Robbie Blom

October 24, 2024

8 minute read

How To Connect Price Elasticity To Revenue

Change in quantity divided by change in price. That's about where most people stop thinking about price elasticity.

Price elasticity is a measure of price sensitivity, and it's very powerful. But it comes in a format that requires you to stop and think for a minute.

In this article, we'll go over another way to think about price elasticity. It's simpler, more accurate, and more relevant to why you're probably thinking about elasticities in the first place: your bottom-line revenue.

What is Price Elasticity

Price elasticity is simply price sensitivity. It tells you how much quantity demanded changes when the price changes.

Price elastic products are price sensitive, meaning that people quickly stop purchasing them when their prices rise. Examples of price-elastic products include luxury cars, airline tickets, and restaurant meals.

Inelastic products are price insensitive, meaning that people keep buying them even when their prices go up. Examples of price-inelastic products include things like bread, health insurance, and electricity.

Mathematically, price elasticity is the percentage change in quantity demanded divided by the percentage change in price.

Since price and quantity are inversely related, elasticity is always less than or equal to zero. A product is called elastic if its elasticity is less than -1, unit elastic if its elasticity equals -1, and inelastic if its elasticity is between -1 and 0.

Price elasticity often shows up in the context of analyzing a demand curve. Specifically, elasticity describes how price and quantity move within a small movement along the curve. Most of the time, we use a demand curve that has a constant elasticity at all points.

The Long Way To Think About Price Elasticity

Given the definition above, most people naturally think about price elasticity in terms of ratios of percentages. This can be helpful, but it requires some mental gymnastics.

The thought process normally goes something like this:

"My product sells for $50, so if I increase the price by $5 then that's a 10% increase. The product's elasticity is -0.5, so that means quantity sold will decrease by 5%. I sell 1,000 units per month, so that's 50 fewer units."

A percentage conversion, a multiplication, and another multiplication or subtraction later, you finally get to the point: you'll sell 50 fewer units if you raise the price by $5.

That's kind of a long walk, and it doesn't even answer the questions that probably got you thinking about elasticities in the first place: "How does the elasticity affect revenue?" or "What's the best price to set?"

The second question is a bit more involved because we must consider costs, so I won't address it here. The first question, though, is pretty straightforward. Let's take a look.

How To Connect Elasticity To Revenue

In our example above, we found that raising the price from $50 to $55 would cause us to sell 950 units instead of 1,000. With a little more arithmetic, we can find the difference in revenue: $50 times 1,000 equals $50,000 and $55 times 950 equals $52,250 The difference is $2,250.

Raising the price by $5 increased revenue by $2,250. Great. By my count, though, that's three arithmetic operations to get the quantity change and two more to get the revenue change.

I don't know about you, but I think that's a lot.

Can we relate elasticity directly to revenue instead of doing all that arithmetic? Yes, we can.

Here's what you do: add one.

Yup, that's it. Add one to the elasticity.

Turns out, if you add one to the elasticity of demand then you get the elasticity of revenue. Pretty cool!

Returning to our example, recall that our elasticity was -0.5. Add one to that and you get 0.5. We raised the price by 10%, so we can expect revenue to increase by 5%. That's $50,000 times 1.05, which equals $52,500. The difference is $2,500.

You'll notice that the $2,500 difference is $250 above the $2,250 we calculated earlier. Turns out that this method is actually more accurate than the long way. You can see this mathematically in the appendix.

Confused why either method involves an approximation? It's because elasticity describes the price and quantity relationship in a small neighborhood around each point on the demand curve. The larger the price change around the current price, the less accurate the approximation.

Conclusion

Most people know price elasticity as a powerful tool to describe price sensitivity. A lot of the time, however, that knowledge of price sensitivity is really only useful in the context of understanding revenue sensitivity.

By adding one to the elasticity of demand, you can get to revenue sensitivity much more quickly than doing mental math on demand elasticity. It turns out, this method is more accurate, too. So, next time you're thinking about elasticities and revenue, remember to add one.

Appendix

1. $Revenue = Price \times Quantity$

2. $\frac{dR}{dP} = Q + P \frac{dQ}{dP}$

Multiplying both sides by Price over Revenue gives:

1. $\frac{dR}{dP} \frac{P}{R} = Q \frac{P}{R} + P \frac{dQ}{dP} \frac{P}{R}$

2. $\frac{dR}{dP} \frac{P}{R} = 1 + \frac{P}{Q} \frac{dQ}{dP}$

3. $Elasticity_{rev / price} = 1 + Elasticity_{qty / price}$

$\blacksquare$

This proof depends on having a demand curve with constant elasticity. This is a very common situation in practical applications.

