Q.Show that average revenue of any firm is always equal to the market price of the good produced by it. Or Prove that marginal revenue of a firm equals zero if the magnitude of price elasticity of demand is unity.
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Start your 14-day free trial to unlock the full solution →AR equals price by simple definition of total revenue; MR becomes zero exactly where price elasticity of demand is unity, by the MR-AR-elasticity identity.
AR always equals price
Total revenue (TR) earned by a firm is the price per unit (P) multiplied by the quantity sold (Q): TR = P x Q. Average revenue (AR) is defined as total revenue per unit of output sold:
AR = TR/Q = (P x Q)/Q = P
This is true for ANY firm in ANY market structure - perfectly competitive, monopoly, or otherwise - because it follows directly from the definitions of TR and AR, as long as every unit is actually sold at price P. This is why a firm's AR curve and its demand curve are one and the same curve.
Or - MR = 0 when |e| = 1
There is a well-known identity linking marginal revenue, price and the price elasticity of demand (e, taken in absolute value):
MR = P (1 - 1/e)
Derivation: TR = P.Q, so MR = d(TR)/dQ = P + Q(dP/dQ). Since e = -(dQ/dP)(P/Q), we get dP/dQ = -P/(e.Q). Substituting:
MR = P + Q x [-P/(e.Q)] = P - P/e = P(1 - 1/e) …
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