# Labor Supply, Demand Unemployment

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```					Labor Supply, Demand &
Unemployment
Mid-term Exam
• Tuesday, October 14th 9AM
• Semi-open Book (Bring 1 A4 size paper
with handwritten notes
• Coverage. Lecture notes including this
one.
Hours per Worker 2005
www.ggdc.net
1,900

1,800

1,700

1,600
Hours

1,500

1,400

1,300

1,200
France   Germany   Italy   Sweden   U.K.   U.S.A
2,600

2,400

2,200

2,000    France
West Germany
Italy
1,800
U.K.
1,600    U.S.A

1,400

1,200

1,000
50

53

56

59

62

65

68

71

74

77

80

83

86

89

92

95

98

01

04
19

19

19

19

19

19

19

19

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20
This section will follow
•   Branson, Chapter 6, 105-121
•   Williamson, Chapter 4, 90-130
•   Williamson, Chapter 15, 539-557
•   Romer, Chapter 4, Section 4 & Chapter 9
Labor Demand

• Output is a diminishing function of labor holding the
capital stock constant.
• The marginal product of labor is the slope of the
production function.
• The slope is diminishing as labor increases. We
can map a function of this slope.
• Firms will hire workers until the marginal benefit of
hiring workers exceeds the marginal cost. To
maximize profits, firms hire workers until the
marginal product of labor equals the real wage.
Marginal Product of Labor =
Slope of Production Function
Q
MPL2

MPL1

MPL

L

L0         L1          L2
Labor Demand Curve

w0

w1

w2

MPL

L
L0   L1   L2
Labor Demand Curve

• A mapping between the marginal product of
labor and real wage is equivalent to the
amount of labor demanded by firms.
• To close out the model, we need a theory of
labor supply.
• Labor demand is based on the profit
maximization of firms. We need a similar
metric to measure the objective of workers.
Utility Function

• When workers make a choice of how many hours
to work, they make a trade-off between the goods
they can buy with their wages and the leisure time
they lose by working.
• Workers will have preferences over a set of
consumption-leisure choices.
• An indifference curve is a set of leisure-
combination choices over which the worker is
indifferent.
Worker Preferences: More is
Better
• We assume workers prefer more to
less. At any level of leisure, lst, it is
always preferred to have more
consumption of goods, Ct.

– This implies that U0 is preferred to U1
which is preferred to U2.
The higher the indifference curve, the
more it is preferred by workers.
C

U0

U1

U2

ls
Preference for Variety

• Consumption and leisure have
diminishing returns.
– The slope of the indifference curve, MRS,
is equal to the amount of consumption
you would have to get in order to make
you just as happy if you had to give up
some leisure.
– Along any indifference curve, the greater
is lst the lower will be MRS
Diminishing Returns to leisure and
Consumption.
C

-MRS1

U2
-MRS0

ls

ls1      ls0
Goods are Normal

• If overall income increases and relative
prices remain the same, households will
want to consume a greater amount of all
goods.
• If you won the lottery, you would probably
would work less hard and consume more
goods, so it is reasonable to assume that
consumption and leisure are normal goods.
Utility Functions
• Economists often act as if preferences
over some choice set can be written as a
mathematical function of the choices.
• Conjecture a utility function which ranks
the different choices giving a higher score
to preferred choices.
• For example, we might write a utility
function in terms of consumption and
leisure U(Ct,lst).
Utility Functions

• More is better   U U
,    0
C ls

• Goods have diminishing returns.

U U
2     2
, 2 0
C ls
2
Normal Goods
• A simple way to insure that the utility
function represents the preferences of a
consumer for whom goods are normal is to
let both goods enter in parallel ways.

U (Ct , lst )  ln Ct   ln lst
Budget Constraint

• Given the resources of the worker, they will
have only a limited set of combinations of
consumption and leisure which they will be
able to choose.
TIME  Lt  lst
• Assume a limited set of time, TIME,
available for workers which must be split
between leisure and work, Nt.
Budget

• Workers will have an income available for
consumption equal to wages plus profits, Πt
(net of investment) minus a poll tax Tt.
Ct  wt Lt   t  Tt
• Given time constraints, we can write
consumption as a negative function of
leisure. The slope of this function is the real
wage rate.       C  w TIME  ls     T
t    t           t    t   t
Budget Constraint.

