Now that we know how to compute marginal revenue, we can figure out the profit-maximizing output level for any firm with market power.

Many people’s first thought is that a firm with market power should sell until the marginal revenue falls to zero and then stop. After all, any more production after that would reduce revenues and wouldn’t be profitable. That idea would be correct if there was no cost of production. With production costs, however, it’s not quite right. A firm with market power should pay attention to its marginal revenue, but one more piece of the puzzle is necessary to figure out the profit-maximizing output level: cost.

In Chapter 8, we discussed the two basic elements of firm profit—revenue and cost—and how each of these is determined by the firm’s choice of how much output to produce. We saw that the profit-maximizing output was the one that set marginal revenue equal to marginal cost. We went on to show that marginal revenue for a perfectly competitive firm equals the market price, so maximizing profit meant producing the quantity at which price equals marginal cost. The logic behind this condition is that if price is above marginal cost, the perfectly competitive firm should produce more because the additional revenue it would earn exceeds the additional cost. If price is below marginal cost, it should cut back on production because it’s losing money on those extra units.

The same underlying logic works for firms with market power except that marginal revenue no longer equals price. To maximize its profit, a firm should choose its quantity where its marginal revenue equals its marginal cost:

MR = MC

If marginal revenue is above marginal cost, a firm can produce more and earn more revenue than the extra cost of production, and increase its profit. If marginal revenue is below marginal cost, a firm can reduce its output, lose less revenue than it saves in cost, and again raise its profit. Only when these two marginal values are equal does changing output not increase profit.

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Thus, we see that the monopolist and the perfectly competitive firm do exactly the same thing. They both produce at the level where MR = MC. But because marginal revenue isn’t the same as the sales price for a firm with market power, it behaves differently than a perfectly competitive firm.

Setting MR = MC gives us the quantity, Q^*, that maximizes the firm’s profit, and from that we figure out the profit-maximizing price. The height of the demand curve at that profit-maximizing quantity Q^* tells us the market price for the firm’s output.

For a firm with market power, we can think of the firm choosing a profit-maximizing quantity as equivalent to choosing a profit-maximizing price. The demand curve ties together price and quantity, so picking one implies the other. The monopolist can either produce the profit-maximizing quantity of output and let the market determine the price (which will be the profit-maximizing price), or it can set the profit-maximizing price and let the market determine the quantity (which will be the profit-maximizing quantity).

An important factor to remember is that even though firms with market power have an ability to set the price for their output, they cannot profitably charge whatever price they want to. A firm with market power can keep raising its price (or equivalently, keep cutting its output), but if the firm raises the price by too much, its customers will stop buying—even if there are no other competitors.

For example, in 2010 the iPad dominated the tablet market and Apple clearly had market power as a result. But Apple could not charge whatever price it wanted. Suppose Apple had charged $20,000 for each iPad. Just about everyone would have stopped buying it, even if it was the only tablet computer on earth. The vast majority of people would simply not find it worth having an iPad at that price. A monopolist doesn’t lose business to direct competitors by raising the price for its product. (A monopolist has no direct competitors.) Instead, it loses business by driving its customers out of the market. A firm is limited by the demand for its product. Because the demand curve is downward-sloping, a rise in price means a decline in quantity demanded. This sensitivity to price means that monopolists can’t (or more precisely, won’t, if they care about profit) charge anything they want. They will charge a higher price than a more competitive firm, however.

We can apply the exact logic of the previous analysis to graphically derive the profit-maximizing output and price of a firm with market power, given the firm’s demand and marginal cost curves. Let’s assume we are again looking at the market for iPads and that marginal cost is constant at $200. Specifically, we will follow these steps:

Step 1:Derive the marginal revenue curve from the demand curve. For a linear demand curve, this will be another straight line with the same vertical intercept that is twice as steep. In Figure 9.3, the marginal revenue curve is shown as MR.

Step 2:Find the output quantity at which marginal revenue equals marginal cost. This is the firm’s profit-maximizing quantity of output. In Figure 9.3, Apple’s profit-maximizing level of output is Q^*, or 80 million iPads.

Step 3:Determine the profit-maximizing price by locating the point on the demand curve at that optimal quantity level. To determine the price Apple should charge consumers to maximize its profit, just follow Q^* up to the demand curve and then read the price off the vertical axis. If Apple produces the profit-maximizing output level of 80 million, the market price will be $600. (Or equivalently, if Apple charges a price of $600, it will sell 80 million iPads.)

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That’s all there is to it. Once we have the firm’s MR curve, we can use the profit-maximization rule MR = MC to find the firm’s optimal level of output and price.

We can also solve for the profit-maximizing quantity and price mathematically, given equations for the firm’s demand and marginal cost curves.

