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LABOR ECONOMICSTheories of Unemployment

Julien Prat

University of Vienna

May 21, 2007

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Part I

Introduction

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Outline

1 MotivationWhy Study Labor Economics?Why Focusing on Unemployment?

2 Modern theories of Unemployment.Search Frictions.Other important specificities of labor markets.

3 Plan of the CoursePart II: Competitive ModelPart III: Search-Matching ModelParts IV and V: Workers’ Motivation and Bargaining

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Why Study Labor Economics?Why Focusing on Unemployment?

Why Study Labor Economics?

One may easily argue that labor markets crucially determinemacroeconomic outcomes:

1 The labor share is quite stable across developed countries andaccount for about two thirds of national income.

2 The vast majority of households draw most of their incomesfrom wage payments.

3 Differences in job creation between the US and Europeexplain a significant share of the differences in output growth.

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Labor Shares. Source: Bentolila and Saint-Paul, 2003.

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Labor Shares.

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Growth Accounting. Source: Pissarides and Vallanti, 2005.

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Why Focusing on Unemployment?

The persistence of unemployment:

1 is at odds with the market-clearing equilibrium conditioncommonly assumed in economics.

2 points to the need for a specific approach to Labor Economics.

3 is often regarded as the single most important problem facingEuropean countries.

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Are Unemployment rates relevant statistics?

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Unemployment rates over time

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Real compensations over time

The positive correlation across countries between Unemploymentand Real Compensation does not square with market adjustmentsto the equilibrium.

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Search Frictions.Other important specificities of labor markets.

Frictions: an important specificity of labor markets

Unemployment cannot be reduced to an ”excess supply of labor”since unfilled vacancies and searching workers coexist. To explainthis phenomenon economists refer to frictions:

1 Due to the high degree of heterogeneity and differentiation ofworkers and jobs, matching is time and resources consumingfor both job seekers and firms.

2 Screening is necessary due to the prevalence of informationalasymmetries between employers and employees (employers donot observe all the characteristics of workers and vice-versa).

3 Due to coordination failures some open vacancies may end upwith more than one applicant and some with none.

4 Workers face important mobility costs which hinder thereallocation process across sub-markets.

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US Average Monthly Labor Flows 1994-97. Source:Blanchard.

Even if Frictions were small, the actual size of the flows wouldgenerate significant Macro Effects.

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Search and Matching Models

1 Search models capture the notion of frictional unemployment.Because it is resources consuming for workers and firms tofind a job, unemployment naturally arises.

2 The search-matching model extends this mechanism tomacro dimension by endogenizing Labor demand.

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Other important specificities of labor markets

1 As opposed to machines, efficient work is not automaticallyextracted from workers. This raises the question of how tomotivate workers?

2 The repartition of revenues occurs through a bargainingprocess between employees and employers.

3 Given their obvious social implications, legislations in thelabor markets are numerous. As a result the labor markets arevery different from ’Walrasian’ markets. Instead, theirfunctioning is strongly determined by social norms.

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Part II: Competitive ModelPart III: Search-Matching ModelParts IV and V: Workers’ Motivation and Bargaining

Part II: Competitive Model

Part II: Competitive Model

1 Labor Supply: Substitution and Income Effects.

2 Labor Demand: Complementarity/Substitutability betweenLabor and Capital.

3 Limits of the competitive model: Counterfactual predictionsabout the business-cycle fluctuations of Employment andWages.

4 Minimum Wage: Does it explain Unemployment?

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Part III: Search-Matching Model

Part III: Search-Matching Model

1 Job Search: Partial Equilibrium.

2 Matching Function: From Partial to General Equilibrium.

3 Search-Matching Model: The Equilibrium Rate ofUnemployment and its determinants.

4 Efficient level of Unemployment and the Minimum Wage.

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Other Topics

Part IV: Efficiency Wage

1 Efficiency Wage Model of Shapiro and Stiglitz and the’Non-shiriking’ Rate of Unemployment.

2 Predictions about the business-cycle fluctuations ofEmployment and Wages.

Part V: Bargaining

1 Union Model of McDonald and Solow.

2 Predictions about the business-cycle fluctuations ofEmployment and Wages.

3 Game Theoretical Foundation of Nash-Bargaining.

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References

Blanchard, O.J., Macroeconomics, Ed: Prentice Hall International.

Blanchard, O.J., (2006), ”European Unemployment: TheEvolution of Facts and Ideas Bentolila”, Mimeo, MIT.

Bentolila, S. and G. Saint-Paul, (2003), ”Explaining Movements inthe Labor Share”, Contributions to Macroeconomics, BerkeleyElectronic Press, vol. 3(1), pages 1103-1103.

Pissarides, C.A. and G. Vallanti, (2005), ”The Impact of TFPGrowth on Steady-State Unemployment”, ForthcomingInternational Economic Review.

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Labor SupplyLabor Demand

Minimum WageBusiness Cycle Fluctuations

References

Part II

Labor Supply and Labor Demand

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References

Outline

4 Labor SupplyIntensive MarginExtensive MarginEffect of Non-wage IncomeEffect of WageEmpirical Estimation of Labor Supply

5 Labor DemandLabor Demand in the Short RunCost MinimizationProfit MaximizationEmpirical Estimation of Labor Demand

6 Minimum Wage

7 Business Cycle Fluctuations

8 References

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References

Intensive MarginExtensive MarginEffect of Non-wage IncomeEffect of WageEmpirical Estimation of Labor Supply

Extensive and Intensive Margins

For our purpose, we are mainly interested in deriving theAggregate supply, i.e. the sum of individuals’ labor supply.Individual decisions can be decomposed as follows:

Extensive Margin: Whether or not to participate in the labormarket?

