Advanced Modern Macroeconomics · Chapter 9: Employment Chapter Summary Dynamic labor market:...

Advanced Modern MacroeconomicsEmployment

Max Gillman

University of Missouri-St. Louis

3 October 2016

Gillman (University of Missouri-St. Louis) Chapter 9 3 October 2016 1 / 45

Chapter 9: EmploymentChapter Summary

Dynamic labor market: supply, demand, comparative static changes.Replicate business cycle change in employment.

Goods productivity factor: wage moves cyclicallybut no change in employment.Substitution, income e¤ects o¤set again in dynamic model

Combine goods productivity and time endowment changes

wage, employment move cyclically.

Add �xed wage to decrease in goods, time endowment

replicate a more severe recession,excess supply of labor, or unemployed labor.With less employment than when �exible wage.

Tax on labor income shifts back supply, demand for output, labor

less output and employment; wage rate the same.Tax policy important for employment level in dynamic model.


Building on the Last Chapters

Last chapter presented baseline dynamic model.

This chapter derives baseline dynamic labor market,

presents business cycle employment theory, and tax e¤ect

Same examples as in Chapter 8:

Examples 8.1, 8.2, 8.3 extended to Examples 9.1, 9.2, 9.3.Business cycle combines Examples 9.2 and 9.3.dynamic extension of static business cycle in Chapter 3;also goods and time endowment changes combined.Capital stock change a¤ects dynamic employment explanation.

Fixed wage, surplus labor, similar to Chapter 3

shows dynamic model gives similar �xed wage analysis.

Dynamic model labor tax compares to static model.

Decreases capital, shifts back labor demand, unlike Chapter 3.Supply, demand both shift back, leave wage unchanged;wage rose in Chapter 3.


Learning Objective

Dynamic equilibrium change in employment

contrasted to unemployment caused when wage cannot adjust.

Business cycle employment change dynamically generated

with same simple comparative static changesin goods and time endowments,but not with change in either one alone.

Tax increases also cause employment to decrease

must be used cautiously in any recessionary policy.

All viewed within dynamic context, with change in capital stock,

and additional shifts in supply, demandnot present in static analysis of Chapter 3.


Who Made It Happen

Pigou 1933 Theory of Unemployment writes (p. 252)"The implication is that such unemployment as exists at any time isdue wholly to the fact that changes in demand conditions arecontinually taking place and that frictional resistances prevent theappropriate wage adjustments from being made instantaneously."

Frictional unemployment of labor today viewed within equilibrium,not with �xed wage, but main alternative is sluggish wage adjustment.

Recessional wage in�exibility: Keynes�s General Theory (1936)1930s Great Depression, excess labor supply, �xed wage.

Modern business cycles: Kydland, Prescott (1982), Long, Plosser.Goods productivity factor causes business cycles,called "real business cycle theory".Labor employment does not move as much as in data

Hansen (1985), Rogerson (1988), rectify with another labor marginenter the labor force or to leave it; "external margin".relates here to exogenous change in time endowment.


Aggregate Labor Supply and Demand

cdt =1

1+ α(wtT + ρkt ) ; xt =

αcdtwt; l st = T � xt .

l st = T �αcdtwt

= T �α1+α (wtT + ρkt )

wt= T � α

1+ α

�T +

�ρ

wt

�kt

�.

ldt =�

γAGwt

� 11�γ

kt .

wt =αρkt

T � (1+ α) l st; wt = γAG

�ktldt

�1�γ

. (1)


Example 9.1: Labor Market Equilibrium

Calibration: Assumed Parameter Values:γ = 1

3 , α = 0.5, ρ = 0.03, T = 1. AG = 0.15;Equilibrium: kt = 2.3148.

Labor Supply : wt =0.5 (0.03) 2.31481� (1.5) l st

, (2)

Labor Demand : wt =13(0.15)

�2.3148ldt

� 23

. (3)


Baseline Labor Market in Example 9.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

Labor Employment

w

Figure 9.1. Labor Market in Baseline Dynamic Model of Example 9.1.


