Unit 7 The firm and its customers

7.5 Demand, elasticity, and revenue

differentiated product: A product produced by a single firm that has some unique characteristics compared to similar products of other firms.

Beautiful Cars is an example of a firm producing a differentiated product. Not all cars are the same. Each make and model is produced by just one firm, and has some unique characteristics of design and performance that differentiate it from the cars made by other firms.

When a firm sells a differentiated product, it faces a downward-sloping demand curve. Section 7.2 gives an empirical example of a demand curve, for Apple Cinnamon Cheerios (another differentiated product). If the price of a Beautiful Car is high, demand will be low because the only consumers who will buy it are those who strongly prefer Beautiful Cars to all other makes. As the price falls, more consumers, who might otherwise have purchased a Ford or a Volvo, will be attracted to a Beautiful Car.

The demand curve

For any product that consumers might wish to buy, the product demand curve is a relationship that tells you the number of items (the quantity) they will buy at each possible price. For a simple model of the demand for Beautiful Cars, imagine that there are 100 potential consumers who would each buy one Beautiful Car today, if the price were low enough.

willingness to pay (WTP): An indicator of how much a person values a good, measured by the maximum amount they would pay to acquire a unit of the good. See also: willingness to accept.

Each consumer has a willingness to pay (WTP) for a Beautiful Car, which depends on how much the customer personally values it (given the resources to buy it, of course). A consumer will buy a car if the price is less than or equal to his or her WTP. Suppose we line up the consumers in order of WTP, with the highest first, and plot a graph to show how the WTP varies along the line (Figure 7.9). Then if we choose any price, say P = $32,800, the graph shows the number of consumers whose WTP is greater than or equal to P. In this case, 18 consumers are willing to pay $32,800 or more, so the demand for cars at a price of $32,800 is 18.

In this diagram, the horizontal axis shows the quantity of cars, and ranges from 0 to 100. The vertical axis shows the price in dollars, and ranges from 0 to 45,000. Coordinates are (quantity, price). A downward-sloping straight line connects points (0, 40,000), (18, 32,800) and (100, 0).

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https://www.core-econ.org/microeconomics/07-firm-and-customers-05-demand-elasticity-revenue.html#figure-7-9

Figure 7.9 The demand for cars (per day).

The Law of Demand dates back to the seventeenth century, and is attributed to Gregory King (1648–1712) and Charles Davenant (1656–1714). King was a herald at the College of Arms in London, who produced detailed estimates of the population and wealth of England. Davenant, a politician, published the Davenant–King Law of Demand in 1699, using King’s data. It described how the price of corn would change depending on the size of the harvest. For example, he calculated that a ‘defect’, or shortfall, of one-tenth (10%) would raise the price by 30%.

If P is lower, there is a larger number of consumers willing to buy, so the demand is higher. Demand curves are often drawn as straight lines, as in this example, although there is no reason to expect them to be straight in reality: the demand curve for Apple Cinnamon Cheerios was not straight. But we do expect demand curves to slope downward: as the price rises, the quantity that consumers demand falls. Conversely, when the available quantity is low, it can be sold at a high price. This relationship between price and quantity is sometimes known as the Law of Demand.

Question 7.6 Choose the correct answer(s)

The diagram depicts two alternative demand curves, D and D′, for a product. Based on this graph, read the following statements and choose the correct option(s).

In this diagram, the horizontal axis shows the quantity of consumers, ranging from 0 to 100, and the vertical axis shows price in pounds, ranging from 0 to 12,000. There are two parallel downward-sloping lines. The bottom line is labelled D and connects the points (0, 6000) and (60, 0). The top line is labelled D-prime and connects the points (0, 10000) and (100, 0).

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On demand curve D, when the price is £5,000, the firm can sell 15 units of the product.
On demand curve D′, the firm can sell 70 units at a price of £3,000.
At a price of £1,000, the firm can sell 40 more units of the product on D′ than on D.
With an output of 30 units, the firm can charge £2,000 more on D′ than on D.

On demand curve D, when the price is £5,000, the firm can sell 10 units.
When Q = 70, the corresponding price on D′ is £3,000.
D′ is a rightward shift of D, by 40 units. So for any price, the firm can sell 40 more units on D′ than on D.
With an output of 30 units, the firm can charge £4,000 more on D′ than on D.

