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Consider a monopoly with a demand function �(�) and the cost function �(�), � ′...

May 17, 2024

Consider a monopoly with a demand function �(�) and the cost function �(�), � ′ (�) > ≥ 0 where � denotes the monopoly output. Deriving the monopoly pricing formula, show that a profit-maximising monopoly increases its price as demand gets inelastic. How does this result compare with that under a perfectly competitive scenario?

Solution

Demand Function: The demand function is denoted as

D(Q)

, where

Q

represents the quantity demanded

Cost Function: The cost function is denoted as

C(Q)

, with

C'(Q) \geq 0

indicating that the marginal cost is non-negative

Revenue Function: The total revenue function is

TR = P(Q) \cdot Q

, where

P(Q)

is the price

Marginal Revenue: The marginal revenue (MR) is derived from the total revenue function,

MR = \frac{d(TR)}{dQ} = P(Q) + Q \cdot \frac{dP(Q)}{dQ}

Profit Maximization: A monopoly maximizes profit where marginal revenue equals marginal cost,

MR = MC

Elasticity of Demand: The price elasticity of demand is

\epsilon = \frac{dQ}{dP} \cdot \frac{P}{Q}

Monopoly Pricing Formula: Using the elasticity of demand, the monopoly pricing formula is

P = \frac{MC}{1 + \frac{1}{\epsilon}}

Inelastic Demand: When demand is inelastic (|\epsilon| < 1), the term

\frac{1}{\epsilon}

is large, leading to a higher price

P

Perfect Competition: In a perfectly competitive market, firms are price takers, and price equals marginal cost,

P = MC

Answer

A profit-maximizing monopoly increases its price as demand becomes more inelastic, whereas in a perfectly competitive market, the price equals the marginal cost.

Key Concept

Monopoly Pricing Formula

Explanation

The monopoly pricing formula

P = \frac{MC}{1 + \frac{1}{\epsilon}}

shows that as demand becomes more inelastic (|\epsilon| < 1), the price

P

increases. In contrast, in a perfectly competitive market, the price is always equal to the marginal cost (

P = MC