macro lecture7

Unemployment and e¢ ciency wages

Ester Faia, Ph.D.

Goethe University Frankfurt

Dec 2010

Ester Faia, Ph.D. (Goethe University Frankfurt) Unemployment and e¢ ciency wages 12/10 1 / 27

Empirical evidence raises the following questions

What determines equilibrium unemployment

If labour supply is bigger than labour demand, why don�t wages fall?

What makes unemployment persistent, at least in Europe (hysteresise¤ect)?

What can be done to reduce unemployment?


E¢ ciency wages1 and hystersis

Bene�ts of higher wages:

better nutrition of workersprevent shirking when monitoring is imperfecthigher average quality of applicant poolbuild loyalty among worker

Hysteresis

The equilibrium level of unemployment depends on the equilibriumlevel in the past.

1Reference: Romer, Ch. 10, Sec. 10.2 - 10.4Ester Faia, Ph.D. (Goethe University Frankfurt) Unemployment and e¢ ciency wages 12/10 3 / 27

Simplest model of e¢ ciency wages

Assumptions:

N identical competitive �rms

L identical workers o¤ering one unit of labor inelastically

Firms maximize pro�t:π = Y � wL (1)

Output depends on labor input L and e¤ort e:

Y = F (eL), F 0 > 0, F 00 < 0 (2)

E¤ort depends positively on the wage rate:

e = (ew) , e 0 > 0 (3)


Problem of the �rm

maxL,w

π (4)

π = F [e (w) L]� wL (5)

(price of output = 1)

FOC w.r.t. L :

∂π

∂L= F 0 [e (w) L] e (w)� w = 0 (6)

FOC w.r.t. w :

∂π

∂w= F 0 [e (w) L] e 0 (w) L� L = 0 (7)

Rearranging:F 0 [e (w) L] e (w) = w (8)

F [e (w) L] e 0 (w) L = L (9)


Problem of the �rm

Dividing (9) by (8) and rearranging, we get:

ηew =we 0 (w)e (w)

= 1 (10)

(10) determines the wage

Given w , (6) determines L.

Labour demand of one �rm is L� (the solution to (6))

Aggregate labor demand is NL�

Hence unemployment is:

U = max f0, L�NL�g (11)


More general e¤ort function

Now we assume the following:

e = e (w ,wa, u) ,∂e∂w

> 0,∂e

∂wa< 0,

∂e∂u> 0 (12)

where wa is average wage in economy and u is unemployment rate

Solution:

ηew =we 0 (w ,wa, u)e (w ,wa, u)

= 1 (13)

w = F 0 [e (w ,wa, u) L] e (w ,wa, u) (14)


Shapiro-Stiglitz model

This is a more complete model:

Endogenous e¤ort decisionsDynamic model

Assumptions:V (t) = e�ρtu (t) dt (15)

ρ = discount rate

Continuous time version of V (t) =∞

∑t=0

�11+ρ

�tu (t)

In fact e�ρt = 1eρt � 1

1+ρ , where

u (t) =�w (t)� e (t) if employed

0 if unemployed(16)


Shapiro-Stiglitz model (con�t)

Two possible e¤ort levels: e = 0 or e = e

Three possible states:

U: unemployedE : employed and exerting e¤ort (e = e)S : employed and shirking (e = 0)

Job breakups: if worker is employed at time t and exerts e¤ort, theprobability of being employed at time t + ∆ (∆ is small) is:

e�b∆ � 1� b∆ (17)

b = "instantaneous" probability of being �red


Shapiro-Stiglitz model (con�t)

If a worker shirks, he or she is caught (and �red) with probability:

1� eq∆ � q∆ (18)

If a worker is unemployed at time t, the probability of being stillunemployed at time t + ∆ is:

e�a∆ � 1� a∆ (19)

The �rm maximizes pro�ts:

π (t) = F (eL (t))� w (t) [L (t) + S (t)] (20)

L : number of employees exerting e¤ort

S : number of employees shirking


Shapiro-Stiglitz model: solution

VE , VS , VU denote the value function for the household if it is instate E , S and U, respectively

V (t) = permanent income = current income+ future value function

Example with no e¤ort, employed forever, discrete time:

V (t) = w (t) +w (t + 1)1+ ρ

+w (t + 2)

(1+ ρ)2+w (t + 3)

(1+ ρ)3+ ... (21)

V (t) = w (t) +1

1+ ρ

"w (1+ t) +

w (t + 2)1+ ρ

+w (t + 3)

(1+ ρ)2+ ...