Excess Labor Demand is Zero at Equilibrium Wage

L (wt ) � ldt � l st ;

L (w) =

�γAGwt

� 11�γ

kt ��1� α

1+ α

�1+

�ρ

wt

�kt

��= 0;

L (w) =

�0.153wt

�1.52.3�

�1� 1

3

�1+

�0.03wt

�2.3��

= 0.

wt = 0.1389.

ldt =

�γAGwt

� 11�γ

kt =�

0.153 (0.1389)

�1.52.3148 = 0.50,

l st = 1� α

1+ α

�1+

�ρ

wt

�kt

�= 1� 1

3

�1+

�0.030.1389

�2.3�= 0.50.

Excess demand L (w) > 0, wt < 0.1389,Excess supply L (w) < 0, wt > 0.1389.


L(w) Function for Example 9.1

0.135 0.136 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144

0.04

0.02

0.00

0.02

0.04

w

L(w)

Figure 9.2. Implicit Solution of the Real Wage from L(w) = 0 in Example9.1.


Productivity Increase: Example 9.2

AG rises 5%; AG = 0.15 ! AG = (0.15) (1.05) = 0.1575;

γ = 13 , α = 0.5, ρ = 0.03, T = 1; .kt rises from 2.3148 to 2.6797.

Labor demand a¤ected directly and through capital increase;

labor supply a¤ected only through capital increase.

wt =(0.03) 2.67971� (1.5) l st

; (4)

wt =13(0.1575)

�2.6797ldt

� 23

. (5)


Labor Market: Supply, Demand Shift Up, Wage Rises,Employment Same

0.3 0.4 0.5 0.6 0.70.05

0.10

0.15

0.20

0.25

0.30

Labor Employment

w

Figure 9.3. Increase in Productivity Raises w and Leaves EmploymentUnchanged in Example 9.2.


Productivity Increase: Excess Demand, Equilibrium Wage

L (wt ) =

�0.15753wt

�1.52.6797�

�1� 0.5

1.5

�1+

�0.03wt

�2.6797

��= 0;

wt = 0.16978;

ldt =

�0.15753 (0.161)

�1.52.68 = 0.50

=

�1� 0.5

1.5

�1+

�0.030.161

�2.68

��= l st .


Graph of L(w): Excess Labor Demand with ProductivityIncrease

0.1607 0.1608 0.1609 0.1610

0.0010

0.0005

0.0000

0.0005

0.0010

w

L(w)

Figure 9.4. Excess Demand and the Real Wage in Example 9.2.


Time Endowment Increase: Example 9.3

T rises 5% from 1 to 1.05. AG = 0.15, γ = 13 , α = 0.5, δk = 0.03,

ρ = 0.03.kt = 2.4306, a 5% increase.

cdt =1

1+ α(1.05wt + ρkt ) , xt =

αcdtwt. l st = 1.05� xt ;

l st = 1.05� αcdtwt

= 1.05� α

1+ α

�1.05+

ρktwt

�

wt =αρkt

1.05� (1+ α) l st=0.5 (0.052632) 2.43061.05� (1.5) l st

. (6)

ldt =�

γAGwt

� 11�γ

kt ,

wt = γAG

�ktldt

�1�γ

=13(0.15)

�2.4306ldt

�1� 13

. (7)


Labor Supply, Demand Shift Out; Wage Unchanged

0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.620.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Labor Employment

w

Figure 9.5. Increase in Time Endowment Increases Employment inExample 9.3.


Equilibrium Wage with Time Endowment Increase

L (wt ) ��0.153wt

�1.52.4306�

[email protected]�h1.05+

�0.03wt

�2.4306

i3

1A = 0,

=) wt = 0.13889.

ldt =

�γAGwt

� 11�γ

kt =�

0.153 (0.139)

�1.5(2.43) = 0.525.

l st = 1.05�αh1.05+

�ρwt

�kti

1+ α

= 1.05��1.05+

� 0.030.139

�(2.43)

�3

= 0.525.


L(w): Excess Demand with Time Endowment Increase

0.1388 0.1389 0.1390 0.1391

0.0010

0.0005

0.0000

0.0005

0.0010

w

L(w)

Figure 9.6. Implicit Solution of the Real Wage in Example 9.3.


Dynamic Model Employment Cycle Explanation: Example9.4

AG = 0.15 �! AG = (0.15) (1.05) = 0.157 5; T = 1 �! T = 1.05.5% increases; and same.γ = 1

3 , α = 0.5, δk = 0.03, ρ = 0.03.k = 2.8137; a 21.6% increase.

l st = 1.05� 0.51.5

�1.05+

�0.03wt

�2.81

�, (8)

ldt =

�0.15753wt

�1.52.81. (9)

wt =0.5 (0.03) 2.813 71.05� (1.5) l st

, (10)

wt =13(0.1575)

�2.813 7ldt

�1� 13

. (11)


Expansionary Shift Out in Demand, Pivot in Supply

0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.620.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Labor Employment

w

Figure 9.7. Expansionary Increase in Time Endowment and GoodsProductivity in Example 9.4.


