Lecture 17: Elasticity and Market Power: From Monopoly to Competition

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What is “Market Power?”

The model presented in the last lecture is usually presented in a textbook chapter called “Monopoly.” However, true monopolies are rare: there are very few firms which are the only sellers of a particular product, and even fewer when you expand the definition of what a product is. For example, you might legitimately argue that Apple has a monopoly on iPhones. However, Apple faces competition: its iPhones compete with Android devices produced by Samsung, Motorola, Ziaomi, Huawei, and others.

What is true is that Apple has market power. A lot of it. This comes from the fact that while an Android device is a substitute for an iPhone, it isn’t a perfect substitute. It doesn’t pair as nicely with AirPods or Apple Watches, or other services. In fact, a lot of what Apple invests in is services which make its phones work less well with Android phones; think about the difficulty of managing a group chat among people with both kinds of devices. Apple wants you in its “walled garden,” and it tries to keep you there not by forcing you to, but by making it feel uncomfortable to use other services. In other words, it wants to make iPhones and Android phones less perfect substitutes for one another.

So, what is market power? In general, it is the ability to set market conditions, rather than take those conditions as given. One specific measure is the ability of a firm to control the price of its goods: i.e., the ability to raise its price above its marginal cost. Its ability to do so depends crucially on how substitutable its product is for other firm’s products, as in the case of Apple and Android phones. And that substitutability is measured by the elasticity of demand.

The main point of today’s lecture, therefore, is to see how the elasticity of demand for a firm’s products affects its ability to raise its price over marginal cost. Generally speaking, the more substitutes there are for a good, the more elastic demand for that good will be, since consumers will respond to a price increase by buying the substitute goods. In the extreme case, which we’ll look at for the second part of class, the firm has no market power: it sells a good which is a perfect substitute for other goods, and so faces a perfectly elastic demand curve. We’ll call such a firm a competitive or price taking firm.

We will first define elasticity, and derive a mathematical formula for it. We will then see how elasticity affects marginal revenue, and therefore profit maximization.

Elasticity

Elasticity measures how responsive an endogenous variable is to changes in an exogenous variable. For example, we may be interested in how responsive the quantity of a good demanded by a consumer is to a change in the price of that good, or in how responsive a firm’s hiring practices might be to an increase in the minimum wage.

First, let’s note that elasticity is a rate: it measures the percentage change in an endogenous (dependent) variable per percentage change in an exogenous (independent) variable.

Intuitively, we might think of elasticity as answering the question: “If some independent variable $X$ increases by $1\%$, what is the resulting percentage change in the dependent variable $Y$?” Mathematically, we may write this as $\epsilon_{Y,X} = \frac{\%\Delta Y}{\%\Delta X}$ The formula (Note: This assumes a small percentage change, so that we don’t have to worry about whether to use the initial or final value as the basis for the percentage change. Of course, a change between $1.00$ and $1.01$ is only approximately a $1\%$ change…but we’ll be using calculus soon enough to determine the elasticity.) for a percentage change in some variable $X$ is $\%\Delta X = 100\% \times \frac{\Delta x}{x}$ Plugging this expression into our formula for elasticity gives us $\epsilon_{Y,X} = \frac{\%\Delta Y}{\%\Delta X} = \frac{100\% \times \frac{\Delta y}{y}}{100\% \times \frac{\Delta x}{x}}$ We can rewrite this as $\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}$

Note that this is related to but not equal to the absolute change in $Y$ per absolute change in $X$ given by $\Delta Y / \Delta X$. In particular, for the special case where $Y = X$, the elasticity and slope are the same thing. But if $Y > X$, then the same absolute change would represent smaller percentage change in $Y$ than it would in $X$; so the elasticity would be less than $\Delta Y/\Delta X$.

