Search-Matching Models of Unemployment - CiteSeerX

Search-Matching Models of Unemployment Yves Zenou May 25, 2006

Matching and Efficiency

1. Introduction 2. The standard matching model Surveys can be found in Mortensen and Pissarides (1999), Pissarides (2000), Cahuc and Zylberberg (2004), Ljungvist and Sargent (2005). 2.1. Model and notations Firms and workers are all (ex ante) identical. The total population is equal to L. A firm is a unit of production that can either be filled by a worker whose production is p units of output or be unfilled and thus unproductive. In order to find a worker, a firm posts a vacancy that can be filled according to a random Poisson process. Similarly, workers searching for a job will find one according to a random Poisson process. In aggregate, these processes imply that there is a number of contacts between the two sides of the market that we assume to be determined by the following matching function: mL = m(uL, vL)

(2.1)

where u and v denote the unemployment rate and the vacancy rate, respectively. We assume that m(.) is increasing both in its arguments, concave and homogeneous of degree 1 (or equivalently has constant return to scale).1 In search-matching models, the search process is such that the transition from a vacant job to a filled job is uncertain and take time. To be more precise, the rate at which a firm fills a vacancy is m(uL, vL)/vL. Because of constant returns to scale, it can be written as: µ ¶ 1 m , 1 ≡ q(θ) θ where θ = v/u is the labor market tightness and q(θ) is a Poisson intensity. By using the properties of m(.), it is easily verified that q0 (θ) ≤ 0: the greater the labor market tightness, the lower rate at which a firm to fill a vacancy. Similarly, for workers, the rate at which the unemployed workers leave unemployment is: m(uL, vL) ≡ θq(θ) uL 1

Here, the number of job contacts as described by (2.1) is equivalent to the number of job matches, i.e., all job contacts lead to job matches.

2

If, in this model, there are no frictions, then unemployment and vacancies disappear. Jobs are found and filled instantaneously. Indeed, lim [θq (θ)] = lim q (θ) = 0

θ→0

θ→+∞

and lim [θq (θ)] = lim q (θ) = +∞

θ→+∞

θ→0

That is, if θ → 0, then the number of unemployed is infinite and thus firms filled their instantaneously (no frictions on the firm’s side) whereas if θ → 0, then the number of vacancies is infinite and thus workers find a job instantaneously (no frictions on the worker’s side). Once the match is made, the wage is determined by the generalized Nash bargaining solution. In each period, a match between a worker and a firm can be destroyed at rate λ. Workers and firms are risk neutral. We can thus write the expected discounted lifetime net income of the unemployed and the employed in steadystate by using the Bellman equations. They are respectively given by: rU = z + θq(θ) (W − U)

(2.2)

rW = w − λ (W − U)

(2.3)

where r is the exogenous discount rate and λ is the separation rate assumed to be exogeneous and constant. Let us comment (2.2). When a worker is unemployed today, he/she enjoys z. Then, he/she can obtain a job at rate θq(θ) and obtains an increase in income of W − U. The interpretation of (2.3) is similar. 2.2. Steady-state equilibrium We can define the labor market equilibrium. We have: Definition 1. A (steady-state) labor market equilibrium (θ, w, u) is such that, given the matching technology defined by (2.1), all agents (workers and firms) maximize their respective objective function, i.e. this triple is determined by a free-entry condition for firms, a wage-setting mechanism and a steady-state condition on unemployment. Free-entry condition and labor demand

3

Let us denote by J and V the intertemporal profit of a job and of a vacancy, respectively. If c is the search cost for the firm per unit of time and p is the product of the match, then J and V can be written as: rJ = p − w − λ(J − V )

(2.4)

rV = −c + q(θ)(J − V )

(2.5)

Because of free entry, firms post vacancies up to a point where: V =0

(2.6)

This means that the number (mass) of jobs is endogenously determined through entry. From this free entry condition, we have the following decreasing relation between labor market tightness and wages: q(θ) =

(r + λ) c p−w

(2.7)

In words, the value of a job is equal to the expected search cost, i.e. the cost per unit of time multiplied by the average duration of search for the firm. Equation (2.7) defines in the space (θ, w) a curve representing the supply of vacancies (what we shall also call a labor demand curve). Let us now determine the endogenous level of wages in the economy. Wage determination The usual assumption about wage determination is that, at each period, the total intertemporal surplus is shared through a generalized Nash-bargaining process between firms and workers. The total surplus is the sum of the surplus of the workers, W − U , and the surplus of the firms, J − V . At each period, the wage is determined by : w = arg max(W − U)β (J − V )1−β First order condition yields: µ ¶ β ∂E ∂U ∂J − =0 J + (E − U) 1 − β ∂w ∂w ∂w