Suppose Apple’s marginal cost of producing iPads is constant at $200, and the demand curve for iPads (where Q is in millions and P in dollars) is Q = 200 – 0.2P. How much should Apple charge for iPads, and how many will it sell at that price? (Again, because of the equivalence of choosing price and choosing output level for firms with market power, we could ask how many iPads Apple should produce and at what price the iPads would sell, and the answer would be the same.)

We can figure this out using the same three-step process described above: Derive the marginal revenue curve, find the quantity at which the marginal revenue equals marginal cost, and then determine the profit-maximizing price by computing the price at that quantity on the demand curve.

Step 1: Derive the marginal revenue curve from the demand curve. Let’s start by obtaining the inverse demand curve by rearranging the demand curve so that price is a function of quantity rather than the other way around:

Q = 200 – 0.2P

0.2P = 200 – Q

P = 1,000 – 5Q

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This is a linear inverse demand curve of the form P = a – bQ, where a = 1,000 and b = 5. Earlier we learned that MR = a – 2bQ is the marginal revenue curve for this type of demand curve.4 So for this demand curve, Apple’s marginal revenue curve is

MR = 1,000 – 2(5Q) = 1,000 – 10Q

Step 2: Find the output quantity at which marginal revenue equals marginal cost. Apple’s marginal cost is constant at $200. Therefore, we just set the marginal revenue curve equal to this value and solve for Q:

MR = MC

1,000 – 10Q = 200

800 = 10Q

Q^* = 80

So, Apple’s profit-maximizing quantity of iPads is 80 million.

Step 3:Determine the profit-maximizing price by locating the point on the demand curve at that optimal quantity level. Find the profit-maximizing price by plugging the optimal quantity into the demand curve. This tells us at what price the optimal quantity (80 million iPads) will be sold:

P^* = 1,000 – 5Q^*

= 1,000 – 5(80)

= 1,000 – 400 = 600

Given this demand curve and a constant marginal cost of $200 per iPad, then, Apple can maximize its profits by charging $600 per unit. It will sell 80 million iPads at this price. Notice that this price is well above Apple’s marginal cost of $200, the price Apple would be charging in a perfectly competitive market. That’s why firms like to have market power. The idea is simple: Reduce output. Raise prices. Make money.

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We can take the logic we’ve just learned even further to come up with a rule-of-thumb for pricing that firms can use to determine profit-maximizing prices and output levels.

Start with the definition of MR from above:

We know that the firm maximizes its profits by setting MR = MC, so plug that in:

Now use a math trick and multiply the second term on the left side of the equation by P/P. This doesn’t change the value of the equation, because multiplying by P/P is just another way of multiplying by 1. This changes our expression to

If the section in parentheses looks familiar to you, it’s because it is the inverse of the elasticity of demand. Remember that in Chapter 2 we defined the price elasticity of demand E^D as . The inverse of this value is Substituting the inverse elasticity into the profit-maximization condition gives us

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A final bit of rearranging yields

The left-hand side of this equation equals the firm’s profit-maximizing markup, the percentage of the firm’s price that is greater than (or “marked up” from) its marginal cost. What this equation indicates is that this markup should depend on the price elasticity of demand that the firm faces. Specifically, as demand becomes more elastic—that is, as E^D becomes more negative, or equivalently, larger in absolute value—the optimal markup as a fraction of price falls. (If you can’t quite see this in the equation, notice that elastic demand means a large negative number for E^D is in the denominator, making the right-hand side of the equation small.) On the other hand, as demand becomes less elastic, E^D becomes smaller in absolute value, indicating that the markup should be a larger fraction of price.

If we stop to think about these implications for a minute, they make perfect sense. When demand is quite inelastic, consumers’ purchases of the firm’s product are not sensitive to changes in price. This makes it easier for the firm to increase its profit by raising its price—it will sell fewer units, but not that many fewer, and it will make a higher margin on every unit it does sell. This is exactly what the equation implies; the firm should mark up its price more with less elastic demand. A firm facing relatively elastic demand, on the other hand, will suffer a greater loss in quantity sold when it raises its price, so high markups over cost benefit them less.

The measure of the markup given by the equation above has a special name: the Lerner index (after Abba Lerner, the economist who proposed it in 1934). As we just discussed, assuming the firm is trying to maximize its profit, the Lerner index tells us something about the nature of the demand curve facing the firm. When the index is high (i.e., when the markup accounts for a large fraction of the price), the demand for the firm’s product is relatively inelastic. When the index is low, the firm faces relatively elastic demand. Because the ability to price above marginal cost is the definition of market power, the Lerner index is a measure of market power. The higher it is, the greater the firm’s ability to price above its marginal cost.

