Application: Price elasticity of demand and total revenue

Section 1.6 Application: Price elasticity of demand and total revenue

In the previous section, we discuss how an understanding of price elasticity of demand can be useful to a firm deciding what price it should charge. If, for example, the consumers of movie tickets have more elastic demand, (say, because there are many alternative activities consumers could buy tickets for,) then a movie theater might anticipate that a price increase is likely to drive many of its highly-price-sensitive consumers away.

Even though we will discuss optimal decision making for firms in much greater detail in a later chapter, let’s explore the relationship between $\epsilon_D$ and firm pricing a little bit here. In particular, we will examine the impact of $\epsilon_D$ on a firm’s total revenue: the amount of money the firm brings in from sales. We calculate a firm’s total revenue by taking the product of the price the firm charges and the quantity of units sold:

\begin{equation*} TR = P\times Q \end{equation*}

If, for example, a movie theater sells 100 tickets at a price of $12 per ticket, then $Q_D = 100\text{,}$¹ $P = 12\text{,}$ and

\begin{equation*} \text{ total revenue } = TR = P\times Q = (12)(100) = \$1200 \end{equation*}

Since a demand function tells us the quantity demanded at every price, we can also simultaneously determine how much revenue will be generated at every price by taking the product of price and quantity. Consider the demand function $Q_D = 200 - 10P\text{.}$ which we saw above. In this application, we interpret the demand function from the firm’s perspective and think of it as the demand for the firm’s product. In other words, it is as if the firm knew that if it charged a price of, say 4, that exactly 160 units would be demanded from the firm.² ³

We know we can use a demand schedule to plug in various prices and find the quantities demanded:

Table 1.6.1. Demand schedule at different prices for $Q_D = 200 - 10P$

$P$	$Q_D = 200 - 10P$

0	200
4	160
8	120
10	100
12	80
16	40
20	0

Now, with the formula for total revenue, the demand function allows us to easily calculate the amount of revenue generate at each different price a firm could charge:

Table 1.6.2. Demand schedule and total revenue at different prices for $Q_D = 200 - 10P\text{.}$

$P$	$Q_D$	$TR = P\times Q$

0	200	0
4	160	640
8	120	960
10	100	1000
12	80	960
16	40	640
20	0	0

With these calculations, we can start to observe the relationship between price and total revenue. Look at the extremes first. If the firm charges a price of 0, then the maximum number of units are demanded (200), because the product is free! However, since the product is free, the firm makes no revenue. Similarly, if the firm charges the high price of 20 (or any price above 20), the quantity demanded will be 0, and the firm will also make revenue. There is a clear tradeoff the firm makes when considering what price to charge to earn more revenue: raising the price will reduce the units demanded, but will increase the amount of revenue brought in per unit; lowering the price will bring in more units but earn the firm fewer dollars per unit. Earning more revenue requires balancing this tradeoff.

We can also view the total revenue calculations on a graph. Notice first that any point along the demand curve $Q_D = 200 - 10P$ gives both the price and the quantity as an ordered pair. Price is the vertical distance from the origin to that point, while quantity is the horizontal distance. As a result, total revenue, which is $P\times Q\text{,}$ can be visualized as the rectangular area under the demand curve at that point, since the area of a rectangle is measured as height×base, or, here, $P\times Q\text{.}$

Figure 1.6.3. Left: $TR$ at $P = 8$ and $Q_D = 120\text{.}$ The area of the rectangle is $(8)(120) = 960\text{.}$ Right: $TR$ at $P = 16\text{,}$ $P = 10\text{,}$ and $P = 8\text{.}$

If total revenue areas are plotted for various prices, we can interpret the tradeoff between price and total revenue as the change in the areas of the total revenue rectangles. At very high prices, the rectangles are tall and narrow with a small area due to the low quantity. As the price comes down, the rectangles widen and get smaller, yet the rectangles grow in size. Then, past the midpoint of the curve - where the rectangle is actually a square - the rectangles start to get shorter, wider and smaller. At extremely low prices, the rectangles are very short and wide, and revenue is very low.

We can even see the firm’s total revenue as its own function of the number of units. Recall the inverse demand function, which solves the demand function for $P\text{.}$ In this application, with $Q_D = 200 - 10P\text{,}$ we get

\begin{equation*} Q_D = 200 - 10P \end{equation*}

\begin{equation*} 10P = 200 - Q_D \end{equation*}

\begin{equation*} P = 20 - \frac{1}{10}Q_D \end{equation*}

From here, since we know that the firm’s total revenue is $P\times Q\text{,}$ substituting in the inverse demand function gives the total revenue solely as a function of Q, since $Q$ is now the only variable left:

\begin{equation*} TR = (P)(Q) \end{equation*}

\begin{equation*} TR = (20 - \frac{1}{10}Q)(Q) \end{equation*}

\begin{equation*} TR = 20Q - \frac{1}{10}Q^2 \end{equation*}

On the graph below, you can see how the upside-down U shape of the total revenue curve mirrors the tradeoff discussed earlier. If the price is extremely high or extremely low, the total revenue is low. Moreover, the total revenue is maximized right at the midpoint of the demand function, when $P = 10$ and $q = 100\text{.}$ Therefore, we can observe the following pattern:

If the price is above 10, then lowering the price will increase the firm’s total revenue.
If the price is below 10, then increasing the price will increase the firm’s total revenue.
If the price is equal to 10, then the firm earns its maximum total revenue.