The elasticity of demand

The demand curve represents the trade-off the firm has to make between price and quantity. To maximize profit, it would like both to be as high as possible—but if it raises the price, fewer consumers will want to buy. So the firm’s choice of price depends on the slope of the demand curve—that is, on how much demand will change if the price changes. If the demand curve is steep, the firm can raise the price without reducing sales very much.

price elasticity of demand: The percentage change in demand that would occur in response to a 1% increase in price. We express this as a positive number. Demand is elastic if this is greater than 1, and inelastic if less than 1.

The price elasticity of demand is a measure of the responsiveness of consumers to a price change. It is defined as the percentage change in demand that would occur in response to a 1% increase in price. For example, suppose that the price of a product increases by 10%, and we observe a 5% fall in the quantity sold. We calculate the elasticity, ε, as follows:

\[\varepsilon = -\frac{\% \text{ change in demand}}{\% \text{ change in price}}\]

ε is the Greek letter epsilon, which is often used to represent elasticity. For a demand curve, quantity falls when price increases. So the change in demand is negative if the price change is positive, and vice versa. The minus sign in the formula for the elasticity ensures that we get a positive number as our measure of responsiveness. In this example, we get:

\[\begin{align*} \varepsilon &= -\frac{-5}{10} \\ &= 0.5 \end{align*}\]

If the demand curve is almost flat, quantity changes a lot in response to a change in price, so the elasticity is high. Conversely, a steeper demand curve corresponds to a lower elasticity. But they are not the same thing. We will explain why the elasticity changes as we move along the demand curve, even if the slope doesn’t.

Suppose Beautiful Cars produces 18 cars per day and sells them at a price of $32,800 (point K in Figure 7.9). To calculate the elasticity of demand at this point, we work out how Q and P would change if the firm moved a small distance along the demand curve—for example to another point, L, where Q = 19. The demand curve has a constant slope of –400 (you can check this from the points where it crosses the axes) so when we move from K to L, the price falls to $32,400.

The table in Figure 7.10 shows the calculation. The move down the demand curve represents a 5.56% increase in Q, and a 1.22% decrease in P. Taking the ratio of the percentage changes gives an elasticity of 4.56.

	Point K	Point L	change	% change	elasticity
Q P	18 32,800	19 32,400	∆Q = 1 ∆P = −400	$$ \frac{100×\Delta Q}{18} = 5.56\% $$ $$ \frac{100× \Delta P}{32,800} = −1.22\% $$	$$ \varepsilon = \frac{5.56}{1.22} = 4.56 $$

Figure 7.10 Calculating the elasticity at a point on the demand curve.

The elasticity tells us that when Beautiful Cars is operating at point K, raising (or lowering) the price by 1% would lead to a 4.56% fall (or rise) in the quantity of cars sold.

There are several different ways to calculate elasticity from the changes in P and Q, summarized in the table in Figure 7.11. They are all equivalent—you can use whichever you like.

Suppose that if price changes by ΔP, demand changes by ΔQ. Then the elasticity of demand can be written in four different ways:
$$ − \frac{\% \text{ change in } Q}{\% \text{ change in } P} $$	$$ \varepsilon = − \frac{100 \Delta Q}{Q} / \frac{100 \Delta P}{P} $$
$$ − \frac{\text{ proportional change in } Q}{\text{ proportional change in } P} $$	$$ \varepsilon = − \frac{\Delta Q}{Q} / \frac{\Delta P}{P} $$
The fraction can be simplified to get:	$$ \varepsilon = − \frac{P}{Q} \frac{\Delta Q}{\Delta P} $$
And since $ \dfrac{\Delta P}{\Delta Q} $ is the slope of the demand curve:	$$ \varepsilon = − \frac{P}{Q} \frac{1}{\text{ slope}} $$

Figure 7.11 Formulas for calculating elasticity.

Figure 7.12 uses the fourth expression to calculate the elasticity at other points on the demand curve for Beautiful Cars. This demand curve has a constant slope; as we move down it P falls and Q rises, so the elasticity falls. When prices are low, demand is less elastic.

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$$ \varepsilon = − \frac{P}{Q} \frac{1}{\text{ slope}} $$
	A	B	C
Q	20	40	70
P	$32,000	$24,000	$12,000
slope	−400	−400	−400
elasticity	4.00	1.50	0.43

Figure 7.12 The elasticity of demand for cars.

We say that demand is elastic if the price elasticity is higher than 1: that is, a 1% increase in price would lead to a fall of more than 1% in the quantity sold. If the elasticity is less than 1, we say that demand is inelastic. For Beautiful Cars, demand is elastic at A and B, but inelastic at C.

