1. FIRMS’ DEMANDS FOR INPUTS
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.

2. 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)

3. 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

4. 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

5. 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

6. 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 profit-maximization become
MRPK = v
MRPL = w

7. 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

8. An Alternative Derivation
The first two equations can be written as

9. 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

10. 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

11. 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 price-taking behavior for the firm in its output market

12. 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

13. Single-Input Case
Taking the total differential of
P  MPL = w
yields

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. Substitution Effect
The substitution effect is shown holding output constant at q0 K  B
As w falls, the firm will substitute L for K in the production process K2 L2 A K1  q0 L L1

21. 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

22. Output Effect
The output effect is shown holding relative input prices constant K C K3 L3 
Since a drop in w leads to a decline in MC, optimal output will rise and the firm will demand more L B q1 K2  q0 L L2

23. 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 C K3 L3  K A K1  B q1 K2  q0 L L1 L2

24. 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

25. 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

26. Substitution and Output Effects
We can now look mathematically at the substitution and output effects of a change in w

27. 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

28. 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

29. 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

30. Output Effects
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

31. Output Effects L/w (from changes in q) < 0 product > 0 < 0
= 1
L/w (from changes in q) < 0

32. Mathematical Derivation
The mathematical conclusion is that L/w < 0
substitution and output effects move in the same direction

33. Decomposing Input Demand
The short-run supply function for a hamburger producer is
The firm’s demand for labor is

34. 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

35. 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

36. 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

37. 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

38. Decomposing Input Demand
Applying Shephard’s lemma yields
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

39. 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

40. 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?

41. 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

42. 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

43. 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

44. 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

45. 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)

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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,

52. Factor Shares and the Elasticity of Substitution
The elasticity of substitution is defined as
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

53. 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

54. 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
In the competitive case, w/L = 0 and MEL = w
If w/L > 0, MEL > w

55. Monopsony in the Labor Market
The firm will set MEL = MRPL to determine its profit-maximizing level of labor (L1)
Wage ME S w1
The wage is determined by the supply curve D
Labor

56. Monopsony in the Labor Market
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*
Wage ME S w*
The wage paid by the firm will also be lower than the competitive level (w*) w1 D
Labor L1

57. 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

58. 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

59. 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

60. 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

61. Monopoly in the Supply of Inputsw1
The monopoly seller would prefer a wage of w1 with L1 workers hired
Wage ME S L2 w2
The monopsony buyer would prefer a wage of w2 with L2 workers hired D MR
Labor

62. 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

63. 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

64. 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

65. 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

66. 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