Managers of firms that have some control over the price they charge should know the fundamentals of the theory of profit maximization by firms with market power. They should also know how to use empirical estimates of the demand for the firm’s product and the cost equations for determining the price and level of output that maximize the firm’s profit. This section describes how to use empirical analysis to find that optimal price and output. We devote most of this section to examining the price and output decision for a monopoly. However, as we stressed previously, the decision-making process for a monopoly is applicable to any firm with market power, with perhaps a few modifications for changes in the form of the demand and marginal revenue functions to account for differences in the market structure.
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
A manager must answer two questions when finding the price and output that maximizes profit. These two questions and the answers forthcoming from the the- oretical analysis are summarized as follows:
1. Should the firm produce or shut down? Produce as long as the market price equals or exceeds minimum average variable cost: P ≥ AVCmin. Shut down otherwise.
2. If production occurs, how much should the firm produce and what price should it charge? Produce the output at which marginal revenue equals marginal cost—MR = SMC—and charge the price from the demand curve for the profit- maximizing output.
I L L U S T R AT I O N 1 2 . 4 Hedging Jet Fuel Prices: Does It Change
the Profit-Maximizing Price of a Ticket?
Many companies engage in a financial practice called
“hedging” when they use large amounts of a key in- put and wish to insure against sharp increases in the price of the important input. While the details of creating a hedge involve rather complex financial instruments—derivatives, futures contracts, and call and put options—the fundamental principle of hedg- ing input prices is not difficult to understand as long as you remember that hedging is a form of insurance for protecting the firm from rising input prices. Hedg- ing lowers the cost of obtaining an input, but it does not lower the cost of using an input.a At any point in time, the cost of using an input is whatever the in- put could be sold for at that time in the input market.
Consequently, the input price that matters for making current decisions about price and output is the price of the input when it will be used in production, not the price created by hedging. This distinction is extremely important for managers to understand, and yet, many managers and business analysts are confused about the role of input price hedging in maximizing the profit and value of firms.
Southwest Airlines, for example, has been very aggressive in hedging the price of jet fuel over the past 20 years, and many airline industry analysts believe Southwest’s successful jet fuel hedges made it possible for the airline to charge lower airfares than their rivals who were not so active or suc- cessful in jet fuel hedging. When a jet fuel hedge works successfully, the savings to Southwest can be substantial. In a typical year, Southwest burns over a billion gallons of jet fuel. In one recent year, when jet fuel prices rose sharply, Southwest’s fuel hedges allowed it to buy jet fuel at a price of $1.98 per gallon while rival American Airlines paid $2.74 per gallon for its fuel. As a result of this successful hedging, Southwest’s financial statement for the year showed a sizeable profit in the section where it reported earnings on hedging activity. We must emphasize, however, that the hedged price of jet fuel had no impact at all on the economic profit from flying passengers. In fact, all airlines incur identical
per gallon costs for jet fuel because they all face the same current or spot price in jet fuel markets. For decision-making purposes, Southwest, as well as all other airlines, will not choose to make a flight unless airfare revenues can cover the total (avoidable) vari- able costs of the flight. The computation of variable fuel costs is done using the spot price of jet fuel at flight time rather than the historic cost of obtaining the jet fuel. Also, when determining air fares, South- west and all other airlines possessing some degree of market power will follow the MR = MC rule to find the profit-maximizing price on their respective demand curves. For profit-maximizing decisions it is the spot price, not the hedged price, of jet fuel that determines the marginal cost of making a flight.
Thus, airfares will not be affected by hedging jet fuel, although successful hedging does increase the por- tion of the airline’s profit attributable to the financial transactions associated with fuel hedging.