Why is the price elasticity of demand important to the firm?

The firm’s price elasticity of demand will depend on how much competition it faces from other firms. If lots of firms sell similar cars that customers would consider as potential alternatives, the demand for Beautiful Cars will be more elastic. Then if it raises the price, consumers will search for alternative sellers, and many of them may decide to buy elsewhere. In this situation, competition from rival products with similar characteristics will limit the firm’s ability to raise its price.

But if Beautiful Cars’ product has unique qualities that appeal to consumers and are not available elsewhere, its price elasticity of demand will be lower (demand will be less elastic). Then it can benefit from a high price. Sales will remain high, and it will make a higher profit on every unit it sells.

total revenue, revenue: A firm’s total revenue is the number of units sold times the price per unit.

There is a direct relationship between the elasticity of demand and how the firm’s revenue changes as quantity increases. Figure 7.13 shows a firm producing at a point on its demand curve where Q = 5 and P = 20. Its revenue (price × quantity) is represented by the area of the rectangle under the demand curve. Work through the figure to understand that if it increases quantity, revenue will rise or fall depending on whether demand is elastic or inelastic.

There are two diagrams. In diagram 1, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points E (5, 20) and (7, 0). The area enclosed by points (0, 0), (0, 10), (6, 10) and (6, 0) is the revenue. The area enclosed by points (0, 10), (0, 20), (5, 20) and (5, 10) is the loss in revenue. The area enclosed by points (5, 0), (5, 10), (6, 10) and (6, 0) is the gain in revenue. The loss in revenue is bigger than the gain in revenue. In diagram 2, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points (0, 30) and E (5, 20). The area enclosed by points (0, 0), (0, 18), (6, 18) and (6, 0) is the revenue. The area enclosed by points (0, 18), (0, 20), (5, 20) and (5, 18) is the loss in revenue. The area enclosed by points (5, 0), (5, 18), (6, 18) and (6, 0) is the gain in revenue. The loss in revenue is smaller than the gain in revenue.

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Figure 7.13 Competition, elasticity, and revenue.

Competition and elasticity: There are two diagrams. In diagram 1, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points E (5, 20) and (7, 0). In diagram 2, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points (0, 30) and E (5, 20).

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https://www.core-econ.org/microeconomics/07-firm-and-customers-05-demand-elasticity-revenue.html#figure-7-13a

Competition and elasticity

In panel A, the firm faces little competition, so its demand curve is steep. If the price rises, many consumers will still want to buy. At point E, demand is inelastic (elasticity = 0.4). In panel B, the firm faces more competition: the demand curve is flatter. Demand at E is elastic (elasticity = 2).

The firm’s revenue: There are two diagrams. In diagram 1, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points E (5, 20) and (7, 0). The area enclosed by the horizontal line at price 20, the vertical line at quantity 5, and the two axes is the revenue. In diagram 2, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points (0, 30), E (5, 20) and (6, 18). The area enclosed by the horizontal line at price 20, the vertical line at quantity 5, and the two axes is the revenue.

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The firm’s revenue

If the firm operates at point E, where P = 20 and Q = 5, its revenue is equal to the area of the rectangle under the curve: revenue = P × Q = 100, in both cases.

If output increases to Q = 6: There are two diagrams. In diagram 1, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points E (5, 20) and (7, 0). The area enclosed by points (0, 0), (0, 10), (6, 10) and (6, 0) is the revenue. The area enclosed by points (0, 10), (0, 20), (5, 20) and (5, 10) is the loss in revenue. The area enclosed by points (5, 0), (5, 10), (6, 10) and (6, 0) is the gain in revenue. In diagram 2, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points (0, 30) and E (5, 20). The area enclosed by points (0, 0), (0, 18), (6, 18) and (6, 0) is the revenue. The area enclosed by points (0, 18), (0, 20), (5, 20) and (5, 18) is the loss in revenue. The area enclosed by points (5, 0), (5, 18), (6, 18) and (6, 0) is the gain in revenue.

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If output increases to Q = 6

In both cases, the firm gains revenue on the extra unit. But the price falls so it will lose revenue on the original five units. A one unit increase in output corresponds to a larger reduction in price with inelastic demand (Panel A) than with elastic demand (Panel B).

