Measures of market power and \epsilon

Section 7.4 Measures of market power and $\epsilon_D$

Earlier, we discussed the importance of price markups. A price markup exists if a firm can charge a price above its marginal cost for its last unit of output. Importantly, the presence of a price markup is an indicator that the firm has market power.

In perfectly competitive markets, firms do not have market power. The firm will continue to increase its production until the point where it no longer marks up its price above its marginal cost, as seen in the graph below.

Figure 7.4.1. Markups in perfect competition. Since all units sell for the same price, and the first few units cost the least to produce, these units’ markup is largest. Later units have a smaller markup, and the last unit produced ($q^*$) experiences zero markup.

As we have seen, the primary measurement of price markups is the ratio $\frac{P}{MC}\text{,}$ which gives the the firm’s price relative to its cost to produce. In perfect competition, though, since the firm will be maximizing its profit at a $q^*$ where $P = MC\text{,}$ its optimal price markup is 1. It is profit maximizing for a competitive firm to not markup its price at all

By contrast, let’s examine the price markup for a monopolist. A monopolist maximizes its profit at a level of output $q^*_M$ where $MR = MC\text{.}$ Additionally, the presence of market power erodes the firm’s marginal revenue: as additional units are sold, the firm’s price on all units decreases! As a result, $MR \lt P\text{.}$ This is typically observed when the marginal revenue curve is graphed below the demand curve.

If we combine, when a monopolist is maximizing its profit,

\begin{equation*} MR = MC \end{equation*}

and

\begin{equation*} MR \lt P \end{equation*}

it must be true that when a firm maximizes its profit,

\begin{equation*} MC \lt P \end{equation*}

When a firm is maximizing its profit, it necessarily charges a price above its marginal cost! This confirms the existence of a price markup when the seller has market power: if $P > MC\text{,}$ then $\frac{P}{MC} > 1\text{.}$

Let’s visualize the markup. Like competitive firms, monopolists can sell the first few units at a markup above marginal cost. This is the result of the diminishing marginal product of labor: the first few units are the cheapest to produce, so price is easily above the low marginal cost. However, the difference arises for the last unit produced. Unlike competitive firms, who do not price markup their last unit sold, a monopolist continues its price markup for every unit produced, even the very last one!

Figure 7.4.2. Left: Markups in perfect competition. The last unit produced ($q^*$) experiences zero markup. Right: Markups in monopoly. The last unit produced ($q^*_M$) still experiences a markup.

A monopolist’s price markup is closely related to the price elasticity of demand for its product. To explore this more deeply, recall that in Chapter 1, we showed that a firm’s total revenue is maximized when demand is unit elastic. Moreover, we showed that the top segment of a linear demand curve - where price is high and quantity is low - is where demand is more relatively elastic.

Figure 7.4.3. From Chapter 1. Notice that when demand is elastic, lowering the price (and increasing $Q$) leads to higher $TR\text{.}$ When demand is inelastic, raising price (and lowering $Q$) leads to higher $TR\text{.}$

In a monopoly, this can be related to the firm’s marginal revenue curve. Since marginal revenue, $MR = \frac{\Delta TR}{\Delta q}\text{,}$ is the slope of the total revenue curve, then:

At the peak of total revenue, $TR$ is flat and marginal revenue should be equal to 0. (This is where $\epsilon_D = -1\text{.}$)
On the left side of the demand curve, $TR$ is increasing and marginal revenue should be greater than 0. (This is where demand is relatively elastic, $\epsilon_D \lt -1$).

This allows us to tie marginal revenue values to their corresponding price elasticity of demand values on the firm’s demand curve. Moreover, this gives important insight on the relationship between $\epsilon_D$ and monopolist choices. A firm with market power will find its profit-maximizing level of output, $q^*_M\text{,}$ where $MR = MC\text{.}$ But if a firm’s marginal cost is positive (a consequence of assumption P1), then at $q^*_M\text{,}$ $MR > 0$ as well. This always puts the monopolist’s choice on the relatively elastic part of the demand curve.

