at low paying jobs and some are employed at high paying jobs. Search theories of employment and wage inequality: Mortensen (1970), Burdett (1978), Burdett and Mortensen (1998). Empirical assessment of the importance of search inequality: Eckstein and Wolpin (1991), Postel-Vinay and Robin (2002).

3 Introduction What is the optimal mechanism to redistribute search inequality? We answer the question in the context of a directed search model of the labor market with homogeneous workers and heterogenous firms which generates both employment and wage inequality.

4 Introduction What is the optimal mechanism to redistribute search inequality? We answer the question in the context of a directed search model of the labor market with homogeneous workers and heterogenous firms which generates both employment and wage inequality. 1. We prove that the equilibrium of the labor market is inefficient because the marginal productivity of applicants is not equated across different firms and because the marginal utility of consumption is not equated across workers in different states.

5 Introduction What is the optimal mechanism to redistribute search inequality? We answer the question in the context of a directed search model of the labor market with homogeneous workers and heterogenous firms which generates both employment and wage inequality. 2. We study the optimal insurance mechanism subject to search frictions and informational frictions (i.e. workers private information about their search): a. we prove that the optimal insurance mechanism mitigates both the productive and the distributive inefficiency of the equilibrium; b. we show that the optimal insurance mechanism is implemented by a positive unemployment benefit, a binding minimum wage and a regressive labor earning tax.

6 Introduction What is the optimal mechanism to redistribute search inequality? We answer the question in the context of a directed search model of the labor market with homogeneous workers and heterogenous firms which generates both employment and wage inequality. 3. Our theory of labor taxes is Pigovian: a. Labor taxes do not insure workers against wage risk. Wage differentials are not caused by luck, but represent compensation for the different employment risk involved with searching for jobs that attract different number of applicants. b. Labor taxes correct an externality that firms impose on one other. The externality is not inherent to the environment (e.g. a matching externality), but it is caused by the introduction of unemployment benefits.

7 Related literature 1. Our findings on labor earning tax differ from those obtained by Mirrlees (1971) and Saez (2001) in the context of a frictionless labor market: a. In Mirrlees and Saez, the labor tax is redistributive and its shape is designed to induce workers to make the socially optimal choices on how many hours to work. b. In our frictional labor market, the labor tax is Pigovian and its shape is designed to induce firms to make the socially optimal choices on how many applicants to attract to their vacancies.

8 Related literature 2. Our paper extends the analysis of Acemoglu and Shimer (1999): a. Acemoglu and Shimer consider a directed search model of the labor market with homogeneous workers and homogeneous firms and show that a small, positive unemployment benefit increases productive efficiency. b. In this paper, we adopt a mechanism design approach and show that the optimal mechanism is implemented by a positive unemployment benefit and, when firms are heterogenous, by a regressive labor earning tax.

9 Environment Workers Firms Labor market

10 Environment Workers: i. continuum of homogeneous workers with measure 1; ii. preferences: u c with u c 0, u c 0; iii. endowment: one job application and one unit of labor.

11 Environment Firms: i. continuum of heterogeneous firms with density f y over support y, y ; ii. technology: 1 unit of labor y units of output.

12 Environment Labor market: Directed search (e.g. Montgomery 1991, Moen 1997, Shimer 1996): 1. firms choose which wage w to offer; 2. workers choose whether to send an application at the cost k and, if so, which wage to seek with it; 3. firms and workers offering and seeking the same wage w come together through a frictional matching process: a. worker matches with a firm w.p. q w where 0, b. firm matches with a worker w.p. q w where 0, 0.

13 Environment Labor market: Directed search (e.g. Montgomery 1991, Moen 1997, Shimer 1996): 1. firms choose which wage w to offer; 2. workers choose whether to send an application at the cost k and, if so, which wage to seek with it; 3. firms and workers offering and seeking the same wage w come together through a frictional matching process: a. worker matches with a firm w.p. q w where 0, b. firm matches with a worker w.p. q w where 0, 0. Information friction: worker s application strategy is private information.

