chapter 16huihe.weebly.com/uploads/1/3/6/1/13611032/ppt16.pdf · • next, we want to consider the...

Chapter 16

Unemployment: Search and Efficiency Wages

Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 16-2

Goal

• Examine the behavior of the unemployment and labor participation rate in U.S.

• Study two models of determination of unemployment – Search model – Efficiency wage model


Unemployment and Labor Participation Rate

• E: number of working age persons who are employed.

• U: number of unemployed. • NL: number of working age persons

who are not in the labor force


• Unemployment rate =

• Participation rate =

UE U+

E UE U NL

++ +


Figure 16.1 The U.S. Unemployment Rate, 1948–2003


Figure 16.2 Deviations from Trend in the Unemployment Rate and Percentage Deviations from Trend in Real GDP for 1948–2003


Determinants of the Unemployment Rate

• Aggregate economic activity: Y↑, u↓ • Demographics: baby boom and baby

bust. Young tend to have higher u. • Government intervention:

Unemployment Insurance (UI). UI↑, u↑ • Sectoral shifts: from manufacturing to

services, structural unemployment.


Stylized Facts of Participation Rate

• Participation rate is procyclical. • After WWII, the increase in the total

participation rate is accounted for solely by an dramatic increase in the participation rate of women.

• Women also had less children, more willing to go to college.


Figure 16.3 The U.S. Participation Rate, 1948–2003


Figure 16.5 Deviations from Trend in the Participation Rate and GDP


Figure 16.4 Labor Force Participation of Women and Men


College Enrollment Rate of Recent HSG: Smoothed Data

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

year

male female

Source: He (2011)


Total Fall Enrollment of Degree-Granting Higher Education Institution

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

10,000,000

1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997

year

male female


Table 16.1 Average Unemployment Rates in OECD Countries, 1983-1996


A Search Model of Unemployment

• In the previous chapters, we assume that consumers typically use their time for only two different purposes: working and leisure.

• In this chapter, we will extend this assumption. People can also use time to search for a job while they are unemployed.


• The infinite-lived worker has preferences given by

00

1( ) ( )1

tt

tE U C

r

∞

= +∑


• We assume the workers are all in the labor force. Let U denote the fraction of workers who are unemployed. 1-U is the employment rate.

• is the value function of being employed at real wage w.

• is the value function of being unemployed.

• Let s denote the separation rate (firing or quitting)

( )eV w

uV


Value of Being Employed

1( ) [ ( ) (1 ) ( )]1e u eV w U w sV s V w

r= + + −

+


• Simplify the equation, we get

• Hence

( ) ( ) [ ( )]e u erV w U w s V V w= + −

( )( ) ue

U w sVV wr s+

=+


• Therefore inherits the properties of U(w). It is strictly increasing and concave. We also have

• Separation rate ↑ → • Tax on wage ↑ →

( )eV w

( ) ( )0, 0e e

w

V w V ws τ

∂ ∂< <

∂ ∂( )eV w ↓

( )eV w ↓


Figure 16.6 The Welfare of an Employed Worker


Value of Being Unemployed

• Next, we want to consider the welfare of an unemployed worker. Let b denote the constant real amount of UI, p be the frequency of receiving job offers.

• Assume wage offer is a random draw from the probability distribution F(w), with the probability density function f(w).


0

1 { ( ) (1 ) max[ , ( )] ( ) }1

w

u u u eV U b p V p V V w f w dwr

−

= + − ++ ∫


• Simplify the equation

0( ) max[0, ( ) ] ( )

w

u e urV U b p V w V f w dw−

= + −∫

0, 0, 0u u u

b

V V Vb p τ

∂ ∂ ∂> > <

∂ ∂ ∂


• Unemployment benefit b ↑ → • Frequency of job offers p ↑ → • Tax on b ↑ →

uV ↑

uV ↑

uV ↓


• The value function of a unemployed worker who receives wage offer w is

( ) max( ( ), )e uV w V w V=


Reservation Wage

• The criteria of choice is based on if accept the offer; otherwise, turn it down • w* is called the reservation wage such that

