# Suppose a monopolist faces the following demand curve: P = 596-6Q. Marginal cost of production is...

## Question:

Suppose a monopolist faces the following demand curve: P = 596-6Q. Marginal cost of production is constant and equal to $20, and there are no fixed costs.

a) What is the monopolists profit maximizing level of output?

b) What price will the profit maximizing monopolist produce?

c) How much profit will the monopolist make if she maximizes her profit?

d) What would be the value of consumer surplus if the market were perfectly competitive?

e) What is the value of the deadweight loss when the market is a monopoly?

## Profit Maximization:

The main aim of any business is to earn profits. In the production process, any firms make the decisions in such a manner that the combination of price, inputs, and outputs should be such that the firm earns the maximum possible profits. This process is known as profit maximization.

## Answer and Explanation: 1

Consider a monopolist whose demand curve and marginal cost are given. The monopolist faces no fixed cost.

{eq}\begin{align*} P &= 596 - 6Q\\ MC &= 20 \end{align*} {/eq}

a) The profit-maximizing condition of the monopolist is given as marginal revenue (MR) must be equal to the marginal cost (MC). To equate these two, first, find the MR from total revenue (TR).

{eq}\begin{align*} TR &= P \times Q\\ &= 596Q - 6{Q^2}\\ MR &= \frac{{dTR}}{{dQ}}\\ &= 596 - 12Q \end{align*} {/eq}

Now, equate MR and MC to find the profit-maximizing level of output (Q).

{eq}\begin{align*} MR &= MC\\ 596 - 12Q &= 20\\ Q &= 48 \end{align*} {/eq}

Thus, the monopolist will produce 48 units of output (Qm) to maximize profit.

b) Given the profit-maximizing output, the price can be calculated by substituting Q in the demand function.

{eq}\begin{align*} P &= 596 - 6Q\\ &= 596 - 6\left( {48} \right)\\ &= \$ 308 \end{align*} {/eq}

Thus, the monopolist will charge a price (Pm) equal to $308 to maximize his profit.

c) Now, given the profit-maximizing P and Q, the monopolist's profit will be:

{eq}\begin{align*} \pi &= P \times Q - MC \times Q\\ &= 308 \times 48 - 20 \times 48\\ &= \$ 13824 \end{align*} {/eq}

Thus, the monopolist's profit is $13,824.

d) In perfect competition, the optimal Pc and Qc are given by the condition price equal to marginal cost.

{eq}\begin{align*} P &= MC\\ {P_c} &= 20\\ {Q_c} &= \frac{{596 - P}}{6}\\ &= 96 \end{align*} {/eq}

The consumer's maximum willingness to pay (WTP) is the price where the quantity demanded (Q) is zero from the given demand curve.

{eq}\begin{align*} P &= 596 - 6\left( 0 \right)\\ P &= 596\\ WTP &= P\\ WTP &= 596 \end{align*} {/eq}

The consumer surplus (CS) under perfect competition is then given as:

{eq}\begin{align*} CS &= \frac{1}{2} \times {Q_c} \times \left( {WTP - {P_c}} \right)\\ &= \frac{1}{2} \times 96 \times \left( {596 - 20} \right)\\ &= \$ 27648 \end{align*} {/eq}

Thus, the consumer surplus will be $27,648 if the market would have been perfectly competitive.

e) The deadweight loss (DWL) due to monopoly is given as:

{eq}\begin{align*} DWL &= \frac{1}{2} \times \left( {{Q_c} - {Q_m}} \right) \times \left( {{P_m} - {P_c}} \right)\\ &= \frac{1}{2} \times \left( {96 - 48} \right) \times \left( {308 - 20} \right)\\ &= \$ 6912 \end{align*} {/eq}

Thus, the DWL due to monopoly is $6912.

#### Learn more about this topic:

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Chapter 7 / Lesson 2Understand the meaning of a monopoly in economics and what it does. Also, know the characteristics of a monopoly and the different types of monopolies.