Chapter 25
Oligopoly
We have thus far covered two extreme market structures – perfect competition where a large
number of small firms produce identical products, and monopoly where a single firm is isolated
from competition through some form of barrier to entry (and through a lack of close substitutes
that could be produced by someone else).1 The models that represent these polar opposites are
incredibly useful because they allow us to develop intuition about important economic forces in the
real world. At the same time, few markets in the real world really fall on either of these extreme
poles, and so we now turn to some market structures that fall in between.
The first of these is the case of oligopoly. An oligopoly is a market structure in which a small
number of firms is collectively isolated from outside competition by some form of barrier to entry.
Just as in the case of monopolies, this barrier to entry may be technological (as, for instance, when
there are high fixed costs) or legal (as when the government regulates competition). We will assume
in this chapter’s analysis of oligopoly that the firms produce the same identical product and will
leave the case where firms can differentiate their products to Chapter 26. Were the firms in the
oligopoly to combine into a single firm, they would therefore become a monopoly just like the one we
analyzed in Chapter 23. Were the barriers to entry to disappear, on the other hand, the oligopoly
would turn to a competitive market as new firms would join so long as positive profits could be
made.
Since there are only a few firms in an oligopoly, my firm’s decision about how much to produce
will have an impact on the price the other firms can charge, or my decision about what price to
set may determine what price others will set. Firms within an oligopoly therefore find themselves
in a strategic setting – a setting in which their decisions have a direct impact on the economic
environment in which they operate. You can see this in how airlines behave as they watch each
other to determine what fares to set or how many planes to devote to particular routes, or in how
the small number of large car manufacturers set their financing packages for new car sales. Below,
we will develop a few different ways of looking at the limited and strategic competition that such
oligopolistic firms face.
1 This chapter builds primarily on Chapter 23 and Section A of Chapter 24. Only Section 25B.3 of this chapter
requires knowledge of Section B from Chapter 24 – and this section can be skipped if you only read Section A of
Chapter 24. The chapter also presumes an understanding of the different types of costs covered in the earlier chapters
on producer theory (as summarized in the first section of Chapter 13) as well as a basic understanding of demand
and elasticity as covered in the first section of Chapter 18.
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25A
Chapter 25. Oligopoly
Competition and Collusion in Oligopolies
While we could think of oligopolies with more than two firms, we will focus here primarily on
the case where two firms operate within the oligopoly market structure (that is then sometimes
called a ”duopoly”). The basic insights extend to cases where there are more than two firms in
the oligopoly – but as the number of firms gets large, the oligopoly becomes more and more like
a perfectly competitive market structure. We will also simplify our analysis by assuming that the
two firms are identical (in the sense of facing identical cost structures) and that the marginal cost
of production is constant. In end-of-chapter exercises, we then explore how our results are affected
by changing these baseline assumptions.
To fix ideas, lets think of the following concrete situation: I am a producer of economist cards
but I recently discovered that you are also producing identical cards. Suppose both of us applied for
a copyright on this idea and, since we both applied at the same time, the government has granted
both of us the copyright but will not grant it to anyone else. For some inexplicable reason that
suggests a general lack of sophistication on the part of the general public, the only people who buy
these cards are economists who attend the annual American Economic Association (AEA) meetings
every January — and you and I therefore have to determine our strategy for selling cards at these
meetings.
Each of our firms in this oligopoly then has, essentially, two choices to make: (1) how much to
produce and (2) how much to charge. It might be that it’s really easy to duplicate the cards at the
AEA meetings, in which case we might decide to simply post a price at our booth and produce the
cards as needed. In this case, price is the strategic variable that we are setting prior to getting to the
meetings as we advertise to the attendees to try to get them to come to our booth. Alternatively,
it might be that we have to produce the cards before we get to the AEA meetings because it’s not
possible to produce them on the spot as needed. In that case, quantity is the strategic variable since
we have to decide how many cards to bring prior to getting to the meetings, leaving us free to vary
the price depending on how many people actually want to buy cards when we get there. Whether
price or quantity is the right strategic variable to think about then depends on the circumstances
faced by the firms in an oligopoly – on what we will call the “economic setting” in which the firms
operate. We will therefore develop two types of models – models of quantity competition and models
of price competition.
