The slides of this lecture are based on: Equilibrium Unemployment Theory

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Equilibrium Unemployment Theory

The Labor Market Aleksander Berentsen

Uni Basel Spring term 2009

Aleksander Berentsen (Uni Basel) Equilibrium Unemployment Theory Spring term 2009 1 / 64

The slides of this lecture are based on:

Pissarides, Christopher A. (2000).

Equilibrium Unemployment Theory,

2nd ed., Cambridge (MA): MIT Press.

This book and the so-called Market Search-Models investigate the consequences of decentralized labor markets.

Structure of this chapter

1 Introduction

2 Trade in the Labor Market

3 Job Creation 4 Workers 5 Wage Determination 6 Steady-State Equilibrium 7 Out-of-Steady-State Dynamics 8 Capital 9 Concluding remarks Introduction

Introduction

Aims of this chapter:

Point out the nature of unemployment in the steady state. Show how wages and unemployment are jointly determined in an equilibrium model.

Central concepts:

Matching: The labor market is decentralized, uncoordinated economic activity.

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Trade in the Labor Market

Matching:

The matching function captures the implications of trade for market equilibrium: It gives the number of jobs formed at any moment in time as a function of the number of workers looking for jobs and the number of rms looking for workers.

The labor market is made up of heterogeneities, information imperfections and other frictions.

Examples of these are: diverse skills, dierent jobs, uncertainty as to the location and timing of job creation, and availability of suitable workers.

This prevents the labor market from clearing automatically, contrary to classical labor market theory.

The matching-function models the frictions mentioned, without investigating their causes explicitly.

It takes the inputs of the matching process (i.e., vacant jobs and job-seekers) and calculates the number of new jobs created.

The matching function can be likened to a production function which calculates the amount of production given specic inputs, without analyzing the process of production.

A steady state produces unemployment, because existing labor relationships are terminated before the unemployed can nd new jobs.

Bargaining:

Trade and production are completely separate activities:

Before jobs can be created, rms and workers have to spend resources. Existing jobs, on the other hand, yield a return.

The return produced by an occupied job is slitted between the rm and worker.

Negotiation: cooperative Nash-bargaining solution.

The model:

Workers and rms are familiar with the matching function. There is no coordination either among workers or rms. Atomistic competition rules.

Workers who have jobs will never go to the labor market to look for work (no on-the-job search).

Similarly, a rm with a job that is occupied will not look for a new worker either.

In equilibrium all parties maximize their utility, given the behavior of all other parties.

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Notation:

There are L workers in the labor market. u is the unemployment rate.

v is the quota of vacant jobs, i.e. the number of vacant jobs as a fraction of the labor force (vacancy rate).

There are uL unemployed workers and vL vacancies.

The number of job matches taking place per unit time (the matching function) is given by

mL=m(uL,vL). (1)

It gives the number of new employment relationships. The number of created jobs per unit of time thus depends on the number of

unemployed workers and the number of job vacancies.

Mathematically, the matching function is monotonically rising for uL and vL, concave and

homogeneous of degree one.

Constant returns to scale (CRS) produce a constant rate of unemployment.

CRS are plausible since a in a growing economy constant returns ensure a constant unemployment rate, not a higher one.

The rate at which vacant jobs are lled is thus: q(θ)≡ m(uL,vL) vL =m u v,1 (2)

θ=v/u is the number of vacant jobs per unemployed worker.

During a small time interval,∆t, a vacant job is matched to an

unemployed worker with probability q(θ)∆t.

Hence, the mean duration of a job is 1/q(θ).

q0₍_θ₎_≤₀

The elasticity of q(θ), isξ(θ) with 0≤ξ(θ)≤ −1. We note |ξ(θ)|=η(θ).

Similarly, the rate at which unemployed workers move into employment is p(θ)≡m(uL,vL) uL =m 1,v u =θq(θ). (3) The elasticity is 1−η(θ)≥0.

The mean duration of unemployment is 1/θq(θ)

The more vacant jobs there are, the larger θ=v/u will be, and the

faster unemployed workers will nd jobs.

Firms, however, can ll a job more quickly whenθ is small, in other

words, when there are few vacancies relative to the number of workers available.

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θ is a measure of labor-market tightness for the rm:

The tightness of the labor market is measured by the relationship of available jobs to job-seekers.

