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Chapter 21
FIRMS’ DEMANDS FOR INPUTS
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
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 = v
TC/L = w
• The first-order conditions for profitmaximization become
MRPK = v
MRPL = w
An Alternative Derivation
• The Lagrangian expression associated
with a firm’s cost-minimization problem is
L = vK + wL + [q0 - f(K,L)]
• First-order conditions are
L/K = v - (f /K) = 0
L/L = w - (f /L) = 0
L/ = q0 - f (K,L) = 0
An Alternative Derivation
• The first two equations can be written as
f

 MPK  v
K
f

 MPL  w
L
An Alternative Derivation
• Since  can be interpreted as marginal
cost in this problem, we have
MC  MPK = v
MC  MPL = w
• Profit maximization requires that MR =
MC so we have
MR  MPK = MRPK = v
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 = v
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 Case
• It is likely that L/w < 0
– this is based on the presumption that the
marginal physical product of labor declines
as the quantity of labor employed rises
• a fall in w must be met by a fall in MPL for the
firm to continue maximizing profits (because P
is fixed)
– this argument is strictly correct for the case
of one input
Single-Input Case
• Taking the total differential of
P  MPL = w
yields
MPL L
dw  P 

 dw
L w
MPL L
1 P 

L w
L
1

w P  MPL / L
Single-Input Case
• If we assume that MPL /L < 0 (MPL
falls as L increases), we have
L/w < 0
• A fall in w will cause more labor to be
hired
– more output will be produced as well
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
Two-Input Case
• If w falls, both L and K will change as a
new cost-minimizing combination of
inputs is chosen
• When K changes, the entire MPL
function changes
– labor has a different amount of capital to
work with
• However, we still expect that L /w < 0
Two-Input Case
• When w changes, we can decompose
the total effect on the quantity of L hired
into two components
– substitution effect
– output effect
Substitution Effect
• If output is held constant and w falls,
there will be a tendency to substitute L
for K in the production process
– cost-minimization requires that RTS = w/v
– a fall in w means that RTS must fall as well
• because isoquants exhibit a diminishing RTS,
the cost-minimizing level of labor hired rises
Substitution Effect
The substitution effect is shown holding
output constant at q0
K
slope  
K1

w1
v
A
As w falls, the firm will substitute L
for K in the production process
B

K2
q0
w
slope   2
v
L1
L2
L
Output Effect
• A change in w will shift the firm’s
expansion path
• This means that the firm’s cost curves
will also shift
– a drop in w will lower MC and lead to a
higher level of output
• This increase in output will lead to a
higher level of L being demanded
Output Effect
K
The output effect is shown holding
relative input prices constant
slope  
w2
v
Since a drop in w leads to a
decline in MC, optimal
output will rise and the firm
will demand more L
C

K3
B

K2
q1
q0
w
slope   2
v
L2 L3
L
Substitution and
Output Effects
Both the substitution effect and the
output effect lead to a rise in the
quantity of L demanded when w falls
K
K1

A
C

K3
B

K2
q1
q0
L1
L2 L3
L
Cross-Price Effects
• No definite statement can be made about
how capital will change when w changes
• The substitution and output effects move
in opposite directions
– a fall in w will lead the firm to substitute
away from K
– a fall in w will lead the firm to increase output
and thus demand more K
Mathematical Derivation
• General input demand functions
generated by the firm’s profit-maximizing
decision are
L = L(P,w,v)
K = K(P,w,v)
• The presence of P in these functions
indicates the close connection between
product demand and input demand
Substitution and
Output Effects
• We can now look mathematically at the
substitution and output effects of a
change in w
L L
L

(q constant) 
(from changes in q )
w w
w
Constant Output
Demand Functions
• Shephard’s lemma uses the envelope
theorem to show that the constant
output demand function for L can be
found by partially differentiating total
cost with respect to w
TC
 L' (q,w ,v )
w
Constant Output
Demand Functions
• Two arguments suggest why L’/w < 0
– in the two-input case, the assumption of a
diminishing RTS combined with the
assumption of cost-minimization requires
that w and L move in opposite directions
when output is held constant
– even in the many-input case, L’/w =
2TC/w2 < 0 if costs are truly minimized
Output Effects
• We can use a “chain rule” argument to
examine the causal links that determine
how changes in w affect the demand for
L through induced output changes
L
L q P MC
(from changes in q ) 



w
q P MC w
Output Effects
L
L q P MC
(from changes in q ) 



w
q P MC w
– P/MC = 1 because P=MC for profit
maximization under perfect competition
– q/P < 0 since there is an inverse
relationship between the firm’s price and its
share of market demand
– L/q and MC/w must have the same sign
Output Effects
L
L q P MC
(from changes in q ) 



w
q P MC w
<0
=1
product > 0
L/w (from changes in q) < 0
Mathematical Derivation
• The mathematical conclusion is that
L/w < 0
– substitution and output effects move in the
same direction
Decomposing Input Demand
• The short-run supply function for a
hamburger producer is
40(10P )
q
(vw )0.5
• The firm’s demand for labor is
(10P )2
L  0. 5 1. 5
v w
Decomposing Input Demand
• If w = v = $4 and P = $1, the firm will
produce 100 hamburgers hire 6.25
workers
• If w rises to $9 while v and P remain
unchanged, the firm will produce 66.6
hamburgers and hire only 1.9 workers
Decomposing Input Demand
• Suppose that the firm had continued to
produce 100 hamburgers even though w
rose to $9
• Cost minimization requires:
K/L = w/v = 9/4
Decomposing Input Demand
• Substituting into the original production
function, we get
q = 100 = 40K0.25L0.25
10 = 4[(9/4)L]0.25L0.25
L = 4.17
• This is the substitution effect
– even if output remained at 100 hamburgers,
employment of L would decline from 6.25 to
4.17
Decomposing Input Demand
• We can compute the constant output
demand for labor using Shephard’s
lemma
• Total costs are
TC = vK + wL
• If we substitute the input demand
functions, we get
TC = [q2v0.5w0.5]/800
Decomposing Input Demand
• Applying Shephard’s lemma yields
TC q 2v 0.5w 0.5
L' 

