1. BASIC PRINCIPLES AND EXTENSIONS
Chapter 12
COSTS
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

2. Definitions of Costs
It is important to differentiate between accounting cost and economic cost
the accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries
economists focus more on opportunity cost

3. Definitions of Costs Labor Costs
to accountants, expenditures on labor are current expenses and hence costs of production
to economists, labor is an explicit cost
labor services are contracted at some hourly wage (w) and it is assumed that this is also what the labor could earn in alternative employment

4. Definitions of Costs Capital Costs
accountants use the historical price of the capital and apply some depreciation rule to determine current costs
economists refer to the capital’s original price as a “sunk cost” and instead regard the implicit cost of the capital to be what someone else would be willing to pay for its use
we will use v to denote the rental rate for capital

5. Definitions of Costs Costs of Entrepreneurial Services
To an accountant, the owner of a firm is entitled to all profits, which are the revenues or losses left over after paying all input costs
Economists consider the opportunity costs of time and funds that owners devote to the operation of their firms
these services are inputs and some cost should be imputed to them
part of accounting profits would be considered as entrepreneurial costs by economists

6. Economic Cost
The economic cost of any input is the payment required to keep that input in its present employment
the remuneration the input would receive in its best alternative employment

7. Two Simplifying Assumptions
There are only two inputs
homogeneous labor (L), measured in labor-hours
homogeneous capital (K), measured in machine-hours
entrepreneurial costs are included in capital costs
Inputs are hired in perfectly competitive markets
firms are price takers in input markets

8. Economic Profits Total costs for the firm are given by
total costs = TC = wL + vK
Total revenue for the firm is given by
total revenue = Pq = Pf(K,L)
Economic profits () are equal to
 = total revenue - total cost
 = Pq - wL - vK
 = Pf(K,L) - wL - vK

9. Economic Profits
Economic profits are a function of the amount of capital and labor employed
we could examine how a firm would choose K and L to maximize profit
“derived demand” theory of labor and capital inputs (see Chapter 21)
But for now we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

10. Cost-Minimizing Input Choices
To minimize the cost of producing a given level of output, a firm should choose a point on the isoquant at which the RTS is equal to the ratio w/v
it should equate the rate at which K can be traded for L in the productive process to the rate at which they can be traded in the marketplace

11. Cost-Minimizing Input Choices
Mathematically, we seek to minimize total costs given q = f(K,L) = q0
Setting up the Lagrangian
L = wL + vK + [q0 - f(K,L)]
First order conditions are
L/L = w - (f/L) = 0
L/K = v - (f/K) = 0
L/ = q0 - f(K,L) = 0

12. Cost-Minimizing Input Choices
Dividing the first two conditions we get
The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices

13. Cost-Minimizing Input Choicesq0
Given output q0, we wish to find the least costly point on the isoquant
TC1
TC2
TC3
Costs are represented by parallel lines with a slope of -w/v
K per period
TC1 < TC2 < TC3
L per period

14. Cost-Minimizing Input Choices
The minimum cost of producing q0 is TC2
This occurs at the tangency between the isoquant and the total cost curve
K per period
TC1
TC3
TC2 K* L*
The optimal choice is L*, K* q0
L per period

15. L = f(K,L) + D(TC1 - wL - vK)
Output Maximization
The dual formulation of the firm’s cost minimization problem is to maximize output for a given level of cost
The Lagrangian is
L = f(K,L) + D(TC1 - wL - vK)
The first-order conditions are identical to those for the primal problem

16. Output Maximization
q00
The maximum output attainable with total cost TC2 is q0 q0 q1
This occurs at the tangency between the total cost curve and isoquant q0
K per period
TC2 = wL + vK K* L*
The optimal choice is L*, K*
L per period

17. Derived Demand
In Chapter 5, we considered how the utility-maximizing choice is affected by the change in the price of a good
we used this technique to develop the demand curve for a good
Can we develop a firm’s demand for an input in the same way?