#(22)

V (t) = w (t) +1

1+ ρV (t + 1) (23)


Shapiro-Stiglitz model: solution (con�t)

If the worker exerts e¤ort:

VE (t) = (w (t)� e)∆+ e�ρ∆he�b∆VE (t + ∆) +

�1� e�b∆

�VU (t + ∆)

i(24)

In our case the solution is time invariant (omit t)

With the approximations (valid for very small ∆):Equation (24) becomes:

VE = (w � e)∆+ (1� ρ∆) [(1� b∆)VE + b∆VU ] (25)

= (w � e)∆+ VE � (ρ+ b)VE∆+ bVU∆+ ρb (VE � VU )∆2

Since ∆ is very small, we can neglect the term in ∆2:

VE = (w � e)∆+ VE � (ρ+ b)VE∆+ bVU∆ (26)



Then we subtract VE on both sides of (26), divide by ∆ and obtain:

ρVE = (w � e)� b (VE � VU ) (27)

Analogously, the value for the worker that shirks is:

ρVS = w � (b+ q) (VS � VU ) (28)

And the value for the unemployed is:

ρVU = a (VE � VU ) (29)

assuming worker exerts e¤ort if she �nds job.



No-shirking condition:VE � VS (30)

! VE = VS (31)

Equations (27) and (28) then imply:

(w � e)� b (VE � VU ) = w � (b+ q) (VS � VU ) (32)

! VE � VU =eq

(33)

Wage: Equations (27) and (29) imply:

w = e + (a+ b+ ρ)eq

(34)



Steady state:

a =NLbL�NL (35)

Substituting (35) in (34):

w = e +�

LL�NLb+ ρ

�eq= e +

�bu+ ρ

�eq

(36)

where u = L�NLL is the unemployment rate.

Labor demand:Conditional on no shirking, (20) becomes:

π = F [eL]� wL (37)

FOC w.r.t. L :∂π

∂L= 0 ! eF 0 (eL) = w (38)

The intersection of (38) and (36) gives w and L in the steady state.This also determines the unemployment rate in equilibrium.



Embedding the wage function (36) into a Keynesian model with rigidgoods prices and imperfect competition (case 3 of Romer, Section5.4), labor demand is determined by aggregate goods demand:

F�eLD

�= Y agg .dem. (39)

Under the condition that

eF 0�eLD

�� w = W

P(40)

So that price is greater than marginal cost:

P � WeF 0 (eLD )

(41)


Hystersis

Hysteresis means that the equilibrium level of unemployment dependson the level of unemployment in the past

There are basically two explanations for hysteresis: the changingnumber of insiders, and changes in search behavior.


Insider-Outsider models2

An insider is a privileged participant in the labor market

Firms �rst hire insiders, and hire outsiders only if labor demand isbigger than the number of insiders

Wage is set by insiders

Employment is chosen by the �rm

Crude assumption: number of insiders is equal to last period�semployment:

NIt = Lt�1 (42)

Firms and insiders are myopic: they maximize current period objectivefunction and neglect impact on future number of insiders

2Reference: Romer, Ch. 10, Sec. 10.6 - 10.7Ester Faia, Ph.D. (Goethe University Frankfurt) Unemployment and e¢ ciency wages 12/10 18 / 27

Insider-Outsider models (con�t)

Firm�s pro�t function:

πt = AtLαt � wrLt (43)

where At is a stochastic productivity parameterThe crucial assumption is that all workers are paid the same wage:the �rms cannot replace insiders by outsiders at a lower wageFirms maximize (43) by choosing Lt , at a time when At is knownThis gives the �rst order condition

AtαLα�1t � wt = 0 (44)