Equilibrium Wage and Employment

L (wt ) ��0.15753wt

�1.52.81�

�1.05� 1

3

�1.05+

�0.03wt

�2.81

��= 0,

ldt =

�(0.157 5)3 (0.161)

�1.5(2.81) = 0.525

= 1.05��1.05+

� 0.030.161

�(2.81)

�3

= l st ;

wt = 0.16078.


Equilibrium Wage and Employment

0.1607 0.1608 0.1609 0.1610 0.1611

0.0010

0.0005

0.0000

0.0005

0.0010

w

L(w)

Figure 9.8. Implicit Solution of the Real Wage in Example 9.4.


Consumption to Output Ratio during Expansion

cdt =1

1+ α(1.05wt + ρkt ) =

11.5

((1.05) 0.161+ 0.03 (2.81)) = 0.169.

21.6%, same increase in capital.

ydt =1

1+ α(1.05wt + kt [ρ+ (1+ α) δk ])

=11.5

((1.05) 0.161+ (0.03+ (1.5) 0.03) (2.8))

= 0.25323,

21.6%, the same as consumption.Consumption to output

cd

yd=0.168820.25323

=23.

Same as before expansion.


Example 9.5 Contraction

AG = 0.15 �! AG = 0.1425; T = 1 �! T = 0.95.

5% decreases; and same.γ = 13 , α = 0.5, δk = 0.03, ρ = 0.03;

k = 1.8854.

l st = 0.95� 0.51.5

�0.95+

�0.03wt

�(1.89)

�,

ldt =

�0.14253wt

�1.5(1.89) .

wt =αρkt

0.95� (1+ α) l st=0.5 (0.03) 1.890.95� (1.5) l st

,

wt = γAG

�ktldt

�1�γ

=13(0.1425)

�1.89ldt

�1� 13

.


Decrease in Employment During Contraction

0.44 0.46 0.48 0.50 0.52 0.54 0.560.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Labor Employment

w

Figure 9.9. Contractionary Decrease in Time Endowment and GoodsProductivity in Example 9.5.


Equilibrium Wage, Excess Labor Demand, with Contraction

L (wt ) =

�0.14253wt

�1.51.89�

[email protected]�h0.95+

�0.03wt

�1.89

i3

1A = 0;

wt = 0.1191.

0.475 = ldt =�

γAGwt

� 11�γ

kt = T �α

1+ α

�T +

�ρ

wt

�kt

�= l st ;

0.475 =

�(0.1425)3 (0.12)

�1.5(1.89) = 0.95� 1

3

�0.95+

�0.030.12

�(1.89)

�


Equilibrium Wage, Excess Labor Demand, with Contraction

0.117 0.118 0.119 0.120 0.121

0.03

0.02

0.01

0.00

0.01

0.02

0.03

w

L(w)

Figure 9.10. Excess Labor Demand during Contraction in Example 9.5.


Fixed Wage Unemployment and Depressions

Big employment decrease: 1930s Depression, 2007-2010 recession.

Depression characterized as having "excess labor supply".

Labor contracts often set a wage rate for some years.

can allow employers to reduce temporarily employmentautomobile, airplane industries: large cyclic swings in output

Model "normal" contraction in dynamic model, with �xed wage.

Chapter 3: �xed wage causes surplus labor, 50% further employmentdecrease;

dynamic model: same 50% result when capital stock arbitrarily held�xed.

Allowing capital to adjust, �xed wage during contraction:

much bigger drop in employment.


Example 9.6: Fixed Wage, Fixed Capital, DuringContraction

Marginal product of labor

MPl = wt = γAG

�ktlt

�1�γ

.

l =

�γAGw̄

� 11�γ

k̄. w = w̄ = 0.1389; kt = k̄ = 2.3148.

lt =

�(0.1425)3 (0.1389)

�1.52.3148 = 0.463.

Fractional employment change: 0.463�0.50.5 = �0.074 , a 7.4% fall,

similar to Chapter 3 �xed wage result;

50% bigger than �exible wage.


