Profit Maximization and Derived Demand

1
Profit Maximization and Derived Demand

• A firms hiring of inputs is directly related to
  its desire to maximize profits
• any firms profits can be expressed as the
  difference between total revenue and total costs,
  each of which can be regarded as functions of the
  inputs used
• ? TR(K,L) - TC(K,L)

2
Profit Maximization and Derived Demand

• First-order conditions for a maximum are

• the firm should hire each input up to the point
  at which the extra revenue yielded from one more
  unit is equal to the extra cost

3
Marginal Revenue Product

• The marginal revenue product (MRP) from hiring an
  extra unit of any input is the extra revenue
  yielded by selling what that extra input produces
  
• MRP MR ? MP

4
Marginal Expense

• If the supply curve facing the firm for the
  inputs it hires are infinitely elastic at
  prevailing prices, the marginal expense of hiring
  a worker is simply this market wage
• If input supply is not infinitely elastic, a
  firms hiring decision may have an effect on
  input prices

5
Marginal Expense

• For now, we will assume that the firm is a price
  taker for the inputs it buys
• ?TC/?K r
• ?TC/?L w
• The first-order conditions for profit-maximization
  become
• MRPK r
• MRPL w

6
An Alternative Derivation

• Profit maximization requires that MR MC so we
  have
• MR ? MPK MRPK r
• MR ? MPL MRPL w

7
Price Taking in theOutput Market

• If a firm exhibits price-taking behavior in its
  output market, MR P
• This means that at the profit-maximizing levels
  of each input
• P ? MPK r
• P ? MPL w
• sometimes P multiplied by an inputs MP is called
  the value of marginal product

8
Comparative Statics ofInput Demand

• We will focus on the comparative statics of the
  demand for labor
• the analysis for capital would be symmetric
• For the most part, we will assume price-taking
  behavior for the firm in its output market

9
Single-Input Demand

• Suppose that the number of truffles harvested in
  a particular forest is

• Assuming that truffles sell for 50 per pound,
  total revenue for the owner is

10
Single-Input Demand

• Marginal revenue product is given by

• If truffle searchers wages are 500, the owner
  will determine the optimal amount of L to hire by

11
Competitive Determination of Income Shares

• If the firm is profit-maximizing, each input will
  be hired to the point where its MRP is equal to
  its price
• Thus,

12
Monopsony in theLabor Market

• In many situations, the supply curve for an input
  (L) is not perfectly elastic
• We will examine the polar case of monopsony,
  where the firm is the single buyer of the input
  in question
• the firm faces the entire market supply curve
• to increase its hiring of labor, the firm must
  pay a higher wage

13
Monopsony in theLabor Market

• The marginal expense of hiring an extra unit of
  labor (MEL) exceeds the wage
• If the total cost of labor is wL, then

• In the competitive case, ?w/?L 0 and MEL w
• If ?w/?L gt 0, MEL gt w

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Monopsony in theLabor Market
Wage
ME
S
D
Labor
15
Monopsony in theLabor Market
Wage
ME
S
w1
D
Labor
L1
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Monopsonistic Hiring

• Suppose that a coal mines workers can dig 2 tons
  per hour and coal sells for 10 per ton
• this implies that MRPL 20 per hour
• If the coal mine is the only hirer of miners in
  the local area, it faces a labor supply curve of
  the form
• L 50w

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Monopsonistic Hiring

• The firms wage bill is
• wL L2/50
• The marginal expense associated with hiring
  miners is
• MEL ?wL/?L L/25
• Setting MEL MRPL, we find that the optimal
  quantity of labor is 500 and the optimal wage is
  10

18
Monopoly in theSupply of Inputs

• Imperfect competition may also occur in input
  markets if suppliers are able to form a monopoly
• labor unions in closed shop industries
• production cartels for certain types of capital
  equipment
• firms (or countries) that control unique supplies
  of natural resources

19
Monopoly in theSupply of Inputs

• If both the supply and demand sides of an input
  market are monopolized, the market outcome will
  be indeterminate
• the actual outcome will depend on the bargaining
  skills of the parties

20
Monopoly in theSupply of Inputs
Wage
ME
S
D
MR
Labor