Elasticity and revenue: There are two diagrams. In diagram 1, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points E (5, 20) and (7, 0). The area enclosed by points (0, 0), (0, 10), (6, 10) and (6, 0) is the revenue. The area enclosed by points (0, 10), (0, 20), (5, 20) and (5, 10) is the loss in revenue. The area enclosed by points (5, 0), (5, 10), (6, 10) and (6, 0) is the gain in revenue. The loss in revenue is bigger than the gain in revenue. In diagram 2, the horizontal axis shows quantity, and ranges from 0 to 10. The vertical axis shows price, and ranges from 0 to 40. Coordinates are (quantity, price). A downward-sloping, straight line passes through points (0, 30) and E (5, 20). The area enclosed by points (0, 0), (0, 18), (6, 18) and (6, 0) is the revenue. The area enclosed by points (0, 18), (0, 20), (5, 20) and (5, 18) is the loss in revenue. The area enclosed by points (5, 0), (5, 18), (6, 18) and (6, 0) is the gain in revenue. The loss in revenue is smaller than the gain in revenue.

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https://www.core-econ.org/microeconomics/07-firm-and-customers-05-demand-elasticity-revenue.html#figure-7-13d

Elasticity and revenue

If demand is inelastic, the loss outweighs the gain: revenue falls. If demand is elastic, the gain is bigger than the loss and revenue rises.

marginal revenue: The change in revenue obtained by increasing the quantity sold by one unit.

The change in revenue when output is increased by one unit is called the marginal revenue. In Figure 7.13:

Marginal revenue is positive when demand is elastic ($\varepsilon>1$); the firm can increase revenue by raising output because prices fall only a little.
Marginal revenue is negative when demand is inelastic; the firm can increase revenue by decreasing output because prices rise a lot.

The extension at the end of this section demonstrates that this result is true for all demand curves. In subsequent sections, we will show that firms facing little competition and less elastic demand curves will set higher prices.

Question 7.7 Choose the correct answer(s)

A shop sells 20 hats per week at $10 each. When it increases the price to $12, the number of hats sold falls to 15 per week. Based on this information, read the following statements and choose the correct option(s).

When the price increases from $10 to $12, demand decreases by 25%.
A 20% increase in the price causes a 25% fall in demand.
The demand for hats is inelastic.
Using these figures, we can estimate the elasticity of demand to be 1.25.

When the price increases from $10 to $12, demand decreases by 100 × (20 –15)/20 = 25%.
The percentage price increase is 100 × 2/10 = 20%. It causes a percentage decrease in demand of 100 × 5/20 = 25%.
Using the figures to estimate the price elasticity of demand gives a value greater than 1, so demand is elastic.
The percentage price increase is 100 × 2/10 = 20%; the percentage decrease in demand is 100 × 5/20 = 25%. So the elasticity can be estimated as 25/20 = 1.25.

Question 7.8 Choose the correct answer(s)

The figure depicts two demand curves, D₁ and D₂.

In this diagram, the horizontal axis displays quantity and the vertical axis displays price. There are two downward-sloping, straight lines. One is labelled D1 and is steeper than the other, which is labelled D2. They cross at point E. Point A is on D1 and is at a lower quantity and higher price than E. Point C is on D1 and is at a higher quantity and lower price than E. Point B is on D2 and is at a higher quantity and higher price than C.

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Based on this figure, read the following statements and choose the correct option(s).

At E, demand curve D₁ is less elastic than D₂.
The elasticity is the same at A and C.
The firm with demand curve D₁ likely faces more competition than the firm with demand curve D₂.
The elasticity is higher at E than at B.

At E, the price and quantity are the same on both demand curves, but D₁ is steeper, so it is less elastic than D₂.
The slope is the same at A and C, but at A the price is higher and quantity is lower, so the elasticity is higher.
Demand curve D₁ is less elastic than D₂, so the firm with demand curve D₁ likely faces less competition than the firm with demand curve D₂.
The slope is the same at E and B. But at E the price is higher and quantity is lower, so the elasticity is higher.

Extension 7.5 The elasticity of demand and marginal revenue

We have examined the elasticity of demand for a linear demand curve in the main part of this section; in that case, elasticity is easy to calculate because the slope of the demand curve is constant. Now, we demonstrate how to measure elasticity when the demand curve is not a straight line, using calculus (differentiation) to determine the slope.

The price elasticity of demand measures the sensitivity of demand to price changes. We have defined it in terms of percentage changes in quantity and price along the demand curve:

\[\varepsilon = \frac{\text{% change in } Q}{\text{% change in } P} = -\frac{P}{Q} \frac{\Delta Q}{\Delta P}\]

For a linear demand curve, $\Delta Q/\Delta P$ is the same whatever value we choose for $\Delta P$. For other demand curves, it is easier to define the elasticity using calculus.

