As is the case when estimating a production function, specification of an ap- propriate equation for a cost function must necessarily precede the estimation of the parameters using regression analysis. The specification of an empirical cost equation must ensure that the mathematical properties of the equation reflect the properties and relations described in Chapter 8. Figure 10.4 illustrates again the typically assumed total variable cost, average variable cost, and marginal cost curves.
user cost of capital The firm’s opportunity cost of using capital.
F I G U R E 10.4 Typical Short-Run Cost Curves
TVC
SMC
AVC
Quantity Quantity
Total variable cost (dollars) Marginal and average variable cost (dollars)
Estimation of Typical Short-Run Costs
Since the shape of any one of the three cost curves determines the shape of the other two, we begin with the average variable cost curve. Because this curve is
∪-shaped, we use the following quadratic specification:
AVC = a + bQ + cQ2
As explained earlier, input prices are not included as explanatory variables in the cost equation because the input prices (adjusted for inflation) are assumed to be constant over the relatively short time span of the time-series data set. In order for the AVC curve to be ∪-shaped, a must be positive, b must be negative, and c must be positive; that is, a . 0, b , 0, and c . 0.4
Given the specification for average variable cost, the specifications for total vari- able cost and marginal cost are straightforward if AVC = TVC/Q, it follows that
TVC = AVC × Q = (a + bQ + cQ2)Q = aQ + bQ2 + cQ3
Note that this equation is a cubic specification of TVC, which conforms to the S-shaped TVC curve in Figure 10.4.
The equation for marginal cost is somewhat more difficult to derive. It can be shown, however, that the marginal cost equation associated with the above TVC equation is
SMC = a + 2bQ + 3cQ2
If, as specified for AVC, a . 0, b , 0, and c . 0, the marginal cost curve will also be ∪-shaped.
Because all three of the cost curves, TVC, AVC, and SMC, employ the same parameters, it is necessary to estimate only one of these functions to obtain estimates of all three. For example, estimation of AVC provides estimates of a, b, and c, which can then be used to generate the marginal and total variable cost functions. The total cost curve is trivial to estimate; simply add the constant fixed cost to total variable cost.
As for the estimation itself, ordinary least-squares estimation of the total (or average) variable cost function is usually sufficient. Once the estimates of a, b, and c are obtained, it is necessary to determine whether the parameter estimates are of the hypothesized signs and statistically significant. The tests for significance are again accomplished by using either t-tests or p-values.
Using the estimates of a total or average variable cost function, we can also ob- tain an estimate of the output at which average cost is a minimum. Remember that when average variable cost is at its minimum, average variable cost and marginal cost are equal. Thus we can define the minimum of average variable cost as the output at which
AVC = SMC
4The appendix to this chapter derives the mathematical properties of a cubic cost function.
Using the specifications of average variable cost and marginal cost presented ear- lier, we can write this condition as
a + bQ + cQ2 = a + 2bQ + 3cQ2 or
bQ + 2cQ2 = 0
Solving for Q, the level of output at which average variable cost is minimized is Qm = −b/2c
Table 10.2 summarizes the mathematical properties of a cubic specification for total variable cost.
Before estimating a short-run cost function, we want to mention a potential problem that can arise when the data for average variable cost are clustered around the minimum point of the average cost curve, as shown in Figure 10.5.
If the average variable cost function is estimated using data points clustered as shown in the figure, the result is that while a is positive and ˆ ˆb is negative, a t-test or a p-value would indicate that c is not statistically different from 0. This ˆ result does not mean that the average cost curve is not ∪-shaped. The problem
T A B L E 10.2 Summary of a Cubic Specification for Total Variable Cost
Cubic total variable cost function
Total variable cost TVC = aQ + bQ 2 + cQ 3 Average variable cost AVC = a + bQ + cQ 2
Marginal cost SMC = a + 2bQ + 3cQ 2
AVC reaches minimum point Qm = −by2c Restrictions on parameters a . 0
b , 0
c . 0
F I G U R E 10.5
A Potential Data Problem
Average variable cost (dollars)
Output
AVC
is that because there are no observations for the larger levels of output, the estimation simply cannot determine whether average cost is rising over that range of output.
Estimation of Short-Run Costs at Rockford Enterprises: An Example
In July 2014, the manager at Rockford Enterprises decided to estimate the total variable, average variable, and marginal cost functions for the firm. The capital stock at Rockford has remained unchanged since the second quarter of 2012. The manager collected quarterly observations on cost and output over this period and the resulting data were as follows:
Average Quarter Output variable cost ($)
2012 (II) 300 $38.05
2012 (III) 100 39.36
2012 (IV) 150 28.68
2013 (I) 250 28.56
2013 (II) 400 48.03
2013 (III) 200 33.51
2013 (IV) 350 45.32
2014 (I) 450 59.35
2014 (II) 500 66.81
Average variable cost was measured in nominal (i.e., current) dollars, and the cost data were subject to the effects of inflation. Over the period for which cost was to be estimated, costs had increased due to the effects of inflation. The manager’s analyst decided to eliminate the influence of inflation by deflating the nominal costs. Recall that such a deflation involves converting nominal cost into constant-dollar cost by dividing the nominal cost by an appropriate price index.