18. Derived Demand
To analyze what happens to K* if v changes, we must know what happens to the output level chosen by the firm
The demand for K is a derived demand
it is based on the level of the firm’s output
We cannot answer questions about K* without looking at the interaction of supply and demand in the output market

19. The Firm’s Expansion Path
The firm can determine the cost-minimizing combinations of K and L for every level of output
If input costs remain constant for all amounts of K and L the firm may demand, we can trace the locus of cost-minimizing choices
called the firm’s expansion path

20. The Firm’s Expansion Path
q00
The expansion path is the locus of cost-minimizing tangencies q0 q1
The curve shows how inputs increase as output increases
K per period E
L per period

21. The Firm’s Expansion Path
The expansion path does not have to be a straight line
some inputs may increase faster than others as output expands
depends on the shape of the isoquants
The expansion path does not have to be upward sloping
if the use of an input falls as output expands, that input is an inferior input

22. Minimizing Costs for a Cobb-Douglas Function
Suppose that the production function for hamburgers is
q = 10K 0.5 L 0.5
Total costs are given by
TC = vK + wL
Suppose that the firm wishes to minimize the cost of producing 40 hamburgers

23. Minimizing Costs for a Cobb-Douglas Function
The Lagrangian expression is
L = vK + wL + ( K 0.5 L 0.5)
The first-order conditions are
L/K = v - 5(L/K)0.5 = 0
L/L = w - 5(K/L)0.5 = 0
L/ = K 0.5 L 0.5 = 0

24. Minimizing Costs for a Cobb-Douglas Function
Dividing the first equation by the second gives us
This production function exhibits constant returns to scale so the expansion path is a straight line

25. Total Cost Function
The total cost function shows that for any set of input costs and for any output level, the minimum cost incurred by the firm is
TC = TC(v,w,q)
As output increases, total costs increase

26. Average Cost Function
The average cost function (AC) is found by computing total costs per unit of output

27. Marginal Cost Function
The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

28. Graphical Analysis of Total Costs
Suppose that K1 units of capital and L1 units of labor input are required to produce one unit of output
TC(q=1) = vK1 + wL1
To produce m units of output
TC(q=m) = vmK1 + wmL1 = m(vK1 + wL1)
TC(q=m) = m  TC(q=1)
TC is proportional to q

29. Graphical Analysis of Total CostsTC
Total costs are proportional to output
Total
costs
AC = MC
Both AC and
MC will be
constant
Output

30. Graphical Analysis of Total Costs
Suppose instead that total costs start out as concave and then becomes convex as output increases
one possible explanation for this is that there is another factor of production that is fixed as capital and labor usage expands
total costs begin rising rapidly after diminishing returns set in

31. Graphical Analysis of Total CostsTC
Total costs rise
dramatically as
output rises
after diminishing
returns set in
Output

32. Graphical Analysis of Total CostsMC
MC is the slope of the TC curve
Average and marginal
costs
If AC > MC, AC must be
falling
If AC < MC, AC must be
rising AC
min AC
Output

33. Shifts in Cost Curves
The cost curves are drawn under the assumption that input prices and the level of technology are held constant
any change in these factors will cause the cost curves to shift

34. Homogeneity
The total cost function is homogeneous of degree one in input prices
if all input prices were to increase in the same proportion (t), the total costs for producing any given output level would also be multiplied by t
a simultaneous increase in input prices does not change the ratio of input prices
cost-minimizing combination of K and L unchanged

35. Homogeneity
The average and marginal cost functions will also be homogeneous of degree one in input prices
In a “pure” inflationary period (one where the prices of all inputs rise at the same rate), there will be no incentive for firms to alter their input choices

36. Change in the Price of One Input
If the price of only one input changes, the firm’s cost-minimizing choice of inputs will be affected
a new expansion path must be derived

37. Change in the Price of One Input
An increase in the price of one input must increase TC for any output level
AC will also rise
If the input is not inferior, MC will also rise

38. Change in the Price of One Input
A change in the price of an input means that the firm must alter its cost-minimizing choice of inputs
in the two-input case, an increase in w/v will be met by an increase in K/L
Define the elasticity of substitution as
s must be nonnegative

