Reading: Elasticity
We know that demand functions are decreasing, so when the price increases, the quantity demanded goes down. But what about revenue = price × quantity? Will revenue go down because the demand dropped so much? Or will revenue increase because demand didn’t drop very much? Elasticity of demand is a measure of how demand reacts to price changes. It’s normalized—that means the particular prices and quantities don’t matter, so we can compare onions and cars. The formula for elasticity of demand involves a derivative, which is why we’re discussing it here.Elasticity of Demand
Given a demand function that gives q in terms of p, The elasticity of demand is [latex] E = |\frac{p}{q} \times \frac{dq}{dp}| [/latex] (Note that since demand is a decreasing function of p, the derivative is negative. That’s why we have the absolute values—so E will always be positive.)- If E < 1, we say demand is inelastic. In this case, raising prices increases revenue.
- If E > 1, we say demand is elastic. In this case, raising prices decreases revenue.
- If E = 1, we say demand is unitary. E = 1 at critical points of the revenue function.
Example
A company sells q ribbon winders per year at $p per ribbon winder. The demand function for ribbon winders is given by: [latex] [/latex] Find the elasticity of demand when the price is $70 apiece. Will an increase in price lead to an increase in revenue?Solution
First, we need to solve the demand equation so it gives q in terms of p, so that we can find [latex] [/latex]: [latex-display] [/latex-display] We need to find q when p = 70: q = 11500. We also need [latex] [/latex] Now compute [latex] [/latex]. E < 1, so demand is inelastic. Increasing the price would lead to an increase in revenue; it seems that the company should increase its price. The demand for products that people have to buy, such as onions, tends to be inelastic. Even if the price goes up, people still have to buy about the same amount of onions, and revenue will not go down. The demand for products that people can do without, or put off buying, such as cars, tends to be elastic. If the price goes up, people will just not buy cars right now, and revenue will drop.Licenses & Attributions
CC licensed content, Shared previously
- Business Calculus. Provided by: Washington State Colleges Authored by: Dale Hoffman and Shana Calaway. Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit. License: CC BY: Attribution.