The higherθis, the tighter the labor market is for the rm.

This expresses the fact that a relatively small number of job-seekers have a large number of vacant jobs to choose from whenθ is high.

It is therefore dicult to ll vacant jobs; and so, the market is said to be tight.

Every job-seeker and every vacant job cause so-called search externalities: An additional job-seeker causes a positive externality for the rm but a negative externality for the other job-seekers.

Each additional job-seeker produces an increased probability of 1−θq(θ)∆t that a co-seeker will not nd a vacancy.

At the same time, the addition increases the probability q(θ)∆t that a

specic vacancy will be lled.

An additional vacancy will have an analogous eect.

The ow into unemployment (job destruction):

A shock (reduction in productivity, fall in the relative price of goods produced, etc.) can make it no longer protable for the rm to oer the job.

This kind of shock occurs with probabilityλ.

In this simple model, every shock leads to immediate job separation (which in this model is equal to job destruction). Hence, the job separation rate isλ.

This process of job separation is exogenous in this version of the model.

The probability of a worker becoming unemployed in a small time interval is given byλ∆t.

Without economic growth (L constant), the workers who enter unemployment in a short time interval ∆t is

λ(1−u)L∆t. (4)

The ow out of unemployment (job creation):

Job creation takes place when a rm and a searching worker meet and agree to form a match at a negotiated wage.

The number of job-seekers who nd a job is

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Equilibrium unemployment:

The evolution of mean unemployment is given by the dierence of the ows into and out of unemployment:

˙

u= du

dt =λ(1−u)−θq(θ)u (6)

In equilibrium (steady state) the mean rate of unemployment is constant; i.e.:

λ(1−u) =θq(θ)u (7)

We derive from this the equilibrium unemployment rate:

u = λ

λ+θq(θ) (8)

The equilibrium unemployment rate is dependent on both transition probabilities; i.e., job creation and job destruction.

The higher the rate of job-separations relative to the rate of job-matchings, the higher unemployment is in equilibrium.

Job Creation

Firms: Job Creation

Job creation takes place when a rm and a worker meet and agree to an employment contract.

For convenience, we assume that rms are very small and employ only one worker.

A rm can re a worker any time and will do so when there is a shock. A rm starts producing only once it has hired a worker.

Products can be sold on the market at a constant price p>0. This

represents the productivity of a worker.

Job Creation

When a job is vacant, the rm is actively engaged in hiring at a xed cost pc >0 per unit time.

The hiring cost is proportional to productivity of the worker, on the grounds that more productive workers are more costly to hire.

The number of jobs oered is endogenous and maximizes the prot of the rm. Any rm is free to open a job vacancy and engage in hiring. There is free market entry with a zero prot condition.

Job Creation

Let J be the present discounted value of expected prot from an occupied job and V the present discounted value of expected prot from a vacant job. The value of a job when a rm enters the market is

V =−δpc+q(θ)δJ+ [1−q(θ)]δV. (9)

This can be rewritten as

(1−δ)V =−δpc+δq(θ)(J−V) (10)

From this follows that

rV =−pc+q(θ)(J−V). (11)

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Job Creation

Equation (11) states that the return on the asset, a vacancy V , is equal in size to the capital costs rV .

The net return (return minus hiring fees) of a job to an employer equals J−V .

In equilibrium rents from vacancies are zero owing to free market entry. Therefore, V =0 which is implying

J= pc

q(θ) (12)

Filled jobs thus yield a return, i.e. J >0.

As 1/q(θ) is the expected duration of a vacancy.

Equation (12) in equilibrium, market tightness is such that the expected prot from a new job is equal to the expected cost of hiring a worker.

Job Creation

The asset value of an occupied job, J, satises a value equation similar to the one for vacant jobs:

J=δ(p−w) + (1−λ)δJ (13)

A net return of p−w is earned, where w is the cost of labor. In

addition, the job runs the exogenous risk of an adverse shock (job destruction).

Note that V is not a part of this equation because it is assumed that a rm aected by a shock disappears from the market. It will not reappear later to oer a job (complete irreversibility).

