Profit Maximization and Derived Demand

• A firm’s hiring of inputs is directly related to its desire to maximize profits– any firm’s 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)

Profit Maximization and Derived Demand

• First-order conditions for a maximum are

0

K

TC

K

TR

K

0

L

TC

L

TR

L

– 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

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

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 firm’s hiring decision may have an effect on input prices

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

An Alternative Derivation

• Profit maximization requires that MR = MC so we have

MR MPK = MRPK = r

MR MPL = MRPL = w

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 input’s MP is called the value of marginal product

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

Single-Input Demand

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

LQ 100

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

LQPTR 000,5

Single-Input Demand

• Marginal revenue product is given by

2/1500,2

L

L

TR

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

2/1500,2500 L

25L

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,

Q

LMP

PQ

LMPP

PQ

wL LL

share slabor'

Q

KMP

PQ

KMPP

PQ

vK KK

share scapital'

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

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

L

wLw

L

wLMEL

• In the competitive case, w/L = 0 and MEL = w

• If w/L > 0, MEL > w

Monopsony in theLabor Market

Labor

Wage

S

ME

D

L1

The firm will set MEL = MRPL to determine its profit-maximizing level of labor (L1)

w1

The wage is determined by the supply curve

Monopsony in theLabor Market

Labor

Wage

S

ME

D

L1

w1

Note that the quantity of labor demanded by this firm falls short of the level that would be hired in a competitive labor market (L*)

L*

w* The wage paid by the firm will also be lower than the competitive level (w*)

Monopsonistic Hiring

• Suppose that a coal mine’s 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

Monopsonistic Hiring

• The firm’s wage bill iswL = 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

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

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

Monopoly in theSupply of Inputs

Labor

Wage

S

ME

DMR

L1

w1

The monopoly seller would prefer a wage of w1 with L1 workers hired

L2

w2

The monopsony buyer would prefer a wage of w2 with L2 workers hired