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Econ 601: Advanced Microeconomic Theory
Spring 2007
Problem Set 9
Firm’s Demands for Inputs
PART A
1. The mowing of lawns requires only labor (gardeners) and capital (lawn mowers).
These factors must be used in the fixed proportions of one worker to one lawn mower,
and production exhibits constant returns to scale. Suppose that the wage rate of
gardeners is $2 per hour, that lawn mowers rent for $5 per hour, and that the price
elasticity of demand for mowed lawns is -2.
a.) What is the wage elasticity of demand for gardeners (that is, what is L/w .wIL)?
b.) What is the elasticity of demand for lawn mowers with respect to their rental rate
(that is, K/v. v/K)?
c.) What is the cross elasticity of demand for lawn mowers with respect to the wage
rate (that is, K/w. w/K.)?
2. Suppose there are a fixed number of 1,000 identical firms in the perfectly competitive
concrete pipe industry. Each firm produces the same fraction of total market output,
and each firm's production function for pipe is given by
q  KL
Suppose also that the market demand for concrete pipe is given by
Q = 400,000- 100,000P,
Where Q is the total concrete pipe.
a. If w =v = $1,in what ratio will the typical firm use K and L? What will be the long-run
average and marginal cost of pipe?
b. In long run equilibrium what will be the market equilibrium price and quantity for
concrete pipe? How much will each firm produce? How much labor will be hired by
each firm and in the market as a whole?
c.
Suppose the market wage, w, rose to $2 while v remained constant at $1. How will
this change the capital-Iabor ratio for the typical firm, and how will it affect its
marginal costs?
d. Under the conditions of part (c), what will the long-run market equilibrium be? How
much labor will now be hired by the concrete pipe industry?
e. How much of the change in total labor demand from part (b) to part (d) represented
the substitution effect resulting from the change in wage and how much represented
the output effect?
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3. The town of Podunk has decided to provide security services to its residents by hiring
workers (L) and guard dogs (D). Security services (S) are produced according to the
production function
S  LD ,
and residents of the town wish to consume 10 units of such services per period.
a. Suppose that Land D both rent for $1 per period. How much of each input should the
town hire to produce the desired services at minimal cost? What will that cost be?
b. Suppose now that Podunk is the only hirer of people who work with guard dogs and
that the supply curve for such workers is given by
L = lOw,
where w is the per-period wage of guard dog handlers. If dogs continue to rent for $1 per
period, how much of each input should the town hire to produce the desired services at
minimal cost? What will those costs be? What will the wage rate of dog handlers be?
4. Carl the clothier owns a large garment factory on an isolated island. Carl’s factory is
the only source of employment for most of the islanders, and thus Carl acts as a
monopsonist. The supply curve for garment workers is given by
L= 80w
where L is the number of workers hired and w is their hourly wage. Assume also that
Carl's labor demand (marginal revenue product) curve is given by
L =400 - 40MRPL
a.) How many workers will Carl hire in order to maximize his profits and what wage
will he pay?
b.) Assume now that the government implements a minimum wage law covering all
garment workers. How many workers will Carl now hire and how much
unemployment will there be if the minimum wage is set at $4 per hour?
c.) Graph your results.
d.) How does a minimum wage imposed under monopsony differ in results as
compared with a minimum wage imposed under perfect competition (assuming the
minimum wage is above the market-determined wage)?
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PART B
Briefly explain why you think the following statements are true, false, or uncertain.
Your grade will depend largely on the quality of your explanations.
1. The elasticity of demand for labor will be higher the greater is the elasticity of
substitution between labor and other inputs and the greater is the elasticity of demand
for the products that use labor as a factor of production.
2. A monopsonist will hire more labor and pay a higher real wage rate if the labor in the
sector is unionized than if the labor market is supplied perfectly competitive.
3. The elasticity of demand for a factor of production will be larger if the prices of all
other factors of production are held constant rather than if the quantities of all other
factors are held constant.
4. Consider a competitive firm’s demand for a factor of production as a function of the
factor price when the prices of other factors are given. Let us consider two cases: (1)
the quantity of output is fixed; (2) the price of the product is fixed. The elasticity of
demand is greater in the second case than in the first.
5. Suppose a union of workers in an industry are attempting to maximize the total wage
bill of its members in the industry. This situation will lead to a quantity of
employment that is greater than the quantity that would be hired in a competitive
market, and the wage earned by the workers will be higher than that resulting from a
perfectly competitive labor market.
6. The elasticity of demand for labor with respect to the wage rate will be larger the
greater is share of labor in the total costs of production of the goods that use labor as
input.
7. The elasticity of demand for a factor of production will be larger if the share of the
factor in the total cost of producing the final good is increased.
8. The imposition of a minimum wage rate can result in increased employment in noncompetitive labor markets.
9. To employ a factor of production so as to maximize profits, a firm selling into a
competitive market should set the value of the marginal product of each factor equal to
the marginal expense of the factor to the firm.
10. In a rapidly-adjusting competitive economy, with constant average wages, wages
would rise in those industries that enjoy the most rapid increases in labor productivity,
would fall in those industries where labor productivity was falling, and would be
constant in industries whose labor productivity was constant.
11. High income women professionals tend to have smaller families than do lower paid
women workers, hence, children must be an inferior good.
12. The elasticity of demand for one factor of production by a competitive industry must
be lower when the supply of other factors is less than perfectly elastic than when the
prices of these other factors of production are held constant.
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