introduction to monopolies and oligopolies

8/4/2019 Introduction to Monopolies and Oligopolies

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Introduction to Monopolies and Oligopolies

In the news lately, stories of the big bad monopolies abound. We constantly hear of

government regulation in software, utilities, transportation, and financial institutions. The

justice department closely scrutinizes mergers and acquisitions so that firms don't end up

with too much market power. All of this begs the question, what's the big deal? Wheredoes our fear of monopolies originate? After all, in a free market economy, there is no

coercion. Presumably, the economy is still driven solely by mutually beneficial exchange

whether one firm exists or many.

Thus far in our treatment of economics, we have assumed that there exist a large number

of firms in a market. This assumption enabled us to treat firms as price takers, since no

firm in particular had any more market power than any other. In this Sparknote, we willinvestigate the impact of a relaxation of the multiple firms assumption on equilibrium.

We will demonstrate the importance of the assumption in our understanding of perfect

competition.

In the first section, we will define a monopoly and walk through the mechanics behindcalculating equilibrium in a monopolistic market. We will also investigate the

monopoly's impact on social welfare. In the second section, we extend the model of a

monopoly to 2 firms and then to n firms. We define the assumptions underlying Cournot,

Bertrand, and Stackelberg duopolies. We then walk through examples designed to clarifythe mechanics behind the various models of duopoly, and generalize to the n firm case in

a Cournot framework.

Terms

Pure monopoly - A firm that satisfies the following conditions:

1. It is the only supplier in the market.

2. There is no close substitute to the output good.3. There is no threat of competition.

Natural monopoly - A firm with such extreme economies of scale that once it begins

creating a certain level of output, it can produce more at a lower cost than any smaller

competitor. Generally characterized by a declining average cost curve.

Economies of scale - Savings acquired through increases in quantity produced.

Oftentimes, large firms in industries with high fixed costs can take advantage of savings

that smaller firms cannot.Price taker - An agent who takes prices as given. For instance, a firm who faces a

perfectly flat demand curve has no choice but to sell at one price. This firm is a price

taker.

Perfect competition - A market operates under perfect competition if it satisfies thefollowing conditions:

1. Numerous firms


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2. Freedom of entry and exit

3. Homogeneous output

4. Perfect information

Deadweight loss - The dollar amount of social surplus that goes unrealized as compared

to the socially optimal solution.Price setter - The opposite of a price taker; a price setter has the power to set prices.

For instance, a firm who faces a downward sloping demand curve can choose price.

Socially optimal - Describes points at which social surplus is maximized, social surplus

being the combined utilities of the firms and the public.

Oligopoly - A market dominated by a small number of firms. At least several of thesefirms are large enough to influence the market price.

Duopoly - A market dominated by two firms. Both firms are large enough to influence

the market price.

Cournot duopoly - A model of duopolies under which two firms simultaneously choose

the quantity to produce.

Stackelberg duopoly - A model of duopolies under which two firms choose thequantity to produce with one firm choosing before the other in an observable manner.

Bertrand duopoly - A model of duopolies under which two firms simultaneously

choose the price for a good.

Cartel - A small number of independent firms who act together to set monopoly pricesand make monopoly profits.

Public information - Information known to everyone.

Reaction curve - A reaction curve is a function that takes as input the moves of theother players and returns the optimal move given the other players' moves.

Nash equilibrium - An equilibrium in which all players are playing their best responses

to everyone else's best response.

Monopolies

Pure Monopolies and Natural Monopolies

Pure Monopolies

A pure monopoly is a firm that satisfies the following conditions:

1. It is the only supplier in the market.

2. There is no close substitute to the output good.3. There is no threat of competition.

In practice, pure monopolies are very rare. For instance, a supermarket may be the onlyfood supplier in a particular town, but if it raises its prices and retains too much of a

profit, a competitor may enter the space. Even the threat of serious competition entering

the market forces the existing firm to act conscionably, and differently from how it would


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act otherwise. A train company may be the only carrier in a particular station, but if cars

are also available in the area, there exists a close substitute to the output good.

