Question

derive the single price monopoly equilibrium using either geometry or first order conditional a single price monopolist maximizes the producers surplus yet why does society stull experiences a deadweight loss at the equilibrium price quantity vector

Answer 1

Derivation of single-price monopoly equilibrium:

A firm is a monopoly if it has no close competitors and hence can ignore the potential reactions of other firms when choosing its output and price.

A monopolist chooses its output to maximize its profit, given the relationship between output and price as embodied in the aggregate demand function for the good it sells.

Denote by TC the monopolist's total cost function and by TR its total revenue function (that is, TR is the product of the firm's output and the price that output fetches, given the demand function). Then the monopolist's profit is

(y) = TR(y) TC(y).

An output y* that maximizes this profit is such that the first derivative of is zero, or

TR'(y*) = TC'(y*)

or, denoting the derivative TR' of total revenue by MR (for marginal revenue),

MR(y*) = MC(y*).

At a maximum, rather than a minimum, the second derivative of profit is nonpositive, or

TR''(y*) TC''(y*) 0

or

MR'(y*) MC'(y*):

the slope of MR is at most the slope of MC at the optimal output.

Finally, the monopolist's profit at y* must be nonnegative (in the long run), otherwise, it would not stay in business.

In summary:

The profit-maximizing output y* of a monopolist is either 0 or is positive and satisfies the following conditions:

• MR(y*) = MC(y*)
• MR'(y*) MC'(y*)
• (y*) 0.

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It shall further be noted that a single price monopolist maximizes the producers surplus yet the society still experiences a deadweight loss at the equilibrium price quantity vector.

In pure competition, an economic surplus which is consumer plus producer surplus is maximized. The industry is allocatively efficient producing where the price is equal to the marginal cost. By restricting output and raising the price, the single price monopolist captures a portion of the consumer surplus. Since the output is restricted, a portion of both the consumer and producer surplus is lost, even when a single price monopolist maximizes the producer surplus.

This loss of economic surplus is known as deadweight loss, that neither the consumer nor the producer enjoys.