Job Creation

The asset value of a lled job satises a value equation, similar to the one for vacancies:

rJ =p−w−λJ (14)

If J =pc/q(θ) is substituted in equation (14) we get

p−w−(r+λ)pc

q(θ) =0 (15)

This equation gives the marginal condition for labor demand. If the rm had no hiring costs, i.e. c =0, then the standard marginal productivity

condition p =w would result.

Job Creation

Equation (15) produces a negative relationship between θ=v/u and the

wage w in theθ, w space.

Figure: The job creation curve (JC)

The downward sloping labor demand curve is also called the job creation condition (JC).

In order to determine equilibrium, the supply side of the market has to be considered. We therefore now turn to workers.

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Workers

In this model the labor force supply L is constant. Moreover, each worker's search intensity is xed. Workers all have the same productivity p.

A worker earns w when employed, and z when searching for a job. Every worker is either employed or searching for employment. z covers unemployment insurance benets or some return from self-employment.

z includes the imputed real return from unpaid leisure activities.

Workers

Let U denote the present-discounted (cash) value of the expected income stream of an unemployed worker.

W is the present-discounted (cash) value of the expected income stream of an employed worker.

The expected income steam of an unemployed worker is thus

U =δz+θq(θ)δW + [1−θq(θ)]δU. (16)

Reformulated, this gives

rU =z+θq(θ)(W −U). (17)

Workers

Equation (17) has the same interpretation as the rm's asset equations (11) and (14).

The asset that is valued is the unemployed worker's human capital. The worker's net return is W −U.

rU is the minimum compensation that an unemployed worker requires to give up search, or the reservation wage.

Workers

Employed workers earn a wage w and lose their jobs at the exogenous rate

λ. Hence, the expected income of a worker is

W =δw+λδU+ (1−λ)δW. (18)

This gives

rW =w+λ(U−W). (19)

rW is not equivalent to the wage w, because it reects the risk of unemployment.

Workers stay in their jobs for as long as W ≥U.

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Workers

Substituting equations (17) and (19) in each other's equations gives rU = (r +λ)z+θq(θ)w

r +λ+θq(θ) , (20)

rW = λz+ [r +θq(θ)]w

r +λ+θq(θ) (21)

TIP: First calculate the dierence

W −U = w−z

r +λ+θq(θ)

Workers

Since w ≥z, it follows from (20) and (21) that with discounting,

employed workers have higher permanent incomes than unemployed

workers (so W ≥U).

Without discounting (r =0), unemployed workers are not worse o

than employed workers.

Reason: Job allocation is random, and every worker is employed sometime (in an innite time horizon).

Wage Determination

In equilibrium, occupied jobs yield a total return that is strictly greater that the sum of the expected returns of a searching rm and a

searching worker.

A lled job yields a pure economic rent that is equal to the sum of expected search costs of a searching rm and a searching worker. It is assumed that the monopoly rent is shared according to the Nash solution to a bargaining problem.

This rent is divided by xing the wage rate.

Since all workers and all jobs are identical in this model, a uniform wage w is established.

This is an atomistic market, i.e. no individual participant is able to inuence the market.

Wage Determination

For a given wage rate w, the rm's expected return from the job, J, satises

rJ =p−w−λJ. (22)

For the worker, it is

rW =w−λ(W −U). (23)

The net return from a job match contract is J−V for the rm and W −U

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Wage Determination

The Nash bargaining solution identies a value for w that maximizes the weighted product of the worker's and rm's net returns from the job match.

In order to form the job match, the worker gives up U for W and the rm gives up V for J.

Therefore the wage rate for the job satises

w =arg max(W −U)β(J−V)1−β. (24)

In symmetric Nash bargaining solutions β= 1₂. A dierentβ implies

dierent measures of bargaining strength or rates of impatience.

Wage Determination

In equation (24), U and V are called thread points. If the two parties are unable to agree, the worker will remain

unemployed and the job vacant (which will not happen in this model given the assumptions on productivity and the arrival process of idiosyncratic shocks).

Wage Determination

The FOC from equation (24) satises

W −U =β(J+W −V −U) (25)

β is labor's share of the total surplus that an occupied job creates

(called the sharing rule).

In order to obtain equilibrium, W and J are substituted from (23) and (22) into (25), and the equilibrium condition V =0 is imposed. The

wage equation is thus

w =rU+β(p−rU) (26)

Workers receive their reservation wage rU and a fractionβ of the net

surplus that they create by accepting the job.