The other feature of oligopoly models is that they either assume that the firms in the oligopoly
make their strategic decision simultaneously or sequentially. Maybe it takes me longer to get my
advertising materials together and I therefore end up posting my price after you do, or maybe I
work in a local market where I have to set the capacity for producing a certain quantity of cards
before you do. As we have seen in our discussion of game theory, we can employ the concept of
Nash equilibrium for the case of simultaneous decision making while we use the concept of subgame
perfect (Nash) equilibrium in the case of sequential decisions. Sometimes, as we will see, it matters
who moves first.
We therefore have four different types of models we will discuss: (1) price competition where
firms make strategic decisions about price simultaneously; (2) price competition where firms make
strategic decisions about price sequentially; (3) quantity competition where firms make strategic
decisions about quantity simultaneously; and (4) quantity competition where firms make strategic
decisions about quantity sequentially. We will begin with price competition and then move to
quantity competition, each time considering both the simultaneous and the sequential case, and
we will see that firms could in principle do better by simply combining forces and behaving like
25A. Competition and Collusion in Oligopolies
967
a single monopoly. Following our discussion of oligopoly price and quantity competition, we will
therefore consider the circumstances under which oligopoly firms might succeed in forming cartels
that behave like monopolies by eliminating competition between the firms in the oligopoly.
25A.1
Oligopoly Price (or “Bertrand”) Competition
Competition between oligopoly firms that strategically set price (rather than quantity) is often referred to as Bertrand competition after the French mathematician Joseph Louis Francois Bertrand
(1922-1900). Bertrand took issue with another French mathematician, Antuine Augustin Cournot
(1801-1877) whose work on quantity competition (which we discuss in the next section) had suggested that oligopolies would price goods somewhere between where price would fall under perfect
competition and perfect monopoly. Bertrand came up with a quite different and striking conclusion:
he suggested that Cournot had focused on the wrong strategic variable – quantity – and that his
result goes away when firms instead compete on price. In particular, Bertrand argued that such
price competition will result in a price analogous to what we would expect to emerge under perfect
competition (price equal to marginal cost) even if only two firms are competing with one another.
25A.1.1
Simultaneous Strategic Decisions about Price
Bertrand’s logic is easy to see in a model with two identical firms that make decisions simultaneously
and face a constant marginal cost of production (with no recurring fixed cost). Suppose we face no
real fixed costs and we can easily adjust the quantity of cards we produce on the spot at the AEA
meetings. We therefore decide to advertise a price and produce whatever quantity is demanded by
consumers at that price. But as we think about announcing a price, we have to think about what
price the other might announce and how consumers might react to different price combinations.
One conclusion is pretty immediate: If we announce different prices, then consumers will simply
flock to the firm that announced the lower price – and the other firm won’t be able to sell anything.
I will therefore want to avoid two scenarios: First, I don’t want to set a price that is so low
that it would result in negative profits if I managed to attract consumers at this price. Since we
are assuming no recurring fixed costs and constant marginal costs, this means I don’t want to set a
price below marginal cost. Second, assuming your firm similarly won’t set a price below marginal
cost, I don’t want to set a price higher than what you set – because then I don’t get any customers.
Put differently, whatever price you set, it cannot be a “best response” for me to set a higher price or
a price below marginal cost. The same is true for you – which means that, in any Nash equilibrium
in which we both do the best we can given the strategy played by the other, we will charge identical
prices that do not fall below marginal cost.
But we can say more than that. Suppose that the price announced by both of us is above
marginal cost. Then I am not playing a “best response” – because, given that you have announced
a price above marginal cost, I can do better by charging a price just below that and getting all the
customers. The only time this is not true is if both firms are announcing the price equal to marginal
cost. Given that you are charging this price, I can do no better by charging a lower price (which
would result in negative profits) or a higher price (which would result in me getting no customers).
The same is true for your firm given that I am charging a price equal to marginal cost. Thus, by
each announcing a price equal to marginal cost, we are both playing “best response” strategies to
the other – and the outcome is a Nash equilibrium.