The analyst used the implicit price deflator for gross domestic product (GDP) pub- lished in the Survey of Current Business, which can be found at the website for the Bureau of Economic Analysis (www.bea.gov/scb). The following values for the price deflator were used to deflate the nominal cost data:
Implicit Price Deflator Quarter (2009 = 100)
2012 (II) 109.16
2012 (III) 109.73
2012 (IV) 110.60
2013 (I) 111.54
2013 (II) 112.22
2013 (III) 113.12
2013 (IV) 114.03
2014 (I) 114.95
2014 (II) 115.89
To obtain the average variable cost, measured in constant (2009) dollars, for the 300 units produced in the second quarter of 2012, $38.05 is divided by the implicit price deflator 104.94 (divided by 100), which gives $36.26
$36.26 = ____________$38.05 104.94 4 100
Note that it is necessary to divide the implicit price deflator by 100 because the price deflators in the Survey of Current Business are expressed as percentages.
Repeating this computation for each of the average variable cost figures, the manager obtained the following inflation-adjusted cost data:
Deflated average Quarter Output variable cost ($)
2012 (II) 300 $36.26
2012 (III) 100 37.33
2012 (IV) 150 27.10
2013 (I) 250 26.89
2013 (II) 400 45.10
2013 (III) 200 31.34
2013 (IV) 350 42.24
2014 (I) 450 55.13
2014 (II) 500 61.73
Given these inflation-adjusted data, the manager estimated the cost functions.
As shown above, it is sufficient to estimate any one of the three cost curves to obtain the other two because each cost equation is a function of the same three parameters: a, b, and c. The manager decided to estimate the average variable cost function:
AVC = a + bQ + cQ2
and obtained the following printout from the estimation of this equation:
DEPENDENT VARIABLE: AVC R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 9 0.9382 45.527 0.0002
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 44.473 6.487 6.856 0.0005 Q 20.143 0.0482 22.967 0.0254 Q2 0.000362 0.000079 4.582 0.0037
After the estimates were obtained, the manager determined that the estimated coefficients had the theoretically required signs: ˆa . 0, ˆ b , 0, and ˆc . 0. To
determine whether these coefficients are statistically significant, the p-values were examined, and the exact level of significance for each of the estimated coefficients was acceptably low (all the t-ratios are significant at better than the 5 percent level of significance).
The estimated average variable cost function for Rockford Enterprises is, therefore,
AVC = 44.473 − 0.143Q + 0.000362Q2
which conforms to the shape of the average variable cost curve in Figure 10.4. As emphasized above, the marginal cost and total variable cost equations are eas- ily determined from the estimated parameters of AVC, and no further regression analysis is necessary. In this case,
SMC = ˆ a + 2 ˆ b Q + 3 ˆ c Q2
= 44.473 − 0.286Q + 0.0011Q2 and
TVC = ˆ a Q + ˆ b Q2 + ˆ c Q3
= 44.473Q − 0.143Q2 + 0.000362Q3
To illustrate the use of the estimated cost equations, suppose the manager wishes to calculate the marginal cost, average variable cost, and total variable cost when Rockford is producing 350 units of output. Using the estimated marginal cost equation, the marginal cost associated with 350 units is
SMC = 44.473 − 0.286(350) + 0.0011(350)2
= 44.473 − 100.10 + 134.75
= $79.12
Average variable cost for this level of output is
AVC = 44.473 − 0.143(350) + 0.000362(350)2
= 44.473 − 50.05 + 44.345
= $38.77
and total variable cost for 350 units of output is TVC = AVC × Q
= 38.77 × 350
= $13,569
The total cost of 350 units of output would, of course, be $13, 569 plus fixed cost.
Finally, the output level at which average variable cost is minimized can be computed as
Qm = −b/2c
In this example,
Qm = ____________0.143 2 × 0.000362 = 197
At Rockford Enterprises, average variable cost reaches its minimum at an output level of 197 units, when
AVC = 44.473 2 0.143(197) + 0.000362(197)2 = 44.473 2 28.17 + 14.05
= $30.35
As you can see from this example, estimation of short-run cost curves is just a straightforward application of cost theory and regression analysis. Many firms do, in fact, use regression analysis to estimate their costs of production.
Now try Technical Problem 3.