(2.8)

Since the wage is negotiated at each period, U does not depend on current w and so ∂U = 0. By using (2.3) and (2.6), we can rewrite (2.8) as: ∂w (W − U )(w) =

4

β pc 1 − β q(θ)

(2.9)

Equation (2.9) defines a positive relationship between wages and labor market tightness that is sometimes interpreted as a wage setting function or a labor supply function. Using (2.9), (2.2) and (2.3), we obtain: w = (1 − β) z + β (p + θ c)

(2.10)

It is easy to verify that, in the plan (w, θ), the wage setting curve described by (2.10) is linear in θ and positively sloped whereas from the free entry condition (2.6), we have a decreasing relation between labor market tightness and wages (2.7). See Figure 1. Combining (2.7) and (2.10) leads to a unique θ∗ determined by: (1 − β) (p − z) =

c [λ + r + βθ∗ q(θ∗ )] q(θ∗ )

(2.11)

Then, plugging the value of θ∗ in (2.10) gives the unique w∗ . As stated above, equation (2.7) determines a unique θ∗ = v/u that gives a relation between v and u. This is an upward sloping curve in the u − v space called the V S curve. Unemployment determination Since each job is destroyed according to a Poisson process with arrival rate λ, the number of workers who enter unemployment is λ(1−u)L and the number who leave unemployment is θq(θ)uL. The evolution of unemployment is thus given by the difference between these two flows, •

u = λ(1 − u) − θq(θ) u

(2.12)

·

where u is the variation of unemployment with respect to time. In steady state, the rate of unemployment is constant and therefore these two flows are equal (flows out of unemployment equal flows into unemployment). We have: u∗ =

λ λ + θq(θ)

(2.13)

which can be rewritten as: λ(1 − u) − v q(θ) = 0

(2.14)

This equation can be mapped in the plane (u, v): it is the so-called Beveridge curve. The two equations (2.7) and (2.10) determine wages and labor tightness parametrized by u. We have three equations (2.10), (2.7) and (2.13) and three unknowns w, θ and u. It is then easy to show that there exists a unique and stable market equilibrium (θ∗ , u∗ , w∗ ). 5

2.3. Welfare and efficiency Let us now proceed to the welfare analysis. In this model, market failures are caused by search externalities. Indeed, the job-acquisition rate is positively related to v and negatively related to u whereas the job-filling rate has exactly the opposite sign. For example, negative search externalities arise because of the congestion that firms and workers impose on each other during their search process. Therefore, two types of externalities must be considered: negative intra-group externalities (more searching workers reduces the job-acquisition rate) and positive inter-group externalities (more searching firms increases the job-acquisition rate). In the context of our matching model, the welfare is given by the sum of the utilities of the employed and the unemployed, the production of the firms net of search costs. The wage w, being pure transfers, are thus excluded in the social welfare function. The welfare function is therefore given by: ½Z ¾ Z +∞ Z −rt Ω= e pdx + zdx − c θ u dt (2.15) employed

0

unemployed

The social planner chooses θ and u that maximize (2.15) under the constraint (2.12). In this problem, the control variable is θ and the state variable is u. Let γ be the co-state variable. The Hamiltonian is thus given by: H = e−rt {(1 − u)p + u z − cθu} + γ [λ(1 − u) − θq(θ)u] The Euler conditions are

∂H ∂θ

= 0 and

∂H ∂u

·

= −γ. They are thus given by:

c e−rt + γ q(θ) [1 − η(θ)] = 0

(2.16) ·

(p − z + pcθ) e−rt + γ [δ + θq(θ)] = γ k

(2.17)

where η(θ) = −q0 (θ)θ/q(θ) is the elasticity of the matching function with respect to unemployment. Let us focus on the steady state equilibrium in ·

·

which θ = 0. By differentiating (2.16), we easily obtain that γ = −rγ.2 By plugging this value and the value of γ from (2.16) in (2.17), we obtain: ∙ ¸ λ + r + θq(θ)η(θ) c (2.18) p−z = q(θ) 1 − η(θ) 2

The transversality condition is given by: lim γ ut = 0

t→+∞

and is obviously verified.