14 Plan of the talk In this paper, we study the optimal insurance mechanism for the economy: 1. Characterize the solution to the mechanism design problem without private information (first best allocation). 2. Characterize the competitive search equilibrium and compare it with the first best allocation. 3. Characterize the solution to the mechanism design problem under private information (second best allocation). 4. Discuss the implementation of the second best allocation.

15 First best allocation Mechanism design problem when the workers application is observable.

16 First best allocation Mechanism design problem when the workers application is observable. 1. The mechanism makes the following prescriptions: a. Worker applies to a firm of type y with probability y. The application probability y implies an applicant-to-firm ratio q y. b. Worker who seeks and finds a firm of type y receives c y. c. Worker who seeks and does not find a firm of type y receives b y.

17 First best allocation Mechanism design problem when the workers application is observable. 2. The mechanism s prescriptions maximize the worker s expected utility max q y,c y,b y q y q y u c y 1 q y u b y k f y dy.

18 First best allocation Mechanism design problem when the workers application is observable. 3. The mechanism s prescriptions are subject to the resource constraints q y f y dy 1, q y yf y dy q y q y c y 1 q y b y f y dy.

19 First best allocation 1. The first-best allocation of applicants, q y, is such that q y y, q y 0, if y y c / 0, if y y c / The first-best allocation of output c y, b y is such that c y b y 1 m q y yf y dy. 3. The multiplier on the application resource constraint is such that q y f y dy 1.

20 First best allocation Comments on the first-best allocation: 1. In the first-best allocation, the marginal productivity of applicants is equalized across different firms. Hence, the first-best allocation maximizes aggregate output. 2. In the first-best allocation, the marginal utility of output is equalized across workers. Hence, the first-best allocation maximizes expected utility given aggregate output.

21 Competitive search equilibrium A competitive search equilibrium is a w y, q w, c y, b, S such that: 1. Profit maximization: w y arg max q w y w ; 2. Optimal search: q w u b w u b S and q w 0 (with c.s.); 3. Market clearing: q w y f y dy 1 and S k (with c.s.); 4. Consumption: b m 1 q w y y w y f y dy, c y b w y.

22 Competitive search equilibrium 1. The equilibrium allocation of applicants q y is such that q y arg max q y p q q, q p q q u 1 S q u b b. 2. The equilibrium allocation of consumption is such that c y u 1 S q y u b. 3. The equilibrium value of searching is such that q w y f y dy 1.

23 Competitive search equilibrium Proposition 1 (Inefficiency of equilibrium) 1. The equilibrium allocation of applicants is inefficient: There is a y 0 y c, y such that q y q y for y y c, y 0, q y q y for y y 0, y. 2. The equilibrium allocation of output is inefficient: c y b c y b y 0 c y b c y b for y y c, y, for y y, y.

24 Competitive search equilibrium Comments on proposition 1: 1. The equilibrium allocation does not equalize the marginal productivity of applicants across firms. In particular, the marginal productivity of applicants is smaller at low productivity firms than at high productivity firms. Hence, the equilibrium allocation does not maximize aggregate output (productive inefficiency).

25 Competitive search equilibrium Comments on proposition 1: 2. The equilibrium allocation does not equalize the marginal utility of consumption across workers in different employment states. In particular, the marginal utility of consumption is lower for workers employed by high-productivity firms than for workers employed by low-productivty firms and for unemployed workers. Hence, the equilibrium allocation does not maximize expected utility given aggregate output (distributive inefficiency).

26 Competitive search equilibrium Comments on proposition 1: 3. Why is the equilibrium inefficient? a. The first best allocation would be decentralized by a competitive market for job applications. In this market, workers would sell their job application at the price and firms would buy job applications at the price. b. The market for labor is not equivalent to the market for job applications when workers are risk averse. In the labor market, the worker s application is rewarded only if successful. Hence, workers face an income risk associated with the application process and need to be compensated for it by firms. Since the risk premium is increasing in the queue length, productive firms choose to attract an inefficiently small number of applicants.