Intensive Margin: For participants, how many hours of laborto supply?

As usual, we analyze the problem backward and so start bycharacterizing decisions at the Intensive margin.

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References


Consumption/Leisure Tradeoff

The problem is largely similar to optimal bundle allocation inConsumer theory. The two goods are consumption C and leisure L.

The wage is the opportunity cost of Leisure.

The determination of the optimal allocation is complicated by thefact that available income is a positive function of the price ofleisure. This generates an additional Income Effect.

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References


Optimization Problem

Definition

w : Wage per unit of time.

R: Non-wage Income.

L0: Time Endowment.


MAXC ,L U(C , L)

s.t. C ≤ w(L0 − L) + R ≡ R0 − wL

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References


First Order Conditions

Lagrangian

L = U(C , L) + λ(R0 − C − wL)

F.O.C.

∂U(C∗,L∗)∂L = w ∂U(C∗,L∗)

∂C

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References


Illustration

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References


Optimization with Participation Constraint

Optimization with Participation Constraint

MAXC ,L U(C , L)

s.t. (i) C ≤ w(L0 − L) + R ≡ R0 − wL

(ii) L ≤ L0

Lagrangian

L = U(C , L) + λ(R0 − C − wL) + µ(L0 − L)

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References


F.O.C. with Participation constraint

F.O.C.

∂U(C∗,L∗)∂C = λ

∂U(C∗,L∗)∂L = λw + µ

1 µ = 0 ⇒ Participation Constraint does not bind. ThenSolution is as before.

2 µ > 0 ⇒ Participation Constraint binds. Then L∗ = L0 andC ∗ = R.

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References


Reservation Wage

Definition

The Reservation wage wr is the minimum wage such that theagent participates to the labor market. Hence:

wr ≡ ∂U(R,L0)/∂L∂U(R,L0)/∂C

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References


Leisure is a normal good

Persons with higher non-wage income work less. This substantiatesthe premise that leisure is a normal good.

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References


Reservation wage and non-wage income

Proposition 1

If leisure is a normal good, the reservation wage is increasing innon-wage income.

Proof:Claim: If leisure is a normal good, ULC UC > UCC UL.Proof: Differentiate F.O.C. and budget constraint with respect to R

C∗R + wL∗R = 1

ULC C∗R + ULLL∗R = w(UCC C∗R + UCLL∗R )

Combine the two equations and replace w by UL/UC to obtain

L∗R

(2UCLULUC − ULL(UC )2 − UCC (UL)2

UC

)= ULC UC − UCC UL (1)

The term in brackets on the left hand side is positive because U(C , L) is quasi-concave. Thus L∗R > 0 iffULC UC − UCC UL, which proves the Lemma.Differentiating the reservation wage yields

∂wr

∂R=

ULC (R, L0)UC (R, L0)− UCC (R, L0)UL(R, L0)

(UC (R, L0))2> 0

where the inequality follows from the Claim.31 / 118



References


Labor Supply and Wage

Proposition 2

If leisure is a normal good, an increase in wages has a positiveSubstitution Effect and negative Income Effect on labor supply.Thus the relationship between wages and hours worked is ingeneral ambiguous.

Proof: Differentiate optimal labor supply with respect to w

∂L∗(w, R0)

∂w= L∗w (w, R0) + L∗R0

(w, R0)∂R0

∂w(2)

Since ∂R0/∂w = L0 and Leisure is a normal good we know that L∗R0∂R0/∂w > 0. This is the Income Effect.

Now, differentiate as before the F.O.C. and budget constraint with respect to w

C∗w + L∗ + wL∗w = 0

ULC C∗w + ULLL∗w = UC + w(UCC C∗w + UCLL∗w )

Rearranging finally yields

L∗w


UC

)= −(UC )2 − L∗(ULC UC − UCC UL)

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References


Labor Supply at Reservation Wage

Proposition 3

If leisure is a normal good, the labor supply is an increasingfunction of wages near the reservation wage.

Proof: Replace L∗R0using equation (1) into equation (2) to obtain

∂L∗(w, R0)

∂w


UC

)= −(UC )2 + (L0 − L∗)(ULC UC − UCC UL)

When w → wr , L∗ → L0 and so limw→wr ∂L∗(w, R0)/∂w < 0.

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References


Self-selection and Estimation

Estimates the following regression

ln hi = αw lnwi + αR lnRi + Xθi + εi

where θi is an [n,1] matrix collecting the individual’s icharacteristics (e.g. age, education, gender...) and X is an [1,n]matrix with the coefficients attached to each characteristic.Self-selection: Restricting the sample to participants or settinghi = 0 for non-participants, limits attention to individuals with ahigher propensity to work ⇒ Biased Estimates.Solution: One should estimate the full-model with Participationconstraint. To do so, a parametric assumption on the utilityfunction (e.g. Cobb-Douglas) is needed.Main difficultly: By definition wages offered to non-participantare not observed ⇒ need to infer them.

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Main Empirical Findings

Elasticity of labor supply is small.

Elasticity of labor supply by women is higher than that ofmen. Estimates for married women lie between 0 and 1,whereas estimates for married men lie between 0 and 0.2.

Most variations in Aggregate labor supply are due toadjustments at the extensive margin, i.e. individuals enteringand/or exiting the labor market.

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References

Labor Demand in the Short RunCost MinimizationProfit MaximizationEmpirical Estimation of Labor Demand

Optimization in the short run

By short run, we mean that the only flexible factor is labor.To simplify matters, we assume that the inverse demand functionis iso-elastic so that P(Y ) = Y η where0 > 1/η = Y ′(P)P/Y (P) > −1 is the elasticity of output demand.