Wage Determination

rU in the equilibrium solution (26) is not particularly interesting. Another method for deriving the wage equation results by following the subsequent steps:

In equilibrium, equation (20) holds. Consequently, q(θ) = pc

J . (27)

The FOC, equation (25), can be rewritten as

J= 1−β

β (W −U). (28)

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Wage Determination

Equations (27) and (28) are inserted into

rU =z+θq(θ)(W −U) (29)

This gives

rU =z+ β

1−βpcθ. (30)

The resulting equation for the reservation wage can now be substituted back into the original wage equation (26):

w = (1−β)z+βp(1+cθ) (31)

Wage Determination

pcθ is the average hiring cost for each unemployed worker (since

pcθ=pcv/u and pcv is total hiring cost in the economy).

Workers are rewarded for the saving of hiring costs that the representative rm enjoys when a job is formed.

θ indicates the tightness of the labor market. With a highθthere is a

large number of jobs relative to number of workers.

In this situation, the workers have a strong bargaining position which has a positive eect on their wages.

This produces an upward-sloping relationship between w andθ

This is the case in spite of a xed labor force size.

Wage Determination

The upward-sloping relation is expressed by the wage setting function (subsequently called wage curve).

Equation (31) replaces the labor supply curve of Walrasian models.

Figure: The wage curve (WC)

Steady-State Equilibrium

We have a triple(u,v,w) that satises the ow equilibrium condition (8)

u= λ

λ+θq(θ), (32)

the job creation condition (15)

p−w−(r +λ)pc

q(θ) =0, (33)

and the wage equation (31)

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For convenience, we will work withθ and instead of v consecutively.

If u andθ are known, the number of lled jobs,(1−u)L, and the

number of vacancies,θuL, are also known.

Equations (33) and (34) determine the wage rate w and the tightness of the labor market θ.

The unemployment rate u can be calculated from equation (32). Equilibrium is unique (which is illustrated with the help of the following two diagrams).

Figure: Equilibrium wages and market tightness.

The job creation curve, equation (33), says that a higher wage rate leads to reduced job vacancies and thus lowers the equilibrium ratio of jobs to workers.

The wage curve, equation (34), says that a tighter labor market increases the bargaining strength of workers and a higher wage is negotiated.

Equilibrium(θ,w) is at the intersection ot the two curves and it is

unique.

The Beveridge diagram (gure on slide 46):

The gure on slide 42 shows that the equilibriumθ is independent of

unemployment.

Equation for thisθ can be explicitly derived by substituting wages

from (34) into (33), to get the job creation line (JC)

(1−β)(p−z)−r+λ+βθq(θ)

q(θ) pc =0. (35)

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The steady-state condition for unemployment, equation (32), is the Beveridge curve (BC).

The Beveridge curve is convex to the origin by the properties of the matching technology: When there are more vacancies, unemployment is lower because the unemployed nd jobs more easily. Diminishing returns to scale to individual inputs in matching imply the convex shape.

Equilibrium vacancies and unemployment are at the unique intersection of the job creation line and the Beveridge curve.

Figure: Equilibrium vacancies and unemployment

Comparative Statics

Productivity p:

Higher productivity p leads to higher wages and lower unemployment (with pc held constant).

Figure: Eects of a higher productivity

Since β <1, JC in gure on slide 42 shifts by more, so both w andθ

increase (see also equation (35))

In gure on slide 46 this rotates the job creation line anticlockwise, increasing vacancies and reducing unemployment.

Remark: This result cannot be maintained if z increases proportionally in p.

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Unemployment income z:

Higher unemployment income z leads to higher wages and higher unemployment (a higher β produces similar eects):

Figure: Eects of a higher unemployment income

Workers claim a higher wage because their reservation wage increases with a higher z. Firms nd it less attractive to create jobs.

It is important that the disincentive eects are ignored here, i.e. in spite of higher z, the unemployed continue to seek employment.

Interest rate r:

A higher interest rate r leads to lower wages and higher unemployment.

Figure: Eects of a higher interest rate

The reason is to be found in the heavy discounting of future revenues in the short-term horizon.

We have shown that workers are indierent between being unemployed and employed with r =0.