6

In order to see if the private and social solutions coincide, we compare (2.11) and (2.18). It is easy to verify that the two solutions coincide if and only if: β = η(θ) This has been showed by Hosios (1990) and Pissarides (2000), who have established that negative intra-group externalities and positive inter-group externalities just offset one another in the sense that search equilibrium is socially efficient if and only if the matching function is homogenous of degree one and the worker’s share of surplus β is equal to η(θ) (this is referred to as the Hosios-Pissarides condition). Of course, there is no reason for β to be equal to η(θ) since these two variables are not related at all and, therefore, the search-matching equilibrium is in general inefficient. However, when β is larger than η(θ), there is too much unemployment, creating congestion in the matching process for the unemployed. When β is lower, there is too little unemployment, creating congestion for firms. 2.4. Dynamics Because of the recursivity of the model, the dynamics of ut and θt are independent and can be treated separately. The dynamics equation for the unemployment rate ut can be written as (see (2.12)): •

ut = λ (1 − ut ) − θq(θ)ut or equivalently dut + [λ + θq(θ)] ut = λ (2.19) dt This is a a first order linear differential equation which solution is given by: ut = Ae−[λ+θq(θ)]t + u∗ where u∗ is given by (2.13). It is easy to see that: lim ut = u∗

t→+∞

Thus, starting from the steady-state, if there is a shock in the economy, ut deviates from u∗ , and then slowly, after some time, goes back to its steadystate value u∗ . This is because ut is a backward looking variable. Let us now determine the dynamics of θt . If one writes the dynamic version of (2.4) and (2.5), we obtain: •

rJ = p − w − λ (J − V ) + J 7

(2.20)

•

rV = −c + q(θ) (J − V ) + V

(2.21)

The free-entry condition is thus given by: •

V = V = 0 From V = 0, one obtains: J= which implies that: •

J = −

c q(θ)

(2.22)

• c 0 2 q (θ)θ [q(θ)]

(2.23)

Moreover, using (2.20) and the fact that V = 0, we have: •

J = (r + λ) J − (p − w)

(2.24)

By combining (2.22) and (2.20), we obtain: •

J = (r + λ)

c − (p − w) q(θ)

By using (2.23), this can be written as: (r + λ)

• c c 0 = (p − w) − 2 q (θ)θ q(θ) [q(θ)]

and using the wage equation (2.10), we have: • c (r + λ) c 0 = (1 − β) (p − z) − βcθ − 2 q (θ)θ q(θ) [q(θ)]

(2.25)

This is the dynamics equation of θ. It is easy to verify that at the steady-state •

, i.e. θ = 0, we found equation (2.11). This differential equation is not linear. So, in order to solve one has to linearize it. Equation (2.25) can be written as: µ ¶ • • c c (r + λ) 0 Φ θ, θ ≡ − (1 − β) (p − z) = 0 (2.26) 2 q (θ)θ + βcθ + q(θ) [q(θ)] •

We will linearize it around the steady-state, i.e. θ = 0 and θ = θ∗ , where θ∗ is the solution to equation (2.11). We have: µ ¶ µ ¶ • • ∂Φ ∂Φ Φ θ, θ ' • |• |• θ− 0 + (θ − θ∗ ) ∗ θ=0,θ=θ θ=0,θ=θ∗ ∂θ ∂θ 8

where

∙

¸ c c 0 • • = q 0 (θ∗ ) < 0 • |θ=0,θ=θ ∗ 2 q (θ) |θ=0,θ=θ∗ = ∗ 2 [q(θ)] [q(θ )] ∂θ

∂Φ

∂Φ ∂θ

| =

•

θ=0,θ=θ∗

"

# − 2 [q 0 (θ)]2 c (r + λ) 0 + βc − q (θ) |• θ=0,θ=θ∗ [q(θ)]3 [q(θ)]2

• q 00 (θ)q(θ)

cθ

= βc −

c (r + λ) 0 ∗ q (θ ) > 0 [q(θ∗ )]2

As a result, we obtain: µ ¶ ∙ ¸ • • c c (r + λ) 0 ∗ 0 ∗ Φ θ, θ ' q (θ )θ + βc − q (θ ) (θ − θ∗ ) = 0 ∗ 2 ∗ 2 [q(θ )] [q(θ )] and thus: ¸ ∙ ¸ ∙ θq(θ∗ )β θq(θ∗ )β + (r + λ) θ = − β + (r + λ) θ∗ θ − β ∗ ∗ η(θ ) η(θ ) •