MAXL Π(L) = P(Y )Y − wL

s.t. Y = F (L)

F.O.C.

F ′(L) =(

11+η

)wP

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Short run Labor demand and wage

Differentiating the F.O.C. with respect to w yields

∂L

∂w=

(1

1 + η

)1

F ′2P ′ + PF ′′ < 0

Hence the short run labor demand is a negative function of laborcost. In terms of elasticity

ηL,w ≡ ∂L

∂w

w

L=

1LP′F ′

P + F ′′LF ′

=1

η LF ′

F + F ′′LF ′

Hence, the higher the market power (i.e. the smaller η), the lesselastic the short run labor demand.

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References


Technology of Production

We now characterize optimal labor demand in the long-run. Werestrict our attention to the following Technology:

Production Technology

2 Factors of Production: Labor,L, and Capital, K .

Production Function F (K , L) : R+ × R+ → R+ is strictlyconcave, so that FK (·) > 0, FL(·) > 0, FKK (·) < 0 andFLL(·) < 0.

F (K , L) is homogenous of degree θ, so thatF (λK , λL) = λθF (K , L) for all λ > 0.

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Conditional Demand

We first consider the problem of a firm that wants to minimize thecost of producing a given quantity of output Y . Let r denote theinterest factor, i.e. 1+interest rate.


MINL,K wL + rK

s.t. F (K , L) ≥ Y

F.O.C.

∂F (K ,L)∂L =

(wr

) ∂F (K ,L)∂K ; where F (K , L) = Y .

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Cost Function

Replacing these optimal conditional demands allows us to definethe minimum cost as a function of input prices and outputproduced.

Properties of Cost Function C (w , r ,Y )

P1 C (w , r ,Y ) is increasing in w and r and homogenous ofdegree 1 in (w , r).

P2 C (w , r ,Y ) is concave in w and r .

P3 It satisfies Shephard’s Lemma: L = Cw (w , r ,Y ) andK = Cr (w , r ,Y ).

P4 C (w , r ,Y ) is homogenous of degree 1/θ in Y , so thatC (w , r ,Y ) = C (w , r , 1)Y (1/θ).

See pp. 234-238, in CZ, Chapter 4, for Proof.40 / 118



References


Properties of conditional demand

Deriving conditional demand for Labor in Shephard’s Lemma (P3)w.r.t w shows that it is decreasing in w

∂L

∂w= Cww ≤ 0

Deriving conditional demand for Labor in Shephard’s Lemma (P3)w.r.t r shows that

∂L

∂r=

∂K

∂w= Cwr

Thus the effect on conditional labor demand of a rise of one euroin the price of capital is similar to the effect on conditional capitaldemand of a rise of one euro in the price of labor.

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References


Unconditional factor demand

We now examine the full problem where the entrepreneurdetermines the optimal level of output.


MAXY Π(w , r ,Y ) ≡ P(Y )Y − C (w , r ,Y )

F.O.C.

P(Y ) = 11+ηCY (w , r ,Y )

S.O.C. is satisfied iff 11+η > θ

The firm sets his price by applying a mark-up 1/(1 + η) to itsmarginal cost CY (·).

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References


Long-run elasticity of labor demand

By Shephard’s Lemma (P3), we have L∗ = Cw (w , r ,Y ∗) where astar superscript indicates long-run optima. Differentiating thisrelation w.r.t. w yields

∂L∗

∂w= Cww︸︷︷︸

Substitution Effect

+ CwY∂Y ∗

∂w︸︷︷︸Scale Effect

The substitution effect: Cww is negative and equal to thederivative of the conditional labor demand.

The scale effect: From the S.O.C. it is easy to show that it isnegative because CwY and ∂Y ∗/∂w must be of opposite sign.

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References


Estimation Results

Estimation procedures either assume a specific ProductionFunction (e.g. Cobb-Douglas, CES...) or even more directlypostulate a particular Cost Function.For obvious technical reasons, most estimates available are for theconditional elasticity of labor demand.

Main findings

Estimates of the elasticity of conditional aggregate labordemand w.r.t. w are negative, between [−0.75,−0.15] andcentered on −0.3.

The scale effect increases (as expected) the absolute value ofthe elasticity to values around 1.

Unskilled labor is more easily substitutable for capital thanskilled labor.

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References

Minimum wage and unemployment

The usual way to introduce unemployment in the competitivemodel is to impose a price floor, i.e. a minimum wage.Empirical evidence, however, does not substantiate the importanceof this explanation:

Over the last 30 years, there has been little change in minimumwage relative to average earnings. Thus it is hard to argue that,everything else equal, it has played an important role in the rise ofEuropean unemployment during this period.

Empirical estimates document a positive and significant correlationbetween the minimum wage and unemployment among workersbelow 24 years of age. But for older workers the effect turns out tobe non-significant.

A ”natural experiment” study by Card and Kruger (1994) on thefast-food industry in the US even exhibits a positive effect onemployment of the minimum wage.

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References

Unemployment rates and Education

No obvious correlation between level of minimum wage and relativeproportion of job seekers among low-skilled workers.If anything, the relative proportion is higher in Anglo-saxoncountries.

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References

Business Cycle Facts

The following properties of Business Cycles fluctuations are welldocumented:

Hours worked and Employment are Pro-cyclical.

Unemployment is Counter-cyclical.

Real Wages are acyclical or mildly Pro-cyclical.

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References

Trend GDP Hours. Source: Prescott.

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References

Correlation Hours GDP US. Source: Prescott.