This gives the labor demand function:

Lt = Ctw�βt (45)

Ct =

�1

αAt

� 1α�1

β =1

1� αEster Faia, Ph.D. (Goethe University Frankfurt) Unemployment and e¢ ciency wages 12/10 19 / 27


We now assume that the shocks on At are such that:

Ct = Ctεt (46)

where Ct is known when the wage is set, and εt is an i.i.d. shockdetermined after wt is set, with E (ε) = 1

We normalize the utility of an unemployed insider to 0, and assumethat the utility of employed insiders is:

U (w) = wγ, 0 < γ < 1 (47)

If insiders�union cares about the expected utility of its representativemember, its utility function is given by:

ut = E�min

�LtNIt, 1��wγt (48)



The term in brackets is the fraction of insiders employed, the secondterm is the utility of insiders conditional on being employed

Substituting labor demand (45) into (48) gives:

ut = E

"min

(Ctεtw

�βt

NIt, 1

)#wγt

Let us now de�ne:

xt =CtNItw�βt (49)

From (45) and (46) we see that xt is the ratio of employment to thenumber of insiders, if εt takes its expected value of 1.



Using (49), we can rewrite the insiders�problem as maximizing:

ut = E [min (εtxt , 1)] x�γ/βt

�CtNIt

�γ/β

over xt . Then it is apparent that the optimal xt , called x�, is a constant(does not depend on Ct and NIt). This means:

wt =�NItCt

��1/β

(50)

andLt = εtNItx

� = εtLt�1x�

In logs we can write this as:

log Lt = const + log Lt�1 + log εt

which shows that the log of L is a random walk with drift. Changes ofemployment have permanent e¤ects, there is no natural rate ofemployment to which Lt would revert.



Explanation: lower imployment implies:

fewer insiders next periodhigher wage next period, because the risk of becoming unemployed islower for an insiderlower employment next periodThe key is that outsiders have no power to bid the wage down toincrease their chance of getting employment


Search behavior3

Another explication of hysteresis is that the search intensity of thelong-term unemployed has decreased, so that the existence of longterm unemployment does not put a downward pressure on wages.

As in the Shapiro-Stiglitz model, we assume that, in each period, bLpeople loose their job, where b is the �job separation rate�. However,the number of people who �nd a job H is not just a function of theunemployment rate, but also of the available vacancies:

H = h (V , cU) (51)

V is the number of vacancies, U the number of unemployed, and c isthe search intensity of the unemployed. We assume that h is linearlyhomogenous.

3For a formal search model see Romer, Ch. 10, Sec. 10.8Ester Faia, Ph.D. (Goethe University Frankfurt) Unemployment and e¢ ciency wages 12/10 24 / 27

Search behavior (con�t)

In a stationary equilibrium with constant unemployment rate, thenumber of people who lose their job is equal to the number of newmatches:

bL = H (52)

Using (52) and the linear homogeneity of (51) we get:

b = h�VL,cUL

�(53)

orb = h (υ, cu) (54)

where υ = V/L is the vacancy rate u = U/L is approximately theunemployment rate (more precisely, the unemployment rate is de�nedas U/(L+ U)).For given m and c , (54) implicitly describes a relationship betweenthe vacancy and the unemployment rate which is called the Beveridgecurve.



Empirical research has found that the matching function h can bewell approximated by a Cobb-Douglas function, so that we can write(53) as:

b = υξ (cu)1�ξ (55)

Empirical studies �nd values of ξ around 0.5

Solving for u we get:

u =1cb1/(1�ξ)υ�ξ/(1�ξ) (56)

From (56) a decrease in search intensity means that the Beveridgecurve shifts outward, something that has been observed in manyEuropean countries in the last decades.



The equilibrium unemployment rate depends on the job separationrate b and also on how many vacancies �rms post. If we assume that,in equilibrium:

V = υU (57)

where υ is a constant that describes �rm behavior, we obtain:

υ = υu (58)

and we can solve (55) for the equilibrium unemployment rate:

u =b

υξc1�ξ(59)

(59) shows how unemployment decreases in search intensity.


macro lecture7

Documents