How does raising prices affect your revenue?

A (quick) look at price elasticity

If you’re in product marketing, the title of this post is the $64,000 question. In my experience, people tend to assume that their prices are too high (especially new entrepreneurs). They fear that if they raise prices, too many customers will leave, and therefore revenue will decline. This (short) post is going to pick at that fear.

The revenue equation

The first order of business is to write down the equation for revenue:

\(R=P\cdot Q\)

R is revenue, P is price, and Q is quantity. So far, no surprises. If you want to know what happens to R when you change P, calculus tells us that we take the derivative of R with respect to P. Q is also variable, so we need to take the partial derivative:

\(\begin{aligned} \frac{\partial R}{\partial P} &= \frac{\partial}{\partial P}\left(P\cdot Q\right)\\[3pt] &= 1\cdot Q+P\cdot\frac{\partial Q}{\partial P}\\[3pt] &=Q\left(1+\frac{P}{Q}\cdot \frac{\partial Q}{\partial P}\right) \end{aligned}\)

The second equality follows from the product rule for derivatives. (Don’t worry just yet about why I factored out a Q.) If you remember your calculus I, for a given price P, R will increase in response to changes in P if the derivative is positive. Since Q and P are both positive, this means that revenue will increase in response to rising prices if

\(\frac{P}{Q}\cdot \frac{\partial Q}{\partial P}>-1.\)

What do we know about this expression? For starters, the law of demand says that dQ/dP < 0: quantity demanded decreases as price increases. This tells us that the above expression is negative. It may or may not be greater than -1.

Price elasticity of demand

Economists came up with the notion of price elasticity (of demand) e to discuss this situation:

\(\begin{aligned} e&=\frac{P}{Q}\cdot \frac{\partial Q}{\partial P}\\[3pt] &\approx\frac{\Delta Q/Q}{\Delta P/P}\\[3pt] &=\frac{\text{%age change in }Q}{\text{%age change in }P} \end{aligned}\)

Going from the instantaneous rate of change dQ/dP to the average rate of change ΔQ/ΔP is a practical matter. In real life, you wouldn’t have a theoretical demand curve Q(P). Instead, you would measure demand at different price levels and use that data to approximate elasticity. As a byproduct, we get a nice interpretation of elasticity in terms of percentage change in each quantity.

Substituting e into the equation for dR/dP tells us how revenue reacts to price changes. Keeping in mind that e < 0 (law of demand), we have the following:

• If |e| > 1, revenue will decrease as price increases. This is called elastic demand.

• If |e| < 1, revenue will increase as price increases. This is called inelastic demand.

• If |e| = 1, revenue has a critical point at this price level. This is called unit elastic or unitary demand.

If demand is elastic, that means that small increases in price lead to large decreases in demand. Consequently, revenue goes down because the increase in revenue per unit (i.e., price) is not enough to compensate for the lost sales. On the flip side, if demand is inelastic, then the quantity doesn’t change that much if you increases prices. This will increase revenue. If e = 1, then you have a critical point, which suggests that revenue is either in a local maximum or minimum. In most cases, it will be a maximum, indicating that you are “optimally priced.”¹

Let’s look at an example. Suppose a company sells 100 units of a product when the price is $10. After increasing the price to $12, the company sells only 75 units. What is the elasticity of demand in that case? Price increases by 20% ($10 → $12). Quantity decreases by 25% (100 units → 75 units). Therefore, e = (-25)/(20) = -1.25. Since this expression is less than -1 (or greater than 1 in absolute value), demand is elastic. Consequently, we expect that revenue should decrease with the increase in price. Sure enough, original revenue is R = P * Q = $10 * 100 = $1,000. With the price increase, revenue decreases to R = $12 * 75 = $900.

Plot of quantity demanded against total revenue (orange) and price (blue). When price = $10, demand is elastic, meaning that an increase in price (i.e., moving up the line, to the left) will decrease revenue. Demand is unit elastic at the point where revenue is maximized.

Some applications and implications

If demand is elastic, it stands to reason that you would want to keep increasing the price until that is no longer the case. (Conversely, if demand is inelastic, you’d want to keep decreasing the price until you reach unitary demand.) The obvious question is how one might go about estimating demand elasticity. I think I’ll save the data-driven approach to that topic for another day. Instead, let’s talk about some factors that impact elasticity. First, the availability of substitutes plays a huge role. If there are many similar products that customers can easily switch to, demand will likely be elastic. For example, if the price of a particular brand of toothpaste increases, customers can quickly switch to a competitor, causing a large drop in demand for the original brand. The same logic would also apply to substitute products (e.g., tea and coffee). Conversely, if there aren’t substitutes, demand tends to be inelastic. For all that people complain about gas prices, gas is known to be nearly perfectly inelastic: changes to price have almost no impact on behavior. (Luckily, high competition keeps prices in check.)

On the other hand, if a product is a necessity or if there are no close substitutes, demand is more likely to be inelastic. Think about prescription medications—when prices go up, patients don’t have the luxury of switching to a different, cheaper brand, so they tend to continue buying despite price increases. Luxury items, on the other hand, tend to be more elastic. You’ve probably been spammed by cruise lines or hotels talking about how they’re having sales or reducing prices. They do this because people don’t really need vacations, so they figure that giving the illusion of lower prices will entice people to purchase. The time frame also matters. In the short run, customers might not be able to change their habits, but given more time, they may find alternatives, making demand more elastic in the long run.

Conclusion

If you remember one thing, it should be that elasticity quantifies the trade-off between increased prices and lost sales. If the percentage change in price is greater than the corresponding percentage decrease in quantity, then you should raise prices. Otherwise, you should lower them. Of course, the reality is a little more complicated. There are many factors beyond price that impact quantity demanded. Nonetheless, elasticity is a critical tool for anyone involved in pricing to leverage. Thank you for reading.

1. However, check out this old post from my pre-Substack days.