Figure 1.6.4. Top: Demand function $Q_D = 200 - 10P\text{.}$ Bottom: $TR$ as a function of $Q\text{.}$ $TR = 20Q - \frac{1}{10}Q^2\text{.}$

Based on our intuition thus far, this pattern matches the explanation that the firm must balance its tradeoff between price and revenue, not charging a price that is too low or too high. But how does this pattern tie in to the concept of price elasticity of demand?

An application in the previous section shows that it is possible to calculate the price elasticity of demand at a point along a linear demand curve. Importantly, with a linear demand, the price elasticity of demand changes at different points along the curve! Demand is more elastic along the top of the curve, where prices are relatively high; conversely, demand is more inelastic along the bottom of the curve, where prices are lower and quantities are larger.

Figure 1.6.5. Price elasticity of demand at points along the curve $Q_D = 200 - 10P\text{.}$

This overlays with the conclusion above price and total revenue. The top half of the demand curve, where price is above the midpoint, is also where demand is relatively elastic; the bottom half of the demand curve, where price is below the midpoint, is also where demand is relatively elastic. When we combine the two, we get the following result:

If demand is relatively elastic, then lowering the price will increase the firm’s total revenue.
If demand is relatively inelastic, then increasing the price will increase the firm’s total revenue.
If demand is unit elastic, then the firm earns its maximum total revenue.

The utility of this result cannot be understated, so let’s explore the intuition. First, the inelastic side. We have shown that when demand is relatively inelastic, a firm can increase its revenue by increasing the price. Intuitively, this makes some sense. For goods with few substitutes (gasoline, certain medicines), consumers are less willing to reduce consumption when the prices go up. As a result, since the price hike does not cost the firm many consumers, revenue is going to grow. This is a major reason why you rarely see a “sale” on gas at the gas station, or on milk at the grocery store. If a business knows that demand for its product is relatively inelastic, it knows that a price reduction will not increase its revenue.

What about goods for which demand is relative elastic? Here, the intuition is flipped. Since these goods are likely to have more substitutes, a price increase will quickly lose the firm customers as they switch to relatively cheaper alternatives. The loss of customers here outweighs the benefit of the price increase, and the firm will lose revenue. So, the way for a firm to increase revenue for a good with elastic demand is to lower prices, making the product relatively cheaper compared to its many substitutes and attracting enough new consumers to outweigh the impact of the lower price. This is a major reason why services with more elastic demand, like movie tickets or fast food restaurants, offer coupons, discounts for matinees, and price reductions for students, seniors. or military personnel.⁴

Figure 1.6.6. Top: Demand function with elasticity ranges. Bottom: $TR$ as a function of $Q\text{.}$ 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 $TR\text{.}$

Checkpoint 1.6.7.

In 2023, EV maker Tesla cut the price of its Model S model by around 6%. By some reports, this helped increase sales by around 20%.⁵ What does this mean for the price elasticity of Model S’s here? Would this have resulted in a decrease or an increase in Tesla’s revenue?

Answer.

We can use the definition of price elasticity to answer the first question: $\epsilon_D = \frac{\%\Delta Q_D}{\%\Delta P}$ so if $Q_{D}$ increased 20% when $P$ decreased 6% we can say that $\epsilon_D = \frac{20}{-6}=-3.33\text{.}$ Since this is less than -1, we know that Model S’s are relatively elastic.

We now can also answer the second part of the question. Total revenue would rise because the percentage increase in quantity demanded was greater than the percentage decrease in price. Hence $TR = PQ_{D}$ must rise.

The knowledge of whether consumer demand for a product is relatively elastic or inelastic can provide a firm with critical information on the relationship between price changes and its total revenue. One important caveat to this result is that it is only relevant for firms who have market power - the ability to impact the price they charge. Many firms have market power, but others, such as perfectly competitive firms, do not. Market power will be central to the discussion of market structures and firm optimization in later chapters.

Finally, a note on the unit elastic point. It is not a coincidence that total revenue is maximized where demand is unit elastic.⁶ For any firm with market power, this must be true. However, we have not discussed what the firm’s objective even is! In optimal decision making for firms (Chapter 3), we will identify the objective for a firm as the maximization of its total profit, NOT its total revenue. Total profit accounts for both the revenue the firm brings in, and the cost of its production, and proves to be a more worthy consideration for firms. (Imagine a firm that just wanted to bring in the most amount of revenue, without caring at all how much it cost them to bring it in! That doesn’t sound like a winning strategy.) We will focus on this objective in greater detail later on, but one surprising fact will come into play. Even though the unit elastic point on the demand curve will maximize the firm’s total revenue, it turns out that this point will never be the point which maximized the firm’s profit, and therefore will never be the optimal choice for the firm! This may seem off or counterintuitive, but we will see why it must be true a bit later on.

Key terms in this section:

total revenue
market power (briefly)
total profit (briefly)

Practice Problems: Subsection 1.7.6

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\(P\)	\(Q_D\)	\(TR = P\times Q\)

0	200	0
4	160	640
8	120	960
10	100	1000
12	80	960
16	40	640
20	0	0