Natural Monopoly

A natural monopoly is a firm with such extreme economies of scale that once it beginscreating a certain level of output, it can produce more at a far lower cost than any smaller

competitor. Natural monopolies exist far more frequently than pure monopolies, mainly

because the requirements are not as stringent.

Natural monopolies occur when, for whatever reason, the average cost curves decline

over a relevant span of output quantities. A firm with high fixed costs relative to its

marginal costs will have declining average costs for a significant span of quantities. A

firm with a decreasing marginal cost structure will also have declining average costs. Forexample, utilities and software are two industries where natural monopolies occur often.

An Example

A monopoly differs from competitive firms in that it is not a price taker. Because it is the

only supplier in the market, it faces a downward sloping demand curve, the marketdemand curve. As a result, the monopoly is free to choose its price and quantity

according to market demand.

Monopolies are still profit maximizing firms and are thus going to satisfy the profit

maximizing condition that marginal cost equal marginal revenue. The key to

understanding monopolies and monopoly power is the marginal revenue calculation. In a

perfectly competitive market, there exists a market price. Marginal revenue is simply

equal to price in this market; every additional unit that is sold brings the market price. Ina monopoly, however, every quantity is associated with a different price. The marginal

revenue is not simply the price.

For example, I may be able to sell 10 guitars at 100each, butinordertosell11guitars,Iwillhavetoofferapriceof 95. Unfortunately, it's very difficult to sell 10 guitars at100andthensellthelastoneat95. In our model of a monopoly, there can only be one price

for a good. If I choose to sell 11 units, I make 95revenueonthe11thguitar, butIlose 5

revenue on each of the first 10 guitars. If it costs me 50toproduceaguitar,mymarginalrevenueisthen 95 - 50 = 45.

Let's generalize. Assume that a monopolistic firm faces a linear, downward- slopingmarket demand curve, described as follows:

Q = 100 - P

Let's further assume its marginal cost curve is constant at a value of 10.MC = 10


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Our firm naturally wants to maximize profits and will therefore aim to satisfy the profit

maximizing condition, MC = MR . Marginal costs are constant at ten, so half of our

equation is easy. To find our marginal revenue, we first look at the total revenue. Totalrevenue is simply:

R = P * Q

Because the monopolist faces the entire market demand curve, price and quantity have a

one-to-one relationship. That is,P= 100 - Q . We can rewrite our total revenue as:

R = (100 - Q) * Q = 100 * Q - Q^2

The marginal revenue is simply the first derivative of the total revenue with respect to Q .

MR = 100 - 2 * Q

If you don't feel comfortable with derivatives, you can convince yourself this MR iscorrect by analyzing its components.

MR = (100 - Q) - Q

(100 - Q) is the price according to our market demand curve. This 100 - Q represents themarginal revenue brought in by selling the next unit. However, in order to sell the next

unit, we had to lower the price by 1 for all units sold (the demand curve has a slope of -1,

so the tradeoff between Q andPis 1 for 1). Therefore, on the margin, we lost 1 unit ofrevenue for all Q units sold. The marginal revenue is then (100 - Q) - Q = 100 - 2*Q .

To solve for the monopolistic equilibrium, we find the quantity at which MR = MC .Solving:

100 - 2 * Q = 10 => Q = 45

At this quantity, the market price would be 100 - 45 = 55 . Assuming no fixed costs, theprofits for this firm would be 45*(55 - 10) = 2025 . Naturally, this is a vast improvement

for the firm over the competitive outcome of zero profits.

Welfare Analysis

So what's wrong with making profits? Certainly, profits are good for the monopolisticfirms. The consumers are willing to pay for the goods at the monopoly price. Nobody is

being forced to do anything, so we have a system of mutually beneficial exchange with

no coercion. I think it would be overstepping our bounds for SparkNotes to say there is

something wrongwith monopoly power, but the foundations for government interventionin monopolistic markets can be found in welfare analysis.


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Let's identify the deadweight loss in the example from the previous section. Let Qm be

the output quantity chosen by the monopolist, 45 in this market. Let Q * be the output

quantity at which the marginal cost curve intersects the market demand curve. Q* = 90 inthis market.