w
1,600
• For q = 100, we have
L’ = 6.25v0.5w -0.5
• If v = w = $4, L’ = 6.25
• If w changes to $9, L’ = 4.17
Decomposing Input Demand
• Note how the constant output demand
function (L’) allows us to hold output
constant in our analysis, while the total
demand function (L) allows output to
change
– there will be a larger impact of a wage
change when using the total demand
function
Responsiveness of Input
Demand to Changes
in Input Prices
• We can now explain the degree to
which input demand will respond to
changes in input prices
• When w rises, will the decline in L be
large or small?
Responsiveness of Input
Demand to Changes
in Input Prices
• Substitution effect
– this will depend on how easy it is to substitute
other inputs for L
• the elasticity of substitution for the firm’s
production function
• the length of time allowed for adjustment
Responsiveness of Input
Demand to Changes
in Input Prices
• Output effect
– this will depend on the size of the decline in
output
• how important labor is in the firm’s total costs
• the price elasticity of demand for the output
Responsiveness of Input
Demand to Changes
in Input Prices
• The price elasticity of demand for any
input will be greater (in absolute value),
– the larger is the elasticity of substitution for
other inputs
– the larger is the share of total cost
represented by expenditures on that input
– the larger is the price elasticity of demand for
the good being produced
Elasticity of Demand for Inputs
• Suppose that the constant output
demand for labor for our hamburger
producer is
L’ = (q2v0.5w -0.5)/1,600
• The constant output wage elasticity of
demand (LL) is
L' w  ln L'
LL 
 
 0.5
w L'  ln w
Elasticity of Demand for Inputs
• Suppose that labor costs represent half
of all variable costs
– sL = 0.5
• This implies that
LL = sL – 1 = -(1 – sL) = -0.5
• This result is a special case of
LL = -(1 – sL)
Elasticity of Demand for Inputs
• In order to quantify output effects, we will
need to use an elasticity form of the chain
of events that occurs when the wage
changes
eL,w (from changes in q) = eL,q  eq,P  eP,MC  eMC,w
• If P is assumed constant and MC is a
linear function of q, any increase in P
must result in a proportional change in q
– this implies that eq,P  eP,MC = -1
Elasticity of Demand for Inputs
• In addition, we know that
– using the TC curve shown earlier, we
can calculate eMC,w = 0.5
– because the production function
exhibits diminishing returns to scale,
eL,q = 1/eq,L = 2 for movements along
the expansion path
Elasticity of Demand for Inputs
• Thus,
eL,w (from changes in q) = (2)(-1)(0.5) = -1
and the total elasticity of demand
(including substitution and output
effects) is
eL,w = -0.5 – 1.0 = -1.5
Elasticity of Demand for Inputs
• When the change in w affects all firms,
the price of the product will change
– In the long run, with constant returns to
scale, eL,q = 1, eP,MC = 1, and eMC,w = sL
• The output effect can be written as
eL,w (from changes in q) = sLeq,P
• The total wage elasticity is
eL,w = LL + sLeq,P = -(1 – sL) + sLeq,P
Competitive Determination
of Income Shares
• Assume there is only one firm producing
a homogeneous output using L and K
• The production function for the firm is
Q = f(K,L) and output sells at a price of P
• The total income received by labor is wL,
while the total income accruing to capital
is vK
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
Factor Shares and the
Elasticity of Substitution
• The elasticity of substitution is defined as
%(K / L )

%(w / v )
– If  = 1, the relative shares of K and L will
remain constant as the capital-labor ratio
rises
– If  > 1, the relative share of K will rise as
the capital-labor ratio increases
– If  < 1, the relative share of K will fall as the
capital-labor ratio increases
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
Important Points to Note:
• The marginal revenue product from
hiring extra units of an input is the
combined influence of the marginal
physical product of the input and the
firm’s marginal revenue in its output
market
Important Points to Note:
• If the firm is a price taker for the inputs it
buys, it is possible to analyze the
comparative statics of its demand fairly
completely
– a rise in the price of an input will cause
fewer units to be hired because of
substitution and output effects
• the size of these effects will depend on the firm’s
technology and on the price responsiveness of
the demand for its output
Important Points to Note:
• The marginal productivity theory of input
demand can also be used to study the
determinants of relative income shares
accruing to various factors of production
– the elasticity of substitution indicates how
these shares change in response to
changing factor supplies
Important Points to Note:
• If a firm has a monopsonistic position in
an input market, it will recognize how its
hiring affects input prices
– the marginal expense associated with hiring
additional units of an input will exceed that
input’s price, and the firm will reduce hiring
below competitive levels to maximize profits
Important Points to Note:
• If input suppliers form a monopoly
against a monopsonistic demander, the
result is indeterminate
– in such a situation of bilateral monopoly, the
market equilibrium chosen will depend on
the bargaining of the two parties