Price Caps and Monopoly Welfare

Introduction

In our last lesson, we explored moral hazard, a market failure arising from hidden actions after a contract is signed. We solved for the optimal incentive contract a principal could offer an agent. Today, we turn our attention back to a market structure we've previously analyzed—monopoly—and investigate a direct form of government intervention designed to mitigate its negative effects.

This lesson focuses on calculating the welfare effects of imposing a price cap on a monopolist. You'll recall from Module 5 that a profit-maximizing monopolist restricts output and charges a price higher than marginal cost, leading to a deadweight loss. A price cap, or price ceiling, is a regulation that sets a maximum price the monopolist can charge. Counterintuitively, and in stark contrast to its effect in a competitive market, a well-designed price cap can compel a monopolist to increase output, lower its price, and reduce or even eliminate the deadweight loss.

Building on your experience with profit maximization and welfare analysis, we will quantitatively determine how consumer surplus, producer surplus (the firm's profit), and total welfare change under different price cap scenarios. This involves a fascinating modification of the monopolist's profit-maximization problem.

How a Price Cap Alters a Monopolist's Behavior

When the government imposes a price cap, $P_{c a p}$ , that is below the monopolist's chosen price, $P_{m}$ , the firm can no longer charge $P_{m}$ . The price cap effectively truncates the monopolist's demand curve.

For any quantity up to the amount consumers demand at $P_{c a p}$ , the price is fixed at $P_{c a p}$ .
In this range, the firm's average revenue (AR) is a horizontal line at $P_{c a p}$ .
Crucially, since the firm can sell additional units in this range without lowering the price on previous units, its marginal revenue (MR) is also equal to $P_{c a p}$ .

This creates a new, kinked average revenue curve and a discontinuous marginal revenue curve. To understand this concept visually, let's examine a document that explains this regulation.

Lec14.pdf

These lecture slides from Avinash Dixit at Princeton University provide a concise explanation of how a price ceiling works.

Please read the short section under the heading Regulation Example. Focus on the description of the "kinked average revenue curve" and "double-kinked marginal revenue curve." This is the theoretical key to everything that follows.

The key insight is that the price cap turns the monopolist into a price-taker up to a certain output level, with $M R = P_{c a p}$ . The firm will therefore expand its production as long as this new marginal revenue ( $P_{c a p}$ ) is greater than or equal to its marginal cost (MC).

The ideal price cap is one set at the level where the marginal cost curve intersects the average revenue (demand) curve. This is the price that would prevail in a perfectly competitive market, $P_{c}$ .

This graph illustrates the effect of an optimal price cap on a monopoly. The unregulated monopolist produces quantity `q` at price `p`. A price cap set at `p1` (the competitive price where AR = MC) forces the firm to lower its price and increase its output to `q1`, eliminating the deadweight loss.

By setting the price at $p_{1}$ , the government can induce the monopolist to produce the allocatively efficient quantity $q_{1}$ , benefiting consumers and increasing total welfare. However, the outcome depends entirely on where the cap is set relative to both the monopoly price and the competitive price.

A Quantitative Analysis of Price Caps

Let's now work through a numerical problem to precisely calculate the welfare effects of different price caps. This exercise will solidify your understanding by applying the theory to a concrete scenario with a non-constant marginal cost.

We will use a problem from a set of solved exercises that aligns perfectly with our learning outcome.

chapter 10 market power: monopoly and monopsony

This document provides a fully solved problem (Question 13) that analyzes a monopolist's response to several different price ceilings. We will walk through it step-by-step.

Please read the following parts of Question 13 on pages 128-130: First, read Part (a) to establish the baseline monopoly equilibrium without regulation. Next, study Parts (b) and (c) to determine the competitive outcome and the monopoly deadweight loss. This gives us the benchmark for welfare improvement. Then, carefully analyze Part (d), which examines a price cap set between the monopoly price and the competitive price. Finally, read Part (e), which considers a price cap set below the competitive price.

Let's consolidate the findings from this problem and reconstruct the calculations to ensure we understand each step.

1. Baseline: Unregulated Monopoly

The problem gives us the following functions:

Demand: $P = 55 - 2 Q$
Total Cost: $C = 100 - 5 Q + Q^{2}$

From these, we derive the marginal revenue and marginal cost:

Marginal Revenue: $M R = 55 - 4 Q$
Marginal Cost: $M C = \frac{d C}{d Q} = 2 Q - 5$

Setting $M R = M C$ to find the profit-maximizing quantity:

55 - 4 Q = 2 Q - 5 ⟹ 6 Q = 60 ⟹ Q_{m} = 10

The monopoly price is $P_{m} = 55 - 2 (10) = $35$ .

