Although it is useful for managers to know the fundamentals of the theory of profit maximization, it is even more useful for them to know how to implement and use the theory to maximize their firms’ profits. A manager should be able to use empirical estimates or forecasts of the relevant variables and equations to de- termine the actual values of the decision variables that maximize the firm’s profit.
You have spent a lot of time learning the techniques of estimating the various demand, production, and cost functions. Now you will learn how to use these empirical skills to answer an important question facing a manager: How can the theory of profit maximization be used in practice to make profit-maximizing deci- sions about production?
We will first outline how managers can, in general, determine the optimizing conditions. This outline gives a pattern for situations in which numerical estimates of the variables and equations are available. Then we present an example of how a firm can use this approach to determine the optimal level of output.
General Rules for Implementation
We emphasized that a manager must answer two questions when choosing the level of output that maximizes profit. These two questions and the answers forth- coming from the theoretical analysis are summarized as follows:
1. Should the firm produce or shut down? Produce as long as the market price is greater than or equal to minimum average variable cost: P $ AVCmin. Shut down otherwise.
2. If production occurs, how much should the firm produce? Produce the output at which market price (which is marginal revenue) equals marginal cost P 5 SMC.
14With a bit of algebra, after substituting the relations MRPL 5 P 3 MPL and MRPK 5 P 3 MPK into these conditions for profit maximization, it can be shown that the amounts of labor and capital that maximize profit are also economically efficient (lie on the expansion path) because they also meet the condition w/r 5 MPL/MPK. While all input combinations that maximize profit lie on the expansion path, only one input combination on the expansion path maximizes profit—the input combination on the isoquant corresponding to the profit-maximizing output.
It follows from these rules that to determine the optimal level of output, a man- ager must obtain estimates or forecasts of the market price of the good produced by the firm, the firm’s average variable cost function, and the firm’s marginal cost function. The steps explained next can be followed to find the profit-maximizing rate of production and the level of profit the firm will earn.
Step 1: Forecast the price of the product To decide whether to produce and how much to produce, a manager must obtain a forecast of the price at which the completed product can be sold. Remember that a perfectly competitive firm does not face a downward-sloping demand curve but simply takes the market price as given. We showed in Chapter 7 how to use two statistical techniques—
time-series forecasting and econometric forecasting—to forecast the price of the product.
Step 2: Estimate average variable cost (AVC) and marginal cost (SMC) As emphasized in Chapter 10, the cubic specification is the appropriate form for estimating a family of short-run cost curves. Thus the manager could estimate the following average variable cost function
AVC 5 a 1 bQ 1 cQ2
As demonstrated in Chapter 10, the marginal cost function associated with this average variable cost function is
SMC 5 a 1 2bQ 1 3cQ2
Step 3: Check the shutdown rule When P is less than AVC, the firm loses less money by shutting down than it would lose if it produced where P 5 SMC. A man- ager can determine the price below which a firm should shut down by finding the minimum point on the AVC curve, AVCmin. As long as price is greater than (or equal to) AVCmin, the firm will produce rather than shut down. Recall from Chapter 10 that the average variable cost curve reaches its minimum value at Qm 5 −b/2c. The minimum value of average variable cost is then determined by substituting Qm into the AVC function
AVC min 5 a 1 b Q m 1 c( Q m)2
The firm should produce as long as P $ AVCmin. If the forecasted price is greater than (or equal to) minimum average variable cost (P $ AVCmin), the firm should produce the output level where P 5 SMC. If the forecasted price is less than the minimum average variable cost (P , AVCmin), the firm should shut down in the short run, and it loses an amount equal to its total fixed costs.
Step 4: If P $ AVC min , find the output level where P 5 SMC A perfectly competitive firm should produce the level of output for which P 5 SMC—
if P $ AVCmin. Thus if the manager decides to produce in the short run, the
manager maximizes profit by finding the output level for which P 5 SMC. In the case of a cubic specification for cost, profit maximization or loss minimization requires that
P 5 SMC 5 a 1 2bQ 1 3cQ2
Solving this equation for Q* gives the optimal output level for the firm—unless P is less than AVC, and then the optimal output level is zero.
Step 5: Computation of profit or loss Once a manager determines how much to produce, the calculation of total profit or loss is straightforward. Profit or loss is equal to total revenue minus total cost. Total revenue for a competitive firm is price times quantity sold. Total cost is the sum of total variable cost and total fixed cost, where total variable cost is average variable cost times the number of units sold.
Hence, total profit, denoted as p is p 5 TR 2 TC
5 (P 3 Q*) 2 [(AVC 3 Q*) 1 TFC]
5 (P 2 AVC)Q* 2 TFC
If P , AVCmin, the firm shuts down, and p 5 −TFC.
To illustrate how to implement these steps to find the profit-maximizing level of output and to forecast the profit of the firm, we now turn to a hypothetical firm that operates in a perfectly competitive market.
Profit Maximization at Beau Apparel: An Illustration
As an example, we use the output decision facing the manager of Beau Apparel, Inc., a clothing manufacturer that produces moderately priced men’s shirts.
Beau Apparel is only one of many firms that produce a fairly homogeneous product, and none of the firms in this moderate-price shirt market engages in any significant advertising.
Price forecasts In mid-December 2015, the manager of Beau Apparel was preparing the firm’s production plan for the first quarter of 2016. The manager wanted to obtain a forecast of the wholesale price of shirts for the first quarter of 2016. This price forecast would subsequently be used in making the production decision for Beau Apparel. The manager requests price forecasts from Beau Apparel’s Marketing/Forecasting Division. The market researchers, using forecasting techniques similar to those described in Chapter 7, provided the man- ager with three wholesale price forecasts based on three different assumptions about economic conditions in the first quarter of 2016
High: $20 Medium: $15 Low: $10
Estimation of average variable cost and marginal cost The manager of Beau Apparel chose a cubic specification of short-run cost for estimating the average variable cost and the marginal cost curves. Using time-series data over the six-year time period 2010(I) through 2015(IV), during which Beau Apparel had the same- size plant, the following average variable cost function was estimated
AVC 5 20 2 0.003Q 1 0.00000025Q2
All the estimated coefficients (20, −0.003, and 0.00000025) had the required signs and were statistically significant. The estimated average cost function provided the information needed for making the decision to produce or shut down. We will return to this decision after we discuss how the manager of Beau Apparel estimated the marginal cost function.
As explained in Chapter 10, the parameter estimates for the average variable cost function can be used to obtain the estimated marginal cost function
SMC 5 a 1 2bQ 1 3cQ2
where a, b, and c are the estimated parameters (coefficients) for the AVC function.
The manager used the estimated coefficients of the average variable cost equation to obtain the corresponding marginal cost function. For the estimate of the aver- age variable cost function given above, the corresponding marginal cost function for shirts was
SMC 5 20 1 2(20.003)Q 1 3(0.00000025)Q2 5 20 2 0.006Q 1 0.00000075Q2
After obtaining forecasts of price and estimates of the average variable cost and marginal cost curves, the manager was able to answer the two production ques- tions: (1) Should the firm produce or shut down? And (2) if production is war- ranted, how much should the firm produce? We now can show how the manager of Beau Apparel made these two decisions and calculated the firm’s forecasted profit.
The shutdown decision Because the estimated average variable cost function for shirts was