Labor Supply, Demand &
• Tuesday, October 14th 9AM
• Semi-open Book (Bring 1 A4 size paper
with handwritten notes
• Coverage. Lecture notes including this
Hours per Worker 2005
France Germany Italy Sweden U.K. U.S.A
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
• Output is a diminishing function of labor holding the
capital stock constant.
• The marginal product of labor is the slope of the
• 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
L0 L1 L2
Labor Demand Curve
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 demand is based on the profit
maximization of firms. We need a similar
metric to measure the objective of workers.
• 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
• An indifference curve is a set of leisure-
combination choices over which the worker is
Worker Preferences: More is
• 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.
Preference for Variety
• Consumption and leisure have
– 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
– Along any indifference curve, the greater
is lst the lower will be MRS
Diminishing Returns to leisure and
Goods are Normal
• If overall income increases and relative
prices remain the same, households will
want to consume a greater amount of all
• 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.
• 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
• More is better U U
• Goods have diminishing returns.
, 2 0
• 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
• 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.
• 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
Optimal Consumption & Leisure:
• 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
Optimal Consumption & Leisure:
• 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
• At any point, you would always willingly give up more
leisure if the slope of indifference curve is less than the
• 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.
w ls w
U C ls
• 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
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
• Write the first order conditions as
dU U U
0 wU1 U 2 0 wt
• Log-log utility function
– Objective Function
max ln(wt TIME lst t Tt ) ln lst
– First Order Conditions
wt TIME lst t T lst
Elasticity of Substitution
• For log,log utility, we can write the first order
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
• Log,log utility is unit elasticity of substitution
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 )
• 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.
lst TIME lst
1 t T
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
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
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
substitute additional income for leisure.
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
C Income Effect
• 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
• Given positive profits, a rise in the real
wage relative to profits will increase the
optimal labor choice.
Upward Sloping Supply Curve &
Capital or Technology Increase/ Labor Demand Curve
Shifts Out/ Equilibrium Wages and Employment Increases
Profit or Tax Increase/ Labor Supply Shifts Out /
Equilibrium Wages Fall and Employment Increases
• 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
Upward Sloping Labor Supply in
the Short Run
• The level of consumption depends on the
level of lifetime income.
• 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
Output per Hour
Labor Supply in Emerging Asia
Hours Worked per Employee
2,600 Hong Kong
2,400 South Korea
Growing Labor Force
Share of Population in 20-55
1950 1955 1960 1965 1970 1975 1980 1985 1990
• The population is split into three
1. (NLt) Not in the Labor Force: those people
who do not have jobs and are not actively
2. (Et) Employed, those people who currently
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
Et U t NLt
• The unemployment rate is that share of
the labor force which is unemployed.
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
• 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
6 Unemployment Rate
USA Japan France Germany Italy Sweden UK
• 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
• Workers separate from their jobs for
idionsyncratic reasons (i.e. they don’t like
their boss, etc.)
• 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)
• 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
• If workers do not have a job, they receive
some unemployment benefits b. Their
utility is an increasing function of the size
Employed Workers Utility
• 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
• 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
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
Unemployment Benefit Falls
• A variety of positions are available. But
some jobs are randomly more profitable
• 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.
• 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 .
• H ( w) dw
Wage Probability Distribution
Probability of Finding a Job
• Given that p is the probability of finding a
– 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
Steady State Unemployment
• The steady state unemployment is:
s pH ( w)
Steady State Unemployment
ur∙p∙H( w* )
• 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*↑
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
• 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
• Reality may be more complicated.
• Within Europe there is no correlation between
high unemployment benefits and high
• In fact, the Netherlands has very low
unemployment rates with very high social
• Some economists emphasize the difference in p,
the rate at which employee receive job offers,
not H, the rate at which they accept them.
• In the mid-1990’s, Holland reformed its
labor market to make it easier to hire and
• 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
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
Efficiency wage models
• Assume that workers productivity depends on
• 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
• 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.
• 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.
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
e '( w) w e 1
2) e '( w) LF '(e( w) L) L
e( w) dw
• Cost of hiring each unit of effective labor is
, which is minimized where elasticity of
effort is 1.
( ) 1
w e '( w) w
e( w) ( ) 1
e( w) w
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 &
• If efficiency wages are high relative to the
market clearing wage we would have
• Firms will not earn profits by hiring
unemployed workers unless the wage they
• Firms will not lower wages to hire
unemployed workers because it will hurt
What determines the
• 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.
• 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
(1 v ur ) w
• 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.
• The firm will offer an efficiency wage of
(1 v ur ) wEQ
• 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.
• In equilibrium, all firms will be paying the
same wage, w = wEQ.
• Here there will be an equilibrium
(1 v ur )
• 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.
Netherlands Unemployment Rate
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
• 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
Unemployment Rate in HK
HK’s Unemployment Rate