The extreme case of perfectly elastic demand is interesting to study in terms of its implications for the Lerner index. When demand is perfectly elastic—the firm faces a horizontal demand curve and any effort to charge a price higher than the demand curve will result in a loss of all sales—then E^d = –∞. As we see in the equation above, the Lerner index is zero in this case, which means the markup is also zero. The firm sells at a price equal to marginal cost, and the firm becomes a price taker.

Another interesting case occurs when E^d falls between 0 and –1—that is, when the firm faces a demand curve that is inelastic or unit elastic. In this case, the Lerner index is greater than 1. But this would imply that P – MC > P, or MC < 0, and marginal cost can’t be negative. Why does the optimal markup equation imply this nonsensical result? There is a mathematical answer that involves calculus, but the basic economic explanation is that a firm should never operate at a point on its demand curve where demand is inelastic or unit elastic. (In the linear demand case, demand becomes less elastic as price falls.) Think about what would happen if a firm was setting a price (or a quantity) that put it on an inelastic or unit elastic portion of its demand curve, and then decided to increase its price. By definition, because demand is inelastic, whatever the percentage increase in price, the percentage drop in quantity will be smaller (or will exactly equal the percentage increase in price if demand is unit elastic). This means that the price increase will raise the firm’s revenue (or not change it if demand is unit elastic). At the same time, because the firm is producing a smaller quantity, its total cost must fall, because costs increase in quantity. So, the price increase raises the firm’s revenue while lowering its cost. In other words, the firm is guaranteed to raise profit by increasing prices as long as demand is inelastic. Thus, it can’t be profit-maximizing to set a price where demand is inelastic.

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The Lerner index can range anywhere from 0 (perfect competition) to just under 1 (barely elastic—almost unit elastic—demand). It is a summary of the amount of market power a firm has. In comparing degrees of market power across firms, the firm with the highest Lerner index has the most market power; the firm with the second-highest Lerner index has the second-most market power, and so on.

Measuring the Lerner Index Firms with market power know their profit-maximizing markups are tied to the price elasticities they face. The difficulty from a practical standpoint is that firms don’t automatically know their Lerner index and ideal markup. So, they often spend considerable effort trying to learn about the shape of the demand and marginal revenue curves they face, because that tells them about the price elasticity of demand of their customers.

Technologies that allow firms to change prices more frequently, and even offer different prices simultaneously to different consumers, have made firms’ processes of feeling out their demand curves easier. This can lead to negative publicity, however. For example, Amazon got into a bit of trouble early in its history for conducting what it called a “pricing experiment” on its customers. Amazon was experimentally offering different prices to different customers for the same products, in an effort to measure the elasticity of demand by seeing how consumers’ purchases responded to price changes. A customer sued when he discovered that when he removed the Amazon cookie from his computer, the price of the product he was shopping for dropped significantly. In the uproar that followed, people assumed that Amazon was price-discriminating, a practice we discuss in the next chapter. Amazon CEO Jeffrey Bezos apologized for the episode and indicated that there was no ongoing systematic price discrimination. He insisted that Amazon was simply randomizing prices to develop a better sense of demand in its market. This chapter shows exactly why Amazon would do this. If a company knows the shape of its own demand curve, it can determine the most profitable price to charge using either the markup formula or the monopoly pricing method.

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We now know how to figure out the profit-maximizing quantity and price for a firm with market power, and we can do so for any given marginal cost and demand curves the firm might face. As you might imagine, we could sketch out all the combinations of the firm’s profit-maximizing quantities and prices implied by any possible set of marginal cost and demand curves.

That sounds a lot like a supply curve—it is, after all, a set of prices and the quantities produced—but it’s not. Firms with market power don’t have supply curves, strictly speaking. Their profit-maximizing price and quantity combinations are not supply curves because those combinations depend on the demand curve the firm faces. As we saw in Chapter 8, competitive supply curves exist completely independently of demand. They depend only on firms’ marginal costs, because a perfectly competitive firm produces the quantity at which the market price (which the firm takes as given) equals its marginal cost. That’s why a perfectly competitive firm’s supply curve is a portion of its marginal cost curve, and a perfectly competitive industry’s supply curve is the industry marginal cost curve. Neither of these supply curves is determined by anything having to do with demand; they are only about costs.

This strict relationship between costs and price isn’t true for a firm with market power. Its optimal output level depends on not only the marginal cost curve, but also the firm’s marginal revenue curve (which is related to the demand curve). Put another way, a supply curve gives a one-to-one mapping between the price and a firm’s output. But for a firm with market power, even holding constant its marginal cost curve, the firm could charge a high price at a given quantity if it faces a steeper demand curve or a lower price at the same quantity if it faced a flatter demand curve. We’ll see an example of this in the next section. Therefore, a simple mapping of price and quantity supplied is not possible for a firm with market power and there would be no supply curve.