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References

Correlation Wage GDP US.

Pro-Cyclical in the 1970’s.Almost acyclical from the 1980’s.

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References

Predictions of competitive model

Reconciling these empirical regularities with theory is one of themain challenge facing Macroeconomists

The cyclical component of real wage excludes Supply shock.

But Demand shocks with a nearly vertical labor supply implythat we should observe strong Procyclicality in real wages andnear Acyclicality in labor input. That is the exact opposite ofwhat is documented in the data.

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References

Reconciling evidence with theory

To address this issue, we basically need to derive an aggregatelabor supply with a higher elasticity than the individual laborsupply. At least two approaches have been proposed:

1 Effect on Aggregation of the prevalence of adjustments at theextensive margin.

2 Imperfection in labor markets and deviation from thecompetitive setting.

In this course, we will focus on the second type of explanation.

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References

References

Labor SupplyCZ, Chapter 1.

Labor DemandCZ, Chapter 4.

Minimum WageCard, D., and Krueger, A., (1994), ”Minimum Wage and Employment: ACase Study of the Fast-food Industry in New Jersey and Pennsylvania”,American Economic Review, 84, pp. 772-793.

Business CyclesPrescott, E.C., (2004), ”The Transformation of Macroeconomic Policyand Research”, Nobel Prize Lecture.

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Job SearchThe Search-matching modelAppendix: Asset Equations

References

Part III

Search-Matching Model

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References

Outline

9 Job SearchThe basic modelWorker TurnoverComparative staticsDiamond’s critique

10 The Search-matching modelThe Matching ProcessWage SettingThe EquilibriumEfficiency

11 Appendix: Asset EquationsPoisson processesAsset Equation

12 References

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References

The basic modelWorker TurnoverComparative staticsDiamond’s critique

The basic model

The model describes the behavior of a worker searching for a job ina world with imperfect information.

The basic model is based on the following hypotheses:

1 Workers are risk neutral.

2 The environment is stationary.

3 The intensity of search cannot be adjusted by the worker.

4 Workers cannot search for alternative job while working.

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References


The basic model

For the moment, we assume that:

Time is discrete.

Worker discount the future at rate 1/(1+r).

When a firm meets a job seeker, it offers him a constant wageover the duration of the job.

Jobs last forever.

At the end of each period:1 Employed workers receive their wages.2 Unemployed workers receive benefits z and sample a wage

offer from the exogenous distribution H(w).

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References


Discounted utilities

A job seeker can reject the wage offer and continue to search.Accordingly, the Bellman equations for employed and unemployedworkers read

Discounted Utilities

Let E denote the discounted utility of an employed worker

E (w) =1

1 + r(w + E (w))

and U the discounted utility of a job seeker

U =1

1 + r

(z +

∫ ∞

0maxU,E (w)dH(w)

)(3)

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References


Optimal search strategy

Job seekers have to solve an optimal stopping problem. SinceE (w) = w/r is strictly increasing, there exists a unique reservationwage wr such that job seekers turn down every job offers if w < wr

and accept every job offers if w ≥ wr .At the reservation wage, workers are indifferent between workingand being unemployed so that: E (wr ) = U. Replacing U = wr/rand E (w) = w/r into (3) yields

Reservation wage

wr = 11+r

(rz +

∫∞0 maxw ,wrdH(w)

)= z + 1

r

(∫∞wr

(w − wr )dH(w))

= z + 1r

(∫∞wr

[1− H(w)]dw))

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References


Continuous time

As shown in Appendix (1), similar solutions can be derivedconsidering the worker’s asset equations in continuous time

Discounted Utilities in Continuous time

rE (w) = w

rU = z + λ

(∫ ∞

0max0,E (w)− UdH(w)

)where job seekers contact firms at rate λ.

In the remainder of the lecture notes, we favor this interpretation.

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References


Job destruction

The basic model cannot account for the fact that job separationsoccur quite frequently (e.g. average job duration in US is close to3 years). Thus we extend the model by considering that jobs aredestroyed at the rate δ. Then the asset equations are

Asset equations with job destruction

rE (w) = w + δ(U − E (w))

rU = z + λ

(∫ ∞

0max0,E (w)− UdH(w)

)The asset equation for E adds the flow income, w, to the capitalloss, U-E, times the rate of job separation, δ. Notice that the gainderived from employment is

E (w)− U =w − rU

r + δ(4)

that is the discounted difference between the wage and theopportunity cost of employment.

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References


Optimal reservation wage

Replacing the worker surplus in (1) and the definition of thereservation wage rU = Wr yields

Optimal reservation wage

wr = z + λr+δ

∫∞wr

(w − wr )dH(w)

= z + λr+δ

∫∞wr

[1− H(w)]dw

It is easy to verify that the reservation wage

is unique.

maximizes the intertemporal utility of a job-seeker.

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References


Interpretation of optimal reservation wage

The optimal reservation wage is higher than the flow income of jobseekers (wr > z) because it also includes the option to search

wr = z︸︷︷︸Flow income

+λ

r + δ

∫ ∞

wr

(w − wr )dH(w)︸︷︷︸Option to search

Intuitively, by accepting a given job offer, the job seeker renouncesto the possibility of finding a better job. Thus, he ”kills” his optionto search.The job seeker takes into account this cost and so ask for apremium over his current flow income.Notice that this effect is less important if workers can also searchwhile being on-the-job.

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References


Comparative statics on the optimal reservation wage

wr ↑ when

z ↑ because the opportunity cost of employment increases.

λ ↑ because the option to search increases.

wr ↓ when

r ↑ because workers value less the future which decreases theoption to search.