Q*

is the socially optimal output quantity. Imagine the firm is trading at a quantity lessthan Q * . At this point, the marginal cost curve is below the demand curve. In other

words, the marginal cost to society is less than the marginal benefit (the demand curve).The society stands to gain by trading at a higher quantity. The opposite is true at

quantities greater than Q* (convince yourself of this).

Remember that Qm is no greater than, and most often less than, Q* . IfQm is less than Q

* , it is suboptimal. The deadweight loss is the area between the demand curve and themarginal cost curve over the quantities between Qm and Q

* . The marginal cost is the

marginal cost to society, and the marginal benefit is the demand curve. Over these

quantities, the marginal benefit is greater than the marginal cost, so the area between the

curves represents social surplus unrealized at the monopolistic equilibrium.

The impact of monopolistic behavior on social welfare varies with the shape of the

demand curve. For example, with a perfectly inelastic demand curve, the market cannot

help but trade at the socially optimal quantity. However, the monopolist has the power toset prices as high as it pleases (for this reason, many of these industries are regulated,

such as suppliers of insulin or water). Therefore, there exists no deadweight loss, but all

social surplus is absorbed by the monopolistic firm.

A monopolist's power is determined by its ability to set prices, which relies completelyon the demand curve a firm faces. In perfect competition, a firm sees a flat demand curve

and therefore does not have a practical choice as to what price to offer. The monopolist'spower comes from facing a downward sloping demand curve.

Duopolies and Oligopolies

Overview

Monopoly power comes from a firm's ability to set prices. This ability is dictated by the

shape of the demand curve facing that firm. If the firm faces a downward sloping demand

curve, it is no longer a price taker but rather a price setter. In our perfect competition

model, we assume there exist multiple participants, and because there are so many

participants, the slice of the demand curve each firm sees is but a flat line. These firmsare price takers.

There is a medium between monopoly and perfect competition in which only a few firmsexist in a market. None of these firms faces the entire demand curve in the way a

monopolist would, but each does have some power to set prices. A small collection of

firms who dominate a market is called an oligopoly. A duopoly is a special case of an

oligopoly, in which only two firms exist.


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Duopolies

We will begin our discussion with an investigation of duopolies. For the following

duopoly examples, we will assume the following:

1. The two firms produce homogeneous and indistinguishable goods.2. There are no other firms in the market who produce the same or substitute goods.

3. No other firms can or will enter the market.

4. Collusive behavior is prohibited. Firms cannot act together to form a cartel.5. There exists one market for the produced goods.

Cournot Duopoly

In 1838, Augustin Cournot introduced a simple model of duopolies that remains the

standard model for oligopolistic competition. In addition to the assumptions stated above,the Cournot duopoly model relies on the following:

1. Each firm chooses a quantity to produce.

2. All firms make this choice simultaneously.

3. The model is restricted to a one-stage game. Firms choose their quantities onlyonce.

4. The cost structures of the firms are public information.

In the Cournot model, the strategic variable is the output quantity. Each firm decides how

much of a good to produce. Both firms know the market demand curve, and each firmknows the cost structures of the other firm. The essence of the model is this: each firm

takes the other firm's choice of output level as fixed and then sets its own production

quantities.

The best way to explain the Cournot model is by walking through examples. Before webegin, we will define the reaction curve, the key to understanding the Cournot model (and

elementary game theory as well).

A reaction curve for Firm 1 is a function Q1*() that takes as input the quantity produced

by Firm 2 and returns the optimal output for Firm 1 given Firm 2's production decisions.

In other words, Q1*(Q2) is Firm 1's best response to Firm 2's choice ofQ2 . Likewise, Q

2*(Q1) is Firm 2's best response to Firm 1's choice ofQ1 .