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Chapter 25. Oligopoly
Exercise 25A.1 Can you see how this is the only possible Nash equilibrium? Is it a dominant strategy
Nash equilibrium?
Exercise 25A.2 Is there a single Nash equilibrium if more than two firms engage in Bertrand competition
within an oligopoly?
25A.1.2
Using “Best Response Functions” to verify Bertrand’s Logic
While the logic behind Bertrand’s conclusion that price competition leads oligopolistic firms to
behave competitively is straightforward, this is a good time to develop a tool that will be useful
throughout our discussion of oligopoly: best response functions. These functions are simply plots of
the best response of one player to particular strategic choices by the other. They are useful when
players have a continuum of possible actions they can take in a simultaneous move game rather than
a discrete number of actions as in most of our game theory development in Chapter 24. When best
response functions for both players are then plotted on the same graph, they can help us identify
the Nash equilibria easily.
Suppose I am firm 1 and you are firm 2. Consider panel (a) of Graph 25.1. On the horizontal
axis, we plot p1 – the price set by me, and on the vertical axis we plot p2 – the price charged by you.
We then plot your best responses to different prices I might announce. We already know that you
will never want to set a price below marginal cost (M C), and if I were to ever be stupid enough to
set a price below M C, any p2 > p1 would be a best response for you (since it would simply result
in you not selling anything and letting me get all the business.) For purposes of our graph, we can
then simply let your best response to p1 < M C be p2 = M C. If I announce a price p1 above M C,
we know that you will want to charge a price just below p1 to get all consumers away from my
booth. Thus, for p1 > M C, your best response is p2 = p1 − ǫ (where ǫ is a small number close to
zero). Since p1 = p2 on the 45-degree line in the graph, this means that your best response in panel
(a) will lie just below the 45-degree line for p1 > M C.
Graph 25.1: Best Response Functions for Simultaneous Bertrand Competition
In panel (b) of Graph 25.1, we do the same for my firm – only now p2 (on the vertical axis) is
taken as given by firm 1, and firm 1 finds its best response to different levels of p2 . If you set your
25A. Competition and Collusion in Oligopolies
969
price below M C, my best response can then be taken to simply be p1 = M C, and if you set your
price p2 above M C, my best response is p1 = p2 − ǫ (which lies just above the 45-degree line).
We defined a Nash equilibrium in Chapter 24 as a set of strategies for each player that are
best responses to each other. In order for an equilibrium to emerge in our price setting model, my
price therefore has to be a best response to your price, and your price has to be a best response to
my price. Put differently, when we put the two best response functions onto the same graph in
panel (c), the equilibrium happens where the two best response functions intersect. This happens
at p1 = p2 = M C just as we derived intuitively above.
25A.1.3
Sequential Strategic Decisions about Price
In the real world, it is often the case that one firm has to make a decision about its strategic variable
before the other – with the second firm being able to observe the first firm’s decision when its turn
to act comes. As we argued in our chapter on game theory, sometimes this makes a big difference –
with the first mover gaining an advantage (or disadvantage) from having to declare its intentions
in advance of the second mover. It’s easy to see that this is not, however, the case for our two firms
engaging in Bertrand competition.
Suppose I move first and you get to observe my advertised price before you advertise your own.
Remember that in such sequential settings, subgame perfection requires that I will have to think
through what you will do for any action I announce. But our discussion above already tells us the
answer: you will choose a price just below p1 whenever p1 > M C, leaving me with no consumers.
Since I will not choose a price below M C, this implies that I will set p1 = M C and you will follow
suit – with our two firms splitting the market by charging prices exactly equal to M C.
Exercise 25A.3 How would you think about subgame perfect equilibria under sequential Bertrand competition with 3 firms (where firm 1 moves first, firm 2 moves second and firm 3 moves third)?
25A.1.4
Real-World Caveats to Bertrand’s Price Competition Result
While Bertrand’s logic is intuitive, few economists believe that his result is one that truly characterizes many real world oligopoly outcomes. There are several real-world considerations that
considerably weaken the Bertrand prediction regarding price competition in oligopolies, and here
we will briefly mention some of them. (In end-of-chapter exercises, we additionally explore how the
Bertrand predictions change with different assumptions about firm costs.)