δ ↑ because the average job spell decreases which in turndecreases the option to search.

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References


Comparative statics on unemployment duration

The average unemployment duration is

Tu =1

λ[1− H(wr )]

Accordingly Tu is an increasing function of z . So:

higher unemployment benefits raise the averageunemployment duration.

λ has an ambiguous effect because it increases the rate ofarrival of job offers but also augments the number of rejectedjob offers (wr ↑). In practice, the former effect dominates sothat ∂Tu/∂λ < 0.

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Unemployment rate

The rate of unemployment evolves as follows

·u = δ(1− u)︸︷︷︸

Flows into Unemployment

− λ[1− H(wr )]u︸︷︷︸Flows out of Unemployment

equilibrium rate of unemployment

u =δ

δ + λ[1− H(wr )]

The comparative statics for unemployment are therefore similar butwith opposite signs to the ones for unemployment duration.

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References


Diamond’s critique

Diamond (1971) raised the following critique: since workers acceptall wage offers above the reservation wage, firms have no incentiveto offer a wage w > wr . Hence the wage distribution should beconcentrated at wr . But then the option to search has no value,and so the equilibrium wage is equal to z . This in turn raises thequestion of why workers are searching in the first place?

Diamond’s critique highlights that the partial model cannot beextended to an equilibrium setting without further assumptions.

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References


On Fishing and Islands

On way to answer this critique is to vary the model’s interpretationby thinking of job seekers

as fishermen looking for lakes. Then H would be thedistribution of fish across lakes and thus can be taken asexogenous.

as searching across islands. On each island there are manyfirms with a constant returns to scale technology using onlylabor. The productivity of a randomly selected island isdistributed according to F. The labor market on each island iscompetitive, so that workers are paid their productivity. Againthe wage distribution is exogenous

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References


From the partial model to the equilibrium model

There are other possible answers (with arguably more economiccontent) to Diamond’s critique:

Worker heterogeneity: under certain conditions, the wagedistribution will correspond to the reservation wages of thedifferent categories. This explanation, however, does notexplain wage dispersion among similar workers.

On the job search: then firms have an incentive to increasetheir wage offers above the reservation wage in order to retaintheir workers and also to attract workers which are alreadyemployed.

Match uncertainty: if the productivity of the job is due tothe quality of the firm-worker match and if workers have somebargaining power, then the wage distribution will reflect theproductivity distribution of jobs.

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References

The Matching ProcessWage SettingThe EquilibriumEfficiency

Motivation

Although the basic search model can be recast as an equilibriummodel, it has the main drawback of not explaining the determinantsof job creation, since the contact rate λ is taken as given.

To devise a full equilibrium framework, we need to take intoaccount the behavior of firms. We have to specify the

Matching process: How workers and firms meet? This willbe done through the introduction of an aggregate matchingfunction.

Wage setting: How wages are determined? This will be doneby considering that wages are set through Nash-bargaining.

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References


Vacancy posting

The search intensity of firms can be quantified using the number ofvacancies posted in the economy.The coexistence of unemployed persons and vacant jobs measuresfrictional unemployment.The Beveridge Curve illustrates the negative relation between theunemployment rate u and the vacancy rate v (the ratio of thenumber of vacant jobs to the size of the labor force).

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References


Beveridge Curve

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Matching function

The relation between vacancies and unemployment can becaptured introducing an aggregate matching function.The matching function m(v , u) gives the number of jobs createdas a function of the number of vacancies and unemployed persons.Obviously, m(·) is nonnegative and increasing in both arguments.It is also commonly assumed to be concave and such thatm(0, u) = m(v , 0) = 0 for all (u, v).In order to rule out size effects, it is standard to assume that thematching function has constant returns to scale, so that

m(v , u)

v= m(1, u/v) = q(θ) ,

where the tightness parameter θ is the vacancy-unemploymentratio (v/u) and q′(θ) < 0.

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References


Foundation of matching function

One can think of the matching function as a production functionwhich, instead of mapping labor and capital into output, maps jobseekers and vacancies into matches.The easiest justification for this reduced-form specification, is torefer to coordination frictions. Assume that:

u unemployed workers know the location of the v vacanciesand send each period one application per period.

If a vacancy receives more than one application it selects anapplicant at random and forms a match.

The remaining applicants return to the unemployment pooland apply again in the next period.

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References


Coordination frictions

Each vacancy receives the application of a worker with probability1/v . Since there are u applicants, the vacancy receives noapplications with probability (1− 1/v)u.Thus the number of matches per period is

m(v , u) = v [1− (1− 1/v)u]

As u and v become large, (1− 1/v)u → e−(u/v) so that

m(v , u) = v [1− e−(u/v)] or q(θ) = 1− e(−1/θ)

This is a CRS matching function with standard properties.

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References


Nash-bargaining

Let p denote the productivity of the firm-worker match. Obviouslythe flow profit of the firm π ≡ p − w . Treating the productivity asa constant, we can define the value for the firm of the job as J(w).Let V denote the value of an open vacancy.

generalized Nash-bargaining solution

w ∈ argmax(E (w)− U)β(J(w)− V )1−β

Taking the natural logarithm and maximizing the Nash-bargainingproblem w.r.t. w yields

β[J(w)− V ]E ′(w) = −(1− β)[E (w)− U]J ′(w) (5)

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References


Foundations of Nash-bargaining

John Nash used an axiomatic approach to prove that, whenβ = 1/2, the solution described before is the only one satisfying:

1 Invariance to affine transformations.

2 Pareto optimality.

3 Independence from Irrelevant Alternatives.

4 Symmetry.

Relaxing the symmetry axiom, yields the generalizedNash-bargaining solution for any β ∈ (0, 1).

