Finding the price elasticity of demand, and the cross price elasticity of demand from a demand function is something that most intermediate microeconomics will require you to know. This idea is related to finding the point price elasticity of demand covered in a previous post. For more information on the process you should review that post. This posting is going to go over an example of calculate both the price elasticity of demand and the cross price elasticity of demand for two related goods from the following demand function to demonstrate how the process is done.
First, we need to have a demand function. The following demand function for hot dogs is given as:
Q = 20 - 4P + Ph - Pb
Where Q is quantity.
P is the price of the product.
Ph is the price of hamburgers. We know that hamburgers are a substitute for hot dogs because it is added to the demand function, meaning that higher prices of hamburgers results in a higher quantity demanded for hot dogs.
And Pb is the price for hot dog buns. We know that hot dog buns are complements with hot dogs because as the price for hot dog buns rises, we see that the quantity demanded of hot dogs will decline, meaning they are complements.
It is necessary to know the equation for calculating the point price elasticity of demand to progress further, the point price elasticity of demand equation is:
e = (change in Q/change in P) * (P/Q) or
Point Price Elasticity of Demand = (P/Q)(∆Q/∆P)
We first need to find what ∆Q/∆P is and we can do this by either taking the derivative of our demand function with respect to P, or by changing P by one unit and finding out how much Q changes by. Either way, the result will be -4.
We now need to know what equilibrium P and Q are to solve the rest of the equation. For this example, I will assume that P = 1, Ph=1, and Pb=1. If you need to brush up on solving for equilibrium mathematically you can check this prior post. Plugging in these values, we are left with Q = 16. We can then plug each of these values into the point price elasticity of demand equation to get:
e = -4*(1/16) = -0.25
This shows us that the good has an inelastic point price elasticity of demand measure because it is greater than -1 or the absolute value of the measure is less than 1.
In order to find out what the cross price elasticity of demand is, we need to do this same process but use the price for the related good. Let's begin with the price associated with the substitute good, or hamburgers. The equation for estimating the point cross price elasticity of demand is:
Point Price Elasticity of Demand = (P2/Q1)(∆Q1/∆P2)
Where Q1 represents the quantity of the good in question (hot dogs) and P2 represents the price of the related good (hamburgers).
We can find (∆Q1/∆P2) using the same method above to get 1.
Q will still be equal to 16 (in equilibrium) and P is 1, and this leaves us with:
e = 1*(1/16) = 1/16
So the cross price elasticity of demand between hot dogs and hamburgers is 1/16, and since it is positive we confirm that they are substitutes.
We can do a similar process with hot dog buns, this will give us a ∆Q1/∆P2 of -1 and a resulting elasticity measure of -1/16. Since it is negative, we confirm that they are complements.
Finally, what if the price of hamburgers increases to 2? The new equilibrium Q will be 15, and calculating the new elasticity will give us:
e = 2*(2/15) = 4/15 = 0.26666
We can see that the number is still positive, but has risen slightly meaning that quantity demanded has become more elastic or sensitive to price changes in the related good.