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									Labor Supply, Demand &
           Mid-term Exam
• Tuesday, October 12th 9AM
• Lecture Theater E
• Semi-open Book (Bring 1 A4 size paper
  with handwritten notes))
• Coverage. Lecture notes including this
Hours per Worker 2005
        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




     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 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
  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
  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




          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
• 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

• Goods have diminishing returns.
           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.
          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.

• 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.

• Given time constraints, we can write
  consumption as a negative function of
  leisure. The slope of this function is the real
  wage rate.
      Budget Constraint.




   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
  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.

• 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

• Rewrite the utility function by inserting the

• Write the first order conditions as
• Log-log utility function
   – Objective Function

   – First Order Conditions
       Elasticity of Substitution
• For log,log utility, we can write the first order
  condition as

• 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
  utility function.
         Labor Supply Curve
• The solution to the first order condition
  maps real wages, profit and tax income
  into an amount of labor

• 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.
 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.

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.
            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



• 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.

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


 Capital or Technology Increase/ Labor Demand Curve

Shifts Out/ Equilibrium Wages and Employment Increases

       w0                      LS




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

Equilibrium Wages Fall and Employment Increases

    w0                       LS



                      L*   L***
• 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
  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
Labor Supply in Emerging Asia
Growing Labor Force
    Share of Population in 20-55
            Age Range
          Unemployment Rate
•   The population is split into three
    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.

• The unemployment rate is that share of
  the labor force which is unemployed.
        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.
• 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
Average Unemployment Rates in
          the 1990’s

• 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
  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


           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




 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





      w*0   w*1
Unemployment Benefit Falls





           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, , the function
  is the percentage of firms with wages
  above .

• F is the cumulative distribution functionof
  the random wage rate.
        Uniform Distribution
• Wages are randomly distributed over the
  range 0 and       .
• We can calculate the probability that job
  searchers are offered a wage less than .

Wage Probability Distribution
   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
  Steady State Unemployment

• The steady state unemployment is:
Steady State Unemployment

                       ur∙p∙H( w* )


            urSS   1
         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**)


       ur*   ur**
 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.
              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
      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.

• Workers willingness to put in effort
  depends on the real wage firms will pay.
             Demand for Labor
• Firms choose wages offered and number of
  workers hired to maximize profits.
• Profits are

• Cost of hiring each unit of effective labor is
       , which is minimized where elasticity of
           effort is 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 &
  w0    Unemployment


        Why unemployment
• 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
  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.
• 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

• However, workers have some probability of not
  getting any job if there is unemployment.
            » Weight placed on unemployment. May be small if
              unemployment benefits are high.
               Wage Offer
• The firm will offer an efficiency wage of

• 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
  unemployment rate.
     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
      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
HK’s Unemployment Rate

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