Example 9.6: Fixed Wage, Variable Capital, DuringContraction

If capital stock readjusts, employment much lower

ldt =�(0.1425)3 (0.1389)

�1.5(1.8854) = 0.377,

a fractional decrease of 0.337�0.50.5 = �0.326 , or a 32% decrease.

l st =

�0.95� 0.5

1.5

�0.95+

�0.030.13889

�1.8854

��= 0.498,

Excess Supply : l st � ldt = 0.498� 0.377 = 0.121 .Excess: 0.498�0.3770.377 = 0.32, or 32% of employment at l = 0.377.Of total "labor force" 0.498, excess is 0.498�0.3770.498 = 0.24, or 24%.Comparable to US unemployment during 1930s depression.

Unemployment reached 22%, compared to 24% in example here.

Chapter 16 : use banking productivity decrease instead of �xed wage,to generate big employment drop.


Contractionary Excess Supply of Labor: 24% of LaborForce

0.30 0.35 0.40 0.45 0.50 0.55 0.600.08

0.10

0.12

0.14

0.16

0.18

0.20

Labor Employment

w

Excess Supply

Figure 9.11. Excess Labor Supply with a Fixed Wage in Example 9.6.


Equilibrium with Labor Income Tax

Tax on labor income, at rate of τl ; with a government budget constraint of

Gt = τlwt lst .

cdt = wt (1� τl ) lst + rtkt + Gt � kt+1 + kt (1� δk ) .

xt =αct

wt (1� τl ).l st = T � xt , rt � δk = ρ.

cdt = wt

�T � αct

wt (1� τl )

�+ ρkt =

wtT + ρkt1+ α

1�τl

.

ydt =wtT + ρkt1+ α

1�τl

+ δkkt =wtT + kt

hρ+ δk

�1+ α

1�τl

�i1+ α

1�τl

.

1wt

=T

ydt�1+ α

1�τl

�� kt

hρ+ δk

�1+ α

1�τl

�i .Gillman (University of Missouri-St. Louis) Chapter 9 3 October 2016 32 / 45

Example 9.7: 20% Labor Income Tax

AG = 0.15, T = 1, γ = 13 , α = 0.5, δk = 0.03, ρ = 0.03;

τl = 0.20 : kt = 2.0576.Fractional decrease of 2.3148�2.05762.3148 = 0.111, or 11.1%.

AS � AD :

1wt=

1ydt�1+ 0.5

1�0.2�� 2.0576

�0.03+ 0.03

�1+ 0.5

1�0.2�� , (12)

1wt=

(y st )1�γ

γ

γA1γ

G (kt )1�γ

γ

=(y st )

2

13 (0.15)

3 (2.0576)2. (13)


AS-AD Both Shift Back, Wage Unchanged

0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.262

4

6

8

10

12

14

Aggregate Output y

1/w

Figure 9.12. Equilibrium with a Labor Income Tax in Example 9.7.


Excess Labor Demand, Equilibrium Wage: 20% Labor Tax

0 = Y (wt ) �wtT + kt

hρ+ δk

�1+ α

1�τl

�i1+ α

1�τl

� (AG )11�γ

�γ

wt

� γ1�γ

kt

Y (wt ) =wt + (2.0576)

�0.03+ 0.03

�1+ 0.5

1�0.2��

1+ 0.51�0.2

� (0.15)1.5

13

wt

! 12

(2.0576) .

wt = 0.13889.


Excess Labor Demand and Equilibrium Wage

0.1387 0.1388 0.1389 0.1390 0.1391 0.1392

0.0004

0.0002

0.0000

0.0002

0.0004

wage rate w

Y(w)

Figure 9.13. Excess Output Demand Y (wt ) in Example 9.7.


Consumption and Output

cdt =wtT + ρkt1+ α

1�τl

=0.13889+ 0.03 (2.0576)

1+ 0.51�0.2

= 0.123 46.

Fractional decrease of 0.13889�0.123 460.13889 = 0.111, or 11.1%,same as percent decrease in capital stock.

ydt =wtT + kt

hρ+ δk

�1+ α

1�τl

�i1+ α

1�τl

=0.13889+ (2.0576)

�0.03+ 0.03

�1+ 0.5

1�0.2��

1+ 0.51�0.2

= 0.18519.