This equation can be written as: •

θ − a θ = −a θ∗ where a≡β

(2.27)

θq(θ∗ )β + (r + δ) > 0 η(θ∗ )

This is a a first order linear differential equation which solution is given by: θt = Aeat + θ∗ Since a > 0, the only convergent path is A = 0.3 This implies that, at each moment, θt = θ∗ and the variable θ jumps immediately to its steady-state value θ∗ . This is due to the fact that θ, firms’ job-creation rate, is a forward-looking variable since when firms want to come into the labor market and create new jobs they look forward by foreseeing their expected profits. Thus, starting from the steady-state, a shock in the economy will implies that θt deviates from θ∗ , but, instantaneously, θt will jump to its steady-state value θ∗ . 3

Indeed, it is easy to verify that: lim θt = +∞

t→+∞

9

3. Endogenous job destruction Mortensen and Pissarides (1994) propose to extend the basic search-matching model exposed above when job destruction is endogenous. In the previous section, there was an exogenous Poisson rate λ that destroyed jobs. Here we go further by assuming that each job is characterized by a fixed irreversible technology. To be more precise, the value of a match is now given by p+ε, where p denotes as before a general productivity parameter and ε is a parameter that is drawn from a distribution G(ε), which has a finite support [ε, ε] and no mass point. It is assumed that a job is created with the highest productivity value, p + ε, capturing the idea that firms with new filled jobs use the best available technology. Then, at a Poisson rate λ, there is an idiosyncratic productivity shock. A job is then destroyed only if the idiosyncratic component of their productivity falls below some critical number b ε < ε. As a result, the rate at which jobs are destroyed are now given by λG(e ε) and not λ as before. Since e ε is endogenous and will be determined in equilibrium by a reservation rule, the job destruction is also endogenous. 3.1. The model The steady-state Bellman equations of workers are given by rU = z + θq(θ) [W (ε) − U ] Z ε W (s)dG(s) + λG(e ε)U − λW (ε) rW (ε) = w(ε) + λ

while, for firms, we have:

ee

rJ(ε) = p + ε − w(ε) + λ

Z

ee

(3.1) (3.2)

ε

J(s)dG(s) − λJ(ε)

rV = −c + q(θ) [J(ε) − V ]

(3.3) (3.4)

Let us interpret these equations. Equation (3.1) shows that an unemployed worker, who enjoys a utility z today, can find a job at rate θ(θ) and, in that case, will start a job at the highest productivity level p + ε and thus at the highest wage level w(ε). Equation (3.4) has a similar interpretation. An employed worker (see (3.2)) who is employed at a certain productivity level ε obtains a wage w(ε) today but then can be “hit” by a productivity shock at rate λ and will continue to work in that job only if the match is still productive, that is the productivity is at least equal to p + b ε. If not, that is if the productivity of 10

the match is equal to p + e ε, which happens with probability G(e ε) = Pr [ε ≤ e ε], the worker becomes unemployed and looses W (ε) − U. The interpretation of (3.3) is similar. 3.2. The steady-state equilibrium We have the following definition, which is as before but has one more variable, i.e. the threshold value of productivity b ε.

Definition 2. A (steady-state) labor market equilibrium (θ, w, u, b ε) is such that, given the matching technology defined by (2.1), all agents (workers and firms) maximize their respective objective function, i.e. this 4−tuple is determined by a free-entry condition for firms, a wage-setting mechanism, a steady-state condition on unemployment and a reservation rule. Free-entry condition and labor demand The free-entry condition is as before and given by V = 0, which using (3.4), gives: c J(ε) = (3.5) q(θ) From this free entry condition, we have the following decreasing relation between labor market tightness and wages: Z ε c p + ε − w(ε) + λ (3.6) J(s)dG(s) − λJ(ε) = rq(θ) ee Wage determination As in the basic model, the total intertemporal surplus is shared through a generalized Nash-bargaining process between firms and workers. The total surplus is now given by S = W (ε) − U + J(ε) − V and the result of the bargaining is equal to: W (ε) − U = βS

and

J(ε) − V = (1 − β) S

which leads to (1 − β) [W (ε) − U ] = β [J(ε) − V ]

(3.7)

Using (3.5) and (3.7), and proceeding as in the previous section, we obtain the following wage for each level of productivity ε: w(ε) = (1 − β) z + β (p + ε + θ c) Of course, the higher the productivity ε, the higher the wage. 11

(3.8)