Figure 7.4.4. Left: Demand is unit elastic where $MR = 0\text{.}$ Demand is relatively elastic where $MR > 0\text{.}$ Right: The monopolist’s profit-maximizing choice of output $q^*_M$ lands on the relatively elastic part of its demand curve.

With this information, we can derive an important measurement for price markups. In practice, it can be quite valuable to know how much a firm marks up its price above its marginal cost. Policy makers might be concerned about the detrimental impacts on consumer welfare if a firm with market power charges an exorbitantly high price relative to its cost. In a 2020 article

De Loecker, J., Eeckhout, J., & Unger, G. (2020). The rise of market power and the macroeconomic implications. The Quarterly Journal of Economics, 135(2), 561-644.

, economists De Loecker, Eeckhout, and Unger find that on average in the U.S. economy, price markups have increased from a 21% markup ($\frac{P}{MC} = 1.21$) to a 61% markup ($\frac{P}{MC} = 1.61$) over the last 40 years. They discuss how this has major implications throughout the economy: the result is correlated with a reduction in consumer welfare, increasing firm profits, and the reduction of firms’ spending on labor which leads to a lower labor share.

However, there is an empirical challenge to this exercise: data on firms’ marginal cost can be extremely difficult to find. How can $\frac{P}{MC}$ be measured without data on marginal cost? Fortunately, there is an alternate approach to finding price markups for firms which relies on price elasticity of demand. How can $\epsilon_D$ data shed light on price markups for firms?

Figure 7.4.5. De Loecker, Eeckhout and Unger (2020) show that on average, firms’ price markups have increased substantially since 1980.

A recalculation

This recalculation can be shown with a little calculus. First, with $TR = P(Q)*Q\text{,}$ $MR = \frac{dP}{dQ}{Q} + P\text{.}$ Then at $Q^*\text{,}$ it must be true that

\begin{equation*} MR = \frac{dP}{dQ}{Q} + P = MC \end{equation*}

Dividing both sides by $P$ yields

\begin{equation*} \frac{dP}{dQ}\frac{Q}{P} + 1 = \frac{MC}{P} \end{equation*}

Flipping both sides then gives

\begin{equation*} \frac{P}{MC} = \frac{1}{\frac{1}{\epsilon_D} + 1} \end{equation*}

since $\frac{dP}{dQ}\frac{Q}{P}$ is the inverse of price elasticity of demand.

tells us that we can express the price markup in the following way:

\begin{equation*} \frac{P}{MC} = \frac{1}{\frac{1}{\epsilon_D} + 1} \end{equation*}

This reformulation allows us to relate the elasticity of demand to the firm’s markup. Keep in mind that the firm will choose where price elasticity of demand is relative elastic, with $\epsilon_D \lt -1\text{.}$ But how can price markups differ based on different elasticities?

As one example, suppose our firm faces demand that is extremely elastic, with $\epsilon_D = -10\text{.}$ Here, demand is wildly sensitive to changes in price: a 1% change in price leads to a 10% change in quantity demanded. We can compute the firm’s price markup directly using the formula above:

\begin{equation*} \frac{P}{MC} = \frac{1}{\frac{1}{(-10)} + 1} = \frac{1}{\frac{9}{10}} = \frac{10}{9} \end{equation*}

The price markup value is $\frac{10}{9} = 1.11\text{,}$ which is a relatively modest 11% markup. In other words, if the last unit produced costs $9, the firm will charge a price of $10 for every unit.

What if, instead, the firm faces demand that is not particularly elastic. Even though demand for the firm cannot be truly inelastic, demand can be on the still-elastic-but-not-very-elastic part of the demand curve, where, say, $\epsilon_D = -1.1\text{.}$ In this case, a 10% change in price leads to an 11% change in quantity demanded. Once again, the firm’s price markup can be calculated using the equation above:

\begin{equation*} \frac{P}{MC} = \frac{1}{\frac{1}{(-1.1)} + 1} = \frac{1}{-\frac{10}{11} + \frac{11}{11}} = \frac{1}{\frac{1}{11}} = 11 \end{equation*}

With demand as nearly inelastic as it can be for a firm with market power, the firm can mark up its price eleven times above its marginal cost. If the firm’s marginal cost for its last unit is $10, it would maximize its profit by charging a price of $110!

