Question

Prove that `E = A/(A-M)` where A = average revenue, M = marginal revenue, and E = elasticity.

Answer 1

Average Revenue (AR): The revenue per unit sold, which is equal to the price (P) of the good:

`A = AR = (TR)/Q = P`

Marginal Revenue (MR): The additional revenue from selling one more unit:

`M = MR = (d(TR))/(dQ)`

TR = P ⋅ Q

Now, differentiate TR with respect to Q to get MR:

`MR = (d(PQ))/(dQ)`

Using the product rule of differentiation:

`MR = P+Qxx(dP)/(dQ)`

Elasticity is given by:

`E = -(dQ)/(dP)xxP/Q`

Take the reciprocal to get:

=> `(dP)/(dQ) = P/Qxx1/E`

Substitute into MR Expression

`MR = P+Qxx(dP)/(dQ) = P+Qxx(P/Qxx1/E)=P(1+1/E)`

Since A = P and M = MR, we rewrite:

`M=A(1+1/E) => M/A = 1+1/E=>M/A -1 = 1/E => 1/E = (M-A)/A => E= A/(A-I)`

`E = A/(A-M)`