C

w

Π-T

ls
TIME
Optimal Consumption & Leisure:
Geometry

• We choose the most preferred consumption/leisure
combo which gives the greatest utility subject to
that combo being feasible.
• Geometrically, this is where an indifference curve
is tangent to the budget line.
• The tangent indifference curve touches the budget
line but does not cross, so it is by definition the top
indifference curve that is part of the budget line.
Optimal Consumption Leisure

C

[C*,ls*]

ls
TIME
Optimal Consumption & Leisure:
Intuition
• The slope of the indifference curve is how much extra
consumption you would need to get to make you just as
well off to give up some leisure.
• The slope of the budget line is the amount of extra
consumption you can actually get if you give up some
leisure
• At any point, you would always willingly give up more
leisure if the slope of indifference curve is less than the
real wage.
• You would also willingly give up consumption if the slope
of the indifference curve was steeper than the real wage.
Optimal Consumption & Leisure: Intuition
• At optimum, the slope of the indifference
curve would be equal to the real wage.
U
U      U
w  ls          w 
U   C      ls
C
• At optimum, the marginal benefit of some
extra leisure is equal to the marginal cost of
leisure. The marginal cost of leisure is the
real wage times the marginal value of the
consumption that the real wage could by.
Optimal Consumption and Leisure: Calculus
• Maximize U(Ct,lst) subject to the constraint
that
Ct  wt TIME  lst   t  Tt

• Rewrite the utility function by inserting the
constraint   max U (w TIME  ls     T , ls )
t           t      t   t   t
ls

• Write the first order conditions as
dU                             U U
 0   wU1  U 2  0  wt   
C ls
t
dls
Example
• Log-log utility function
– Objective Function
max ln(wt TIME  lst   t  Tt )   ln lst

– First Order Conditions
wt                

wt TIME  lst    t  T lst
Elasticity of Substitution
• For log,log utility, we can write the first order
condition as
wt         Ct
       wt
Ct lst       lst
• The real wage is the opportunity cost/price of
leisure in terms of consumption.
• A 1% increase in the relative price of leisure
leads to a 1% increase relative demand for
leisure.
• Log,log utility is unit elasticity of substitution
utility function.
Labor Supply Curve
• The solution to the first order condition
maps real wages, profit and tax income
into an amount of labor
Ls  TIME  ls ( wt ,  t  Tt )
t

• The labor supply curve is a mapping of the
real wage into an optimal amount of labor
provided by workers at a given amount of
profit income and taxes.
Example
              t  T 
lst    TIME  lst         
               wt 
           t  T 
lst        TIME         
1           wt 
1              t  T 
Lt       TIME              
1         1    wt 
Increase in Lump-sum Income
• If leisure & consumption are normal goods, an
increase in profit or a cut in taxes will increase
the amount of leisure that is desired. This will in
turn cut the labor supply.
• Income Effect: At a constant wage rate, an
increase in income would increase consumption.
This would reduce the marginal value of any
wage income earned because consumption has
diminishing returns. Thus, the marginal cost of
leisure would drop inducing workers to take
more leisure.
Effect of Wages on Labor Supply
•   There are two channels through which an a
change in the wage rate affects the marginal
cost of leisure.      U
w
C
1. Substitution Effect: An increase in the wage
rate directly increases the cost of not working
because it increases the pay-off to each hour
worked. This will tend to make the worker
Income Effect
2. Income Effect: An increase in wages
increases income & consumption,
decreases the marginal utility of
consumption, and decreases the welfare
value of wages. The income effect will
tend to make the worker choose to enjoy
more leisure time.
Income vs. Substitution Effect
• In theory, there are no clear assumptions
about preferences that would make us
think that either the income or the
substitution effect would be stronger.
• In theory, either effect could be stronger
and an increase in wages could have
either effect.
Wages Rise