Let's assume the two firms face a single market demand curve as follows:

Q = 100 - PwherePis the single market price and Q is the total quantity of output in the market. For

simplicity's sake, let's assume that both firms face cost structures as follows:

MC_1 = 10MC_2 = 12


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Given this market demand curve and cost structure, we want to find the reaction curve for

Firm 1. In the Cournot model, we assume Q 2 is fixed and proceed. Firm 1's reaction

curve will satisfy its profit maximizing condition, MR = MC . In order to find Firm 1'smarginal revenue, we first determine its total revenue, which can be described as follows

Total Revenue = P * Q1 = (100 - Q) * Q1 = (100 - (Q1 + Q2)) * Q1= 100Q1 - Q1 ^ 2 - Q2 * Q1

The marginal revenue is simply the first derivative of the total revenue with respect to Q1(recall that we assume Q2 is fixed). The marginal revenue for Firm 1 is thus:

MR1 = 100 - 2 * Q1 - Q2\

Imposing the profit maximizing condition of MR = MC , we conclude that Firm 1's

reaction curve is:

100 - 2 * Q1* - Q2 = 10 => Q1* = 45 - Q2/2

That is, for every choice of Q 2 , Q 1* is Firm 1's optimal choice of output. We can

perform analogous analysis for Firm 2 (which differs only in that its marginal costs are

12 rather than 10) to determine its reaction curve, but we leave the process as a simple

exercise for the reader. We find Firm 2's reaction curve to be:

Q2* = 44 - Q1/2

The solution to the Cournot model lies at the intersection of the two reaction curves. We

solve now forQ 1* . Note that we substitute Q 2

* forQ2 because we are looking for a

point which lies on Firm 2's reaction curve as well.

Q1* = 45 - Q2*/2 = 45 - (44 - Q1*/2)/2 = 45 - 22 + Q1*/4= 23 + Q1*/4 => Q1* = 92/3

By the same logic, we find:

Q2* = 86/3

Again, we leave the actual computation ofQ2* as an exercise for the reader. Note that Q

1* and Q 2

* differ due to the difference in marginal costs. In a perfectly competitive

market, only firms with the lowest marginal cost would survive. In this case, however,Firm 2 still produces a significant quantity of goods, even though its marginal cost is 20%

higher than Firm 1's.

An equilibrium cannot occur at a point not in the intersection of the two reaction curves.If such an equilibrium existed, at least one firm would not be on its reaction curve and

would therefore not be playing its optimal strategy. It has incentive to move elsewhere,

thus invalidating the equilibrium.


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The Cournot equilibrium is a best response made in reaction to a best response and, by

definition, is therefore a Nash equilibrium. Unfortunately, the Cournot model does not

describe the dynamics behind reaching equilibrium from a non-equilibrium state. If thetwo firms began out of equilibrium, at least one would have an incentive to move, thus

violating our assumption that the quantities chosen are fixed. Rest assured that for the

examples we have seen, the firms would tend towards equilibrium. However, we wouldrequire more advanced mathematics to adequately model this movement.

Stackelberg duopoly

The Stackelberg duopoly model of duopolies is very similar to the Cournot model. Like

the Cournot model, the firms choose the quantities they produce. In the Stackelbergmodel, however, the firms do not move simultaneously. One firm holds the privilege to

choose production quantities before the other. The assumptions underlying the

Stackelberg model are as follows:

1. Each firm chooses a quantity to produce.2. A firm chooses before the other in an observable manner.

3. The model is restricted to a one-stage game. Firms choose their quantities only

once.

To illustrate the Stackelberg model, let's walk through an example. Assume Firm 1 is thefirst mover with Firm 2 reacting to Firm 1's decision. We assume a market demand curve

of:

Q = 90 - P

Furthermore, we assume all marginal costs are zero, that is:

MC = MC1 = MC2 = 0

We calculate Firm 2's reaction curve in the same way we did for the Cournot Model.Verify that Firm 2's reaction curve is:

Q2* = 45 - Q1/2

To calculate Firm 1's optimal quantity, we look at Firm 1's total revenues.