*, ( )e uw w V w V≥ ⇒ ≥ ⇒

( *)e uV w V=


Figure 16.7 The Reservation Wage


Effects on Reservation Wage

• UI↑ → → w*↑, unemployed workers are more picky

• Tax on wage ↑ → → w*↑, unemployed workers are more picky

uV ↑

( )eV w ↓


Figure 16.8 An Increase in the Unemployment Insurance Benefit b


Figure 16.9 An Increase in the Taxes on the Wage Income of the Employed


The Determination of the Unemployment Rate

• The flow of workers from employment to unemployment is s(1-U)

• Let H(w) denote the fraction of unemployed workers receiving a wage offer is greater than w. H(w) (hazard rate) is decreasing in w.

( ) 1 ( )H w F w= −


• The flow of workers from unemployment to employment is UpH(w*).


Long-run Equilibrium

• The labor market equilibrium is determined by

(1 ) ( *) (1 ( *))s U UpH w Up F w− = = −


• The equilibrium unemployment rate U* is given by

1*[1 ( *)] [1 ( *)] 1

sU pp F w s F ws

= =− + − +


• We have

* * *0, 0, 0*

U U Uw s p

∂ ∂ ∂> > <

∂ ∂ ∂


Figure 16.10 The Fraction of Unemployed Workers Receiving a Wage Offer Greater than w


Figure 16.11 The Determination of the Unemployment Rate U in the Search Model


Figure 16.12 The Determination of the Reservation Wage and the Unemployment Rate in the Search Model


Policy Experiments

* ( *) *

* ( *) , ( *)? *?

( ) * ( *) *

, ( ) , * *

u

u

w e

w b e u

b V w H w Up V w H w UpH w U

V w w H w U

V w V w U

τ

τ τ− −

• ↑⇒ ↑⇒ ↑⇒ ↓⇒ ↑

• ↑⇒ ↑⇒ ↑⇒ ↓ ⇒

• ↑⇒ ↓⇒ ↑⇒ ↓⇒ ↑

• ↑ ↑⇒ ↓ ↓⇒ ⇒


Figure 16.13 An Increase in the Unemployment Insurance Benefit b


Figure 16.14 An Increase in the Job Offer Rate p


Figure 16.15 An Increase in Taxes on Labor Income


Figure 16.16 Taxes on Labor Income and Unemployment Benefits

Optimal UI

• Shavell and Weiss (1979) and Hopenhayn and Nicolini (1997)

• A worker chooses consumption c and search effort α to maximize life-time utility


[ ]ttt

t acuE −∑∞

=

)(0β

• We assume the prob. Finding a job (α=0 if employed)


0)0(,0)('',0)('

],1,0[)(

=<>

∈

papap

ap

Autarky Problem • If an unemployed worker does not

access to UI

• FOC (with equality if α>0, IC constraint)

• Solution:


{ }]))(1()([)0(max1

)(

0 ueau

e

VapVapauV

wuV

−++−=−

=

≥ ββ

1])[(' ≤− ue VVapβautu VVaa == ,*

UI with Full Information

• An insurance agency can observe and control

• Insurance agency wants to design an optimal UI to minimize the discounted cost C(V) of giving the worker expected discounted utility

• Given V, contract is if the worker is unemployed Copyright © 2005 Pearson Addison-Wesley. All rights reserved. 16-50

),( ac

),,( uVacautVV >

Optimal UI with Full Information


{ }

})](1[)({)()constraint keeping-(promise ..

)()](1[)( min,,,

ue

uVac

VapVapacuVts

VCapcVCu

−++−≤

−+=

β

β

FOCs

• Let θ be the multiplier for PK constraint


θβ

θ

θ

=

−−=

=

)('

)()('

1)(

)('1

u

ueu

VC

VVap

VC

cu

• Given the envelope condition , we have

• The worker’s consumption is “fully smoothed” under unemployment spell

• Both c and search effort α are constant during the unemployment spell


θ=)(' VC

VVVCVC uu =⇒= )(')('

Optimal UI under Asymmetric Information • Insurance agency can control c, but

cannot observe α • The fact that contract is costly

and the worker does not take the cost of insurance into account induces the worker will choose a search effort below the social optimal one.


