Optimal Output and Price for a Monopolist: A Professor's Guide

The demand curve for a monopolist is often a fascinating area of study in microeconomics. This article delves into the specifics of how to determine the optimal level of output q and price p for a monopolist given the demand and cost functions. We will explore the process step-by-step using both calculus and an intuitive approach to find the profit-maximizing point for the monopolist.

Setting the Stage: The Demand and Cost Functions

The demand curve for the monopolist is given by Dq 100 - 2p. Its cost is cy 2y. We need to determine the optimal level of output and price for the monopolist to maximize profit. Let's explore this step-by-step using both traditional and alternative methods.

The Traditional Method: Derivatives and Critical Points

1. Find the inverse demand curve: The demand curve is given by Dq 100 - 2p. Rearrange this to express p in terms of q:
2p 100 - Dq implies p 50 - frac{Dq}{2}
Since Dq q at the monopolist's output level, we can write:
p 50 - frac{q}{2}

2. Total revenue (TR): Total revenue is given by:
TR p cdot q left(50 - frac{q}{2}right) q 50q - frac{q^2}{2}

3. Marginal revenue (MR): Marginal revenue is the derivative of total revenue with respect to q:
MR frac{dTR}{dq} 50 - q

4. Marginal cost (MC): The cost function is given by cy 2y. Therefore, the marginal cost is:
MC frac{dc}{dy} 2

5. Set MR MC: To maximize profit, set marginal revenue equal to marginal cost:
50 - q 2

Solving for q:
q 48

6. Find the optimal price p: Substitute q 48 back into the inverse demand equation to find the price:
p 50 - frac{48}{2} 50 - 24 26

So, the optimal level of output and price for the monopolist are:

Optimal output q 48 Optimal price p 26

An Alternative Approach Using the Lerner Index

In economics, the Lerner index is a measure that indicates the degree to which a firm is price-discriminating. The Lerner index is defined as:
L frac{p - Cy}{p}

Consider a firm that faces a demand curve py and a cost curve Cy. The firm's problem is to maximize profit:
max_{y} pi ypy - Cy

The first-order condition is:
pi_y p_y y_p - Cy 0)

Which simplifies to:
p - Cy -ypy

Divide both sides of the equation by py:
frac{p - Cy}{py} -frac{ypy}{py} -frac{Dp}{p} frac{dp}{dDp}

Notice that the right-hand side is the price elasticity of demand #948; and the left-hand side is the Lerner index. Thus:
L -frac{1}{#948;}

For the given demand curve py 100 - 2p, we can compute the right-hand side as -frac{50 - p}{p}. Therefore:
frac{p - 2}{p} -frac{50 - p}{p}

Multiplying both sides by p, transposing all p terms to the left and all constants to the right, dividing both sides by 2, and solving for p, we get:
p 26

Substitute p 26 into the demand curve to get y 48.

The profit-maximizing price is 26 units, and the profit-maximizing output is 48 units. This alternative approach using the Lerner index is not only instructive but also provides an intuitive feel for the problem.

Conclusion

The optimal level of output and price for the monopolist can be found using both the traditional method of derivatives and the Lerner index method. By using calculus, we can determine the profit-maximizing output and price for the monopolist. Intuitively, profit is maximized when marginal revenue equals marginal cost, which, given the linear nature of these functions, is straightforward to determine.

Key Takeaways

The optimal output for the monopolist is 48 units. The optimal price for the monopolist is 26 units. The Lerner index provides an alternative method to find the optimal price and output. The traditional method using derivatives also confirms these results.

Understanding these concepts is essential for any student or professional studying microeconomics, particularly within the context of monopolistic markets.