Firm 1's Total Revenue = P * Q1 = (90 - Q1 - Q2) * Q1 = 90 * Q1 - Q1 ^ 2 - Q2 * Q1

However, Firm 1 is not forced to assume Firm 2's quantity is fixed. In fact, Firm 1 knowsthat Firm 2 will act along its reaction curve which varies with Q1 . Firm 2's quantity very

much relies on Firm 1's choice of quantity. Firm 1's Total Revenue can thus be rewritten

as a function ofQ1 :

R1 = 90 * Q1 - Q1 ^2 - Q1 * (45 - Q1/2)


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Marginal revenue for firm 1 is thus:

MR1 = 90 - 2 * Q1 - 45 + Q1 = 45 - Q1

When we impose the profit maximizing condition (MR = MC) , we find:

Q1 = 45

Solving forQ2 , we find:

Q2 = 22.5

Although much of the logic behind the Stackelberg model is used in the Cournot model,

the two outcomes are radically different: being the first to announce creates a credible

threat. In the Cournot model, both firms make their choices simultaneously and have no

communication beforehand. In the Stackelberg model, Firm 1 not only announces first,

but Firm 2 knows that when Firm 1 announces, Firm 1's actions are credible and fixed.This demonstrates how a slight change in the flow of information can drastically impact

the outcome of a market.

Bertrand Duopoly

The Bertrand duopoly Model, developed in the late nineteenth century by French

economist Joseph Bertrand, changes the choice of strategic variables. In the Bertrand

model, rather than choosing how much to produce, each firm chooses the price at whichto sell its goods.

1. Rather than choosing quantities, the firms choose the price at which they sell thegood.

2. All firms make this choice simultaneously.3. Firms have identical cost structures.

4. The model is restricted to a one-stage game. Firms choose their prices only once.

Although the setup of the Bertrand Model differs from the Cournot model only in the

strategic variable, the two models yield surprisingly different results. Whereas theCournot model yields equilibriums that fall somewhere in between the monopolistic

outcome and the free market outcome, the Bertrand model simply reduces to the

competitive equilibrium, where profits are zero. Rather than take you through a series of

convoluted equations to derive this result, we will simply show there could be no otheroutcome.

The Bertrand equilibrium is simply the no profit equilibrium. First, we will demonstrate

that the Bertrand outcome is indeed an equilibrium. Imagine a market in which twoidentical firms sell at market price P, the competitive price at which neither firm earns

profits. Implicit in our argument is our assumption that at equal price, each firm will sell

to half the market. If Firm 1 were to raise its price above the market price P, Firm 1


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would lose all its sales to Firm 2 and would have to exit the market. If Firm 1 were to

lower its price below P, it would be operating below cost and therefore at a loss overall.

At the competitive outcome, Firm 1 cannot increase profits by changing its price in eitherdirection. By the same logic, Firm 2 has no incentive to change prices. Therefore, the no

profit outcome is an equilibrium, in fact a Nash equilibrium, in the Bertrand model.

We now demonstrate uniqueness of the Bertrand equilibrium. Naturally, there can be no

equilibrium where profits are negative. In this case, all firms would operate at a loss andexit the market. It remains to be shown that there is no equilibrium where profits are

positive. Imagine a market in which two identical firms sell at market price P, which is

greater than cost. If Firm 1 were to raise its price above the market price P, Firm 1 wouldlose all its sales to Firm 2. However, if Firm 1 were to lower its price ever so slightly

below P (while still remaining above MC), it would capture the entire market at a profit.

Firm 2 is faced with the same incentives, so Firm 1 and Firm 2 would undercut each otheruntil profits are driven to zero. Therefore no equilibrium exists when profits are positive

in the Bertrand model.

Collusion

You may ask yourself why firms don't agree to work together to maximize profits for all

rather than competing amongst themselves. In fact, we will show that firms do benefitwhen cooperating to maximize profits.

Assume both Firm 1 and Firm 2 face the same total market demand curve:

Q = 90 - P

where P is the market price and Q is the total output from both Firm 1 and Firm 2.

Furthermore, assume that all marginal costs are zero, that is:MC = MC1 = MC2 = 0

Verify that the reaction curves according to the Cournot model can be described as:

Q1* = 45 - Q2/2Q2* = 45 - Q1/2

Solving the system of equations, we find:

Cournot Equilibrium: Q1* = Q2* = 30

Each firm produces 30 units for a total of 60 units in the market place. Pis then 30 (recallP= 90 - Q ). Because MC= 0 for both firms, the profit for each firm is simply 900 for atotal profit of 1,800 in the market.

However, if the two firms were to collude and act as a monopoly, they would act

differently. The demand curve and the marginal costs remain the same. They would act


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together to solve for the total profit maximizing quantity Q . Revenues in this market can

be described as:

Total Revenue = P * Q = (90 - Q) * Q = 90 * Q - Q^2

Marginal Revenue is therefore:

MR = 90 - 2 * Q

Imposing the profit maximizing condition (MR = MC) , we conclude:

Q = 45

Each firm now produces 22.5 units for a total of 45 in the market. The market price P is

therefore 45. Each firm makes a profit of 1,012.5 for a total profit of 2,025.

Notice that the Cournot equilibrium is much better for the firms than perfect competition(under which no one makes any profits) but worse than the collusive outcome. Also, the

total quantity supplied is lowest for the collusive outcome and highest for the perfectly

competitive case. Because the collusive outcome is more socially inefficient than thecompetitive oligopoly outcome, the government restricts collusion through anti-trust

laws.

Extension of the Cournot Model

We now extend the Cournot Model of duopolies to an oligopoly where n firms exist.

Assume the following:

1. Each firm chooses a quantity to produce.

2. All firms make this choice simultaneously.

3. The model is restricted to a one-stage game. Firms choose their quantities onlyonce.

4. All information is public.

Recall that in the Cournot model, the strategic variable is the output quantity. Each firm

decides how much of a good to produce. All firms know the market demand curve, andeach firm knows the cost structures of the other firms. The essence of the model: each

firm takes the other firms' choice of output level as fixed and then sets its own production

quantities.

Let's walk through an example. Assume all firms face a single market demand curve asfollows:

Q = 100 - P

wherePis the single market price and Q is the total quantity of output in the market. For

simplicity's sake, let's assume that all firms face the same cost structure as follows:


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MC_i = 10 for all firms I

Given this market demand curve and cost structure, we want to find the reaction curve for

Firm 1. In the Cournot model, we assume Q i is fixed for all firms i not equal to 1. Firm1's reaction curve will satisfy its profit maximizing condition, MR1 = MC1 . In order to

find Firm 1's marginal revenue, we first determine its total revenue, which can bedescribed as follows

Total Revenue = P * Q1 = (100 - Q) * Q1 = (100 - (Q1 + Q2 +...+ Qn)) * Q1= 100 * Q1 - Q1 ^ 2 - (Q2 +...+ Qn)* Q1

The marginal revenue is simply the first derivative of the total revenue with respect to Q1(recall that we assume Qi fori not equal to 1 is fixed). The marginal revenue for firm 1 is

thus:

MR1 = 100 - 2 * Q1 - (Q2 +...+ Qn)

Imposing the profit maximizing condition of MR = MC , we conclude that Firm 1's

reaction curve is:

100 - 2 * Q1* - (Q2 +...+ Qn) = 10 => Q1* = 45 - (Q2 +...+ Qn)/2

Q1* is Firm 1's optimal choice of output for all choices ofQ2 to Qn . We can perform

analogous analysis for Firms 2 through n (which are identical to firm 1) to determinetheir reaction curves. Because the firms are identical and because no firm has a strategic

advantage over the others (as in the Stackelberg model), we can safely assume all would

output the same quantity. Set Q1* = Q2

* = ... = Qn* . Substituting, we can solve forQ1

*

.

Q1* = 45 - (Q1*)*(n-1)/2 => Q1* ((2 + n - 1)/2) = 45

=> Q1* = 90/(1+n)

By symmetry, we conclude:

Qi* = 90/(1+n) for all firms I

In our model of perfect competition, we know that the total market output Q = 90 , the

zero profit quantity. In the n firm case, Q is simply the sum of all Qi* . Because all Qi

*

are equal due to symmetry:

Q = n * 90/(1+n)

As n gets larger, Q gets closer to 90, the perfect competition output. The limit of Q as n

approaches infinity is 90 as expected. Extending the Cournot model to the n firm case

gives us some confidence in our model of perfect competition. As the number of firms

grow, the total market quantity supplied approaches the socially optimal quantity.

introduction to monopolies and oligopolies

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