U(w) = w + βu(w) U(w) =

Transcription

1 Economics 250a Lecture 11 Search Theory 2 Outline a) search intensity - a very simple model b) The Burdett-Mortensen equilibrium age-posting model (from Manning) c) Brief mention - Christensen et al (2005) d) DMP search and matching model References Kenneth Burdett and Dale Mortensen (1998) Wage Differentials Employer Size and Unemployment International Econ Revie 39, pp Alan Manning (2003). Monopsony in Motion. Princeton: PU Press. Bent Christensen, Rasmus Lentz, Dale Mortensen and George Neumann (2005). On the Job Search and the Wage Distribution. JOLE 23 (January 2005), pp Christopher Pissarides Equilibrium Unemployment Theory. Cambridge MA: MIT Press, 1990 (Second edition 2000). a) A Very Basic Model ith Search Intensity The folloing model is a simplification of the one presented in Card-Chetty- Weber. It assumes that a given orker ill receive a age if employed (so there is no uncertainty about ages offers). With no risk of job destruction the value function for employment at age is: U() = + βu() U() = 1 β A orker ho is unemployed chooses search intensity s hich is just the probability of finding a job. The value function for unemployment at benefit b for a orker ho ill get a age hen she/he finds a job is: V (b, ) = max s {b + β[su() + (1 s)v (b, )] ψ(s)} here ψ(s) is the (convex) cost of job search effort. The optimal level of search effort s solves the FOC: ψ (s ) = β[u() V (b, )] And V (b, ) = b + βs U() ψ(s ) 1 β(1 s ) 1

2 Notice that e can get the derivatives of V (b, ) directly from this expression, ignoring the endogeneity of s, from the envelope theorem. So e have: U = V = V b = β βs 1 β + βs U < U 1 1 β(1 s ) So ψ (s ) s b = βv b ψ (s ) s = β[u V ] Since ψ (s ) > 0 e have that s decreases ith b and increases ith. You can port this same basic structure to a job ladder model of the type explored in Lecture 10, and get similar basic results. b) Manning s simplification of Burdett-Mortensen s age posting model The BM model is an equilibrium search model in hich atomistic employers each choose (or post) a age, and employees search on the job. The model generates an equilibrium distribution of age offers, F, an associated distribution of accepted ages G, an equilibrium rate of unemployment (or strictly speaking non-employment) u. Firms ith higher ages are larger and have a loer attrition rate (because feer orkers are bid aay). Firms ith loer ages are smaller and have higher attrition. All age choices yield the same profit. The model is an extension of a famous paper by Burdett (1978) hich first proposed the idea of job ladders on the job search by employed ho are looking for a age at a higher age firm. Economists differ in their enthusiasm for the BM model and related age posting models. Some argue that it does not make sense for firms to refuse to re-negotiate ith orkers ho find a better-paying job (i.e., a firm that posts a single age is committing to no offer-matching). The empirical importance of offer matching is an open subject for research. As a prologue, some evidence of the importance of age posting comes from a survey by Hall and Krueger (NBER #16033, May 2010): they find that only about 30% of orkers report there as some bargaining in setting the age for their current job. The rate is especially lo for blue collar orkers (5%) but much higher for knoledge orkers (86%). We ill follo Manning s exposition (MM, chapter 2). This derivation underlines the statistical mechanics of BM - behavior is summarized by the strategies of unemployed and employed searchers. 2

3 Key notation: -M orkers, all equally productive, each has non-ork option b -M f firms, each has constant productivity per orker p. - M = M f /M - each firm offers age for all its orkers -F () = distribution of ages across firms (to be determined) -employed and nonemployed orkers receive offers randomly (from F ) at rate λ. A GE variant could make λ endogenous. -employed orkers leave for nonemployment at rate δ Steady State Behaviors a) As in our basic model ith on the job seach, non-employed orkers accept any job offer paying more than b. Employed orkers accept any job paying more than their current age. b) Steady-state profits π(; F ) = (p )N(; F ) here N(; F ) is the steady state level of employment at a firm paying given F. We are going to assume π is the same for all firms, hich means small firms have loer ages and make higher profit per orker. c) Balancing flos: firm has separation rate s(; F ) and recruiting flo R(; F ). So in steady state s(; F )N(; F ) = R(; F ). d) (no spikes). If 0 < λ/δ <, F () has no spikes (atoms). Why? If there is a spike at some age 0 < p then another firm could guarantee a higher level of profit by offering a age 0 +ɛ. This has only slightly loer profit per orker but a discretely higher recruiting rate, contradicting the equal profit condition. (If 0 = p then a firm could make strictly positive profit by loering its age). e) Separation rate for firm paying is: s(; F ) = δ + λ(1 F ()). f) Steady state non-employment. The outflo from non-emp is λum, the inflo is δ(1 u)m, balance gives: u = δ δ + λ u 1 u = δ λ This relationship holds in any steady state flo model and is useful to remember. 3