Fractional decrease: 0.208 32�0.185190.20832 = 0.111 , or 11.1%, same asconsumption.Consumption to output ratio stays same:

cd

yd=0.123 460.18519

=23.


Labor Market with 20% Labor Income Tax

l st = T �α

�wtT+ρkt1+ α

1�τl

�wt (1� τl )

. (14)

wt =

0@ αρkth1+ α

1�τl

i(1� τl ) (T � l st )� αT

1A , wt = γAG

�ktldt

� 23

,

wt =

0.5 (0.03) (2.058)�

1+ 0.51�0.2

�(1� 0.2) (1� l st )� 0.5

!, wt =

(0.15)3

�(2.058)ldt

� 23

,

wt = 0.1389; ldt =�

(0.15)3 (0.1389)

�1.5(2.0576) = 0.444. (15)

Ffractional decrease in employment of 0.5�0.444390.5 = 0.111, or 11.1%,same as decrease in capital stock, consumption, output.Government spending G = wtτl lt = (0.13889) (0.20) (0.444) = 0.012.


Labor Market with Tax: Supply, Demand Shift Back, SameWage

0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.580.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Labor Employment

w

Figure 9.14. Increase in Time Endowment Shifts out Labor Supply inExample 9.7.


Isocost, Isoquant, Input Ratio with Labor Tax

Budget Line

0.18519 = yt = wt lt + rtkt (0.13889) lt + (0.06) kt ,

kt =0.185190.06

� (0.13889) lt0.06

. (16)

Isoquant curve

0.18519 = y st = AG�ldt�γ(kt )

1�γ = 0.15�ldt� 13(kt )

23 ;

kt =

0@ (0.18519)0.15

�ldt� 13

1A 32

=

� 0.185190.15

� 32�

ldt� 12

. (17)

Input ratioktlt=2.05760.444

= 4.6342. (18)


Isocost, Isoquant Shift Up, Same Wage and Input Ratio

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01.5

2.0

2.5

3.0

3.5

4.0

l Labor

k Capital

Example 9.15. Factor Market Equilibrium with Labor Income Tax τl , inExample 9.7 (lower) and Baseline (upper).


Labor Tax: Production Function, Utility Level, Budget Line

cdt = y st � it = AG�ldt�γ(kt )

1�γ � δkkt ,

cdt = (0.15)�ldt� 13(2.0576)

23 � (0.03) (2.0576) ; (19)

u = ln ct + α ln xt = ln ct + α ln (1� lt ) ,�2.385 3 = ln 0.123 46+ 0.5 ln (1� 0.444) ,

ct =e�2.3853

(1� lt )0.5. (20)

cdt = wt l st (1� τl ) + ρkst + G

cdt = (0.13889) (1� 0.20) l st + (0.03) (2.0576) + 0.012. (21)


Tax: Intersection, Not Tangency, Between Utility,Production

0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.600.115

0.120

0.125

0.130

0.135

0.140

l Labor

c

Figure 9.16. General Equilibrium Goods and Labor with Labor Tax inExample 9.7.


Application: Hours per Week, Taxes and Regulations

Marginal productivity rises, but hours per week stay the same.Implies with rising productivity, work week remains same.International data not consistent constant hours of work per week.

Post WWII hours declined: UK, France, Germany, Australia, Canada.But rose in USA: yet all countries have upward trend in productivity.

Standard dynamic model cannot explain.Many alternative explanations: open question of active research.

Case of France: restricted workweek to 35 hours in Feb 2000further restriction from 39 hour week by President Mitterrand.Idea (Meade, 1937): allow more employed; each employed works less,Many consider it is regulation on labor market like an implicit tax.Restriction relaxed May 2002 to May 2005, and since May 2007.France�s hours dropped sharply from 2000 to 2002; up in 2003.

Implicit taxes in US down as union membership has trended down.Ohanian, Ra¤o and Rogerson (2008) well explain European-US hours

by including taxes, implicit taxes on labor market, in standard model.


Hours Worked Per Week: US, Canada, Europe, Australia

Average usual weekly hours worked on the main job

32.0

33.0

34.0

35.0

36.0

37.0

38.0

39.0

40.0

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

year

wee

kly

hour

s w

orke

d

AustraliaFranceGermanyUnited KingdomUnited StatesCanada

Figure 9.17.Gillman (University of Missouri-St. Louis) Chapter 9 3 October 2016 45 / 45

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