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References


Non-cooperative Foundations of Nash-bargaining

The axiomatic approach proposed by John Nash is a cooperativesolution of the bargaining game. Thus it does not provide insightson how the bargaining process actually unfolds.A game-theoretical description of the bargaining process has beenproposed by Ariel Rubinstein. Each agents makes successive offersand counter-offers. The surplus is shared only after an agreementhas been reached. Since agents discount the future they have anincentive to agree as soon as possible. Thus the bargaining poweris given by the ratio of their discount factors. The subgame perfectequilibrium converges to the Nash solution when the time intervalbetween offers and counteroffers becomes small.

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References


Analytical solution of Nash-bargaining

The asset value of the job for the firm and worker satisfy

rJ(w) = p − w + δ(V − J(w))

rE (w) = w + δ(U − E (w))

We assume free entry so that the expected value of posting avacancy is zero, i.e. V = 0. In next sub-section we will show thatfree-entry is satisfied in equilibrium , but for the moment we take itas given. Using equation (5) yields

w = βp + (1− β)rU (6)

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References


Set-up

We now combine matching and bargaining to endogenize vacancyposting decisions. We assume that:

Firms and workers are both infinitely lived and risk neutral

Workers cannot search on-the-job.

Each firm employs only one worker.

All jobs have the same output p which remains constantduring the whole lifetime of the job.

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References


Job creation

Let c denote the flow cost of vacancy posting. Then

rJ(w) = p − w + δ(V − J(w))

rV = −c + q(θ)(J(w)− V )

since q(θ) is the rate at which a vacancy is filled by a worker.Free entry is satisfied iff V = 0. Thus, in equilibrium. it must bethe case that

J(w) =p − w

r + δ=

c

q(θ)(7)

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References


Value of unemployment

Let z denote the flow value of non-market activity (unemploymentbenefit, household production...). Then

rE (w) = w + δ(U − E (w))

rU = z + θq(θ)(E (w)− U)

since θq(θ) is the rate at which a job seeker finds a job (why?).Combining the two equations, we find that the surplus from beingemployed is

E (w)− U =w − z

r + δ + θq(θ)

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References


Wage curve

To derive the wage curve, we replace (7) into the asset equationfor U

rU = z + θq(θ)(E (w)− U)

= z + θq(θ)(

β1−β

)J(w) = z +

(β

1− β

)cθ

Reinserting this expression into (6) yields

w = βp + (1− β)z + βcθ

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References


The equilibrium

The values of the three endogenous variables u, w and θ followsfrom the system of equation

Equilibrium conditions

(BC ) : u =δ

δ + θq(θ)

(JC ) :p − w

r + δ=

c

q(θ)

(W ) : w = βp + (1− β)z + βcθ

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References


The equilibrium

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References


The equilibrium

We now discuss under which conditions the allocation is efficient.We define the social optimum as the allocation which maximizesthe sum of agents’ utility; i.e. output net of the disutility of workand search costs.We simplify the problem by setting the discount rate r to zero.Then the problem of the social planner is static.

Planner’s problem

Maxθ,u Ω ≡ p(1− u) + zu − cv = p + u(z − p − cθ)

s.t. u = δδ+θq(θ)

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References


The Hosios condition

Reinserting the constraint into the objective function reduces thedimension of the optimization problem

Maxθp +δ

δ + θq(θ)(z − p − cθ)

The F.O.C. reads

(1− η(θ))(p − z)

δ + θq(θ)η(θ)=

c

q(θ)

where η(θ) = −q′(θ)(θ/q(θ)) is the elasticity of the matchingfunction.

Hosios condition

The decentralized and optimal allocation coincide when β = η(θ).

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References


Policy implications

Search frictions implies that there exists an optimal positive levelof unemployment.

In general, there are no reasons for the Hosios condition to besatisfied. This implies that there is room for optimal policyintervention when the workers’ bargaining power is too high or toolow.

For example, when β < η(θ), the introduction of a bidingminimum wage can restore efficiency.

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References

Poisson processesAsset Equation

Poisson process

The first step for deriving asset equations consists in defining thestochastic process that changes the state of the agents. Forexample, consider an unemployed worker. Suppose that on averageshe receives 3 job offers per year. Given this expected value, whatis the probability that she will receive 5 offers in one year? Can wealso compute the probability that she will receive a job offer in onemonth?It is shown in this section how these two statistics can be easilycomputed using the Poisson distribution. In order to rigorouslydefine Poisson processes, we need to introduce first Binomialdistributions.

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References


Binomial distribution

The binomial distribution is a discrete probability distributionwhich describes the number of successes in a sequence of nindependent yes/no experiments, each of which yielding successwith probability p.If the random variable X follows the binomial distribution withparameters n and p, we write X ∼ B(n, p). The probability ofgetting exactly k successes is given by

Pr X = k =

(n

k

)pk(1− p)n−k

for k = 1, 2, ..., n and where(n

k

)=

n!

k!(n − k)!

is the binomial coefficient.90 / 118


References


Pascal’s triangle

The Binomial distribution can be described visually using Pascal’striangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

...

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References


Binomial and Poisson distribution

If X ∼ B(n, p), then the expected value of X is

E [X ] = np

and the variance is

var(X ) = np(1− p)

Poisson distribution

The Poisson distribution arises when n is much larger than theexpected number of successful outcomes. In other terms, Poissonprocesses are limits of binomial processes where n →∞ and p → 0while np remains constant.