As a final cautionary note about hedging input prices, we must stress that not all hedging is suc- cessful, because input price changes are typically difficult to predict accurately. Southwest Airlines has on several occasions suffered substantial losses on its jet fuel hedges when crude oil prices fell unex- pectedly, causing the price of jet fuel to fall as well.b Hedging input prices can add to the value of a firm when input prices rise sharply, but it can also be a burden when input prices fall. Either way, manag- ers wishing to maximize profit from the production of goods and services should always use the actual market price of the input for making output and pricing decisions.
aYou learned in Chapter 8 that the cost of using any input is what the firm gives up to employ the input in its own operation rather than selling, leasing, or renting the input to someone else. Even if the cost of obtaining an input is zero, the cost of using the input is its current market price. Recall from Illustration 8.2 that Jamie Lashbrook obtained two Super Bowl tickets for free and he learned that the cost for him to use the tickets was equal to $1,200 for each ticket, the amount other sport fans offered to pay him for his two tickets.
bIn 2008, for example, Southwest Airlines reported a quarterly loss of $247 million on its fuel hedging activity, even as it reported positive profit from airline operations.
It follows from these rules that to determine the optimal price and output, a man- ager will need estimates or forecasts of the market demand of the good produced by the firm, the inverse demand function, the associated marginal revenue function, the firm’s average variable cost function, and the firm’s marginal cost function. We now set forth the steps that can be followed to find the profit- maximizing price and output for a firm with market power.
Step 1: Estimate the demand equation To determine the optimal level of output, the manager must estimate the marginal revenue function. Marginal revenue is derived from the demand equation; thus the manager begins by estimating de- mand. In the case of a linear demand specification, the empirical demand function facing the monopolist can be written as
Q = a + bP + cM + dPR
where Q is output, P is price, M is income, and PR is the price of a good related in consumption. To obtain the estimated demand curve for the relevant time period, the manager must have forecasts for the values of the exoge- nous variables, M and PR, for that time period. Once the empirical demand equation has been estimated, the forecasts of M and PR (denoted Mˆ and PˆR) are substituted into the estimated demand equation, and the demand function is expressed as
Q = a′ + bP where a′ = a + c Mˆ + dPˆR.
Step 2: Find the inverse demand equation Before we can derive the marginal revenue function from the demand function, the demand function must be expressed so that price is a function of quantity: P = f(Q). This is accomplished by solving for P in the estimated demand equation in step 1
P = −____ba ′ + __1 b Q
= A + BQ
where A = −____ba ′ and B = 1 __b . This form of the demand equation is the inverse demand function. Now the demand equation is expressed in a form that makes it possible to solve for marginal revenue in a straightforward manner.
Step 3: Solve for marginal revenue Now recall that when demand is expressed as P = A + BQ, marginal revenue is MR = A + 2BQ. Using the inverse demand function, we can write the marginal revenue function as
MR = A + 2BQ = ____−ab ′ + 2 __b Q
Step 4: Estimate average variable cost (AVC ) and marginal cost (SMC ) In Chapter 10 we discussed in detail the empirical techniques for estimating cubic cost functions. There is nothing new or different about estimating SMC and AVC for a monopoly firm. The usual forms for the AVC and SMC functions, when TVC is specified as a cubic equation, are
AVC = a + bQ + cQ2
SMC = a + 2bQ + 3cQ2
You may wish to review this step by returning to Chapter 10 or to Chapter 11.
Step 5: Find the output level where MR 5 SMC To find the level of output that maximizes profit or minimizes losses, the manager sets marginal revenue equal to marginal cost and solves for Q
MR = A + 2BQ = a + 2bQ + 3cQ2 = SMC
Solving this equation for Q* gives the optimal level of output for the firm—unless P is less than AVC, and then the optimal level of output is zero.
Step 6: Find the optimal price Once the optimal quantity, Q*, has been found in step 5, the profit-maximizing price is found by substituting Q* into the inverse demand equation to obtain the optimal price, P*
P* = A + BQ*
This price and output will be optimal only if price exceeds average variable cost.