The equation of the demand curve

We can think of the equation of the demand curve in two different ways. We drew the demand curve for Beautiful Cars showing the price on the vertical axis and the quantity on the horizontal axis. In other words, we have shown price as a function of quantity:

\[P = f(Q)\]

We call $f(Q)$ the inverse demand function: it is the highest price at which the firm can sell exactly $Q$ cars. To define the elasticity, it is more convenient to write the demand function in its direct form:

\[Q = g(P)\]

$g(P)$ is the quantity of Beautiful Cars demanded if the price is $P$. The function, $g$, is the inverse function of $f$; mathematically, we can write $g(P)=f^{-1}(P)$.

To define the elasticity using calculus, we treat $Q$ as a continuous variable, just as we did in Extension 7.4 to calculate marginal costs. Then we find the elasticity at a point $Q$ by taking the limit of the expression, $-\frac{P}{Q} \frac{ \Delta Q}{ \Delta P}$, as $\Delta P$ tends to 0:

\[\varepsilon = - \frac{P}{Q} \, \frac{ dQ}{dP}\]

This gives us the elasticity in terms of the derivative of the demand function: $dQ/dP =g'(P)$. The value of the elasticity is normally positive, since according to the Law of Demand, the derivative of the demand function will be negative.

If you calculate the elasticity as we did earlier, using the changes between two points on the demand curve, you will not normally get the same answer as when you use the derivative. In the case of a linear demand curve, the answer is the same, because the slope is constant. But otherwise the calculus method corresponds to using two points infinitesimally close together; taking points further apart gives a different answer because they give a less accurate estimate of the slope at a particular point.

You may have wondered why we don’t just use the slope of the demand function $dQ/dP$ to measure responsiveness to price. The problem with $dQ/dP$ is that it depends on the units in which $P$ and $Q$ are measured: for example, we would get different answers if we measured price in dollars rather than euros. Elasticity, defined in terms of proportional changes, is independent of the units of measurement.

Two ways of writing the elasticity

The elasticity as expressed above depends on both $P$ and $Q$. But substituting for $Q$ using the demand function, $Q=g(P)$, we can write it entirely in terms of price:

\[\varepsilon = - \frac{P}{Q} \, \frac{ dQ}{dP} =-\frac{Pg'(P)}{g(P)}\]

And if we use the inverse demand function, $P = f(Q)$, we can write it entirely in terms of quantity. To understand why you need to remember the inverse function rule:

\[\frac{dP}{dQ} = 1 \left/ \frac{dQ}{dP} \right.\]

Then:

\[\varepsilon = - \frac{P}{Q} \left/ \frac{dP}{dQ} \right. = - \frac{f(Q)}{Qf'(Q)}\]

Example 1: The elasticity of a linear demand function

The demand curve for Beautiful Cars shown in Figure 7.12 corresponds to the inverse demand function:

\[P= f(Q) \text{ where } f(Q) = 400(100-Q)\] \[\Rightarrow P=400(100-Q)\]

Rearranging this to find $Q$ in terms of $P$ gives us the demand function:

\[Q=g(P) \text{ where } g(P) = 100 - \frac{P}{400}\]

Using the expression for elasticity in terms of $P$ we get:

\[\varepsilon =-\frac{Pg'(P)}{g(P)} =-\frac{P \times \frac{-1}{400}}{100-\frac{P}{400}} =\frac{P}{40,000 - P}\]

Using the expression for elasticity in terms of $Q$, we get:

\[\varepsilon =- \frac{f(Q)}{Qf'(Q)} =-\frac{400(100-Q)}{Q\times -400} =\frac{100-Q}{Q}\]

Each of the two expressions for $\varepsilon$ shows that the elasticity falls as we move to the right along the demand curve, which increases $Q$ and reduces $P$. (This is true for all linear demand functions.) For example:

If Beautiful Cars sets a price so high that it sells only five cars per day, $\varepsilon=(100-5)/5 = 19$.
If it sets a price low enough to sell 95 cars per day, $\varepsilon=(100-95)/95\approx 0.053$.

Example 2: A demand function with constant elasticity

Consider the demand function:

\[Q=100 P^{-0.8}\]

Here,

\[\varepsilon=-\frac{P}{Q}\,\frac{dQ}{dP} = -\frac{P}{100P^{-0.8}}\times -80P^{-1.8} = 0.8\]

In this special case, the elasticity of demand is constant—it is equal to 0.8 at all points on the demand curve.