By plugging this wage w(ε) in (3.3), we obtain: (r + λ) J(ε) = (1 − β) (p + ε − z) − β θ c + λ

Z

ε

J(s)dG(s)

(3.9)

ee

The reservation rule To determine the reservation rule, we have to solve the following equations J(b ε) = 0 W (b ε) = 0 The wage determination through bargaining, as given by (3.7), make these two conditions identical. So we can use one of them to determine the reservation rule. First, evaluating (3.9) at ε = b e, we obtain: Z ε (r + λ) J(b ε) = (1 − β) (p + b ε − z) − β θ c + λ J(s)dG(s) ee

Now, substracting this equation from (3.9) and noting that J(b ε) = 0, we get: (r + λ) J(ε) = (1 − β) (ε − b ε)

(3.10)

Replacing this value into the J(s) in the integral in (3.9) yields: Z λ(1 − β) ε (s − b ε) dG(s) (3.11) (r + λ) J(ε) = (1 − β) (p + ε − z) − β θ c + r+λ ee

Evaluating (3.10) at ε = b ε, we obtain:

(r + λ) J(ε) = (1 − β) (ε − b ε)

Combining this equation with (3.5), we obtain: (1 − β)

ε) c (ε − b = (r + λ) q(θ)

(3.12)

This is the job-creation condition. This equation states that the firm’s expected gain from a new job is equal to the expected hiring cost paid by the firm. Let us now determine the job-destruction condition. For that, we evaluate (3.11) at ε = b ε and using the reservation rule J(b ε) = 0, we get: Z ε z β λ b ε− − (s − b ε) dG(s) = 0 (3.13) cθ + p 1−β r + λ ee Unemployment determination

12

The number of workers who enter unemployment is λG(b ε)(1 − u)L and the number who leave unemployment is θq(θ)uL. The evolution of unemployment is thus given by the difference between these two flows, •

u = λG(b ε)(1 − u) − θq(θ) u

(3.14)

·

where u is the variation of unemployment with respect to time. In steady state, the rate of unemployment is constant and therefore these two flows are equal (flows out of unemployment equal flows into unemployment). We have: u∗ =

λG(b ε) λG(b ε) + θq(θ)

(3.15)

3.3. Efficiency The welfare function is still given by (2.15) and the constraints are (3.14) and Z ε • y = pθq(θ)u + λ(1 − u) (p + s) dG(s) − λy (3.16) ee

which is the dynamics equation of the productivity. The social planner maximizes (2.15) under the constraints (3.14) and (3.16). Proceeding as before, we obtain: (ε − eb) c [1 − η (θ)] = (3.17) (r + λ) q(θ) Z ε η (θ) λ z (s − eb)dG(s) = 0 (3.18) cθ + e− − b p 1 − η (θ) r + λ ee

If we compare (3.12) and (3.13), and (3.17) and (3.18), the market and the social efficient solutions coincide again if β = η(θ). Here the reservation productivity b ε is maximized at β = η(θ).

4. Directed search and competitive search equilibrium

In this section we derive a model where wages are not anymore bargained between the firm and the worker but instead firms post a wage and the worker who searches it either takes it or leave it. We will focus on workers and firms that direct their search, which mean that they do not search the whole market but only a subset of it that we called a submarket. Using the paper by Moen (1997), we will show that a directed search equilibrium with wage posting can in fact be efficient.

13

4.1. Model and notations We use the same model and notation as in the previous section. We just need to define a submarket. A submarket i = 1, ..., n consists of a subset of unemployed workers and firms with vacancies that are searching for each other. The number of matches in submarket i is simply m(ui Li , vi Li ), where ui and vi respectively denote the unemployment and vacancy rates in submarket i and Li the total labor force in submarket i. As before, we can thus define a labor market tightness in submarket i as θi = vi /ui and thus the arrival rates for firms and workers are respectively given by q(θi ) and θi q(θi ). Both workers and firms are free to move between submarkets and it is assumed that each submarket contains a continuum of searching workers and firms. In each submarket, all vacancies offer the same wage. However, across submarkets wages differ. These two statements are not assumptions but will be the result of the wage determination (wage posting). Thus, if there are n ≥ 1 submarkets, then there is a distribution of wages denoted by (w1 , ..., wn ), where wi is the wage in submarket i and wi ≥ wj for i > j. The key question from the unemployed worker’s point of view is to decide to which submarket to enter. The Bellman equations are still given by (2.2) and (2.3), with an subscript i for each endogenous variable, that is Ui , Ei , θi , wi . Combining (2.2) and (2.3), it is convenient to write rUi as follows: rU =