C                  Income Effect

Pure
Substitution
Effect

[C*,ls*]

ls
TIME
Example
• If there were no profit income and no taxes
in the log-log case, then the income and
substitution effects would exactly cancel
out and labor supply would not depend on
real wages.                        1
lst    TIME  Lt    TIME
1           1 

• Given positive profits, a rise in the real
wage relative to profits will increase the
optimal labor choice.
Upward Sloping Supply Curve &
Equilibrium
w0             LS

w*

LD
L
L*
Capital or Technology Increase/ Labor Demand Curve

Shifts Out/ Equilibrium Wages and Employment Increases

w0                      LS

w**

w*

LS
LD’

LD
L
L*   L**
Profit or Tax Increase/ Labor Supply Shifts Out /

Equilibrium Wages Fall and Employment Increases

w0                       LS
LS’

w*

w***
LD’

LD
L
L*   L***
Taxes
• Raising income tax rates has counter-
veiling impacts on supply.
• Lump sum taxes/poll taxes have a positive
impact on labor suppy.
Is an Upward Sloping Labor Supply
Consistent with Long Term?
• Over the very long-term, productivity and
real wages have risen by a large amount
and labor supply per person has been
falling, if anything.
• Over the long-run, non-labor income
should be rising along with real wages.
Thus, this income effect may be driving
labor down.
Upward Sloping Labor Supply in
the Short Run
• The level of consumption depends on the
• A temporary rise in real wages will not
have a strong effect of lifetime income.
• Thus, a temporary increase in real wages
will have a strong substitution effect and a
weak income effect.
Rise in Labor Productivity Over
Time

Output per Hour

25.00

20.00
Hong Kong
15.00
Singapore
South Korea
10.00
Taiwan
5.00

0.00
60

63

66

69

72

75

78

81

84

87

90

93

96

99

02
19

19

19

19

19

19

19

19

19

19

19

19

19

19

20
1990 US\$
Labor Supply in Emerging Asia

Hours Worked per Employee

2,900
2,800
2,700
2,600                               Hong Kong
Hours

2,500                               Singapore
2,400                               South Korea
2,300                               Taiwan

2,200
2,100
2,000
50

54

58

62

66

70

74

78

82

86

90

94

98

02
19

19

19

19

19

19

19

19

19

19

19

19

19

20
Growing Labor Force
Participation
60.00%

50.00%

40.00%
Hong Kong
Singapore
30.00%
South Korea
Taiwan
20.00%

10.00%

0.00%
60

63

66

69

72

75

78

81

84

87

90

93

96

99

02
19

19

19

19

19

19

19

19

19

19

19

19

19

19

20
Share of Population in 20-55
Age Range
•
0.6

0.55

0.5
Korea
Singapore
0.45

0.4

0.35
1950   1955   1960   1965   1970   1975   1980   1985   1990
Unemployment Rate
•   The population is split into three
categories:
1. (NLt) Not in the Labor Force: those people
who do not have jobs and are not actively
seeking employment.
2. (Et) Employed, those people who currently
have jobs.
3. (Ut) Unemployed, those people seeking jobs.
• The labor force participation rate is the
share of the population which are
employed or unemployed.
Et  U t
lpft 
Et  U t  NLt
• The unemployment rate is that share of
the labor force which is unemployed.

Ut
urt 
Et  U t
Types of Employment

• Frictional – Unemployment that results from the
standard dynamic nature of the labor market.
When people change jobs, there is frequently
some period when they are looking for work.
• Structural – Unemployment that results from some
big change in the economy caused by new trade
competition or new technology.
• Cyclical – Unemployment associated with the
business cycle. We will concentrate on type 1 in
this section.
Data
• The Bureau of Labor Statistics of the USA
Department of Labor maintains a large
database of international unemployment
rates, wage rates, inflation rates, and
productivity levels (in addition to extensive
US labor market and inflation measures).
• The web-site http://www.bls.gov
Average Unemployment Rates in
the 1990’s
Unemployment Rate