Step 7: Check the shutdown rule For any firm, with or without market power, if price is less than average variable cost, the firm will shut down (Q* = 0) because it makes a smaller loss producing nothing than it would lose if it produced any positive amount of output. The manager calculates the average variable cost at Q* units
AVC* = a + bQ* + cQ*2
If P* ≥ AVC*, then the monopolist produces Q* units of output and sells each unit of output for P* dollars. If P* < AVC*, then the monopolist shuts down in the short run.
Step 8: Computation of profit or loss To compute the profit or loss, the manager makes the same calculation regardless of whether the firm is a monopolist, oli- gopolist, or perfect competitor. Total profit or loss is
π* = TR − TC
= (P* × Q*) − [(AVC* × Q*) + TFC]
If P < AVC, the firm shuts down, and π = −TFC.
To illustrate how to implement these steps to find the profit-maximizing price and output level and to forecast profit, we now turn to a hypothetical firm that possesses a degree of market power.
Maximizing Profit at Aztec Electronics: An Example
By virtue of several patents, Aztec Electronics possesses substantial market power in the market for advanced wireless stereo headphones. In December 2015, the manager of Aztec wished to determine the profit-maximizing price and output for its wireless stereo headphones for 2016.
Estimation of demand and marginal revenue The demand for wireless headphones was specified as a linear function of the price of wireless head- phones, the income of the buyers, and the price of stereo tuners (a complemen- tary good)
Q = f (P, M, PR)
Using data available for the period 2005–2015, a linear form of the demand func- tion was estimated. The resulting estimated demand function was
Q = 41,000 − 500P + 0.6M − 22.5PR
where output (Q) is measured in units of sales and average annual fam- ily income (M) and the two prices (P and PR) are measured in dollars. Each estimated parameter has the expected sign and is statistically significant at the 5 percent level.
From an economic consulting firm, the manager obtained 2016 forecasts for income and the price of the complementary good (stereo tuners) as, respectively,
$45,000 and $800. Using these values— Mˆ = 45,000 and P ˆR = 800—the estimated (forecasted) demand function for 2016 was
Q = 41,000 − 500P + 0.6(45,000) − 22.5(800) = 50,000 − 500P
The inverse demand function for the estimated (empirical) demand function was obtained by solving for P
P = 100 − 0.002Q
From the inverse demand function, the manager of Aztec Electronics obtained the estimated marginal revenue function
MR = 100 − 0.004Q
Figure 12.9 illustrates the estimated linear demand and marginal revenue curves for Aztec Electronics.
Estimation of average variable cost and marginal cost The manager of Aztec Electronics obtained an estimate of the firm’s average variable cost function using
Now try Technical Problem 15.
a short-run quadratic specification (as described in Chapter 10). The estimated average variable cost function was
AVC = 28 − 0.005Q + 0.000001Q2
For this estimation, AVC was measured in dollar units and Q was measured in units of sales. Given the estimated average variable cost function, the marginal cost function is
SMC = 28 − 0.01Q + 0.000003Q2
As you can see, the specification and estimation of cost functions are the same regardless of whether a firm is a price-taker or a price-setter.
The output decision Once the manager of Aztec obtained estimates of the mar- ginal revenue function and the marginal cost function, the determination of the optimal level of output was accomplished by equating the estimated marginal revenue equation with the estimated marginal cost equation and solving for Q*.
Setting MR equal to SMC results in the following expression 100 − 0.004Q = 28 − 0.01Q + 0.000003Q2
Solving this equation for Q, the manager of Aztec finds two solutions:
Q = 6,000 and Q = 24,000. Inasmuch as Q = 24,000 is an irrelevant solution—
negative outputs are impossible—the optimal level of output is Q* = 6,000. That is, the profit-maximizing (or loss-minimizing) number of wireless stereo headphones to produce and sell in 2016 is 6,000 units—if the firm chooses to produce rather than shut down.