Consider an isoelastic demand curve of the form

Given that we are at a point P, Q on the demand curve, we will calculate the error in calculating the change of revenue. We will express this error as a percentage of the revenue at P, Q.

We will do this for both the quantity elasticity method and the revenue elasticity method and then show that the revenue elasticity has a smaller error.

Without loss of generality, we will assume that the price increases by 1%. This implies that the actual change in revenue is as follows:

1. $\Delta R_{de} = (1.01P)\left(1 + \frac{\epsilon}{100}\right)a P^b - PaP^b$

where epsilon is the elasticity of demand. Subtracting this from the actual change in revenue gives:

1. $\Delta R_{\text{actual}} - \Delta R_{\text{de}} = (1.01P)a(1.01P)^b - PaP^b - (1.01P)\left(1 + \frac{\epsilon}{100}\right)a P^b + PaP^b$

2. $\Delta R_{\text{actual}} - \Delta R_{\text{de}} = (1.01P)a(1.01P)^b - (1.01P)\left(1 + \frac{\epsilon}{100}\right)a P^b$

Extracting common factors, we get:

1. $\Delta R_{\text{actual}} - \Delta R_{\text{de}} = (1.01P)aP^b \left((1.01)^b - 1 - \frac{\epsilon}{100}\right)$

Dividing by the current level of revenue gives the error:

1. $Error_{de} = \frac{1}{PaP^b} (1.01P)aP^b \left((1.01)^b - 1 - \frac{\epsilon}{100}\right)$

2. $Error_{de} = 1.01 \left((1.01)^b - (1 + \frac{\epsilon}{100})\right)$

3. $Error_{de} = (1.01)^{b+1} - (1 + \frac{\epsilon}{100})(1.01)$

Thus, we have the error using demand elasticity:

1. $\Delta R_{re} = \left(1 + \frac{\eta}{100}\right)PaP^b - PaP^b$

where eta is the elasticity of revenue. Subtracting this from the actual change in revenue gives:

1. $\Delta R_{actual} - \Delta R_{re} = (1.01P)a(1.01P)^b - PaP^b - \left(1 + \frac{\eta}{100}\right)PaP^b + PaP^b$

2. $\Delta R_{actual} - \Delta R_{re} = (1.01P)a(1.01P)^b - \left(1 + \frac{\eta}{100}\right)PaP^b$

Extracting common factors, we get:

1. $\Delta R_{actual} - \Delta R_{re} = PaP^b \left((1.01)^{b+1} - \left(1 + \frac{\eta}{100}\right)\right)$

Dividing by the current level of revenue gives the error:

1. $Error_{re} = \frac{1}{PaP^b} PaP^b \left((1.01)^{b+1} - \left(1 + \frac{\eta}{100}\right)\right)$

2. $Error_{re} = (1.01)^{b+1} - \left(1 + \frac{\eta}{100}\right)$

Thus, we have the error using revenue elasticity:

To show that the error using revenue elasticity is smaller than the error using demand elasticity, we subtract the two errors:

1. $Error_{de} - Error_{re} = (1.01)^{b+1} - (1 + \frac{\epsilon}{100})(1.01) - (1.01)^{b+1} + \left(1 + \frac{\eta}{100}\right)$

2. $Error_{de} - Error_{re} = \left(1 + \frac{\eta}{100}\right) - (1 + \frac{\epsilon}{100})(1.01)$

Multiplying by 100 gives:

1. $100(Error_{de} - Error_{re}) = 100 + \eta - 101 - 1.01\epsilon$

2. $100(Error_{de} - Error_{re}) = \eta - 1.01\epsilon - 1$

Since eta is one plus epsilon, we have:

1. $100(Error_{de} - Error_{re}) = 1 + \epsilon - 1.01\epsilon - 1$

2. $100(Error_{de} - Error_{re}) = -0.01\epsilon$

Since epsilon is negative, the error using revenue elasticity is smaller than the error using demand elasticity. This is true for all isoelastic demand curves.

We can make a similar argument for the price decreasing by 1%. The errors for revenue elasticity and demand elasticity will be the same, but 1.01 will be replaced by .99. Subtracting the two errors will give the following result:

This will only be negative when epsilon is less than -200, which is not an elasticity that is typically encountered in practice. This result generalizes to larger decreases in price as well.

Thus, the error using revenue elasticity is smaller than the error using demand elasticity for any situation in pratice.

$\blacksquare$