For example: suppose the price of a good increased from $€1.00$ to $€1.01$, and this resulted in the quantity demanded decreasing from 50 million units to 48 million units. In this case we consider price to be the exogenous variable $X$ and quantity to be the endogenous variable $Y$; so using the formula above we have $\begin{aligned} \Delta x &= €1.01 - €1.00 = +€0.01\\ x &= €1.00\\ \Delta y &= 48 \text{ million units} - 50 \text{ million units} = -2 \text{ million units}\\ y &= 50 \text{ million units} \end{aligned}$ We calculate the percentage change in price as $\text{\% change in price} = 100\% \times \frac{\Delta x}{x} = 100\% \times \frac{+€0.01}{€1.00} = 1\%$ and the percent change in quantity as $\text{\% change in quantity} = 100\% \times \frac{\Delta y}{y} = 100\% \times \frac{-2 \text{ million units}}{50 \text{ million units}} = -4\%$ Therefore we would say that a $1\%$ increase in the price led to a $4\%$ decrease in the quantity demanded, so the elasticity (written $\epsilon$) would be $-4$: $\epsilon_{Q,P} = \frac{\text{\% change in quantity}}{\text{\% change in price}} = \frac{-4\%}{+1\%} = -4$ We could also just plug this directly into the formula to obtain $\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y} = \frac{-2 \text{ million units}}{+€0.01} \times \frac{€1.00}{50 \text{ million units}} = -4$ Note that in each case all the units cancel, leaving the measure of elasticity as a unitless measure.

Types of elasticity

The sign of an elasticity indicates whether the two variables are positively related or negatively related. For example, we generally assume that an increase in the price of a good results in a decrease in the quantity demanded of that good; therefore demand elasticity of the sort we looked at above will generally be negative. However, when we think about how an increase in the price of a good affects the demand for some other good, the sign becomes important: depending on whether the two goods are complements or substitutes, it might be negative or positive.

The magnitude of elasticity — specifically, whether it is greater or less than one — describes whether a change in the exogenous variable results in a proportional, less than proportional, or greater than proportional change in the endogenous variable. That is, we can characterize elasticity as follows:

Perfectly inelastic ($|\epsilon| = 0$ or $|\% \Delta Y| = 0$): the endogenous variable does not change when the exogenous variable increases
Inelastic ($|\epsilon| < 1$, or $|\% \Delta Y| < |\% \Delta X|$): the percentage change in the endogenous variable is less than the percentage change in the exogenous variable.
Unit elastic ($ |\epsilon| = 1$, or $|\% \Delta Y| = |\% \Delta X|$): the percentage change in the endogenous variable is the same as the percentage change in the exogenous variable
Elastic ($|\epsilon| > 1$, or $|\% \Delta Y| > |\% \Delta X|$): the percentage change in the endogenous variable is greater than the percentage change in the exogenous variable
Perfectly elastic ($|\epsilon| = \infty$, or $|\% \Delta X| = 0$): any change in the exogenous variable would cause the exogenous variable to change “infinitely” in percentage terms (usually this means going from a positive number to zero or vice versa)

Elasticity with Linear Relationships

We’ve established that the general formula for elasticity is $\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}$ Let’s examine the case in which there is a constant slope: that is, where $\Delta y/\Delta x$ is constant. For example, if we have $y = mx + b$ In this case $\Delta y/\Delta x = m$, so $\begin{aligned} \epsilon_{Y,X} &= \frac{\Delta y}{\Delta x} \times \frac{x}{y}\\ &= m \times \frac{x}{mx + b}\\ &= \frac{mx}{mx + b}\end{aligned}$ Notice that we plug in $y = mx + b$ for $y$ in order to express elasticity as a function of the independent variable $X$.

Example

A common example of this in economics is a linear demand curve; for example, $Q = 80 - P$ Applying the formula, we have that the demand elasticity is $\epsilon_{Q,P} = -\frac{P}{80-P}$ You can try moving the price up and down to see the elasticity at different points along this demand curve:

(Graph: point_slope)

See interactive graph online here.