F I G U R E 12.9 Demand and Marginal Revenue for Aztec Electronics
Price and marginal revenue (dollars)
50,000 40,000
30,000 20,000
20 80
Output (Q)
P = 100 – 0.002Q
40 60 100
MR = 100 – 0.004Q
0 10,000
The pricing decision Once the manager of Aztec Electronics has found the optimal level of output, determining the profit-maximizing price is really nothing more than finding the price on the firm’s demand curve that corresponds to the profit-maximizing level of output. The optimal output level Q* is substituted into the inverse demand equation to obtain the optimal price. Substituting Q* = 6,000 into the inverse demand function, the optimal price P* is
P* = 100 − 0.002(6,000) = $88 Thus Aztec will charge $88 for a set of headphones in 2016.
The shutdown decision To see if Aztec Electronics should shut down produc- tion in 2016, the manager compared the optimal price of $88 with the average variable cost of producing 6,000 units. Average variable cost for 6,000 units was computed as
AVC* = 28 − 0.005(6,000) + 0.000001(6,000)2 = $34
Obviously, $88 is greater than $34; so if these forecasts prove to be correct in 2016, all the variable costs will be covered and the manager should operate the plant rather than shut it down. Note that Aztec’s expected total revenue in 2016 was
$528,000 (= $88 × 6,000) and estimated total variable cost was $204,000 (= $34 × 6,000). Because total revenue exceeded total variable cost (TR > TVC), the man- ager did not shut down.
Computation of total profit or loss Computation of profit is a straightforward process once the manager has estimated total revenue and all costs. The manager of Aztec has already estimated price and average variable cost for 2016, but total fixed cost is needed to calculate total profit or loss. On the basis of last year’s data, the manager of Aztec Electronics estimated that fixed costs would be $270,000 in 2016. The profit was calculated to be
π = TR − TVC − TFC
= $528,000 − $204,000 − $270,000
= $54,000
Figure 12.10 shows the estimated equations for 2016 and the profit-maximizing price and output. At point A, MR = SMC, and the profit-maximizing level of out- put is 6,000 units (Q* = 6,000). At point B, the profit-maximizing price is $88, the price at which 6,000 units can be sold. At point C, ATC is $79, which was calcu- lated as
ATC = TC/Q = ($204,000 + $270,000)/6,000
= $79
The total profit earned by Aztec is represented by the area of the shaded rectangle.
The firm makes a loss Now suppose that per capita income falls, causing the demand facing Aztec to fall to
P = 80 − 0.002Q so marginal revenue is now
MR = 80 − 0.004Q Average variable and marginal costs remain constant.
To determine the new level of output under the new estimated demand condi- tions, the manager equates the new estimated marginal revenue equation with the marginal cost equation and solves for Q*
80 − 0.004Q = 28 − 0.01Q + 0.000003Q2
Again there are two solutions: Q = −3,167 and Q = 5,283. Ignoring the negative level of output, the optimal level is Q* = 5,283. Substituting this value into the inverse demand function, the optimal price is
P* = 80 − 0.002(5,283) = $69.43
To determine whether to produce or shut down under the reduced-demand situation, the manager calculated the average variable cost at the new level of output and compared it with price
AVC = 28 − 0.005(5,283) + 0.000001(5,283)2 = $29.49
F I G U R E 12.10 Profit Maximization at Aztec Electronics
Price and cost (dollars)
10,000 8,000
6,000 4,000
34
Output
D: P = 100 – 0.002Q 76
100
MR = 100 – 0.004Q
28 79 88
A B C
AVC = 28 – 0.005Q + 0.000001Q2 SMC = 28 – 0.01Q + 0.000003Q2
ATC = AVC + AFC
0 2,000
Clearly if Aztec produces in 2016, total revenue will cover all of total variable cost since
P = $69.43 > $29.49 = AVC Aztec’s profit or loss is
π = TR − TVC − TFC
= $69.43(5,283) − $29.49(5,283) − $270,000
= $366,799 − $155,796 − $270,000
= −$58,997
Despite the predicted loss of $58,997, Aztec should continue producing. Losing
$58,997 is obviously better than shutting down and losing the entire fixed cost of
$270,000.