First, the pure Bertrand model assumes that firms are able to produce any quantity demanded
at the price that they announce. This might in fact be true in some markets but typically does not
hold. As a result, real world firms have to set some “capacity” of production as they think about
announcing a price, and this capacity choice, as we will again mention in Section 25A.2.2, then
introduces quantity as a strategic variable. In cases where capacity choices are in fact binding on
the Bertrand competitors, the model predicts that each firm will again announce the same price
but that this price will be above marginal cost in much the way that it is under strict quantity
competition (as we will demonstrate in the next section).2 Second, we have assumed throughout
that the two firms in our oligopoly interact only one time, whether simultaneously or sequentially.
But in the real world, firms typically interact repeatedly – which implies that price competition
2 This “solution” to the Bertrand “Paradox” of p = M C was first developed by Francis Edgeworth (1845-1926) at
the end of the 19th century and has since been formalized using modern economic tools.
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Chapter 25. Oligopoly
of the type envisioned by Bertrand occurs in a repeated game context. Again, we would expect an
equilibrium in which the firms in the oligopoly announce the same price in each period. In the
non-repeated game, we concluded that the only such equilibrium price has to be equal to marginal
cost because, were this not the case, neither firm is “best responding” to the strategic choice of
the other. But now suppose that firms are engaged in repeated price competition and consider
whether p > M C could emerge in a given period. A “strategy” for each firm must then specify
a price for any possible previous price history, which opens the possibility of “trigger strategies”
of the following form: I will begin our repeated interactions by charging a price p > M C and
will continue to do so in future periods as long as that price has been played by both of us in all
previous periods; otherwise, I will charge p = M C forever. Suppose we both play this strategy.
Then, in any given period, I have to weigh whether the short run gain from charging a price slightly
below p (which results in me getting all the customers this period) outweighs the long-run cost of
reverting to p = M C in all future periods. It is quite plausible that this short run benefit is smaller
than the long run cost – which would make my strategy a best response to yours (and yours a
best response to mine). In infintiely repeated interactions, or in interactions where there is a good
chance we will meet again, we can therefore see how p > M C can emerge as an equilibrium under
price competition.
Exercise 25A.4 Suppose our two firms know that we will encounter each other n times and never again
thereafter. Can p > M C still be part of a subgame perfect equilibrium in this case assuming we engage in
pure price competition?
Finally, Bertrand assumed that firms are restricted to producing identical products. If we allow
for the possibility that consumers differ somewhat in their tastes for how economist cards look and
what exactly they say on the back, we might however decide to produce slightly different versions of
economist cards – and through such product differentiation become able to charge p > M C. This
is because consumers that have a strong preference for my type of card will still buy from me at
a somewhat higher price, and similarly those with a preference for your type of card will continue
to buy yours at a somewhat higher price. Product differentiation therefore also introduces the
possibility of p > M C emerging under price competition. We will develop this more in Chapter 26.
25A.2
Oligopoly Quantity Competition
The implicit assumption that underlies Bertrand competition is that firms can easily adjust quantity
once they set price. In our example, we assumed that we can both just produce the required cards
on the spot at the AEA meetings. But, as we just mentioned, many firms have to set capacity
for their production and, once they have done so, cannot easily deviate from this in terms of how
much they will produce. It might be hard for us to have our card factory at our booth at the AEA
meetings, which means we will have to produce our cards ahead of time and bring them with us
to our booths. In such circumstances, it is more reasonable to assume that firms choose capacity
(or “quantity”) first and then sell what they produce at the highest price they can get. This is the
scenario that Cournot had in mind when he investigated competition between oligopolistic firms,
and it is the scenario we turn to next. As we will see, this model, known as the Cournot Model, has
very different implications regarding the equilibrium price at which oligopolistic firms produce. As
in the previous section, we will continue by assuming that firms in our oligopoly are identical and
face constant marginal cost.
