How to calculate point price elasticity of demand with examples

Point elasticity is the price elasticity of demand at a specific point on the demand curve instead of over a range of the demand curve. It uses the same formula as the general price elasticity of demand measure, but we can take information from the demand equation to solve for the “change in” values instead of actually calculating a change given two points.  Here is the process to find the point elasticity of demand formula:

Point Price Elasticity of Demand = (% change in Quantity)/(% change in Price)

Point Price Elasticity of Demand = (∆Q/Q)/(∆P/P)
Point Price Elasticity of Demand = (P/Q)(∆Q/∆P)

Where (∆Q/∆P) is the derivative of the demand function with respect to P.  You don’t really need to take the derivative of the demand function, just find the coefficient (the number) next to Price (P) in the demand function and that will give you the value for ∆Q/∆P because it is showing you how much Q is going to change given a 1 unit change in P. Finding the point elasticity solution is best demonstrated with examples...


Example 1:

To find the point price elasticity of demand we begin with an example demand curve:
Q = 15,000 - 50P

Imagine that given this demand curve we are asked to figure out what the point price elasticity of demand is at two different prices, P = 100 and P = 10.

First we need to obtain the derivative of the demand function when it's expressed with Q as a function of P.  Since quantity (Q) goes down by 50 each time price (P) goes up by 1,

This gives us (∆Q/∆P)= -50

Next we need to find the quantity demanded at each associated price and pair it together with the price: (100; 10,000), (10; 14,500)

Then we plug those values into our point elasticity of demand formula to obtain the following:

e = -50(100/10,000) = -.5
e = -50(10/14,500) = -.034

And these results make sense, first, because they are negative (which demonstrates a downward sloping demand relationship) and second, because the higher level results in a relatively more elastic price elasticity of demand measure.

Example 2:

How to find the point price elasticity of demand with the following demand function:

Q = 4,000 – 400P

We know that ∆Q/∆P in this problem is -400, and we need to find the point price elasticity of demand at a price of 10 and at a price of 8.

At a price of ten, we demand 0 of the good, so the measure is undefined.  At a price of 8 we will demand 400 of the good, so the associated measure is:

e = -400(8/400) = -8

Example 3:

What about a demand function of:

Q = 8,800 – 1,000P

Here our ∆Q/∆P will be -1,000 and we will need to find the associated measure at prices of 0, 2, 4, and 6.

This means we will end up with:

e = -1,000(0/8,800) = 0

e = -1,000(2/6,800) = -0.294

e = -1,000(4/4,800) = -0.8333

e = -1,000(6/2,800) = -2.14

Sometimes you may be required to solve for quantity or price and are given a point price elasticity of demand measure. In this case you need to backwards solve by rearranging the point price elasticity of demand formula to get the quantity or price you need for the problem.

Summary:

The trick to solving point price elasticity of demand problems is to find the coefficient on the price (P) and then to plug the corresponding price and quantity values in to the point price elasticity of demand formula. After that you can simplify using algebra.