Consider a demand function P = 100=2x The total cost function C = x and MC = 1 S

Question

Consider a demand function P = 100=2x The total cost function C = x and MC = 1 Show

(a) the provite maximizing quantity x can never exceed 25 (solve for x)

(B) The revenue maximizing quantity x must be greater than x*

(c) Show that price elasticity of demand at the profit maximizing quantity x* is greater than 1 and at the revenue maximizing quantity is equal exactly to 1

(d) for the above demand function show that the marginal revenue is negative for all x > 25

(e) show that the profit maximizing price at x* must be higher than the marginal cost

Explanation / Answer

P = 100 - 2x [2x = 100 - P, or x = 50 - 0.5P]

Total revenue, TR = P.x = 100x - 2x2

Marginal revenue, MR = dTR / dx = 100 - 4x

(a) Profit is maximized when MR = MC

100 - 4x = 1

4x = 99

x = 24.75

So x is lower than 25.

(b) Revenue is maximized when dTR / dx = 0, i.e. MR = 0

100 - 4x = 0

4x = 100

x = 25 > 24.75

(c)

When x = 24.75, P = 100 - (2 x 24.75) = 100 - 49.5 = 51.5

Price elasticity of demand = (dx / dP). (P / x) = - 0.5 x (51.5 / 24.75) = - 1.04

So, (absolute value of) elasticity is higher than 1.

When x = 25, P = 100 - (2 x 25) = 100 - 50 = 50

Price elasticity of demand = (dx / dP). (P / x) = - 0.5 x (50 / 25) = - 1

So, (absolute value of) elasticity is 1.

(d)

MR = 100 - 4x

When MR < 0, 100 - 4x < 0

100 < 4x

25 < x, Or x > 25

(e) When profit is maximized, P = 51.5 > 1 (MC)

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