# Maths with Price Elasticity

Below is an example of how to use the formulae for PED and PES to work out the supply and demand functions.

Firstly we have to be able to re-arrange PED (the same applies for the respective formula for PES):

PED = ΔQD / ΔP (where Δ is change, QD is quantity demanded, and P is price)

We know that the formula for percentage change is:

((Change/Original)*100)

Therefore the formula for nominal change (a change in numbers, not percentages) is (Change / Original)

Hence, using this, we can change the PED formula to:

PED = ΔQD/QD ÷ ΔP/P

We can change the division symbol to a multiplication symbol by flipping the term on the right hand side:

PED = ΔQD/QD x P/ΔP

PED = ΔQD/ΔP x P/QD

A change in QD divided by a change in P can be written as the differential of QD in terms of P:

PED = dQD/dP x P/Q

We can then work out dQD/dP by differentiating the demand function and substituting this value in:

QD = a – bP

dQD/dP = -b (differentiate a – bP)

Substitute:

PED = -b(P/Q)

We can now use this formula to find out demand functions (and if you use the same process with PES, supply functions).

Here is an example:

The price of a tonne of wheat last year was £20. At this price 200,000 tonnes were sold. An economist firm calculated that the PED of wheat was -0.2 and that the PES was 0.6. Calculate the demand and supply functions based on this information.

First we substitute the information into PED = -b(P/Q):

-0.2 = -b(20/200,000)

-0.2 = -b(0.0001)

b = 2000

This can then be substituted into the demand function (QD = a – bP):

200,000 = a – 2000(20)

200,000 = a – 40,000

a = 240,000

Therefore QD = 240,000 – 2000P which can be simplified to:

QD = 240 – 2P

We now apply the same process to find out the supply function

PES = d(P/Q)

0.6 = d(20/200,000)

d = 6000

QS = c + dP

200,000 = c + 6000(20)

200,000 = c + 12,000

188000 = c

We can then substitute and simply to get QS = 188 + 12P