39. Size of Shifts in Costs Curves
The increase in costs will be largely influenced by the relative significance of the input in the production process
If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs

40. Cobb-Douglas Cost Function
Suppose that the production function is
q = 10K 0.5L0.5
First-order conditions for cost minimization require that
w/v = K/L
Dividing by K yields
q/k = 10(v/w)0.5

41. Cobb-Douglas Cost Function
This means that
K = (q/10)w 0.5v -0.5
Multiplying both sides by v, we get
vK = (q/10)w 0.5v 0.5
A similar chain of substitutions yields
wL = (q/10)w 0.5v 0.5

42. Cobb-Douglas Cost Function
Because
TC = vK + wL
we have
TC = (2/10)qw 0.5v 0.5
If w = v = $4, then
TC = 0.8q

43. Cobb-Douglas Cost Function
Because the production function exhibits constant returns to scale, AC and MC will be constant
AC = TC/q = 0.8
MC = TC/q = 0.8
If v rises to $9, TC, AC, and MC rise
TC = (2/10)qw 0.5v 0.5 = 1.2q
AC = TC/q = 1.2
MC = TC/q = 1.2

44. Short-Run, Long-Run Distinction
In the short run, economic actors have only limited flexibility in their actions
Assume that the capital input is held constant at K1 and the firm is free to vary only its labor input
The production function becomes
q = f(K1,L)

45. Short-Run Total Costs Short-run total cost for the firm is
STC = vK1 + wL
There are two types of short-run costs:
short-run fixed costs (SFC) are costs associated with fixed inputs
short-run variable costs (SVC) are costs associated with variable inputs

46. Short-Run Total Costs
Short-run costs are not minimal costs for producing the various output levels
the firm does not have the flexibility of input choice
to vary its output in the short run, the firm must use nonoptimal input combinations
the RTS will not be equal to the ratio of input prices

47. Short-Run Total Costs Because capital is fixed at K1,
K per period
Because capital is fixed at K1,
the firm cannot equate RTS
with the ratio of input prices K1 q2 q1 q0
L per period L1 L2 L3

48. Short-Run Marginal and Average Costs
The short-run average total cost (SATC) function is
SATC = total costs/total output = STC/q
The short-run marginal cost (SMC) function is
SMC = change in STC/change in output = STC/q

49. Short-Run Average Fixed and Variable Costs
Short-run average fixed costs (SAFC) are
SAFC = total fixed costs/total output = SFC/q
Short-run average variable costs are
SAVC = total variable costs/total output = SVC/q

50. Relationship between Short-Run and Long-Run Costs
STC (K2)
Total costs
STC (K1) q0 q1 q2 TC
The long-run
TC curve can
be derived by
varying the
level of K
STC (K0)
Output

51. Relationship between Short-Run and Long-Run Costs
The geometric
relationship
between short-
run and long-run
AC and MC can
also be shown
Costs q0 q1 AC MC
SATC (K0)
SMC (K0)
SATC (K1)
SMC (K1)
Output

52. Relationship between Short-Run and Long-Run Costs
At the minimum point of the AC curve:
the MC curve crosses the AC curve
MC = AC at this point
the SATC curve is tangent to the AC curve
SATC (for this level of K) is minimized at the same level of output as AC
SMC intersects SATC also at this point
the following are all equal:
AC = MC = SATC = SMC

53. Important Points to Note:
A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices

54. Important Points to Note:
Repeated application of this minimization procedure yields the firm’s expansion path
the expansion path shows how input usage expands with the level of output
The relationship between output level and total cost is summarized by the total cost function [TC(q,v,w)]
the firm’s average cost (AC = TC/q) and marginal cost (MC = TC/q) can be derived directly from the total cost function

55. Important Points to Note:
All cost curves are drawn on the assumption that the input prices and technology are held constant
when an input price changes, cost curves shift to new positions
the size of the shifts will be determined by the overall importance of the input and by the ease with which the firm may substitute one input for another
technical progress will also shift cost curves

56. Important Points to Note:
In the short run, the firm may not be able to vary some inputs
it can then alter its level of production only by changing the employment of its variable inputs
it may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs