Multi-Market Antitrust Economics pp 15-33 | Cite as

# Pure Monopoly Model

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## Abstract

The classical model of the anti-competitive and socially undesirable monopoly is the pure monopoly model. In this model, the monopolist provides less output—at a higher price—than would be provided by competitive firms facing the same production costs. The output drop and price hike are both anti-competitive negative consequences to consumers of monopoly. The model accommodates other measures of anti-competitive harm, including changes in consumer surplus and total surplus.

## Keywords

Pure monopoly Anti-competitive Marginal cost Demand curve Consumer surplus Total surplus## 2.1 Antitrust and Economic Models

Laws, or statutes, exist that can be put to use by the government or aggrieved parties when confronted with offensive antitrust or anti-competitive behavior. The “how” of putting the statutes to work is where the courts come in, as discussed earlier in the US Supreme Court cases *United States v. E. C. Knight Company* and *United States v. United Shoe Machinery Co.* Recall that Sherman Act, as a written document, is only about two pages long. Applying the Sherman Act in particular cases requires the government—courts and government agencies—to take a close look at markets and the economics going on in markets. Consider, for example, the following key part of the Sherman Act:

Every person who shall monopolize, or attempt to monopolize, or combine or conspire with any other person or persons, to monopolize any part of the trade or commerce among the several states, or with foreign nations, shall be deemed guilty of a felony…

The US federal government, mainly the Department of Justice, has the job of enforcing this part of the Sherman Act, and for this the government needs to know what markets exist for trade “among the several states” and needs to keep tabs on businesses that may try to monopolize trade in some good or service. With many goods and services produced and distributed in and among US states each year, the government has a big job keeping track of them.

The brevity of some antitrust statutes makes possible absurd outcomes. For example, suppose a company L-Scratch, based in Oregon, is the nation’s only producer of left-handed back scratchers.^{1} If the company starts selling its products in California, has it committed a felony? Hopefully not, but it does monopolize Oregon-California trade in left-handed back scratchers, so why does the Sherman Act not make L-Scratch a criminal enterprise? One reason may be that if no other company has bothered to make left-handed back scratchers, then L-Scratch is automatically a monopoly and does not “monopolize” offensively or obnoxiously. Another may be that, if L-Scratch has a patent on the left-handed back scratcher, then it is legally entitled to monopolize. Here antitrust law may bump up against patent law, awkwardly.

US federal statutes covering antitrust law, specifically Chapter 1 of Title 15 in the US code, currently run about 46 pages, not a long read. The range of possible situations in which a lack or loss of market competition might be harmful is far greater than the range of actual cases considered by Congress when writing the antitrust statutes. The Justice Department and Federal Trade Commission (FTC) have to apply antitrust law but also figure out what it should mean in particular cases.

To successfully apply antitrust law, courts and government agencies need some leeway in interpreting it, as in *United States v. E. C. Knight Company* and *United States v. United Shoe Machinery Co.* They also need access to information or data on markets. In addition to broad market data, they can compel businesses to report prices charged, customer names, and goods quantities produced. Market and company data shed light on the extent of possible monopoly or other sorts of anti-competitive arrangements or behavior.

For example, if in the industry for personal computer operating systems one company—call it M—has a 90 percent market share (selling 90 percent of all pc operating systems that customers buy), then that company is dominant.^{2} Whether company M “monopolizes” in some offensive or injurious way is unclear without more data, but what kind of data? Without some economic model of injurious monopoly, the court may not know.

## 2.2 The Pure Monopoly Model

^{3}In the graph, market outcomes are shown for some consumer good or service, such as sugar, with the amount sold being a variable attached to the horizontal axis and the sale price being a variable attached to the vertical axis. Implicitly, market outcomes are those that occur during some time period, like year 2016, and in some location, like the United States. For example, in the United States, the price of retail refined sugar in 2016 was about 65 cents, and about 25 million pounds were delivered for domestic food and beverage use.

^{4}

All else equal, consumers want to purchase greater quantities, and want lower prices. The line labeled “Demand Curve” is the consumers’ demand curve, showing the quantity that all consumers would buy of a good—at each given price—assuming that all consumers pay the same price for the good and that each acts as if their quantity choice does not affect price. The line labeled “MR” is the monopolist’s marginal revenue—the additional revenue received from each successive unit of good sold— and the horizontal line “MC” is the monopolist’s marginal cost, the additional cost incurred from successive units sold, assumed to be the same at each level of output.

The monopolist, as the sole provider of the good, chooses a quantity to supply so as to maximize profit. The monopolist’s profit is the difference between its revenue and cost, and maximum profit is reached at that quantity *Q*_{ m } where marginal revenue equals marginal cost—represented by the crossing point *A* of the *MR* and *MC* curves. The monopolist sells this quantity *Q*_{ m } at the highest price *P*_{ m } that consumers are willing to pay. At the monopoly price *P*_{ m }, the quantity demanded and quantity supplied each equal *Q*_{ m }, and the economy is in pure monopoly equilibrium. In Fig. 2.1, the monopoly equilibrium point is *E*_{ m }, which represents the quantity-price pair (*Q*_{ m }, *P*_{ m }). There is another point, labeled *E*_{ c }, which represents the quantity-price pair (*Q*_{ c }, *P*_{ c }), with *Q*_{ c } the competitive equilibrium quantity and *P*_{ c } the competitive equilibrium price.

In the pure monopoly model, competitive equilibrium is a benchmark point of reference—the economic outcome that would arise if there were many sellers in the market, each having the same marginal cost (*MC*) and each choosing a quantity for sale so as to maximize their individual profit—while assuming that their quantity choice does not change market price. As shown in Fig. 2.1, competitive equilibrium (*E*_{ c }) is a market outcome in which consumers get more output and a lower price than they get in monopoly equilibrium (*E*_{ m }). *E*_{ c } would be a better deal for consumers than *E*_{ m }, if they could get it.

### Example 2.1

In the US market for cellular phones, let the market demand curve takes the form of the straight line *P* = 1000 − 4*Q*, with price *P* on the vertical axis and quantity *Q* (in millions of phones) on the horizontal axis. The marginal revenue curve is then a straight line *MR* = 1000 − 8*Q*. Let the marginal cost (*MC*) of producing a cellular phone be 200 dollars. In competitive equilibrium, *MC* and demand curves cross: *MC* = 200 = 1000 − 4*Q*, in which case competitive quantity is *Q*_{ c } = 200 and competitive price is *P*_{ c } = 1000 − 4*Q*_{ c } = 200. In monopoly equilibrium, *MC* and *MR* curves cross: *MC* = 200 = 1000 − 8*Q*, so the monopolist’s quantity is *Q*_{ m } = 100 and their price is *P*_{ m } = 1000 − 4*Q*_{ m } = 600. In monopoly, consumers pay an extra *$*400 per phone, and across all 100 phones sold in monopoly equilibrium, customers pay an overcharge of 400 × 100 = *$*40, 000.

In pure monopoly consumers get a worse deal than with competition, to an extent that is quantified via a higher price paid and lower quantity consumed. To apply the model, the court can try to collect data on the amount of per-unit overcharge (monopoly equilibrium price minus competitive equilibrium price, *P*_{ m } − *P*_{ c } in Fig. 2.1) and undersupply (competitive equilibrium quantity minus monopoly equilibrium quantity, *Q*_{ c } − *Q*_{ m } in Fig. 2.1). If such data are available, and suggest a significant per-unit overcharge and/or undersupply, the court might view the result as evidence of injurious monopoly.

In the Sugar Trust case *United States v. E. C. Knight Company*, the US Supreme Court had available price and quantity data before and after sugar refining companies banded together to form a monopoly. If the “before” data represent a competitive equilibrium market outcome, and the “after” data correspond to monopoly, then a comparison of “before” and “after” is also a comparison of competition and pure monopoly. As discussed earlier, the Court found that quantity actually rose over time, and price rose but modestly.

The pure monopoly model, with its built-in competitive market benchmark, gives a particular sense in which monopoly can be a bad deal for consumers.^{5}

Economic theory provides some additional ways of evaluating the desirability of competitive and monopoly outcomes, in terms of consumer surplus, producer surplus, and total surplus. In Fig. 3.1, *consumer surplus* is the area of the triangle which is above market price (horizontal) line and below the demand curve, *producer surplus* is the area below the market price line and above the marginal cost curve, and *total surplus* is the sum of consumer surplus and producer surplus.

Consumer surplus measures the benefit to consumers of a market outcome, producer surplus measures benefit to producer(s), and total surplus measures benefit to society as a whole. The terminology and graphs associated with these three concepts have changed some since the days of the Sherman Act (1890), but the essential ideas were already in textbook form then—via Alfred Marshall’s *Principles of Economics* (1890).^{6}

Consumer surplus is higher in competitive equilibrium than in monopoly equilibrium, while producer surplus is higher in monopoly equilibrium. Overall, total surplus is higher in competitive equilibrium, and the drop in total surplus in moving from competitive to monopoly equilibrium is the *deadweight loss* associated with the monopoly—which in Fig. 2.1 is the area of the triangle with corner points *E*_{ m }, *E*_{ c }, and *A*.

### Example 2.2

Consider again the cell phone market in Example 2.1. The following table shows values for consumer surplus, producer surplus, and total surplus, for competitive and monopoly equilibrium outcomes.^{7}

| | | |
---|---|---|---|

| | | |

Monopoly | 2000 | 4000 | 6000 |

Competition | 8000 | 0 | 8000 |

Difference | −6000 | 4000 | −2000 |

As indicated, consumers are better off in competitive equilibrium, with greater consumer surplus there, while the producer is better off in monopoly equilibrium. Society, on the whole, is better of in competitive equilibrium, and the deadweight loss of monopoly equilibrium is the total surplus drop from going from competition to monopoly, here equal to −2000, or just 2000 when measured in absolute terms.

## 2.3 The Model’s Assumptions

The pure monopoly model depicted in Fig. 2.1 includes a linear demand curve and a flat marginal cost curve. Figure 2.1 focuses on these modeling assumptions, labeling the endpoints of the demand curve, and using *MC* as the numerical value of marginal cost and also the label of the marginal cost curve.

*a*and

*b*positive parameters, and

*Q*in the range (0,

*a*∕

*b*).

^{8}Parameter

*a*represents consumers’ maximum willingness to pay for the good, and parameter

*b*represents consumers’ required price reduction to purchase one more unit of the good. Both

*a*and

*b*are assumed to be positive. Also, to allow demand and marginal cost curves to cross at some positive quantity

*Q*and price

*P*, it’s necessary to suppose that

*a*>

*MC*.

^{9}

The demand curve linearity and flat marginal cost curve flatness assumed here are not necessary to the theory of natural monopoly but are a simple specification often presented in economic texts—see, for example, Mas-Colell et al. (1995, Example 12.B.1) and Nicholson and Snyder (2012, Chapter 18).^{10} Also, with the idea of discussing monopoly and related issues in a way that is accessible to both economists and others interested in antitrust, a particularly simple modeling approach makes sense, and the assumptions made here match those in the antitrust works (Posner 1976, Appendix and Blair and Kaserman 2009, Section 3–6).

*Q*

_{ c }and price

*P*

_{ c }that have the following formulas

^{11}:

*Q*

_{ m }, and at that

*Q*

_{ m }the corresponding price

*P*

_{ m }is determined by the demand curve, in which case

^{12}:

*a*is higher and when marginal cost

*MC*is lower.

^{13}The overcharge associated with monopoly is (

*P*

_{ m }−

*P*

_{ c })

*Q*

_{ m }. Expressed in terms of demand and cost parameters via formulas (2.2) through (2.5) the overcharge takes the form:

*a*−

*MC*between willingness to pay and marginal cost is higher, and when the price drop

*b*, needed to get one more unit of good purchased by consumers, is smaller.

^{14}

^{,}

^{15}

^{,}

^{16}

Surplus effects of pure monopoly

| | | |
---|---|---|---|

| | | |

Monopoly | \(\frac{1} {8} \frac{(a-MC)^{2}} {b}\) | \(\frac{1} {4} \frac{(a-MC)^{2}} {b}\) | \(\frac{3} {8} \frac{(a-MC)^{2}} {b}\) |

Competition | \(\frac{1} {2} \frac{(a-MC)^{2}} {b}\) | 0 | \(\frac{1} {2} \frac{(a-MC)^{2}} {b}\) |

Difference | \(-\frac{3} {8} \frac{(a-MC)^{2}} {b}\) | \(\frac{1} {4} \frac{(a-MC)^{2}} {b}\) | \(-\frac{1} {8} \frac{(a-MC)^{2}} {b}\) |

The effect of monopoly is, according to Table 2.1, to cut total surplus by 25 percent relative to the competitive market outcome, regardless of the demand parameters *a* and *b*, and production’s marginal cost *MC*. The fact that the 25 percent surplus drop is invariant to parameter values is a result of assuming linear demand and flat marginal cost curves.

A 25 percent reduction in total surplus or welfare may or may not sound like a big anti-competitive problem. On the other hand, from Table 2.1 consumer surplus falls by 75 percent under monopoly, a bigger effect. Changes in surplus, price, and quantity give a range of ways in which to describe pure monopoly’s anti-competitive effects, with no obviously best choice among them. If the goal is to promote consumer welfare, then the monopoly overcharge and drop in consumer surplus are relevant measures of anti-competitive harm.

*a*−

*MC*)

^{2}∕(

*b*), while the fall in consumer surplus is (3∕8)((

*a*−

*MC*)

^{2}∕(

*b*). The change \(\Delta PS\) in producer surplus is therefore related to the change \(\Delta CS\) in consumer surplus and to the change \(\Delta TS\) in total surplus via:

In Posner (1975, 1976), Richard Posner argues that an increase in producer surplus triggered by monopoly should not be counted as an offset to consumer surplus loss, in determining societal harm. From Eqs. (2.9)–(2.10), under the assumptions maintained here, the inclusion or exclusion of producer surplus is a big deal: monopoly profits, that is, the gain in producer surplus, are half the size of the deadweight loss defined by \(-\Delta TS\). Deducting producer surplus as a relevant component of total surplus, the remainder is consumer surplus. The idea that consumer surplus is a relevant measure of societal good, when evaluating anti-competitive harm, has gained wide support in antitrust law.^{17}^{,}^{18}

The foregoing characterization of pure monopoly effects relies on a linear demand curve and flat marginal cost curve, and results can differ greatly if instead the demand curve is linear or marginal cost curve is increasing. Nicholson and Snyder (2012, Example 18.2) presents the case of a nonlinear constant elasticity demand curve and flat marginal cost curve, showing, for example, that monopoly shrinks consumer surplus to a degree commensurate with demand elasticity. At the extreme, as elasticity approaches infinity, monopoly causes a reduction in consumer surplus of 1 − (1∕*e*) = 0. 6322…or about 63 percent, with *e* Euler’s constant.^{19} By comparison, with a linear demand curve monopoly causes a 75 percent drop in consumer surplus. The fact that different specifications of the pure monopoly model generate different conclusions is important to keep in mind.^{20}

## 2.4 Applying the Model

The pure monopoly model is an insightful reference point when interpreting possible harm caused by monopoly and related anti-competitive phenomena. Government agencies, and courts, may refer to the pure monopoly model when determining if given market situation is offensively anti-competitive, offensive enough to break antitrust law. In this way, economic models can help to determine whether or not a business is guilty or liable for anti-competitive harm.

If antitrust guilt or liability is found, economic models may also help determine the form and extent of punishment and payments required of antitrust violators. The monetary loss to consumers from a monopoly overcharge on a quantity of goods purchased is the per-unit overcharge *P*_{ m } − *P*_{ c } times the quantity *Q*_{ m } sold. With *P*_{ m } the monopoly price charged to customers, *P*_{ c } is the benchmark price—whose value depends on some economic model. Earlier we interpreted *P*_{ c } as the competitive equilibrium price and expressed overcharge in terms of demand and cost parameters via Eq. (2.8).

In the Sugar Trust case discussed earlier, market price is observed before and after a (near) monopoly takeover, and a relevant benchmark price is the “before” price, with no need to discern relevant values for demand and cost parameters when evaluating overcharge effects.^{21}

Fines levied on an offending monopoly firm via the Sherman Act might be set equal to the monopolist’s overcharge, assuming a benchmark price *P*_{ c } can be determined.

Courts can award economic damages and relief to individuals, businesses, or society, in antitrust civil lawsuits against businesses, via the Clayton Act. The FTC government agency can prevent firms from merging or acquiring each other, via the FTC Act and Clayton Act.

## 2.5 Model’s Limitations and Extensions

The pure monopoly model is an elegant and insightful simplification of the real world. It’s likely not 100 percent accurate in any market situation. The most accurate “model” of any observable phenomenon is the phenomenon itself, but that “model” has no formal content or general theme. It’s easy to add more realistic details to the pure monopoly model, much as one would add decals or stickers as a details on a model airplane. More substantively, the pure monopoly model can be modified to allow for fundamental departures from key underlying assumptions. This is like adding or switching parts on a model airplane.

Not every monopoly is well-described as “pure monopoly.” The term itself appeared originally in Edward Chamberlain’s book on monopolist competition (Chamberlain 1933), but appears earlier in Alfred Marshall’s *Principles of Economics* (1890), and Marshall also sketches a variety of extensions to the model—to allow for efficiencies in the combining of firms, business goals other than short-term profit, and price discrimination.

- 1.
Efficiency: In becoming a monopolist, companies are merged and efficiencies gained, with higher productivity and lower marginal costs.

- 2.
Competitive fringe: The monopolist cannot significantly overcharge without attracting competitors who enter the market, sell goods, and drive price toward its competitive equilibrium level.

- 3.
Bad benchmark: The benchmark equilibrium market outcome in the pure monopoly is infeasible, making “overcharge” and “undersupply” irrelevant too.

*#*1 on the above list, with a big merger-induced drop in marginal cost (

*MC*). The pre-merger competitive equilibrium, labeled

*E*

_{ c }, has lower quantity and higher price than the post-merger monopoly equilibrium

*E*

_{ m }. If the drop in marginal cost is instead small, monopoly may end up lowering output and raising price,

^{22}like in the pure monopoly model.

^{23}

### Example 2.3

In the US market for railroad passenger transportation, let the market demand curve be the straight line *P* = 1000 − *Q*. The marginal revenue curve is then a straight line *MR* = 1000 − 2*Q*. Let the marginal cost (*MC*) equal 800 for individual, competing firms, and let it be 400 for a monopoly firm. In competitive equilibrium, *MC* = 800 = 1000 − *Q*, and competitive quantity is *Q*_{ c } = 200 and competitive price is *P*_{ c } = *MC* = 800. In monopoly equilibrium, *MC* and *MR* curves cross: *MC* = 400 = 1000 − 2*Q*, so the monopolist’s quantity is *Q*_{ m } = 300 and their price is *P*_{ m } = 1000 − *Q*_{ m } = 700. So, the monopoly equilibrium provides more output, at a lower price, than competitive equilibrium. In terms of Marshallian surplus, monopoly and competition compare as in Table 2.2, with a gain in total surplus or welfare of 11,500.^{24}

Surplus effects of pure monopoly

| | | |
---|---|---|---|

| | | |

Monopoly | 4500 | 9000 | 13,500 |

Competition | 2000 | 0 | 2000 |

Difference | 2500 | 9000 | 11,500 |

In the “competitive fringe” scenario, (*#*2, above), possible anti-competitive harm from a dominant firm is limited by free and willing entry of other firms into the market. It’s possible that only one firm actually sells the good but charges a price that keeps economic profits close to zero, discouraging entry by other firms. In Fig. 2.1, the argument is that the monopoly firm will not go for pure monopoly outcome *E*_{ m } because in doing so other firms will want some of the economic profits and enter the market, raising market quantity from *Q*_{ m } toward *Q*_{ c }.^{25}

In the “bad benchmark” scenario (*#*3, above), competitive equilibrium—with many firms facing the same costs—is infeasible. To see how this scenario might play out, let competing firms each face a price *P*_{ c } equal to the marginal cost *MC* of producing the good. Suppose, in addition, each firm faces a positive fixed setup cost *F* of getting into the business. Then economic profit of any one firm *f*—which is its revenue *P*_{ c }*Q*_{ f } minus its total cost *F* + *MC* × *Q*_{ f }—is negative for each positive quantity *Q*_{ f } it might produce. Assuming that firms refuse business opportunities that lose money, competitive equilibrium becomes infeasible and so irrelevant as a benchmark.^{26}

## 2.6 Problems

- 1.In the pure monopoly model, two anti-competitive effects are a quantity drop and a price hike.
- (a)
Which effect—quantity drop or price hike—seems to you the worse sort of anti-competitive effect on consumers? Explain.

- (b)
Using Eqs. (2.6) and (2.7) in the text, for what values of the demand curve’s slope parameter

*b*is the quantity drop bigger (in absolute terms) than the price hike? For what values of*b*is the reverse true?

- (a)
- 2.In the market for sugar, suppose that the market demand curve takes the form of the straight line
*P*= 10 − 2*Q*, with price*P*on the vertical axis and quantity*Q*(in millions of pounds) on the horizontal axis. The marginal revenue curve is then a straight line*MR*= 10 − 4*Q*. Let the marginal cost (*MC*) of producing a pound of sugar be 20 cents.- (a)
Graph the demand curve, marginal revenue curve, and marginal cost curve, all on the same graph as in Fig. 3.1.

- (b)
Find the competitive equilibrium price and quantity, at which the demand curve and marginal cost curve intersect.

- (c)
Find the pure monopoly equilibrium quantity, at which the marginal revenue and marginal cost curves intersect, then find the monopoly price.

- (d)
Compare the competitive equilibrium and monopoly equilibrium outcomes, in terms of monopoly overcharge.

- (e)
Compute consumer surplus, producer surplus, and total surplus, in competitive equilibrium and monopoly equilibrium, similar to Example 2.1 in the text.

- (a)
- 3.In the US market for high-performance personal computers, let the market demand curve be the straight line
*P*= 2000 −*Q*. Let the marginal cost (*MC*) equal 1800 for individual, competing firms, and let it be 600 for a monopoly firm.- (a)
Graph the demand curve, marginal revenue curve, and marginal cost curves, all on the same graph as in Fig. 3.2

- (b)
Find the competitive equilibrium price and quantity, at which the demand curve and competitive marginal cost curve intersect.

- (c)
Find the monopoly equilibrium quantity, at which the marginal revenue curve and monopoly marginal cost curve intersect, and then find the monopoly price.

- (d)
Compare the competitive equilibrium and monopoly equilibrium outcomes, in terms of monopoly overcharge.

- (e)
Compute consumer surplus, producer surplus, and total surplus, in competitive equilibrium and monopoly equilibrium, similar to Example 2.3 in the text.

- (a)
- 4.
In an industry with a single seller and a competitive fringe of potential competitors, monopoly’s extreme market concentration need not create anti-competitive effects. Explain.

- 5.In the market for computer operating systems, suppose that each firm that supplies systems faces a cost
*F*+*MCQ*of producing*Q*systems, with positive fixed cost*F*and marginal cost*MC*= 2. Suppose also that consumers’ demand curve is*P*= 3 −*Q*.- (a)
Find the competitive equilibrium quantity

*Q*_{ c }and price*P*_{ c }, and show that industry profit*P*_{ c }*Q*_{ c }− (*F*+*MCQ*_{ c }) equals −*F*, which is negative. Will any firms choose to produce in this competitive industry? Explain. - (b)
Find the monopoly equilibrium quantity

*Q*_{ m }and price*P*_{ m }, and show that industry profit*P*_{ m }*Q*_{ m }− (*F*+*MCQ*_{ m }) equals \(\frac{1} {4} - F\). If \(F <\frac{1} {4}\), will the monopolist choose to produce in this industry? Explain.

- (a)
- 6.Suppose that the demand curve for a good is:
*P*=*βQ*^{−1∕ɛ }, with*β*a positive parameter and*ɛ*the elasticity of demand—a number greater than 1. Firms produce the good with no fixed cost and with marginal cost*MC*which is the same for each unit produced.- (a)
Show that the competitive equilibrium quantity is

*Q*_{ c }= (*MC*∕*β*)^{−ɛ }, and competitive equilibrium price is*MC*. - (b)
Suppose now that a monopoly takes over the industry. Show that the monopolist’s marginal revenue is \(MR =\beta (1 -\frac{1} {\varepsilon } )Q^{-1/\varepsilon }\), and profit-maximizing quantity and price are \(Q_{m} = \left ( \frac{MC} {\beta (1-\frac{1} {\varepsilon } )}\right )^{-\varepsilon }\), \(P_{m} = \frac{MC} {1-\frac{1} {\varepsilon } }\).

- (c)
Show that the Lerner Index, defined as (

*P*−*MC*)∕*P*, equals 0 in competitive equilibrium and equals 1∕*ɛ*in monopoly equilibrium. - (d)
Show that \(Q_{m}/Q_{c} = (1 -\frac{1} {\varepsilon } )^{\varepsilon }\) and \(P_{m}/P_{c} = (1 -\frac{1} {\varepsilon } )^{-1}\), each increasing in

*ɛ*. Does monopoly have anti-competitive effects on quantity and price? Explain. - (e)
If

*ɛ*= 2, show that monopoly causes price to double and quantity to fall by 75 percent. - (f)
As elasticity

*ɛ*approaches infinity, show that*P*_{ m }∕*P*_{ c }→ 1 while*Q*_{ m }∕*Q*_{ c }→ 1∕*e*, with*e*Euler’s constant. In the limit, does monopoly have anti-competitive effects? Explain.

- (a)

### Notes

1. Being left-handed, and raised in Oregon, I know of no such company or way to make a left-handed back scratcher better than a regular back scratcher, but I beg the reader’s indulgence.

2. Currently, Microsoft has a market share of about 90 percent.

3. This model appears in Marshall (1890) and was later termed “pure monopoly” – in contrast to pure or perfect competition – by Chamberlain (1933).

4. For retail sugar price data, see “Table 6—US retail refined sugar price, monthly, quarterly, and by calendar and fiscal year” provided by the USDA online, and for sugar quantity data, see USDA “Table 24a—US sugar: supply and use, by fiscal year.”

5. Pure monopoly has the same consumer impact as the collusion of firms to fix price at a collectively profit-maximizing level, so pure monopoly is obviously bad for consumers. Schemes or agreements among firms that aim to fix prices, or limit price flexibility, can be declared illegal even without courts’ reliance on market data, see *United States v. Trans-Missouri Freight Association,* 166 U.S. 290 (year 1897).

6. This classic text went through eight editions, you can browse/search the first online at babel.hathitrust.org, and download the last (8th) edition at libertyfund.ort.

7. Consumer surplus values equal areas of triangles framed by the demand curve/line and the price line. Producer surplus is zero in competitive equilibrium since there is an assumed constant marginal cost (*MC*) which coincides with the price line, while in monopoly equilibrium it is the area of the rectangle with (*Q*, *P*) corners (0, 600), (0, 200), (100, 600), (100, 200), also equal to (*P*_{ m } − *MC*)*Q*_{ m }.

8. For values of *Q* greater than *a*∕*b*, price is negative—impossible under usual circumstances.

9. With a flat marginal cost curve, a firm’s total cost of producing a quantity *Q* is *F* + *MC* × *Q*, with fixed cost *F* which here is assumed equal to zero. This sort of cost function is said to have constant returns.

10. A linear demand curve has constant slope and two parameters—an intercept and a slope. Another two parameter demand model is the constant elasticity model, which can be written *P* = *βQ*^{−1∕ɛ }, with *β* positive and *ɛ* the elasticity of demand—assumed greater than 1. See discussion below and Problem 2.6 for more on this approach.

11. Demand and marginal cost curves cross at: *MC* = *P* = *a* − *bQ*, in which case *Q* = (*a* − *MC*)∕*b*.

12. Revenue is *PQ* = (*a* − *bQ*)*Q*, and marginal revenue is *MR* = *d*∕*dQ*(*PQ*) = *a* − 2*bQ*, in which case condition *MR* = *MC* is the same as *a* − 2*bQ* = *MC*, yielding *Q*_{ m } = (*a* − *MC*)∕(2*b*). Price is then *P*_{ m } = *a* − *bQ*_{ m } = *a* − *b*((*a* − *MC*)∕(2*b*)), which is (*a* + *MC*)∕2.

13. We can also express the quantity drop relative of initial (competitive) quantity, and likewise express price increase relative to initial price, the latter known as the Lerner Index: (*P*_{ m } − *P*_{ c })∕*P*_{ c } which here takes the form ((*a*∕*MC*) − 1)∕2.

14. Consumer surplus, in competitive equilibrium = (1∕2)(*a* − *P*_{ c })*Q*_{ c }, which is (1∕2)(*a* − *MC*)((*a* − *MC*)∕*b*), in turn equal to (*a* − *MC*)^{2}∕(2*b*). By comparison, in monopoly consumer surplus is (1∕2)(*a* − *P*_{ m })*Q*_{ m }, which is (1∕2)(*a* − (*a* + *MC*)∕2)(*a* − *MC*)∕(2*b*), equal to (1∕2)((*a* − *MC*)∕2)(*a* − *MC*)∕(2*b*), also equal to (*a* − *MC*)^{2}∕(8*b*).

15. Producer surplus equal 0 in competitive equilibrium, while in monopoly it is (*P*_{ m } − *MC*)*Q*_{ m }, equal to ((*a* + *MC*)∕2 − *MC*)(*a* − *MC*)∕(2*b*), or just (*a* − *MC*)^{2}∕(4*b*).

16. Total surplus, in competition, is (1∕2)(*a* − *MC*)^{2}∕(*b*), and in monopoly it is (*a* − *MC*)^{2}∕(8*b*) plus (*a* − *MC*)^{2}∕(4*b*), which is (3∕8)(*a* − *MC*)^{2}∕*b*.

17. Posner (1975) measures the social cost of monopoly as the deadweight loss \(-\Delta TS\) plus the producer surplus increase \(\Delta PS\), which by definition equals the consumer surplus loss \(-\Delta CS\) in going from competitive equilibrium to monopoly equilibrium. He shows empirically that such losses may be substantial in comparison to sales in industries like sugar, rubber, and electric bulbs.

18. For discussion see Whinston (2008, Chapter 1).

19. Equation (18.18) of Nicholson and Snyder (2012), with *ɛ* used here to denote demand elasticity in absolute (positive) terms, expresses the ratio of consumer surplus in monopoly and competition situations as \(\frac{CS_{m}} {CS_{c}} = \left ( \frac{1} {1-\frac{1} {\varepsilon } } \right )^{1-\varepsilon }\), and as *ɛ* approaches infinity this ratio approaches 1∕*e*, with *e* Euler’s constant.

20. Which specifications are most appropriate, for drawing conclusions about actual markets, should be informed by data on buyers, sellers, and markets; see Posner (1975, 1976), for example.

21. For price-fixing, Section 15d of US Code Chap. 1 gives considerable leeway in the court’s determination of economic damages: “…damages may be approved and assessed in the aggregate by statistical or sampling methods, by the computation of illegal overcharges, or by such other reasonable system of estimating aggregate damages as the court in its discretion may permit…”

22. Improved efficiency, and lower marginal cost, cannot rationalize the increased output *and* (slightly) higher price described in the sugar trust case. Some additional factor, such as increased sugar demand—and a shifting out of demand and marginal revenue curves—may also play a role.

23. See Williamson (1968) for a classic discussion of the tradeoffs involved in such cases.

24. As in Example 2.2, consumer surplus is the area of the triangle above the market price line and below the demand curve, while producer surplus is *Q*_{ m }(*P* − *MC*).

25. If each firm has identical costs—with the same marginal cost *MC* at each output level, and no fixed cost—then to discourage entry by fringe firms, the dominant firm will be forced to raise quantity to *Q*_{ c }. If fixed cost is instead positive, then fringe firms cannot enter the market costlessly, and the dominant firm need not raise quantity all the way to *Q*_{ c }.

26. In this situation there the average cost of production is (*F* + *MC* × *Q*_{ f })∕*Q*_{ f } = *F*∕*Q*_{ f } + *MC*, which is falling in *Q*_{ f }. With decreasing average cost, the industry has increasing returns to scale, making monopoly a “natural” economic outcome. See Chap. 6 for more discussion of natural monopoly.

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