27 Competitive search equilibrium Sketch of the proof of proposition 1: The first best queue q y solves max q y q. The associated first order condition is q y y, which implies q y q y y q y.

28 Competitive search equilibrium Sketch of the proof of proposition 1: The equilibrium queue q y solves q y arg max q y p q q, q p q q u 1 S q u b b. The associated first order condition is q y y p q y q y p q y, which implies q y q y y q y 2p q y p q y q y.

29 Competitive search equilibrium Sketch of the proof of proposition 1: The efficient and equilibrium queues are such that q y q y q y y q y, q y y q y 2p q y p q y q y. 1. Using these derivatives, we show that q y q y q y q y. 2. There is at most one y 0 s.t. q y 0 q y Since q and q integrate to 1, there is a y 0 s.t. q y 0 q y 0.

30 Second best allocation Mechanism design problem when the workers application is unobservable.

31 Second best allocation Mechanism design problem when the workers application is unobservable. 1. The mechanism makes the following prescriptions: a. Worker applies to a firm of type y with probability y. The application probability y implies an applicant-to-firm ratio q y. b. Worker who is employed at a firm of type y receives c y. c. Worker who is unemployed receives b.

32 Second best allocation Mechanism design problem when the workers application is unobservable. 2. The mechanism s prescriptions maximize the worker s expected utility max q y,c y,b,s q y q y u c y 1 q y u b k f y dy

33 Second best allocation Mechanism design problem when the workers application is unobservable. 3. The mechanism s prescriptions are subject to the workers incentive compatibility constraints k S, and to the resource constraints q y f y dy 1, q y u c y u b S and q y 0, q y yf y dy q y q y c y 1 q y b f y dy.

34 Second best allocation 1. The second best allocation of applicants q y is such that q y y q y ĉ y 1 q y b q y q y u ĉ y u b u ĉ y ĉ y b The second best value of searching Ŝ is Ŝ k. 3. The second best consumption b, ĉ y is such that ĉ y u 1 Ŝ q y u b.

35 Second best allocation 1. The second best allocation of applicants q y is such that q y y q y ĉ y 1 q y b q y q y u ĉ y u b u ĉ y ĉ y b The second best value of searching Ŝ is Ŝ k. 4. The multipliers 1, 2 are such that q yf y dy 1, q y yf y dy q y q y ĉ y 1 q y b f y dy.

36 Second best allocation Proposition 2 (Properties of the second best allocation) 1. There is a y 1 ŷ c, y such that q y q y for y ŷ c, y 1, q y q y for y y 1, y. There is a y 2 y c, y such that q y q y for y y c, y 2, q y q y for y y 2, y.

37 Second best allocation Proposition 2 (Properties of the second best allocation) 2. Let L be defined as L c, b, q u q c 1 q b q u c 1 q u b. For all q 0, we have L ĉ q, b, q L c q, b, q L ĉ q, b, q L c q, b, q 0.

38 Second best allocation Comments on proposition 2: 1. In the second best allocation, the marginal productivity of applicants is smaller at low-productivity firms than at high-productivity firms. However, the marginal productivity of applicants across firms is less unequal than in the equilibrium. Hence, in the second best allocation, output is smaller than in the first best allocation, but higher than in the equilibrium allocation. That is, the second best allocation mitigates the productive inefficiency of the equilibrium.

39 Second best allocation Comments on proposition 2: 2. In the second best allocation, the marginal utility of consumption is not equalized across workers in different employment states. However, the marginal utility of consumption across workers is less unequal than in the equilibrium. Hence, in the second best allocation, utility is lower than in the first best allocation, but higher than in the equilibrium allocation (for given output). That is, the second best allocation mitigates the distributive inefficiency of the equilibrium.

40 Second best allocation Comments on proposition 2: 3. Why is the equilibrium constrained inefficient? a. The second best allocation would be decentralized by a labor market and a competitive insurance market. In the insurance market, insurance companies would offer incentive-compatible insurance policies that give workers a positive transfer if they remain unemployed and a negative transfer if they find a job. b. The labor market alone does not provide workers with insurance against the risk associated with the application process. Firms need to compensate workers for this risk. Since the risk premium is increasing in the queue length, productive firms choose to attract an inefficiently small number of applicants.

