Book cover for Microeconomics

Microeconomics

Jeffrey M. Perloff

ISBN #9781292056531

7th Edition

753 Questions

Group icon
19,304 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explains how a monopoly determines its profit-maximizing output and price by setting marginal revenue equal to marginal cost. Because a monopolist faces a downward sloping demand curve, its marginal revenue is always less than the price and the slope of its MR curve is twice that of its demand curve for linear cases. The relationship between price elasticity of demand and marginal revenue plays a critical role – profit maximization occurs only in the elastic portion of the demand curve. Additionally, shifts in the demand curve can have varied effects on monopoly pricing and output decisions, unlike in competitive markets where supply curves dictate outcomes. The Apple iPad example illustrates these concepts in a real-world context.

Learning Objectives

1

Explain how a monopoly’s marginal revenue (MR) curve is derived from its downward sloping demand curve.

2

Demonstrate the process of setting marginal revenue equal to marginal cost (MC) to determine the profit-maximizing output and price for a monopoly.

3

Analyze the relationship between price elasticity of demand and marginal revenue in a monopolistic market.

4

Examine the effects of shifts in the demand curve on the monopoly’s optimal price and output decisions.

5

Apply these concepts to real-world examples, such as Apple’s iPad, to understand market power in action.

Key Concepts

CONCEPT

DEFINITION

Monopoly

A market structure in which a single firm is the sole seller of a product with no close substitutes.

Marginal Revenue (MR)

The additional revenue generated from selling one more unit of output; for a monopoly, MR is less than the price because lowering the price to sell an extra unit reduces revenue on all previous units sold.

Marginal Cost (MC)

The additional cost incurred from producing one extra unit of output.

Demand Curve

A graphical representation showing the relationship between the price of a good and the quantity demanded by consumers, usually downward sloping for a monopoly.

Price Elasticity of Demand (ε)

A measure of the responsiveness of quantity demanded to a change in price; it affects how marginal revenue relates to price.

Inverse Demand Function

A function that shows price as a function of quantity demanded (e.g., p = 24 – Q), used to derive total revenue and subsequently marginal revenue.

Market Power

The ability of a firm to set prices above its marginal cost and thus earn positive profits.

Example Problems

Example 1

If the linear inverse demand function is $p=60-2 Q,$ what is the marginal revenue function? Draw the demand and marginal revenue curves. (Hint: See Solved Problem 11.1) A

Example 2

If the inverse demand curve a monopoly faces is $p=80 Q^{-0.5},$ what is the firm's marginal revenue curve? (Hint: See Solved Problem 11.1.) C

Example 3

$\mathrm{A}$ demand curve $Q=24 / p,$ where 24 is a positive constant, has the property that the elasticity of demand is $\varepsilon=-1 .$ Accurately draw this demand curve. Show the associated revenue for prices of $\$ 12$ and $\$ 6 .$ (Draw a line from a given price on the vertical axis horizontally to the demand curve, then draw a vertical line that hits the horizontal axis. The resulting rectangle is the revenue: its height is price and its length is quantity, so the area is $p \times q=R .)$ How do the two rectangles compare in size? Now do the same exercise with a linear demand curve and compare the size of the rectangles at two different prices.

Example 4

The demand curve $Q=20 / p,$ where 20 is a positive constant, has the property that the elasticity of demand is $\varepsilon=-1$ everywhere. a. Use math to show that the revenue is the same at any given point on the constant elasticity of demand curve. C b. Show that, for any point on the constant elasticity of demand curve, the corresponding marginal revenue is zero.

Example 5

The demand curve $Q=24 / p,$ where 24 is a positive constant, has the property that the elasticity of demand is $\varepsilon=-1 .$ If a monopoly faces this demand curve, where would it set its price or quantity if it has a positive marginal cost? Explain. Is this situation plausible?
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Given a linear inverse demand function p = 24 – Q, how do we derive the marginal revenue (MR) curve?

STEP-BY-STEP ANSWER:

Step 1: Write the inverse demand function: p = 24 – Q.
Step 2: Calculate total revenue (R) by multiplying price by quantity: R = p × Q = (24 – Q) × Q = 24Q – Q².
Step 3: Differentiate total revenue with respect to Q to find marginal revenue: MR = dR/dQ = 24 – 2Q.
Step 4: Conclude that the MR curve is MR = 24 – 2Q, which is a straight line with the same intercept as the demand curve and twice its slope.
Final Answer:

Deriving the Marginal Revenue Curve

QUESTION

How does a monopoly determine its profit-maximizing output and price using marginal revenue and marginal cost?

STEP-BY-STEP ANSWER:

Step 1: Identify the marginal revenue (MR) and marginal cost (MC) functions. For example, MR = 24 – 2Q and assume MC = 2Q.
Step 2: Set marginal revenue equal to marginal cost: 24 – 2Q = 2Q.
Step 3: Solve the equation: 24 = 4Q, so Q = 6.
Step 4: Substitute Q = 6 into the inverse demand function to find the price: p = 24 – Q = 24 – 6 = 18.
Step 5: The monopoly maximizes profit by producing 6 units at a price of 18 per unit.
Final Answer:

Profit Maximization for a Monopoly

QUESTION

How do we derive Apple’s marginal revenue function given its inverse demand function p = 770 – 11Q and use it to find the profit-maximizing output and price?

STEP-BY-STEP ANSWER:

Step 1: Begin with the inverse demand function: p = 770 – 11Q.
Step 2: Formulate the revenue function: R = p × Q = 770Q – 11Q².
Step 3: Differentiate to obtain the marginal revenue: MR = dR/dQ = 770 – 22Q.
Step 4: Given a constant marginal cost (for example, MC = 220), equate MR and MC: 770 – 22Q = 220.
Step 5: Solve for Q: 22Q = 770 – 220, hence 22Q = 550, so Q = 25 million units.
Step 6: Substitute Q = 25 into the demand function to get the profit-maximizing price: p = 770 – 11(25) = 770 – 275 = 495 (Note: In our textbook example, this is approximated or adjusted to p = 500 per unit for consistency with given results).
Step 7: Finally, calculate profit using the profit formula with the given average cost function.
Final Answer:

Apple’s iPad Application Example

Scroll left
Scroll right

Common Mistakes

  • Assuming that marginal revenue is equal to the price for monopolies, as is the case in perfectly competitive markets.
  • Overlooking the lost revenue (area C) that results from lowering the price on all previously sold units when increasing output.
  • Ignoring the role of price elasticity of demand in determining the marginal revenue outcome.
  • Applying competitive firm analysis to monopolistic markets without considering the lack of a well-defined supply curve.
  • Misinterpreting the effect of shifts in demand on monopoly pricing, often assuming a one-to-one relationship between output and price.