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A monopolist faces a demand curve: P = 100 - Q for its product. The monopolist has fixed costs of...

Question:

A monopolist faces a demand curve: {eq}P = 100 - Q {/eq} for its product. The monopolist has fixed costs of 1000 and a constant marginal cost of 4 on all units.

Find the profit-maximizing price, quantity, and profits if this monopolist charges a single price for all units.

Monopolist Profit Maximization:

A monopolist is the sole producer in a market and has the market power to set prices. Facing a downward sloping demand curve, the monopolist must set a lower price in order to sell more units. The equilibrium price a monopolist sets is higher than the price in a competitive market.

Answer and Explanation: 1

To maximize profit, the monopolist will produce until the marginal revenue is equal to the marginal cost. We know that the marginal cost is 4, which is constant. To determine the profit-maximizing price, we need to first compute the marginal revenue.

Given the inverse demand function, the total revenue function is given by:

  • Total revenue = {eq}PQ = (100 - Q)Q{/eq}

Differentiating the total revenue function with respect to quantity yields the following marginal revenue function:

  • Marginal revenue = {eq}100 - 2Q{/eq}

To maximize profit, the firm produces until marginal cost is equal to marginal revenue, i.e.,

  • {eq}100 - 2Q = 4{/eq}

which yields

  • {eq}Q = 48{/eq}

The corresponding price = 100 - 48 = 52, maximized profit = 52 * 48 - (1000 + 4*48) = 1304.


Learn more about this topic:

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Monopolistic Competition: Definition, Theory, Characteristics & Examples

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Chapter 3 / Lesson 56
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Learn the monopolistic competition definition with examples. Study monopolistic competition vs. perfect competition and other market types to learn the differences.


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