You can check the “show regions of elasticity” box in the diagram above to see which part of the demand curve is perfectly elastic, elastic, unit elastic, inelastic, and perfectly inelastic. You might notice that, regardless how you move the intercepts of the demand curve:

the demand elasticity (in absolute value) increases as $P$ increases. That is, as price increases and quantity decreases, even though a $€1$ increase in price always leads to the same decrease in quantity, that same $€1$ increase in price represents a smaller and smaller percent of the price, while the quantity decrease represents a larger and larger percent decrease in quantity.
in the extreme case when $Q = 0$ (along the vertical axis), demand is perfectly elastic ($\epsilon = -\infty$). This is because even the smallest decrease in price will result in an infinite percentage change in quantity (from zero to some positive number).
in the other extreme case when $P = 0$ (along the horizontal axis), demand is perfectly inelastic ($\epsilon = 0$). This is because it would take an infinite percentage change in price (from zero to some positive number) to cause any change in quantity.
demand is unit elastic at the midpoint of the demand curve. This is because, if demand is given by $Q = a - bP$, then the demand curve extends from $(a, 0)$ to $(0, \frac{a}{b})$, and the formula for elasticity is $\epsilon_{Q,P} = -\frac{bP}{a-bP}$ This is equal to $-1$ when $bP = a - bP$, which occurs at the the point $(\frac{a}{2}, \frac{a}{2b})$ – i.e., the midpoint of the demand curve.

Putting this all together, we can see that the upper (i.e., left) half of the demand curve is elastic, and the lower (i.e. right) half of the demand curve is inelastic.

Calculating Elasticity using Calculus: The Point Elasticity Method

We can rewrite our general formula $\epsilon_{Y,X} = \frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}$ as $\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}$ If we take the limit as the change in the exogenous variable $\Delta x$ gets smaller, we write this in the standard calculus way: $\lim_{\Delta X \rightarrow 0} \epsilon_{Y,X} = \frac{dy}{dx} \times \frac{x}{y}$ If we know the formula for the relationship between the exogenous variable $X$ and the endogenous variable $Y$ – that is, if $y = f(x)$ – then we can write the instantaneous elasticity (or “point elasticity) as a function of $x$ as $\epsilon(x) = f^\prime (x) \times \frac{x}{f(x)}$ Note that this is all written as a function of the exogenous variable: for example, if we were to think of the price elasticity of demand, we would ask what the elasticity is at a certain price $P$ (the exogenous variable), not at a certain quantity $Q$ (the endogenous variable).

Example

Suppose that the relationship between $X$ and $Y$ may be given by the function $y = f(x) = 10 + 2x + \frac{1}{4}x^2$. Then the elasticity of $Y$ with respect to $X$ is $\epsilon_{Y,X} = \frac{dy}{dx} \times \frac{x}{y} = (2 + \tfrac{1}{2}x) \times \frac{x}{10 + 2x + \frac{1}{4}x^2} = \frac{2x + \frac{1}{2}x^2}{10 + 2x + \frac{1}{4}x^2}$ This isn’t constant; it varies as $x$ changes. However, there is another special case worth examining: the case of constant elasticity.

A Special Case: Relationships with Constant Elasticity

We’ve just established the general formula for elasticity using calculus is $\epsilon = \frac{dy}{dx} \times \frac{x}{y}$ There’s a special case of this in which elasticity is constant: specifically, when the relationship between the exogenous and endogenous variables is multiplicative. For example, consider the relationship $y = ax^b$ In this case $\frac{dy}{dx} = abx^{b-1}$ so the elastictiy is $\begin{aligned} \epsilon &= abx^{b-1} \times \frac{x}{ax^b}\\ &= \frac{abx^b}{ax^b}\\ &= b \end{aligned}$ For example, consider a demand function given by $Q(P) = 4P^{-\frac{1}{2}}$ The price elasticity of demand for this function is $\begin{aligned} \epsilon_{Q,P} &= -2P^{-\frac{3}{2}} \times \frac{P}{4P^{-\frac{1}{2}}}\\ &= -\frac{2P^{-\frac{1}{2}}}{4P^{-\frac{1}{2}}}\\ &= -\frac{1}{2} \end{aligned}$ This extends to more than one independent variable! (Note: This is perhaps the most important result for this topic! Read it carefully!) If you see some function of the form $y = x_1^ax_2^bx_3^c$the elasticity of $Y$ with respect to $X_1$ is going to be $a$, with respect to $X_2$ is going to be $b$, and with respect to $X_3$ is going to be $c$.