This massive difference in markups highlights an intuitive relationship between price markups and price elasticity of demand. Market power is not immune to the characteristics of demand for products, and that impact comes through in our analysis here. If the firm sells a good with many substitutes, for example, demand will be more elastic and the firm will mark their price up relatively less. The smaller markup is an indicator of less market power. However, if the firm sells a necessity, or a good with very few substitutes, demand will be more inelastic and the firm will be able to mark up its price considerably higher. A firm facing inelastic demand has much more market power!

The Lerner index can illustrate a similar relationship between price elasticity of demand and measures of market power. Recall that the Lerner index gives the percentage of a firm’s price that is markup, and is denoted by

\begin{equation*} \ell = \frac{P - MC}{P} \end{equation*}

A rearrangement of the new price markup formula above

We can first simplify the Lerner index to

\begin{equation*} \ell = \frac{P - MC}{P} = 1 - \frac{MC}{P} \end{equation*}

then substitute the inverse of the price markup expression $\frac{P}{MC} = \frac{1}{\frac{1}{\epsilon_D} + 1}$ to get

\begin{equation*} \ell = \frac{P - MC}{P} = 1 - \frac{MC}{P} = 1 - (\frac{1}{\epsilon_D} + 1) = -\frac{1}{\epsilon_D} \end{equation*}

allows us to express the Lerner index in terms of price elasticity of demand:

\begin{equation*} \ell = \frac{P - MC}{P} = -\frac{1}{\epsilon_D} \end{equation*}

Earlier, we saw the lowest possible Lerner index values at 0 - indicating no market power - and Lerner index values increase as firms markup their price higher. Now, we can relate this to price elasticity of demand for firms with market power using our examples from above. First, if a firm faces demand that is not very elastic ($\epsilon_D = -1.1$), the firm has a huge price markup of 11, suggesting that if the firm’s marginal cost for its last unit is $10, it would maximize its profit by charging a price of $110. To find its Lerner index, then, we can use the price elasticity of demand to show

\begin{equation*} \ell = \frac{P - MC}{P} = -\frac{1}{\epsilon_D} = -\frac{1}{-1.1} = \frac{10}{11} = .909 \end{equation*}

This tells us that over 90% of the firm’s price is markup! This makes sense with our prior calculations: with a marginal cost of $10 and a price of $110, 100 dollars out of the firm’s $110 price are markup, making the overwhelming majority of what consumer’s pay markup.

If, however, the firm faces very elastic demand ($\epsilon_D = -10$), the firm’s markup will be lower. This will lead to a smaller Lerner index value:

\begin{equation*} \ell = \frac{P - MC}{P} = -\frac{1}{\epsilon_D} = -\frac{1}{-10} = \frac{1}{10} = .1 \end{equation*}

The firm’s Lerner index is $\ell = 0.1\text{,}$ suggesting only 10% of the firm’s price is markup. Most of what consumers pay represents the firm’s marginal cost! This aligns with our previous calculations as well. We showed that if the last unit produced costs $9, this firm will charge a price of $10 for every unit. Therefore, out of a price of $10, 9 dollars are marginal cost, and only one dollar is markup.

Keep in mind that both the price markup value and the Lerner index can only take on values in a given range:

Price markup: $\frac{P}{MC} \geq 1\text{.}$ With no market power, $\frac{P}{MC} = 1$
Lerner index: $0 \leq \ell \leq 1\text{.}$ With no market power, $\ell = 0\text{.}$

The table below gives an efficient comparison of how market power measures can differ:

	high market power	low market power

Concept	price is significantly	price is minimally
	above marginal cost	above marginal cost
price markup ($\frac{P}{MC}$)	large $\frac{P}{MC}$	small $\frac{P}{MC}\text{,}$ close to 1
Lerner index ($\ell$)	high $\ell\text{,}$ close to 1	low $\ell\text{,}$ close to 0
price elasticity of demand	not very elastic, $\epsilon_D$ close to -1;	very elastic, $\epsilon_D$ larger negative values

In Depth 7.4.6. Can a price ceiling combat price markups?