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References


Poisson random variable

Let’s go back to the example considered at the beginning of this section.We know that on average an unemployed worker receives λ = 3 job offersper year. Given this expected value, how can we model the actualdistribution of offer arrivals in a second? One possible model is to breakup each year into tiny intervals of size δ > 0 years, so there are a verylarge number, n = 1/δ, of intervals in one year.Then we declare that in each interval, a job offer arrives with probabilityp = λδ (this gives the right expected number of arrivals). So it is as if theworker were drawing a sequence of n independent yes/no experiments,each of which yielding success with probability λδ. Note that this is notquite the same as counting the number of arrivals, since more than onejob offer may arrive in a given interval. But if the interval is smallenough, this is so unlikely that we can ignore the possibility. Hence theapproximation holds in the limit where: δ → 0 ⇒ n = 1/δ →∞.

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References



Under this model, the number X of job offers which actually arrivein a year has a binomial distribution:

Pr X = k =

(1/δ

k

)(λδ)k (1− λδ)1/δ−k (8)

Now, let δ become infinitesimally small (while holding k fixed) andmake use of three approximations:(

1/δ

k

)≈ (1/δ)k

k!

(1− λδ)1/δ ≈ e−λ

1− δk ≈ 1

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References



Plugging the last equations in (8) yields

Pr X = k =

(1/δ

k

)(λδ)k (1− λδ)1/δ−k

=

(1/δ

k

)(λδ)k (1− λδ)(1−δk)/δ

≈ (1/δ)k

k!(λδ)k(1− λδ)1/δ

=λk

k!(1− λδ)1/δ

≈ λk

k!e−λ (9)

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References



What happens if we want to answer a question that involves aninterval of t > 0 units of time rather than 1 unit of time? Then wehave a new random variable Xt that is Poisson distributed withmean λt. The probability of seeing exactly k counts in t units oftime is

Pr Xt = k =(λt)k

k!e−λt

The probability distribution (9) is known as the Poissondistribution. When system events appear according to a Poissondensity, the system is called a Poisson process.

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References


Properties of the Poisson distribution

As a sanity check on our distribution, the probability values (9)had better sum to 1. Using the Taylor expansion for eλ , we canverify that they do:∑

k∈N

PrX = k =∑k∈N

e−λ λk

k!= e−λ

∑k∈N

λk

k!= e−λeλ = 1

A further sanity check is that the expected number of arrivals in ayear is indeed λ, namely

E [X ] = λ

Similarly, the binomial distribution B(n, p) has variance np(1− p).Since the Poisson distribution is the limit of B(1/δ, λδ) as δvanishes, it ought to have variance

Var [X ] = (1/δ)(λδ)(1− λδ) = λ(1− λδ) ≈ λ

The final approximation holds since 1− λδ ≈ 1 for vanishing δ. 97 / 118


References


Pr X = k for k = 1, 2, ..., 10. when E [X ] = 3

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References


Waiting time and exponential distribution

Now we consider the waiting time W , which is the time onemust wait to see the next count. The following two events areequivalent: W > t = Xt = 0. Both conditions say that nocounts arrive in the first t units of time. Hence

Pr W > t = Pr Xt = 0 = e−λt (10)

Furthermore, since Pr W > t = 1− Pr W ≤ t , the cumulativedistribution of W is F (t) = Pr W ≤ t = 1− e−λt .Finally, the density function f (t) of W can be found bydifferentiating F (t) with respect to t. We obtainf (t) = F ′(t) = λe−λt , for t > 0. This is the density function ofthe so-called exponential distribution with rate λ. It has meanE [W ] = 1/λ. Intuitively, it means that the worker waits 1/3 = 4months before receiving an offer.

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References


Waiting time and exponential distribution

Notice that a Taylor approximation of Pr W ≤ t aroundt = 0 yields Pr W ≤ t →

t→0λt. This is why people often refer to

λ as the “instantaneous rate of arrival”.An important property of the exponential distribution is that it is”memoryless.” This means that, for 0 < s < t,Pr W > t|W > s = Pr W > t − s . Given that there havebeen no counts up to time s, the conditional probability that thereare no counts up to the later time t is just the probability of seeingno counts in a period of length t − s.For example, if a machine breaks down according to a Poissonprocess, it doesn’t matter if it is 10 years old or just new.Surprisingly, the exponential law has remarkable explanatory powerfor the breakdown probability of many devices, one example beinglight bulbs.

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References


Asset Equation

We want to derive the expected income of an unemployed worker.We begin by setting up the Bellman equation as though we weresolving a discrete time problem. Let ε denote the length of a “timeperiod”.While unemployed, the worker gets unemployment benefits equalto b at every instant. He consumes all his income. This payoff mustbe discounted back to the beginning of the “period,” which is now,and that is the role played by the“instantaneous” discount factor

1

1 + rε

Thereforeu(b)ε

1 + rεis the discounted value of the current period return over the shortinterval ε.

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References


Asset Equation

Now, we assume that the worker randomly meet firms and thenchanges state from unemployed to employed. This event ismodeled using a Poisson process: so in any small interval of timeε, the worker meets at most one potential employer. From(10) , we know that the likelihood of of meeting an employerduring the time interval ε is equal to e−λε.Let U and E denote the expected lifetime utility of an unemployedand employed worker

U (ε) =

(1

1 + rε

) (u(b)ε +

(1− e−λε

)E (ε) + e−λεU (ε)

)⇒ rεU (ε) = u(b)ε +

(1− e−λε

)(E (ε)− U (ε))

⇒ rU (ε) = u(b) +

(1− e−λε

ε

)(E (ε)− U (ε)) (11)

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References


Asset Equation

Now consider the limit of (11) when ε → 0. The numerator and

denominator of(

1−e−λε

ε

)tend to zero. Hence we can use

l’Hospital’s rule to obtain the limit

limε→0

1− e−λε

ε=

limε→0

(∂(1−e−λε)

∂ε

)limε→0

(∂ε∂ε

) = limε→0

λe−λε = λ (12)

Substituting (12) into (11) yields

rU︸︷︷︸flow return

= u(b)︸︷︷︸flow income

+ λ (E − U)︸︷︷︸capital gain

(13)

where U ≡ limε→0

U(ε) and E ≡ limε→0

E (ε).