4 g) Distribution function of ages across orkers (i.e., fraction of orkers earning less than some age is G(), hich is not the same as F (), because firm size varies ith. In fact: G(; F ) = δf () δ + λ(1 F ()) = F () 1 + λ < F () ( ) δ (1 F ()) To prove this: consider the set of jobs ith. The size of the pool of orkers in these jobs is (1 u)g()m (by definition of G()). - net entry to this set is from the pool of unemployed. The inflo is um λf () since a share λf () of the unemployed get offers from b to. - net exit from this set has to parts: job destruction flo=δ(1 u)g()m and exit to higher age jobs= λ(1 u)g()(1 F ())M. - equating inflo and outflo e get: uλf () = δ(1 u)g()+λ(1 u)g()(1 F ()) = (1 u)g()(δ+λ(1 F ()) Simplifying yields ( ). Note that ( ) implies as λ 0 G F. h) Flo of recruits to the firm. A firm that pays gets ne orkers from 2 sources. It gets a share of the non-employed ho received an offer that does not depend on its age: the flo rate from the non-employed pool is λu M M f = λu M here M = M f M. It also gets a share of all those ho are employed at a age less than. flo rate from this group is The (1 u)λg(; F ) M (1 u)λg(; F ) = M f M Thus R(; F ) = λ (u + (1 u)g(; F )) M = λ ( δ M δ + λ + λ ) δf () δ + λ δ + λ(1 F ()) = λδ ( ) 1. M δ + λ(1 F ()) Finally, using sn = R and substituting e get N(; F ) = R(; F ) s(; F ) = λδ M[δ + λ(1 F ())] 2 Thus a firm that pays a higher age ill have more employees. this model: R(; F ) = λδ 1 M s(; F ) Note that in 4

5 hich implies that the elasticity of the recruiting rate ith respect to the age is just the negative of the elasticity of the separation rate ith respect to the age. Manning (p. 97) presents another derivation of this relation, hich has to hold quite generally. We are finally ready for the last step: substituting N into the equation for profits e get π(; F ) = (p )N(; F ) = λδ(p ) M[δ + λ(1 F ())] 2 In equilibrium all offered ages give the same level of profit, and no other possible ages yield higher profit. To solve for the equilibrium level of profit, BM (and Manning) sho that the loest age offered in equilibrium is b. The basic point of the proof is that if the loest equilibrium age offered is above b, then a firm at this position could loer its age and get the same flo of recruits (all of hom are coming from non-employment) and have a loer age - so there ould be a contradiction. Using this fact, e get π(; F ) = π(b; F ) = λδ(p b) u(1 u)(p b) = M[δ + λ] 2 M Finally, then, once can solve for F (). The results are: ( ) 2 δ b p (p b) δ + λ F () = δ + λ λ ( G() = δ λ E[] = ( 1 ) p p b ) p b p 1 δ δ + λ b + λ δ + λ p Notice that as λ, E[] p (search frictions disappear), hereas as λ 0, E[] b hich is the so-called Diamond paradox. A problem for the model is that the implied shape of the age distribution is very unrealistic. c) Christensen et al. (2005) CLMN use a variant of the BM setup, and estimate some of the underlying parameters using a relatively simple database from Denmark that contains ages earned by orkers at each firm in 12-month period, the exit rate of orkers from each firm, and the number of people hired at the firm ho ere previously not orking. The tist is that no each orker chooses a search 5

6 intensity. In general, search intensity ill be decreasing ith the current age. They do not try to explain ho higher and loer age firms can co-exist. Nevertheless the paper has some interesting ideas for thinking about mapping a BM type orld to real data. Some basic features: a) They use the age distribution for people ho ere hired from nonemployment to estimate the distribution F (). b) they use the exit hazard rate of orkers at firms ith different ages to infer the (normalized) seach intensity λ(). As in the baseline BM model, net exit from a firm that pays a age has to components: job destruction hich is independent of ages at rate δ; and exit to higher age jobs, at rate λ()(1 F ()). Thus the exit hazard is d() = δ + λ()(1 F ()) The hard ork in the paper is to get an expression for λ(), hich depends on the optimal choice of search intensity by a orker ho is receiving a age and gets offers from a distribution F (). This is derived as follos. Let W () denote the value function for a job paying age, and let U denote the value of unemployment. The continuous time Bellman equation is: ˆ rw () = max s c(s)+λ 0 s (max[w (x), W ()] W ()) df (x)+δ(u W ()) x Note that the net arrival rate of offers is λ 0 s here λ 0 is some scale factor. Re-organizing this can be ritten as ( c(s) + δu + λ0 s ) max[w (x), W ()]df (x) x W () = max s r + δ + λ 0 s Next, using the envelope theorem e can sho (as in the simple model at the start of the lecture) that: W () = 1 r + δ + λ 0 s()(1 F ()) Finally, the FOC for optimal search (using integration by parts) is: c (s) = λ 0 = λ 0 = λ 0 ˆ ˆ ˆ [W (x) W ()]df (x) [W (x)(1 F (x))dx (1 F (x))dx r + δ + λ 0 s()(1 F ()) 6

