Q.Prove that marginal revenue of a firm equals zero if the value of price elasticity of demand is unity.
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Start your 14-day free trial to unlock the full solution →Using the formula MR = P(1 − 1/e), setting e = 1 gives MR = P(1−1) = 0 — marginal revenue is exactly zero at the point of unit elasticity, which is also where total revenue is at its maximum.
Derivation: Let the demand function be Q = f(P), so Total Revenue, TR = P × Q.
Differentiating TR with respect to Q:
MR = d(TR)/dQ = P + Q × (dP/dQ)
Factoring out P:
MR = P [ 1 + (Q/P)(dP/dQ) ]
Price elasticity of demand is conventionally defined (taking the absolute/positive value, since price and quantity demanded move in opposite directions) as:
e = − (P/Q)(dQ/dP), which rearranges to give (Q/P)(dP/dQ) = −(1/e)
Substituting this into the MR expression:
MR = P [ 1 − 1/e ]
Proof for e = 1: Substitute e = 1 into MR = P(1 − 1/e):
MR = P (1 − 1/1) = P (1 − 1) = P × 0 = 0
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