Marginal revenue and elasticity

OK, now that we’ve defined elasticity, let’s see how it’s related to marginal revenue.

Recall the diagram we saw last time, with the “price effect” and the “output effect” of a price change:

See interactive graph online here.

Try dragging the $(q,p)$ point to the right along the demand curve. Note that even though this demand curve is linear, so $dp/dq$ is constant, the relative sizes of the red and green areas change as quantity increases. Indeed, beyond a certain quantity, the red area is greater than the green area, meaning that revenue decreases if you increase the quantity. This happens when

$\begin{aligned} \text{Price effect} &> \text{Output effect}\\ |dp| \times q &> dq \times p\\ {|dp| \over p} &> {dq \over q}\\ \text{\% change in price} &> \text{\% change in quantity}\\ \end{aligned}$ Recall that our definition of the price elasticity of demand was $\epsilon_{q,p} = {\text{\% change in quantity} \over \text{\% change in price}} = {dq \over dp} \times {p \over q}$ This says that marginal revenue will be positive when demand is elastic, and negative when demand is inelastic.

In fact, we can express marginal revenue in terms of elasticity. Let’s start with the formula for marginal revenue, $MR(q) = {dp \over dq}\times q + p$ If we multiply the first term by $p/p$, we get $MR(q) = \left[{dp \over dq} \times {q \over p}\right] \times p + p = p \left(1 + {1 \over \underbrace{\frac{dq}{dp} \times {p \over q}}_\epsilon}\right)$ Notice that the denominator of the last term is just our expression for the price elasticity of demand, which we know is negative because the demand curve is downward sloping ($dq/dp < 0$). Therefore we can write this as $MR = p \left(1 - {1 \over |\epsilon|}\right)$ Note that the more elastic demand is (i.e., the higher $|\epsilon|$ is), the closer marginal revenue is to the price; and that marginal revenue will be negative if $|\epsilon| < 1$ (that is, if demand is inelastic.)

Elasticity, profit maximization, and markup pricing

Now let’s see how elasticity is related to profit maxmization. Recall that the profit-maximizing condition was $MR = MC$

If we plug in the elasticity definition of MR into the profit-maximization condition, we can rewrite that condition as $p \times \left(1 - {1 \over |\epsilon_{q,p}|}\right) = MC$ or $p = {MC \over 1 - {1 \over |\epsilon_{q,p}|}}$ This is known as the inverse elasticity pricing rule. We can confirm, for example, that it held in our example of profit maximization from last time: for that case, we had the demand function $D(p) = 20 - p$ and found that the optimal quantity to produce was $q^\star = 8$, which it sold for $p^\star = 12$, as shown in the diagram below:

See interactive graph online here.

At the optimal choice, therefore, its price elasticity of demand would be $\epsilon_{q,p} = {dq \over dp} \times {p \over q} = -1 \times {12 \over 8} = {3 \over 2}$ Its marginal cost at $q = 8$ was $4$; so the IEPR would say that its optimal price would be $p = {4 \over 1 - {1 \over 3/2}} = {4 \over 1 - {2 \over 3}} = {4 \over {1 \over 3}} = 12$ which is indeed the optimal price.

When to use the IEPR

The IEPR is particularly useful in cases where both the marginal cost and elasticity are constant. For example, if a firm has the cost function $c(q) = 200 + 4q$ and faces the demand function $D(p) = 6400p^{-2}$ then its MC is constant at 4, and its price elasticity of demand is constant at $-2$, so the IEPR says it should charge $p = {4 \over 1 - {1 \over |-2|}} = 8$ Plugging this back into the demand function gives us the optimal quantity of $q = D(8) = 6400 \times 8^{-2} = 100$ Of course, it would be possible to go the long way round — solve for the inverse demand, find total and marginal revenue, yada yada yada — but this is much faster in cases like this.