12

10

8

6                                                         Unemployment Rate

4

2

0
USA   Japan   France Germany   Italy   Sweden   UK
Unemployment

• Supply and demand may be best
representation of auction-style market
which clear quickly.
• Labor markets are more specialized with
workers trying to find a good fit for their
skills.
• Workers separate from their jobs for
idionsyncratic reasons (i.e. they don’t like
their boss, etc.)
Dynamics

• The unemployment rate, ur, is the % of
workers who are trying to find jobs.
• The share of workers with jobs is (1-ur)
• The percentage of workers who separate
from their jobs every period is s.
• The percentage of labor force who lose their
jobs can be written as s∙(1-ur)
Separation

s(1-ur)

ur

1
Workers
• To simplify things, we write workers utility
(if they have a job) as an increasing,
diminishing function of the real wage they
receive Ve(w) . Taxes on labor income will
reduce utility.
• If workers do not have a job, they receive
some unemployment benefits b. Their
utility is an increasing function of the size
of benefits.
Employed Workers Utility

Ve(w)

w
Reservation wage
• Workers accept jobs if their utility as workers
exceeds their utility as the unemployed.
• There exists a wage level (called the reservation
wage) such that the workers utility is equal to the
unemployed’s utility.
• If an unemployed worker, searching for a job
receive no job offer above the reservation wage,
they will remain unemployed.
Reservation Wage w*
Employed Utility = Unemployed Utility

Ve(w)

Vu

w

w*
Reservation Wage and Tax Policy
• If taxes on workers increase, the
reservation wage rate will rise.
• If government benefits for the unemployed
rise, the reservation wage rate will rise.
Worker Taxes Rise

Ve(w)

Ve’(w)

Vu

w

w*0   w*1
Unemployment Benefit Falls

Ve(w)

Vu1

Vu

w

w*0   w*1
Idiosyncratic Jobs
• A variety of positions are available. But
some jobs are randomly more profitable
than others.
• For a given wage rate,w , the function
is the percentage of firms with wages
above w .
H (w)  1  F (w)
• F is the cumulative distribution functionof
the random wage rate.
Example
Uniform Distribution
• Wages are randomly distributed over the
range 0 and w     .
• We can calculate the probability that job
searchers are offered a wage less than w .
w w
w
1
•        H ( w)   dw 
w
w     w
Wage Probability Distribution
pdf

H (w)

w      w
Probability of Finding a Job

• Given that p is the probability of finding a
job offer:
– The percentage of people who find a job is
ur∙p∙H( w* )
– If in the long-run, the percentage of people
losing a job is equal to the percentage of
people getting a job is
ur∙p∙H( w*) = (1-ur)s

• The steady state unemployment is:

s
ur 
ss

s  pH ( w)

ur∙p∙H( w* )

s(1-ur)

ur
Reservation Wage
• An increase in either unemployment
benefits or an increase in worker taxes will
increase the reservation wage.
• Either will decrease the probability that
workers find jobs that they find acceptable.
• A decline in the job finding rate will
increase equilibrium employment
Reservation Wage rises w*↑
ur∙p∙H( w*)

ur∙p∙H( w**)

s(1-ur)