(r + λ) z + wi θi q(θi ) , r + λ + θi q(θi )

i ≥ bi

(4.1)

for wi ≥ z. If wi < z, the worker do not search and obtain z/r. Workers will enter the submarket that yields the highest expected income, that we denote by U . Since all workers are identical, all submarkets that attract workers gives the same expected income. Expressing the relation above in terms of θi q(θi ), we easily obtain: rU − z (r + λ) (4.2) θi q(θi ) = wi − rU For a given U this gives a unique relationship between wi and θi in each submarket i = 1, ..., n. This relationship is denoted by θi (wi ; U), where θi is decreasing in wi and increasing in U. The intuition is straightforward. In a submarket with a low wage wi , the gain from finding a job is low but workers are compensated by a high arrival rate θi q(θi ). On the contrary, when the submarket is defined by high wages, few firms enter the market and thus the arrival rate is low. Let us now study the firm’s decision to enter a submarket. In this model, firms incur a sunk cost k ≥ 0 for opening a vacancy. Then, after the payment 14

of the sunk cost, the productivity of a vacancy is determined as follows. It is drawn from a discrete probability distribution F (·) with mass points at p1 , ..., pn (which corresponds to each submarket i = 1, ..., n). If the productivity is too low, the vacancy is destroyed immediately and without cost. Otherwise, the firm decides which submarket to join and search for a worker. Since the firm knows its productivity when making this decision, the expected discounted lifetime utilities of a vacant and a filled job can be written as follows V (pi , w, θ) and J (w, θ). Indeed, the firm knows pi but does not know in which submarket it will enter. The Bellman equations are still given by (2.4) and (2.5) but replacing J and V by J (w, θ) and V (pi , w, θ), respectively. By combining these two equations, we obtain: [r + q(θ)] V (pi , w, θ) = q(θ)

pi − w −c r+λ

(4.3)

Each firm with a vacancy enters a submarket that maximizes V (pi , w, θ). 4.2. Wage posting A set Wa = w1 , ..., wn of wages is announced in equilibrium by a measure v1 , ..., vn of vacancies. As stated above, the unemployed only search a fraction of jobs and thus choose to search in only one submarket (apply for only a subset of jobs). The number of matches in submarket i is thus m(ui Li , vi Li ). The labor market tightness in each submarket is given by θi (wi ; U), which is defined by (4.2). Here is the way it works. Firms announce the wages that maximize the expected value of their vacancy, given their beliefs, qe (w), about the relationship between the wage announced and the arrival rate of workers. Thus, as in (4.4), we have: wi = arg maxV (pi , w, θ(w, U)) , w

i ≥ bi

where, using (4.3), V (pi , w, θ(w, U)) can be written as: ∙ ¸ 1 pi − w e V (pi , w, θ) = q (w) − pc [r + qe (w)] r+λ Rational expectations are assumed, which implies that, in equilibrium, q e (w) = q (θ(w, U)) for all wages w actually announced in equilibrium. 4.3. Competitive search equilibrium We have the following definition. 15

Definition 3. A competitive equilibrium allocation is a no-surplus allocation. That is, if W∗ = w1∗ , ..., wn∗ is the equilibrium set of wages, then there does not exist a wage w0 such that, for some i, V (pi , w0 , θ(w0 )) > V (pi , w∗ , θ(w∗ )), for all w∗ ∈ W∗ . This definition implies that the wage of each submarket has to solve the following program: wi = arg maxV (pi , w, θ(w, U)) , w

i ≥ bi

(4.4)

where θ(w, U) is defined by (4.2). This a well-defined problem, which has a unique solution that is characterized by the tangency point between firms’ iso-profit curve (V = constant) and the indifference curve for workers θ(w, U). The free-entry condition is similar to that of the standard model (V = 0) but has to be calculated for expected value of V since a firm does not know ex ante which submarket it enters. Since not all vacancies are maintained (some are destroyed because not enough productive), denote by bi the lowest vacancy type that is maintained. In that case, the expected utility of opening a vacancy is equal to: i=n X V (U ) = fi max V (pi , w, θ(w, U)) (4.5) i=ei

w

where fi is the density function which is such that fi = Pr (p = pi ). Firms enter in the market up to the point that V (U) equals the creation cost k, i.e. V (U) = k

(4.6)

