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 it. 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.

Example 1:
Here is an example demand curve: Q = 15,000 - 50P

Given this demand curve we have to figure out what the point price elasticity of demand is at 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 goes down by 50 each time price 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)

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

And these results make sense, first, because they are negative (downward sloping demand) and second, because the higher level results in a relatively more 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 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

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

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