Labor Supply, Demand & Unemployment Mid-term Exam

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Labor Supply, Demand & Unemployment

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 one.

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

Labor Demand Curve

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.

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.

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

  • 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: 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

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.

  • 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

  • Rewrite the utility function by inserting the constraint

  • 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 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

  • 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.

  • 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


  • 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 & Equilibrium

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 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 Time

Labor Supply in Emerging Asia

Growing Labor Force Participation

Share of Population in 20-55 Age Range

Unemployment Rate

  • The population is split into three categories:

    • (NLt) Not in the Labor Force: those people who do not have jobs and are not actively seeking employment.
    • (Et) Employed, those people who currently have jobs.
    • (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 skills.

  • 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

Unemployment Benefit Falls

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.

Example 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

Steady State Unemployment

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*↑

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 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.

  • 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

  • FOC

  • Cost of hiring each unit of effective labor is , which is minimized where elasticity of effort is 1.


Upward Sloping Supply Curve & Equilibrium

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.


  • 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.

  • 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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