ur
ur*   ur**
1
High European Unemployment
• Unemployment and welfare benefits are higher
than Europe than US.
• Search theory explain why European
unemployment is on average higher than USA
unemployment.
• May also explain why it is precisely those
countries with high unemployment that have
high productivity. Only high wage jobs are
accepted by workers with high unemployment
benefits.
Reality
• Reality may be more complicated.
• Within Europe there is no correlation between
high unemployment benefits and high
unemployment.
• In fact, the Netherlands has very low
unemployment rates with very high social
benefits.
• Some economists emphasize the difference in p,
the rate at which employee receive job offers,
not H, the rate at which they accept them.
Dutch Treat
• In the mid-1990’s, Holland reformed its
labor market to make it easier to hire and
fire workers.
• Making it easier to fire workers might
increase, the separation rate s and
increase frictional unemployment.
• However, it seemed to have an even
stronger indirect effect by increasing p and
reducing unemployment.
Models of p
• Some economic models emphasize that
taxes on firms or labor market restrictions
may make firms less likely to offer jobs.
• This would reduce p and increase
unemployment.
Efficiency wage models
• Assume that workers productivity depends on
workers effort.
• When production shifts to large scale mechanize
manufacture, firms must make efforts to insure
that workers are working hard.
• Monitoring workers may be costly and there may
be some incentive for carrots to incentivize
worker effort.
• Henry Ford and the \$5 a week salary.
Efficiency Wages: Implications
• Firms will have an incentive to pay
workers above market wages in order to
keep them happy.
• But high wages offered to workers will
have the effect of being less able to pay
workers reducing the demand for labor.
Efficiency Wages
• Fixed supply of labor but…
• Output depends on labor hours and
worker effort, e.                 1
Y  F (eL)  (eL)
• Workers willingness to put in effort
depends on the real wage firms will pay.
w        
e  e( w)  (          )

Demand for Labor
• Firms choose wages offered and number of
workers hired to maximize profits.
• Profits are   F (e(w) L)  wL
• FOC 1) e( w) F '(e( w) L)  w

de
e '( w) w      e 1
2)   e '( w) LF '(e( w) L)  L            
e( w)      dw
w
• Cost of hiring each unit of effective labor is
w
e( w)
, which is minimized where elasticity of
effort is 1.
Example
w        w 
(          )  1
w           e '( w) w      
e( w)  (          )                                      1
             e( w)        w  
(     )


w
1 

The more sensitive effort is to wages (i.e. β close to 1) the higher will
be the efficiency wage.
The higher is the threat point (i.e. the minimum wage necessary to get
any effort at all), the higher will be real wage.
Upward Sloping Supply Curve &
Equilibrium
w0    Unemployment
LS


w
1 

w*

LD
L
L*
Why unemployment
• If efficiency wages are high relative to the
market clearing wage we would have
unemployment.
• Firms will not earn profits by hiring
unemployed workers unless the wage they
pay drops.
• Firms will not lower wages to hire
unemployed workers because it will hurt
overall effort.
What determines the
unemployment rate?
• The minimum wage that workers demand
to put in effort is likely determined by
– Likelihood they will lose their job if they don’t
put in effort.
– Welfare if they are unemployed
– Probability of quickly finding a new job.
– Wages in new job.
Example
• Aggregate wages: Individual workers when
considering their effort, compare their wages
with those they could obtain elsewhere if they
were fired after being observed not giving the full
effort.
  (1  v  ur )  w
EQ

• However, workers have some probability of not
getting any job if there is unemployment.
» Weight placed on unemployment. May be small if
v(benefits)  1     unemployment benefits are high.

Wage Offer
• The firm will offer an efficiency wage of
   (1  v  ur )  wEQ
w      
1          1 

• A lower unemployment rate or a higher
unemployment benefit increases the
wages that firms will have to offer to
induce the cost efficient amount of labor.
Equilibrium
• In equilibrium, all firms will be paying the
same wage, w = wEQ.
• Here there will be an equilibrium
unemployment rate.
(1  v  ur )        
1                  ur 
1               v
Unemployment Benefits
• In this model, high unemployment benefits
reduce the disincentive for effort causing
firms to increase the amount of wages
paid to incentivize effort.
• This reduces the demand for labor.
• Contrast with search models.
Dutch Treat
Netherlands Unemployment Rate

8

7

6

5

4                                                                      Netherlands

3

2

1

0
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Cyclical Unemployment
• Overtime, as capital and technology have
grown, real wages have also grown.
• Labor hours per capita have tended to
remain stable or even decrease.

• How can we explain variations in
employment levels that occur at short-run.
– Wage stickiness
%

0
1
2
3
4
5
6
7
8
9
10
Oct-81

Oct-83

Oct-85

Oct-87

Oct-89

Oct-91

Oct-93

Oct-95
Unemployment Rate in HK

Oct-97

Oct-99

Oct-01
HK’s Unemployment Rate

Oct-03

```
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