Question

4. Consider the demand function D(p;m) = me−p, where m > 0 is consumers’ average income. The supply consists of a monopoly, whose revenue from sales is given by R(p; m) = pD(p; m).

(a) Compute the elasticity function, E(p; m) = àD0(p; m) p àD(p;m)

(b) Find the value of p such that E(p; m) = 1.

(c) Compute the marginal revenue function, MR(p; m) = R0(p; m).

(d) What is the solution to the equation MR(p;m) = 0? Compare your answer to your answer from part (a).

Answer 1

Solution:

Demand function: D(p, m) = me - p, m > 0

R(p, m) = p*D(p, m)

a) Price elasticity, e =

= -1

aD(p, m)

e = -1*p/(me - p) = -p/(me - p)

b) To find value of p for which |e| = 1

p/(me - p) = 1

p = me - p

p + p = me

p = me/2

So, for p = me/2, price elasticity is 1 (in absolute terms)

c) q = D(p, m) = me - p

So, p = me - q

R(p, m) = p*D(p, m)

So, R(q, m) = (me - q)q = meq - q²

MR(q, m) = = me - 2q

So, MR(p, m) = me - 2(me - p) = - me + 2p

d) Solution to the equation: MR(p, m) = 0

- me + 2p = 0

2p = me

p = me/2

On comparison with part (a), clearly, this is the value of p for which price elasticity of given demand function is 1. So, we can conclude that since here solution to MR(p, m) = 0, gives p = me/2, it means here elasticity is 1.