Finally, we need to close the model with a steady-state condition between unemployment and the labor flows into and out of the various submarkets. It is given by fi ui θi q(θi ) = (1 − u)λ , i ≥ bi (4.7) 1 − F (pei ) u=

i=n X i=ei

ui

(4.8)

where F (pei ) = Pr [p ≤ pi ]. The first equation equates the inflows and the outflows of vacancies in each submarket. Indeed, in submarket i, the inflow of vacancies is equal to ui θi q(θi ) while the outflow of vacancies is given by fi (1−u)λ, that is (1−u)λ the total outflows of the economy times 1−Ffi(pe) , 1−F (pei ) i the percentage of vacancies in each submarket i. The second equation is just an identity equation. 16

Definition 4. A (steady-state) competitive equilibrium (U; w1 , ..., wn ; θ1 , ..., θ n ; u1 , ..., un ; u) is such that (4.6) (free-entry condition), (4.4) (wage equation in each submarket), (4.1) (expected discounted lifetime utility of the unemployed), (4.7) (flows in and out vacancies in each submarket) and (4.8) (identity equation of unemployment) are satisfied. 4.4. Efficiency As in the basic model, the optimality (efficiency) of the model is such that the planner maximizes the discounted aggregate production net of search costs and creation costs of vacancies. The welfare function is defined as follows: Z +∞ i=n X Ω= e−rt [Ni pi + z ui − c vi − a k] dt (4.9) 0

i=ei

where a is the flow of new vacancies created and Ni the measure of workers working in firms with productivity pi . The social optimum maximizes (4.9) with respect to a, bi and uei , ..., un , given that the paths of the state variables Nei , ..., Nn , vei , ..., vn are governed by the following differential equations •

N i = vi q (vi /ui ) − λ (1 − Ni ) , •

vi = a fi − vi q (vi /ui ) , and given the constraint i=n X

i ≥ bi

i ≥ bi

(ui + Ni ) = 1

i=ei

Proposition 1. The equilibrium allocation (U ; w1 , ..., wn ; θ1 , ..., θn ; u1 , ..., un ; u) as defined in Definition 4 is optimal. Proof. See Moen (1997).

5. Equilibium with wage posting and on-the-job search This section is based on Mortensen (2000). 5.1. The model We assume that the total population L is normalized to 1, L = 1, so that the number of workers is equal to u + e = 1. There is on-the-job search so that 17

employed workers can search for a job in order to improve their wage. The matching function with on-the-job search can now be written as: m = m(u + e, v) As above, we assume that the matching function m(., .) is well-behaved. Contrary to the standard matching model described in section 2, we do not assume a complete specialization in exchange or production. Even though only vacant jobs are part of the exchange process, the unemployed and employed workers look for a job in an asymmetric way. As a result, flows from employment to employed are not excluded. At each period, the meeting between workers and firms leads to a flow from a search activity to a productive activity. In equilibrium, unemployment prevails since, at each period, some jobs are destroyed and it will take some time for the resulting unemployed workers to find a job (because of search frictions). Here, the search process is such that the transition from a vacant job to a filled job as well as from a filled job to another filled job is uncertain and takes time. The rate at which firms fill a vancant job is given by: µ ¶ m 1 q(θ) = =m ,1 v θ where the labor market tightness θ is now given by v/(u + e). Because of the Poisson process, the average duration of a vacant job is thus 1/q(θ). Similarly, the arrival rate of wage offers is equal to: θq(θ) =

m = m (1, θ) u+e

The average duration time before getting a wage offer is thus 1/θq(θ). In steady-state, the number of workers leaving unemployment is θq(θ)u while the number of workers leaving employment is equal to λ(1 − u). In steady state, these two flows are equal and we obtain: θq(θ)u = λ(1 − u)

(5.1)

This is the same equation as (2.13) with the difference that θ has a different value since it takes into account both the unemployed and employed workers looking for a job. The only control variable is the number of vacancies, which is the result of the trade off between entry and exist of firms. The number of unemployed u is an adjustment variable. 18

There exist a priori flows from unemployment to employment and from employment to employment for all possible wage offers. Specifically, the flows of workers leaving a job with a wage less or equal than w is: (1 − u) [λ + θq(θ) (1 − F (w))] G(w) where F (.) is the wage distribution available in the economy whereas G(.) is the wage distribution of the employed workers in the economy. Since the employed workers only change jobs to obtain better-paid ones, the flows of workers employed in a job with a wage less or equal than w is equal to: θq(θ)uF (w) By equalazing these two flows, we can calculate the steady-state equilibrium wage distribution, which is given by: G(w) =