Relationship to market power

If we rearrange the IEPR, a little algebra shows we can write it as ${p - MC \over p} = {1 \over |\epsilon_{q,p}|}$ The left-hand side of this equation is the fraction of the price which represents a markup above marginal cost. For example, our firm facing the linear demand curve $D(p) = 20 - p$ charged $p^\star = 12$, but only had a marginal cost of $MC = 4$; hence its markup over marginal cost was $12 - 4 = 8$, which represented $8/12 = 2/3$ of its price. As we saw above, the price elasticity of demand at $p = 12$ was $3/2$.

This ratio of markup to price is known as the Lerner Index, and has historically been used in antitrust cases as a measure of market power. Clearly, the less elastic the demand curve faced by a firm, the more it will be able to raise its price above marginal cost. However, the inverse is also true: as the demand curve becomes more elastic, the firm’s ability to raise price over marginal cost gets smaller and smaller; and in the extreme case of a competitive firm, vanishes entirely. We’ll conclude our analysis of profit maximization by looking at this special case of the competitive firm.

Perfectly elastic demand and the competitive firm

The IEPR implies that the more elastic demand is (i.e., the higher $|\epsilon|$ is), the closer marginal revenue is to the price; and that marginal revenue will be negative if $|\epsilon| < 1$ (that is, if demand is inelastic.)

An extreme case of this is that of perfectly elastic demand. Intuitively, a firm faces a perfectly elastic demand curve at some price $p$ if it can sell as much as it likes at that price, but would sell zero if it raised the price above that price. Visually, this is a horizontal demand curve, and we call a firm that faces such a demand curve a perfectly competitive or price-taking firm:

See interactive graph online here.

The idea here is that a firm is a “price taker” if they sell a good which is a perfect substitute for a good its competitors are selling – say, a commodity like corn – for which there is an established market price $p$. If we think of its inverse demand curve as a constant function $p(q) = p$, then its total revenue is $r(q) = pq$ and its average and marginal revenues are just $p$: $AR(q) = {pq \over q} = p$ $MR(q) = {dr \over dq} = p$

Visually, the total revenue function is just a line with slope $p$; and $AR = MR = p$, so both of those curves are horizontal lines with a height of $p$:

Visually, we can see that the marginal revenue is composed only of the output effect, not the price effect:

See interactive graph online here.

The fact that $MR = p$ for a competitive firm simplifies its profit maximization problem to just $\overbrace{\pi(q)}^\text{PROFIT} = \overbrace{p \times q}^\text{REVENUE} - \overbrace{c(q)}^\text{COST}$ The firm’s marginal profit is $\pi^\prime(q) = \overbrace{p}^{MR} - \overbrace{c^\prime(q)}^{MC}$ and so its optimal quantity is found by setting $p = MC$. Note that this is really just a special case of $MR = MC$, since in this case the firm’s marginal revenue from each unit sold is just $p$.

For example, if we use the same cost function as in Lecture 14 ($c(q) = 64 + {1 \over 4}q^2$), and suppose that the firm can sell as much quantity as it likes at price $p = 12$, the firm’s profit will be given by $\pi(q) = 12q - (64 + \tfrac{1}{4}q^2)$ which is maximized when $\begin{aligned} \pi^\prime(q) = \overbrace{12}^p - \overbrace{\tfrac{1}{2}q}^{MC} &= 0\\ 12 &= \tfrac{1}{2}q\\ q^\star &= 24 \end{aligned}$ We can see this visualized in the graph below. As with the case of a firm facing a downward-sloping inverse demand curve, the profit may be represented as the area $(AR - AC) \times q$. You can drag the quantity back and forth to see that this profit is indeed maximized at $q = 24$:

See interactive graph online here.

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