The Functional Form of the Total Cost Function

ATC=TC/Q MC=dTC/dQ

FIGURE 6.2 Marginal,average total,average variable,and average fixed cost curves. AVC= TVC/Q

0 Q1 Q3 Q2 AFC=TFC/Q

Q

and group health insurance,amounted to $5,000.David’s monthly variable costs,including wages and salaries,telecommunications services (telephone, fax,e-mail,conference calling,etc.),personal computer rentals and maintenance,mainframe computer time sharing,and office supplies,amounted to$20,000. a.Assuming that David does not pay himself a salary,what are the total monthly explicit costs of Computer Compatriots? b.What are the firm’s total monthly economic costs?

Solution

a.Total explicit cost refers to all “out-of-pocket”expenses.The total monthly explicit costs of Computer Compatriots are

b.“Total economic cost”is the sum of total explicit costs and total implicit costs.In this case,the total monthly implicit (opportunity) cost to David

Ricardo of founding Computer Compatriots is $10,000,which was the monthly salary earned by him while working for IMC.The new firm’s total monthly economic costs are

TC TFC TVCexplicit explicit= + = + =$, $ , $ , 5000 20000 25000

TC TFC TVC TVC economic explicit implicit = + + = + + = $, $ , $ ,5000 20000 120000 12 $ ,35000

THE FUNCTIONAL FORM OF THE TOTAL COST FUNCTION

Figure 6.1 illustrated a short-run total cost function exhibiting both increasing and diminishing marginal product.In the diagram,increasing

marginal product occurs over the range of output from 0 to Q1,while diminishing marginal product characterizes output levels greater than Q1.The inflection point,which occurs at output level Q1,corresponds to minimum marginal cost.The short-run total cost function in Figure 6.1 may be characterized by a cubic function of the form TCQ b bQ bQ bQ ( ) = + + + 0 1 2 (6.10)

For Equation (6.10) to make sense,the values of the coefficients (bi s) must make economic sense.The value of the constant term,for example, must be restricted to some positive value, b0 > 0,since total fixed cost must be positive.

To determine the restrictions that should be placed on the remaining coefficient values,consider again Figure 6.1.Note that as illustrated in Figure 6.2,in which marginal cost is always positive,total cost is an increasing function of output.Since the minimum value of MC is positive,it is necessary to restrict the remaining coefficients so that the absolute minimum value of the marginal cost function is also positive.These restrictions are3

2 3 3

3 To see this,consider output level at which MC is minimized.Taking the first derivative of the total cost function to derive the marginal cost function,we find that

Taking the first derivative of the marginal cost function (the second derivative of the total cost function) and setting the results equal to zero,we get

which reduces to

The second-order condition for a minimum is

That is,the value of b3 must be positive.Since we expect the optimal value of Q (Q*) to be positive,then by implication b2 must be negative.Upon substituting Q* into the marginal cost function,the minimum value of MC becomes

The equation for MCmin indicates that the restrictions b3 > 0 and b2 < 0 are not sufficient to guarantee that the absolute minimum of marginal cost will be positive.This requires the additional condition that (3b3b1 - b2 2) > 0.The last restriction implies that b1 > 0 and that 3b3b1 >

dTC dQ = MCQ b bQ bQ ( ) = + +1 2 3 2 3

2

dMC dQ 2 dTC dQ b bQ = = + = 2 2 3 2 6 0

Q = -b2 b33

2 dMC dQ2 b3 6 0= >

2 MC b b min = +1 2 Ê Ë -b2 b33 ˆ ¯ + b33 Ê Ë -b2 b33 ˆ ¯

2

2b b

2 b =- + = 33 1 bb b 3 1 23 3b3

2

b2 2 > 0.