The less patient workers are (r increases), the more important it is for them to nd a job. Hence, wages drop.

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Beveridge curve (BC):

An exogenous fall in the matching function can cause the BC to shift out. This may be caused by frictions in the labor market.

A worsening in the matching function leads to lower wages and higher unemployment.

Figure: Eects of a shift in the Beveridge curve

Another cause of changes that can shift the Beveridge curve is an increase in exogenously occurring shocksλ:

A higher λshifts the Beveridge curve out because at a given

unemployment rate u a higherλ implies a bigger ow into

unemployment than out of it. Unemployment needs to increase to bring the ow out of unemployment into equality with the higher inow.

Out-of-Steady-State Dynamics

The model's dynamics:

Changes in the parameters have an immediate eect on the wage w (through the assumption that wages can be renegotiated any time). By the assumption that the expected prot from the creation of a new job vacancy is zero, rms have to be able to adjust their vacancies immediately, i.e. v and thusθare both jump variables.

However, the unemployment rate u does not jump because it is tied to the matching function that matches unemployed workers with

vacancies over time.

How does the economy react to an increase in productivity p?

Immediate reaction: w andθjump instantly to their new values. There is

no adjustment dynamic.

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This direct eect immediately rotates the JC in the Beveridge diagram. Vacancies jump immediately to their new value. The unemployment rate however moves along the JC to the new equilibrium point C.

Figure: Eects of an increase in productivity

Vacancies have the tendency to overshoot. Firms create a lot of new jobs which are then closed in the matching process over a period of time until equilibrium is re-established.

Capital

Assumptions:

There is a perfect second-hand capital market. The interest rate is exogenous.

The rm can buy and sell capital at the price of output. Since capital is costly, vacancies do not own capital.

Capital

Productivity p is reinterpreted as a labor-augmenting productivity parameter that measures the eciency units of labor.

K is aggregate capital. N is aggregate employment.

F(K,pN)is an aggregate production function with positive but

diminishing marginal products and constant returns to scale. k is the ratio K/pN.

f(k) =F(K/pN,1)is the output per eciency unit of labor.

f(k)satises f0(k)>0 and f00(k)<0.

Capital

Asset value of a vacant job: Still given by equation (11),

rV =−pc+q(θ)(J−V).

Asset value value of an occupied job:

The asset value of a job is now given by J+pk.

The real capital cost of the job is r(J+pk).

The job yields the net return pf(k)−δpk−w.

The job runs a risk of an adverse shock λ, which leads to a loss of J.

J is hence determined by the condition

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Capital

A rearrangement of equation (36) gives

rJ =p[f(k)−(r +δ)k]−w−λJ, (37)

which generalizes equation (14). It can be seen that only the job product p is aected by the introduction of capital.

Therefore the model can be solved as before but with the generalization that product p is multiplied by[f(k)−(r +δ)k]:

f0₍_k_{) =}_r ₊_δ, ₍₃₈₎ p[f(k)−(r +δ)k]−w−(r+λ)pc q(θ) =0, (39) w = (1−β)z+βp[f(k)−(r +δ)k+cθ], (40) u = λ λ+θq(θ). (41)

Capital

This equilibrium system is recursive:

With knowledge of r, equation (38) gives the capital-labor ratio. With knowledge of r and k, equations (39) and (40) give wages and market tightness.

With knowledge ofθ, equation (41) determines unemployment.

Notice: The essential features of the unemployment model remain unaltered and the capital decision is unaected by the existence of matching frictions. Hence, the eects described with the gures on slides 42 and 46 stay the same with capital.

Concluding remarks

This model demonstrates how both unemployment and vacancies can exist concurrently in labor market equilibrium.

The key reason for this is that the searching activities of the unemployed and job-seekers bear frictions that prevent the labor market from clearing automatically.

Search externalities play a role in the derivation of the results. They are the reason that price is not the sole allocation mechanism. For every price, there is always a positive probability that a vacancy will not be lled or an unemployed worker will not nd a job.

Concluding remarks

Existing frictions in the labor market are formed by the matching function.

The job creation rate is the same as the job destruction rate in the steady state. If there is a deviation from it, there will be a dynamic readjustment that equalizes both rates.

Since the job destruction rate is given as exogenous withλ, q(θ)