0)( >uVC

soue

soue

so

aaaVVapaVVap

<⇒

>−=

>−>−

−

0 optimal privatefor )()]('[

0 optimal socialfor )()]('[

*

*1*

1

β

β

Agency Problem under AI


{ }

(IC) 1])[('(PK) })](1[)({)(

..

)()](1[)( min,,,

≤−−++−=

−+=

ue

ue

uVac

VVapVapVapacuV

ts

VCapcVCu

ββ

β

FOCs


)(1)(')('

)()(')('')(

)(')('')(

)('1)(

)('1

apapVC

VVapapVV

apapVV

apVC

cu

u

ueueueu

−−=

−−=−−

−−=

=

ηθ

ηηβ

θ

θ

Optimal UI under AI

• FOCs implies since • So we have

• c↓ as the duration of unemployment spell increases

• IC constraint also implies α ↑ as

16-57

0>η

VVVCVC

u

u

<⇒< C(.) ofconvexity strict theplus ),(')('

↓uV

0)( >uVC

↓↓→⇒−

−= cVap

apcu

VC uu )(1)('

)('1)(' η

Optimal UI under AI

• “Carrot-and-stick” strategy • UI replacement ratio UI c/w should

decrease as the duration of unemployment spell increases

• Carrot: high UI benefit and low search effort in an early spell

• Stick: low UI compensation and high search effort later in the spell



Efficiency Wage Model • An alternative model of unemployment • Key assumption: workers’ effort

depends on their wage • To induce workers to work harder, a firm

may be willing to pay a real wage higher than the competitive market wage

• This higher efficiency wage thus creates unemployment


Effort of the Worker

• A worker’s effort increases with the wage

• Effective labor input is working hours N

multiplied by effort

( ), '( ) 0e w e w >

( )EN e w N=


Why e(w) increases with w? • Adverse Selection: Firm does not

know the productivity of workers. Higher wage will more likely attract high-ability workers

• Moral Hazard: Firm has difficulty in monitoring the on-the-job effort of its workers. Higher efficiency wage means the opportunity cost of shirking increases


Firm’s Optimization Problem • Production function • Choose N and w to maximize profits

( , ( ) )Y zF K e w N=

( , ( ) )F K e w N wNπ = −


FOCs • FOC w.r.t. N

• FOC w.r.t. w

( ) 0 ( )ENF e w w MP e w w

N ENπ∂ ∂= − = ⇒ =

∂ ∂

'( ) 0 '( ) 1ENF e w N N MP e w

w ENπ∂ ∂= − = ⇒ =

∂ ∂


Determination of Efficiency Wage • Combining two FOCs, we have

efficiency wage is determined by

**

*

( ) '( )e w e ww

=


Figure 16.17 Effort of the Worker as a Function of His or Her Wage


Figure 16.18 The Demand for Labor in the Efficiency Wage Model


Figure 16.19 The Ratio of Effort to the Real Wage


Figure 16.20 Determination of the Efficiency Wage


Labor Market Equilibrium • When efficiency wage > market wage,

there is unemployment. • When efficiency wage < market wage,

there is labor shortage. Firms bid up to the market wage to attract workers.


Figure 16.21 Unemployment in the Efficiency Wage Model


Efficiency Wage and Business Cycle

• Output supply curve is vertical (Why?) • Shock on demand side will not affect

output, e.g., change in G • Only shock on output supply curve can

affect aggregate output, e.g., TFP z


• Equilibrium effect of z↑: Y↑, r↓, w* no change, N↑, u↓, C↑, I↑, • Consistent with most of facts except

wage


Figure 16.22 The Output Supply Curve in the Efficiency Wage Model


Figure 16.23 An Increase in G in the Efficiency Wage Model


Figure 16.24 An Increase in Total Factor Productivity in the Efficiency Wage Model


Can We Fix It?

• Assume z affects effort, e(w)↑ as z↑ • Efficiency wage w*↓ • Wage is countercyclical, contrast to the

data! • Conclusion: Efficiency wage model is

not consistent with all business cycle facts


Figure 16.25 An Increase in the Effectiveness of Monitoring by the Firm

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