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 = (% change in Quantity)/(% change in Price)

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

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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## 2 comments:

shouldn't quantity demanded what price equals 8 be 4000 - 3200 equals 800.

Yes, you are right! the equations should be adjusted to include 800 instead of 400 (but the methods/calculations are all correct except this).

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