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References


Asset Equation

There is a clear analogy with the pricing of financial assets (whichyield periodic dividends and whose value may change over time), ifwe interpret the left-hand side of (13) as the flow return(opportunity cost) that a risk neutral investor demands if sheinvests an amount in a risk free asset with return r . Theright-hand side of the equation contains the two components ofthe flow return on the alternative activity “unemployment”: theexpected dividend derived from consumption, and the expectedchange in the asset value of the activity or capital gain due to theswitch from unemployment to employment.This interpretation justifies the term “asset equations” forexpressions like (13) .

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References

References

CZ, Chapter 3 and 9.

P, Chapter 1.

Rogerson, R., Shimer, R. and Wright, R., (2005),”Search-Theoretic Models of the Labor Market: A Survey”,Journal of Economic Literature, Vol. XLIII, pp. 959.98

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No-shirking conditionEmployers

EquilibriumReferences

Part IV

Efficiency Wage

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Outline

13 No-shirking condition

14 Employers

15 Equilibrium

16 References

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Unemployment as a discipline device

The efficiency wage model of Shapiro and Stiglitz (1984) showsthat if employers cannot perfectly and costlessly monitor the effortsof their employees, the economy will exhibit involuntaryunemployment in equilibrium.The intuition is straightforward. The only risk induced by shirkingis to be fired:

If there is no unemployment, shirkers immediately find a newjob and thus pay no penalty. Moreover, when monitoring isimperfect, shirkers are also employed for some periods. Henceshirkers have the same lifetime income than non-shirkers but ahigher utility since they provide no effort.

If there is unemployment, workers have an incentive not toshirk. For if they are fired, they will incur some losses byremaining unemployed for some time.

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Workers’ utility function

The worker’s utility U(w , e) depends positively on the wage w andnegatively on the effort e so that Uw (w , e) > 0 and Ue(w , e) < 0.

For simplicity, we assume that the utility function is separable andthat workers are risk neutral so that: U(w , e) = w − e.

Again, for simplicity, we consider that efforts can take one of onlytwo values 0, e where e > 0.

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Workers’ asset values

We assume that:

Workers discount the future at rate r .

Jobs are destroyed at the exogenous rate δ.

Shirkers are detected at rate d .

Notice that when d →∞, detection is immediate. In other words,monitoring is perfect and the equilibrium converges to the one in astandard economy without moral hazard problem.

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No-shirking condition

Let ES denote the asset value of a shirker and ENS that of anon-shirker. Decomposing the asset values yields

rES = w + (δ + d)(U − ES)

rENS = w − e + δ(U − ENS)

No-Shirking Condition

ENS ≥ ES ⇒ w ≥ rU +

(r + δ + d

d

)e ≡ w (14)

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No-shirking condition

To interpret the NSC, notice that it is equivalent tod(ES − U) ≥ e. Thus when ES = U, so that there is no penaltyassociated with being unemployed, everyone will shirk.

w ↑ when

e ↑, since workers require more compensation for their efforts.

U ↑, since the cost of being fired is lower.

d ↓, since the likelihood of being detected shirking decreases.

δ, r ↑, since the expected returns from the job are moreheavily discounted.

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Labor demand

There are M identical firms, each with a production functionQi = f (Li ) where Li is efficient labor.

Without loss of generality, we assume that workers that do notshirk provide one unit of efficient labor and 0 unit if they shirk.

Naturally, firms offer the lowest wage consistent with the NSCcondition (14), so that f ′(Li ) = w . This yields an aggregate labordemand F ′(L) = w .

Finally we assume that F ′(N) = e to ensure that full employmentis efficient.

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Equilibrium wage

The asset value for U is

rU = z + λ(E − U)

where λ is the job finding rate and E = ENS since workers do notshirk in equilibrium.

Combining this equation with the asset value for ENS , we obtain

w = e + z + (r + λ + δ)(E − U)

As explained before, the NSC condition is equivalent tod(ES − U) ≥ e. In equilibrium, ES = ENS = E , so we can replacethis equality into the previous equation to obtain

w = e + z +r + λ + δ

de (15)

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Equilibrium unemployment

As explained in Part III, the equilibrium rate of unemployment setsthe flows in and out of the unemployment pool equal. Hence,u = λ/(δ + λ).Replacing the equilibrium wage into the No-shirking wage (15)yields

w = e + z +e

d

(δ

u+ r

)Hence when u → 0, w diverges to infinity. This shows that itbecomes virtually impossible to motivate workers as unemploymentdisappears.

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Equilibrium employment

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Welfare Analysis

Solving the social planner problem shows that the optimum occursat the point where the NSC curve intersects the Average productof labor locus.

When returns to labor are decreasing, this implies that wages aretoo low and unemployment too high in the decentralized economy.This is because the private cost w is superior to the social cost e.

Hence, there is room for beneficial policy intervention. Namely,wages should be subsidized by taxing away all pure profits.

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References

Shapiro, C. and Stiglitz, J.E., 1984, ”Equilibrium Unemploymentas a Worker Discipline Device”, American Economic Review, Vol.74(3), p.433-444.

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