Key Relationships:Average Total Cost,Average Fixed Cost,Average Variable Cost,and Marginal Cost

total variable cost is an increasing function of the level of output. Total cost is the sum of total fixed and total variable cost.

KEY RELATIONSHIPS: AVERAGE TOTAL COST, AVERAGE FIXED COST, AVERAGE VARIABLE COST, AND MARGINAL COST

The definitions of average fixed cost,average variable cost,average total cost,and marginal cost are,appropriately,as follows:

Average fixed cost: AFC = TFC Q (6.5)

Average variable cost: AVC = TVC Q (6.6)

Average total cost: ATC TC Q = = + TFC TVC Q AFC AVC= + (6.7)

Marginal cost: C= M dTC dQ dTVC dQ (6.8)

Average total cost is the total cost of production per unit.It is the total cost of production divided by total output.Average total cost is a short-run production concept if total cost includes fixed cost.It is a long-run production concept if all costs are variable costs. Average fixed cost,which is a short-run production concept,is total fixed cost per unit of output.It is total fixed cost divided by total output. Average variable cost is total variable cost of production per unit of output.Average variable cost is total variable cost divided by total output.

Marginal cost is the change in the total cost associated with a change in total output.1 Contrary to conventional belief,this is not the same thing as the cost of producing the “last”unit of output.Since it is total cost that is changing,the cost of producing the last unit of output is the same as the

1 Strictly speaking,it is incorrect to assert that marginal cost is the rate of change in total cost with respect to a change in output unless total cost has been properly specified.The total cost equation TC = PKK + PLL is a function of inputs K and L,and factor prices PK and PL.Mathematically,it is incorrect to differentiate total cost with respect to the nonexistent argument, Q. As we saw from our discussion of isoquants in Chapter 5,it is possible to produce a given level of output with numerous combinations of inputs.For the profit-maximizing firm,however,only the cost-minimizing combination of inputs is of interest.Marginal cost is not,therefore,an arbitrary increase in total cost given an increase in output.Rather,marginal cost is the minimum increase in total cost with respect to an increase in output.The appropriate total cost equation is

where L* and K* represent the optimal input levels for a cost-minimizing firm.Once the total cost function has been properly defined,marginal cost is MC =∂TC*(PK, PL, Q)/∂Q.For a more detailed discussion,see Silberberg (1990,Chapter 10,pp.226–227).

TC PK P P Q PL P P Q TC P P Q K K L L K L K L= ( ) + ( ) = ( )* , , * , , * , ,

per-unit cost of producing any other level of output.More specifically,the marginal cost of production for a profit-maximizing firm is equal to average total cost plus the per-unit change in total cost,multiplied by total output.2 Equation (6.8) shows that marginal cost is the same as marginal variable cost,since total fixed cost is a constant.

Related to marginal cost is the more general concept of incremental cost. While marginal cost is the change in total cost given a change in output, incremental cost is the change in the firm’s total costs that result from the implementation of decisions made by management,such as the introduction of a new product or a change in the firm’s advertising campaign.By contrast, sunk costs are invariant with respect to changes in management decisions.Since sunk costs are not recoverable once incurred,they should not be considered when one is determining,say,an optimal level of output or product mix.Suppose,for example,that a textile manufacture purchases a loom for $1 million.If the firm is able to dispose of the loom in the resale market for only $750,000,the firm,in effect,has permanently lost $250,000. In other words,the firm has incurred a sunk cost of $250,000.

Related to the concept of sunk cost is the analytically more important concept of total fixed cost.Total fixed cost,which represents the cost of a firm’s fixed inputs,is invariant with respect to the profit-maximizing level of output.This is demonstrated in Equation (6.8).Changes in total cost with respect to changes in output are the same as changes in total variable cost with respect to changes in output.In other words,marginal variable cost is identical to marginal cost.

The distinction between sunk and fixed cost is subtle.Suppose that when the firm operated the loom to produce cotton fabrics,the rental price of the loom was $100,000 per year.This rental price is invariant to the firm’s level of production.In other words,the firm would rent the loom for $100,000 per year regardless of whether it produced 5,000 or 100,000 yards of cloth during that period.A sunk cost is essentially the difference between the purchase price of the loom and its salvage value.

2 As discussed in footnote 1,the appropriate total cost equation for a profit-maximizing firm is TC = PKK*(PK, PL, Q) + PLL*(PK,PL, Q) = TC*(PK, PL, Q).Thus,appropriate average total cost function is

Differentiating this expression with respect to output yields

Rearranging,and noting that MC* =∂TC*/∂Q,yields

For a more detailed exposition and discussion,see Silberberg (1990,Chapter 10,pp.229–230).

ATC P P Q K L * , ,( ) = TC P P Q K L * , ,( ) Q

∂ATC ∂Q Q TC Q TC * * *

Q2

MC ATC* *= +Ê Ë ∂ ATC Q *ˆ ¯Q

TC=f(Q)

0

Q1 Q2 Q

FIGURE 6.1 Total cost curve exhibiting increasing then diminishing marginal product.

Additional insights into the relationship between total cost and production may be seen by examining Equation (6.8).Recalling that Q = f(K,L) and applying the chain rule we obtain

Recalling from Equation (6.3) that TC = PKK0 + PLL,then marginal cost may be written as

(6.9)

where PL is the rental price of homogeneous labor input and MPL is the marginal product of labor.Equation (6.9) establishes a direct link between marginal cost and the marginal product of labor,which was discussed in Chapter 5.Since PL is a constant,it is easily seen that MC varies inversely with MPL.Recalling Figure 5.2 in the preceding chapter,the shapes of the TC, ATC, AVC,and MC curves may easily be explained.When MPL is rising (falling),for example, MC will be falling (rising).These relationships are illustrated in Figures 6.1 and 6.2.

Equation (6.9) indicates clearly the relation between the theory of production and the theory of cost.The cost curves are shaped as they are because the production function exhibits the properties it does,especially the law of diminishing marginal product.In other words,underlying the short-run cost functions are the short-run production functions. Problem 6.1. For most of his professional career,David Ricardo was a computer programmer for International Megabyte Corporation (IMC). During his last year at IMC,David earned an annual salary of $120,000. Last year,David founded his own consulting firm,Computer Compatriots, Inc.His monthly fixed costs,including rent,property and casualty insurance,

MC dTC dQ = = Ê Ë dTC dL ˆ ¯Ê Ë dL dQ ˆ ¯

MC PL= Ê Ë dL dQ ˆ ¯ =

LP MPL