41 Second best allocation Sketch of the proof of proposition 2: The second best queue q y solves q y arg max q y p q k, b q q 1/ 2 q, p q Ŝ, b q u 1 Ŝ q u b b. The associated first order condition is q y y p q y Ŝ, b q y p q y Ŝ, b 1/ 2, which implies q y q y y q y 2p q y Ŝ, b p q y Ŝ, b q y.

42 Second best allocation Sketch of the proof of proposition 2: The equilibrium queue q y solves q y arg max q y p q q, q p q S, b q u 1 S q u b b. The associated first order condition is q y y p q y S, b q y p q y S, b, which implies q y q y y q y 2p q y S, b p q y S, b q y.

43 Second best allocation Sketch of the proof of proposition 2: The second best and equilibrium queues are such that q y q y q y y q y 2p q y Ŝ, b p q y Ŝ, b q y, q y y q y 2p q y S, b p q y S, b q y. 1. Using these derivatives, we show that q y q y q y q y. 2. There is at most one y 1 s.t. q y 1 q y Since q and q integrate to 1, there is a y 1 s.t. q y 0 q y 0.

44 Implementation of the second best Proposition 3. The second best allocation can be implemented in a labor market with the following policies: 1. the minimum wage e ŷ c, 2. the unemployment benefit B u 1/ the increasing and concave labor earning tax T e e q q q 0.

45 Implementation of the second best Intuition for proposition 3: 1. In equilibrium, the value of searching S plays two roles. First, S guarantees that demand and supply of applicants are equal. Second, S determines the income risk that workers face when joining a queue of length q. 2. The mechanism sets Ŝ k so as to minimize the worker s income risk. This can be accomplished with a positive B u. 3. However, for Ŝ k, there is no guarantee that demand and supply of applicants will be equal. Indeed, for Ŝ k, there is excess demand. In order to lower demand, it is necessary to increase the price firms pay for applicants without increasing Ŝ. This is accomplished with a positive, increasing and concave T e.

46 Implementation of the second best Sketch of the proof of proposition 3: The second best queue q y solves max q y w q 1/ 2 q s.t. q u b w q u b k. If B u, T e implements the second best in a labor market, q y must also solve max q y w q B u T q s.t. q u b w q u b k, T q T e e q, e q w q B u T q.

47 Implementation of the second best Sketch of the proof of proposition 3: If B u, T e implements the second best in a labor market, q y must also solve q B u T q 1/ 2 q T q q B u

48 Implementation of the second best Sketch of the proof of proposition 3: In order to recover the labor earning tax T e, note that T e e q T q, e q w q B u T q, T q 1/ 2 / q B u. Since the previous equations must hold for all q, we have T e e q w q T q T q T e e q q q 2 T e e q 1 2 k u ĉ q q 1 q k u ĉ q 1. q 1 q 2 2

49 Implementation of the second best Sketch of the proof of proposition 3: In order to find the optimal unemployment benefit B u, note that q y q y T q y 1 q y B u f y dy 0. Using the optimality condition for T q, we can rewrite this as q y q y q y B u 1 q y B u f y dy 0, which implies B u

50 Conclusion 1. When labor trade is subject to search frictions, decentralizing the efficient allocation would require setting up a competitive market for search inputs. Unfortunately setting up a market for search inputs requires a great deal of information (e.g. which workers did actually search and how hard they searched...). 2. In some special cases, a market for labor is equivalent to a market for search inputs: a. risk neutrality, bargaining and Mortensen rule (Mortensen 1982). b. risk neutrality and posting (Moen 1997, Shimer 1996). 3. Apart from those special cases, a labor market is not equivalent to a market for search inputs. Then, labor market policies (unemployment benefits and labor taxes) can be used to achieve constrained efficiency.