Download Market for Inputs.SU4

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
 TR TC


0
K
K
K
 TR TC


0
L
L
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 profitmaximization 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 the
Output 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 of
Input 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 pricetaking behavior for the firm in its output
market
Single-Input Demand
• Suppose that the number of truffles
harvested in a particular forest is
Q  100 L
• Assuming that truffles sell for $50 per
pound, total revenue for the owner is
TR  P  Q  5,000 L
Single-Input Demand
• Marginal revenue product is given by
TR
 2,500L1/ 2
L
• If truffle searchers’ wages are $500, the
owner will determine the optimal
amount of L to hire by
1/ 2
500  2,500L
L  25
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,
wL P  MPL  L MPL  L
labor' s share 


PQ
PQ
Q
vK P  MPK  K MPK  K
capital' s share 


PQ
PQ
Q
Monopsony in the
Labor 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 the
Labor Market
• The marginal expense of hiring an extra
unit of labor (MEL) exceeds the wage
• If the total cost of labor is wL, then
wL
w
MEL 
w L
L
L
• In the competitive case, w/L = 0 and
MEL = w
• If w/L > 0, MEL > w
Monopsony in the
Labor Market
Wage
ME
S
The firm will set MEL =
MRPL to determine its
profit-maximizing level
of labor (L1)
The wage is determined
by the supply curve
w1
D
L1
Labor
Monopsony in the
Labor Market
Wage
ME
S
w*
w1
D
L1
L*
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*)
The wage paid by the
firm will also be lower
than the competitive
level (w*)
Labor
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 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
Monopoly in the
Supply 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 the
Supply 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 the
Supply of Inputs
The monopoly seller
would prefer a wage of
w1 with L1 workers hired
Wage
ME
S
w1
The monopsony buyer
would prefer a wage of
w2 with L2 workers hired
w2
MR
L1 L2
D
Labor