θq(θ)uF (w) λF (w) = (1 − u) [λ + θq(θ) (1 − F (w))] λ + θq(θ) (1 − F (w))

(5.2)

where the second equality uses (5.1) to eliminate u. So, the only difference with the standard model exposed in section 2 is that the rate at which workers have job offers θq(θ) depends on the decision of firms. 5.2. Firms’ decisions and wage distribution: The case of homogenous workers and firms A firm is a job. In order to produce, this job has to be filled. Firms can hire workers or “poach” workers by proposing them a higher wage than the current one. In steady-state, the value functions J(w) and V are respectively given by: rJ(w) = p − w − θq(θ)[1 − F (w)](J(w) − V ) − λ(J(w) − V )

(5.3)

rV = max {−c + q(θ)a(w)(J(w) − V )} w≥R

For a contact between a firm and a worker to become a match, it has to be that the wage offer is greater than R. The latter is equal to G(w) is the person is employed. As a result, we have-, a(w) ≡ [u + (1 − u)G(w)] λ = λ + θq(θ)[1 − F (w)] 19

(5.4)

where this last equation is obtained by using (5.1) and (5.2). Regarding the filled jobs, they can be destroyed at rate λ or employed workers can quit them at rate θq(θ)[1 − F (w)], i.e. the contact rate θq(θ) times the probability that the wage offer is better [1 − F (w)]. As long as there exit strictly positive profits in the economy, firms can enter the labor market. This free-entry condition implies that V = 0. This condition together with equations (??), (5.3) and (5.4) give: ¶µ ¶¾ ½µ c p−w λ = max (5.5) w≥R q(θ) λ + θq(θ)[1 − F (w)] r + λ + θq(θ)[1 − F (w)] The left-hand-side of this equation gives the expected cost of a vancant job, which has to be equal in equilibrium to the expected value of the future profits. Determining the steady-state labor market equilibrium consists in finding θ and F (.), i.e. the labor market tightness and the wage offer distribution. The lowest wage that will be accepted in the labor market will be by an unemployed worker: this is the reservation wage R. As a result, the solution to equation (5.5) is: µ ¶µ ¶ λ (p − R)θq(θ) cθ = (5.6) λ + θq(θ) r + λ + θq(θ) since F (R) = 0 by definition of the reservation wage R. In order to exclude the trivial case where there is no exchange in the labor market, we set p > R. Given the properties of the matching function, there exist two solutions to equation (5.6): The first one is unstableand is such that θ = 0; the second one is stable and is such that v > 0. From this, we can calculate the equilibrium value of v. There are two reasons for the firms to offer wages higher than the reservation wage R = z: • the acceptation rate of wage offers is higher, the higher the wage, which reduces search costs; • higher wages allow firms to keep their employed workers longer, which reduce future search costs. These two ideas explain why the two right members of equation (5.5) are increasing functions of w. Thus, some firms will offer wages higher than the reservation wage R in order to hire more rapidly workers. In equilibrium, these wage differences are such that the reduction in terms of costs at each period 20

are compensated reductions in hiring costs. The discounted sum of profits is thus the same whatever the filled job: p−w p−R = (λ + θq(θ))(r + λ + θq(θ)) (λ + θq(θ)[1 − F (w)])(r + λ + θq(θ)[1 − F (w)]) This allows us to calculate the maximal wage offer w. Indeed, for w = w, we have F (w) = 1, and thus: ¸ ∙ (r + λ)λ (p − z) w =z+ 1− (r + λ + θq(θ))(λ + θq(θ)) We can also deduce from equation (??) the analytical expression of the wage offer distribution F (w).

6. Conclusion References [1] A. Hosios (1990), “On the efficiency of matching and related models of search and unemployment,” Review of Economic Studies, 57, 279-298. [2] Moen, E.R. (1997), “Competitive search equilibrium,” Journal of Political Economy, 105, 385-411. [3] Mortensen, D.T. (2000), “Equilibrium unemployment with wage posting: Burdett-Mortensen meet Pissarides,”. [4] Mortensen, D.T. and C.A. Pissarides (1999), “New developments in models of search in the labor market”, in Handbook of Labor Economics, D. Card and O. Ashenfelter (Eds.), Amsterdam: Elsevier Science, ch.39, 2567-2627. [5] C.A. Pissarides (2000), Equilibrium Unemployment Theory, Second edition, M.I.T. Press, Cambridge.

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