When market power is present, profit-maximizing firms will mark up prices more than in the absence of market power. In general, we have seen how markups lead to lower consumer surplus. Importantly, as we have seen, the markup is likely to be especially high for products whose demand is more inelastic. This can be extra problematic for natural monopolies. In natural monopolies, such as public water or electricity markets, the excessively high cost to provide the service restricts the market to only one seller: the government. Public water and electricity services have fairly inelastic demand, which could put these markets in jeopardy for large price markups.

In part, there is reason to think a city government may not fully exercise its ability to mark up its price, since the government’s objective is often not to maximize its profit (like a private firm). Nevertheless, we can use this situation to explore how a price ceiling can soften the consequences of market power and prevent firms from implementing excessive price markups. Whether the price ceiling is used to check a private firm, or to operate as a public safeguard, it can play a key role.

As we know, a price ceiling would set a maximum price in the market, and limit the potential of the firm to mark its price high. To see how this works, consider how a price ceiling would change the residual demand curve for the firm, below:

Left: Residual demand and marginal revenue before price ceiling. Right: With a price ceiling at $P_C\text{,}$ both residual demand and marginal revenue are limited above. At low quantities of output, $MR = P_C\text{.}$ At high quantities of output, $MR \lt P\text{,}$ as is typical in the presence of market power.

On the left, we see the typical demand and marginal revenue (we can call this the residual marginal revenue curve) that a firm with market power faces. But with a price ceiling in place at price $P_C\text{,}$ no legal price can be above the price ceiling. As a result, for the first group of units sold, the firm will charge as high as possible: $P_C\text{.}$ Moreover, in this region of quantity, each additional unit sold will sell for the same $P_C$ price! This makes marginal revenue equal to price $P_C$ for the first quantities of output.

Because the firm’s price and marginal revenue are identical here, the firm has a strong incentive to increase its output, much like a perfectly competitive firm! Even though the firm still has market power, the price ceiling effectually blocks the primary impact of market power - the disincentive to increase output to keep prices high - from taking effect since the price cannot go beyond $P_C\text{.}$ Once the quantity of output increases enough to push the price below $P_C\text{,}$ the monopolist’s problem looks like it normally does, since the price ceiling is no longer binding.

Now we examine how the firm chooses to maximize its profit under a price ceiling. Let’s set the price ceiling $P_C$ to be identical to the market price under the perfect competition approximation, where marginal cost and demand intersect. The firm will still choose to maximize its profit at $q^*_M$ where $MR = MC\text{,}$ per usual. In the graph below, notice how the as profit is maximized, the firm’s price will not exceed the price ceiling (since it is illegal for it to do so), and the quantity of output it produces will vary depending on its marginal cost.

If its marginal cost is fairly high (left on the graph below), the firm will maximize its profit where $MC = MR = P_C\text{.}$ The firm still produces a lower quantity than it would produce in perfect competition. But its price is $P_C$ and its price markup is $\frac{P}{MC} = 1\text{,}$ meaning there is no price markup. If instead, the marginal cost is a bit lower (right on the graph below), the firm will maximize profit where $MC = MR \lt P_C\text{.}$ The firm still marks up its price a bit, but the firm’s price and quantity match the perfect competition approximation exactly!

2 scenarios: Left: Profit maximization with high marginal costs. The price charged is $P_C$ and the output produced is reduced; (2) Right: Profit maximization with low marginal costs. The price charged is $P_C$ and the output produced is the perfect competition approximation.

In either scenario, the firm charges a price at the price ceiling exactly, which minimizes the price markup for a firm with market power. In doing so, the price ceiling can neutralize the inefficiency market power generates, and mitigate the negative impacts of market power on consumers. A price ceiling can be an effective price control strategy to guide a monopoly approximately closer to a perfectly competitive market!

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Behavior and Choice: EC 102

Section 7.4 Measures of market power and \(\epsilon_D\)

In Depth 7.4.6. Can a price ceiling combat price markups?