The competitive outcome occurs where $P = M C$ :

55 - 2 Q = 2 Q - 5 ⟹ 4 Q = 60 ⟹ Q_{c} = 15, P_{c} = $25

The deadweight loss (DWL) from monopoly is the triangle between the competitive and monopoly outcomes:

D W L = \frac{1}{2} \times (P_{m} - M C (Q_{m})) \times (Q_{c} - Q_{m}) = \frac{1}{2} \times (35 - (2 (10) - 5)) \times (15 - 10) = \frac{1}{2} \times 20 \times 5 = $50

2. Case 1: Price Cap at $27

Here, the price cap is set between the monopoly price and the competitive price ( $P_{m} = $35 > $27 > P_{c} = $25$ ).

The firm's new marginal revenue is $27. It will produce up to the quantity demanded at this price, provided that for the last unit, $P_{c a p} \geq M C$ .

Quantity Demanded at $27: $27 = 55 - 2 Q ⟹ 2 Q = 28 ⟹ Q = 14$ .
Marginal Cost at Q=14: $M C (14) = 2 (14) - 5 = $23$ .
Since $P_{c a p} = $27 > M C = $23$ , the firm is willing to produce all 14 units.

Welfare Effects:

Producer Surplus (Profit): $π = (27) (14) - (100 - 5 (14) + 1 4^{2}) = 378 - 226 = $152$ . (Down from $200)
Consumer Surplus: The area under the demand curve above $27, up to Q=14. This is a trapezoid: $C S = \frac{1}{2} \times (55 - 27) \times 14 = $196$ . (Up from $100)
Deadweight Loss: $D W L = \frac{1}{2} \times (P (Q = 14) - M C (Q = 14)) \times (Q_{c} - Q) = \frac{1}{2} \times (27 - 23) \times (15 - 14) = $2$ . (Down from $50)

Total surplus has increased significantly, as the deadweight loss has been almost entirely eliminated.

3. Case 2: Price Cap at $23

Now the price cap is below the competitive price ( $P_{c} = $25 > $23$ ).

The firm's new MR is $23. It will produce up to the point where $M C = $23$ , because it cannot be forced to produce at a loss on the marginal unit.
Find Quantity Supplied: $M C = 2 Q - 5 = 23 ⟹ 2 Q = 28 ⟹ Q = 14$ .
At this price, consumers demand $23 = 55 - 2 Q ⟹ Q = 16$ . However, the firm is only willing to supply 14 units. The quantity traded is therefore 14.

Welfare Effects:

Producer Surplus (Profit): $π = (23) (14) - (100 - 5 (14) + 1 4^{2}) = 322 - 226 = $96$ . (Further decrease)
Consumer Surplus: This is the area under the demand curve above the price of $23, for the 14 units that are actually sold. This area is a trapezoid. The price on the demand curve at Q=14 is $P = 55 - 2 (14) = $27$ .
$C S = Area above $27 \frac{1}{2} (55 - 27) (14) + Rectangle from $27 to $23 (27 - 23) (14) = 196 + 56 = $252$
Deadweight Loss: The deadweight loss calculation is identical to the previous case because the final quantity (14) is the same. $D W L = $2$ .

In this specific example, the quantity produced happens to be the same under both price caps, but the distribution of surplus is different. The lower price cap transferred an additional $56 of surplus from the producer to consumers. Setting the price too low can reduce output further and increase DWL compared to the optimal cap. For instance, a price cap of $12 (Part f) would reduce output to 8.5 units, causing profit to become negative and DWL to balloon to $84.50, which is worse than the unregulated monopoly.

Conclusion

In this lesson, you've analyzed the powerful effects of a price cap as a tool for regulating monopolies. By transforming the monopolist's revenue structure, a price cap can guide the firm towards a more socially efficient outcome.

Key Takeaways:

A price cap on a monopoly alters the firm's marginal revenue curve. For outputs below the quantity demanded at the cap, MR becomes equal to the capped price.
A price cap set between the monopoly price and the competitive price will cause the monopolist to lower its price and increase its quantity, reducing deadweight loss.
The welfare-maximizing (or output-maximizing) price cap is typically set where price equals marginal cost, mimicking the competitive outcome and eliminating deadweight loss entirely.
Setting a price cap too low (below the competitive price) can cause the quantity supplied to fall, potentially creating shortages and increasing deadweight loss relative to the optimal cap.
By calculating consumer surplus, producer surplus, and deadweight loss before and after the intervention, we can precisely quantify the welfare impact of the policy.

In our next and final lesson for this module, we will explore another major market failure—externalities—and quantitatively compare two different policy instruments used to correct them: tradeable permits and corrective taxation.

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