7 CLMN assume c(s) is a convex function ith constant elasticity, so c (s) = c 0 s 1/γ so this solves out to γ ˆ λ() λ 0 s() = K (1 F (x))dx r + δ + λ()(1 F ()) (here K depends on c 0 and λ 0 ). This is a functional equation for λ(). So CLMN actually estimate the exit hazard d() and given F () solve for (δ, K, γ) assuming r is knon. d) DMP This is a very important class of models that essentially endogenize the arrival rate of offers. The ne constuct is the idea of vacancies, hich represent the employer side of the market. The economy is in steady state. There are L orkers, ul are unemployed, here u is the unemployment rate. There are also vl job vacancies. There is a match function M(uL, vl) that gives the rate of job match creation. It is assumed that M has CRS so e can rite: M(uL, vl) = vm( u v, 1). Define θ = v/u as tightness and q = M/vL as the vacancy filling rate, so q = M(uL, vl)/vl = M( u v, 1) = q(θ) Notice that q(θ) = M( 1 θ, 1) q (θ) = 1 θ 2 M 1 < 0, so hen there are more vacancies per searcher, the rate of v-filling ill fall. On the other hand the rate that U s find jobs is M/uL = θq(θ) = θm( 1, 1) = M(1, θ) θ hich is increasing in θ. This is the equivalent of λ the arrival rate of offers. Job creation and destruction The total rate of job creation is θq(θ) ul. The rate of job destruction is δ(1 u)l. As usual the steady state rate of unemployment is u = δ δ + θq(θ) = δ δ + M(1, v u ) hich is a relation beteen u and v this is donard sloping and depends on the match function and the rate of job destruction. For example, set M = U.5 V.5 then e get v = δ2 (1 u) 2 u 7

8 hich is plotted at the end of the lecture. This is called the Beveridge Curve. Next e need to get a model for vacancy creation. Each firm decides to post a vacancy or not. There is a flo cost c of a vacancy. Letting V represent the value of an unfilled vacancy and J the value of a filled vacancy e have: rv = c + q(θ)(j V ) If e assume jobs are created until V = 0 e get an equilibrium condition J = c/q(θ) so the value of a job equals the flo cost c times 1/q hich is the expected time to fill the job - so c/q(θ) is expected cost to fill a job. No if output of a job is p and the age is and the destruction rate is δ e also have a Bellman equation: so in equilibrium e must have or rj = p δj J = p r + δ p = (r + δ)c/q(θ) = p (r + δ)c/q(θ) Wages have to be less than p by an amount that is the amortized cost of creating the job. This is the equilibrium inverse labor demand equation. Finally e have to determine ho ages are set, once a match is formed. (Note that ages do not affect the job creation rate). The standard ay is to figure out the value of a nely created job to orkers, and assume that the age is determined to split the net value to the orker PLUS the firm. Defining U as the value of unemployment and W () as the value of a job e get ru = b + θq(θ)(w () U) rw () = + δ(u W ()) (so e are ignoring on the job search). Note that the second eq. implies W () = r + δ + δ r + δ U and using this ith the expression for ru e get ru = b(r + δ) + θq(θ) r + δ + θq(θ) ( ). hich in equilibrium depends on the age that ill be recieved by orkers. Holding constant U the loest age the orker ill accept,, has W ( ) = U hich implies ru = 8

9 And then W () U = r + δ r r + δ U = r + δ No hen the orker and the firm match the total surplus created (holding constant U) is hich does not depend on. Nash problem: the solution is S = W () U + J = r + δ = p r + δ + p r + δ The standard assumption is ages solve the max ( r + δ )β ( p r + δ )1 β = + β(p ) = (1 β)ru + βp This is the age that ill be negotiated ith a given U. No notice that ru is endogenous, and actually depends on. In equilibrium e have to use (*) and this equation to get. My calculation is: = (1 β)(r + δ)b r + δ + βθq(θ) β(r + δ) + βθq(θ) + p r + δ + βθq(θ) Substituting this into the inverse labor demand gets a second, upard-sloping relation beteen u and v. hich is the job creation curve. 9

10 Beveridge Curve - Cobb-Douglas Match Function Vacancy Rate Unemployment Rate