This constant-elasticity property holds for any demand curve of the form, $Q = aP^{-b}$, where $a$ and $b$ are positive constants: you can check that the elasticity of demand is equal to $b$. This is the only class of demand functions for which the elasticity is constant.

Elasticity and marginal revenue

marginal revenue: The change in revenue obtained by increasing the quantity sold by one unit.

A firm’s revenue is given by $\text{price} \times \text{quantity}$: that is, $R=PQ$. The inverse demand function, $P=f(Q)$, tells us the maximum price, $P$, at which $Q$ cars can be sold, so we can write revenue as a function of $Q$ alone. We call this the revenue function, denoted by $R(Q)$:

\[R(Q) = f(Q) \times Q\]

Previously, we defined the marginal revenue as the change in revenue when output is increased by one unit: $\text{MR}=\Delta R/\Delta Q$. Treating $Q$ as a continuous variable and using calculus, we write:

\[\text{MR}=\frac{dR}{dQ}\]

That is to say, marginal revenue is the rate at which revenue increases in response to a small (infinitesimal) change in $Q$. Using the rule for differentiating a product to differentiate $R(Q) = Qf(Q)$:

\[\text{MR} = \frac{d}{dQ} (Qf(Q)) = f(Q) + Qf'(Q) = P+Qf'(Q)\]

Rewriting this expression using the formula $\varepsilon=-\dfrac{f(Q)}{Qf'(Q)}$ and using the fact that $P=f(Q)$, we derive a relationship between marginal revenue and the elasticity of demand:

\[\text{MR} = f(Q) - \frac{f(Q)}{\varepsilon} = P\left(1 - \frac{1}{\varepsilon}\right)\]

This implies that marginal revenue will be positive if $\varepsilon > 1$, and negative if $\varepsilon < 1$.

Remember that demand is said to be elastic if $\varepsilon > 1$ and inelastic if $\varepsilon < 1$, and that (except in the particular case in Example 2 above) it changes as we move along the demand curve. What we have just shown is that marginal revenue is positive if, and only if, the firm is operating on a part of the demand curve where demand is elastic. This is the result illustrated for the linear demand curve in Figure 7.13; writing MR in terms of elasticity, as we have done here, shows that it is true for all demand curves.

Exercise E7.2 Linear demand function: Elasticity and marginal revenue

A firm faces the following demand function: $Q = 800 - 2P$.

For this demand function:

Find the inverse demand function ($P$ as a function of $Q$) and use this function to derive an expression for the elasticity of demand (as a function of $Q$).
Use your answer to Question 1 to draw a diagram showing how the elasticity of demand changes with $Q$ ($Q$ on the horizontal axis, $\varepsilon$ on the vertical axis).
Describe the shape of the elasticity function. At which quantities is demand elastic?
Derive an expression for marginal revenue (as a function of $Q$). Sketch the marginal revenue function and demand function on the same diagram, with $Q$ on the horizontal axis, and price and marginal revenue on the vertical axis.
Describe the shape of the marginal revenue curve and use your answers to Questions 2 and 3 to verify that marginal revenue is positive if $\varepsilon > 1$ and negative if $\varepsilon < 1$.

Exercise E7.3 Constant elasticity of demand: Elasticity and marginal revenue

A firm faces the following demand function: $Q = 5P^{-1.4}$.

Do the following for this demand function:

Find the inverse demand function and use a diagram (price on the vertical axis, quantity on the horizontal axis) to sketch its shape. (Hint: Choose values of $Q$ at regular intervals and plot the corresponding ($Q$, $P$) coordinates, then connect these points to approximate the function.)
Use the elasticity of demand formula to find the elasticity of demand (remember, for this type of function, the elasticity of demand will be the same at all points on the curve). Explain why demand curves with constant elasticity must have the general shape of the function you have drawn in Question 1. (Hint: Use the elasticity of demand formula in your interpretation.)
Derive an expression for marginal revenue (as a function of $Q$) and sketch this function in a new diagram, with $Q$ on the horizontal axis and marginal revenue on the vertical axis. Describe the shape of the marginal revenue curve and relate it to the elasticity you calculated in Question 2.

Read more: Sections 6.4 (on marginal revenue and elasticity) and 7.4 (on inverse functions and the inverse function rule) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.