25A. Competition and Collusion in Oligopolies
25A.2.1
971
Simultaneous Strategic Decisions about Quantity: Cournot Competition
We can again use best response functions to see what Nash equilibrium will emerge when two firms
in an oligopoly choose capacity simultaneously. In panel (a) of Graph 25.2, we begin by considering
firm 2’s best response to different quantities x1 set by firm 1. If I set x1 = 0, then you would
know that you will have a monopoly on economist cards at the AEA meetings. From our work in
Chapter 23, we can then easily determine the optimal quantity for you by solving the monopoly
problem. This is depicted in panel (b) of the graph where D is the market demand curve and M R
is your monopoly’s marginal revenue curve that has the same intercept (as D) but twice the slope.
Your firm, firm 2, would then produce the monopoly quantity xM where M R = M C (and charge
the monopoly price pM ). The quantity xM therefore becomes your best response to x1 = 0 and
determines the intercept of your best response function in panel (a).
Graph 25.2: The Best Response Function for Firm 2 under Simultaneous Cournot Competition
Now suppose I set x1 = x1 > 0. You then know that you no longer face the entire market demand
curve because I have committed to filling x1 of the market demand. Put differently, you now face a
demand curve that is equal to the market demand curve D minus x1 . In panel (c) of Graph 25.2, we
therefore shift the demand D by x1 to get the new “residual” demand Dr that remains given that I
will satisfy a portion of market demand. From this, we can calculate the residual marginal revenue
curve M Rr that now applies to your firm. Once again, you will maximize profit where marginal
revenue equals marginal cost; i.e. M Rr = M C. This results in a new optimal quantity given x1 –
denoted x2 (x1 ), which in turn becomes your best response to me having set x1 = x1 . Note that
x2 (x1 ) necessarily lies below xM — i.e. your best response quantity decreases as x1 increases. We
can imagine doing this for all possible quantities of x1 to get the full best response function for
your firm 2 as depicted in panel (a).
Exercise 25A.5 Can you identify in panel (b) of Graph 25.2 the quantity that corresponds to the horizontal
intercept of firm 2’s best response function in panel (a)?
Exercise 25A.6 What is the slope of the best response function in panel (a) of Graph 25.2? (Hint: Use
your answer to exercise 25A.5 to arrive at your answer here.)
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Chapter 25. Oligopoly
We can then do what we did for Bertrand competition by putting the best response functions of
the two firms together into one graph to see where they intersect. Since our two firms are identical,
my best response function can be similarly derived. This is done in panel (a) of Graph 25.3, which
is just the mirror image of the best response function for your firm that we derived in the previous
graph. The two best response functions then intersect at x1 = x2 = xC in panel (b), with xC the
Cournot-Nash equilibrium output for each of our firms in the oligopoly.
Graph 25.3: Simultaneous Move Cournot-Nash Equilibrium
25A.2.2
Comparing and Reconciling Cournot, Bertrand and Monopoly Outcomes
In panel (c) of Graph 25.3 we can then see how the quantities produced under monopoly, Cournot
and Bertrand competition compare. As illustrated in panel (b), C represents each firm’s output
under Cournot (or quantity) competition. From constructing the best response functions, we know
that the vertical intercept of firm 2’s best response function is the monopoly quantity, as is the
horizontal intercept of firm 1’s best response function. When we connect these (with the dashed
magenta line in panel (c)), we get all combinations of firm 1 and firm 2 production that sum to
the monopoly quantity. Were the two firms to collude, for instance, and simply split the monopoly
quantity, they would produce half of xM at the point labeled M . Thus, production is unambiguously
higher under Cournot competition than it would be under monopoly production.
We can also see how Cournot production compares to Bertrand production. From our work in
the last section we know that Bertrand or price competition results in both firms charging a price
equal to M C. At such a price, market demand will be equal to x∗ in panel (b) of Graph 25.2. Now
suppose that, under Cournot competition, firm 2 determines its best response to firm 1 setting its
quantity to x∗ . This would imply that firm 2’s residual demand is equal to D shifted inward by
x∗ , leaving it with a residual demand curve that has a vertical intercept at M C. Thus any output
that firm 2 would produce given that firm 1 is producing x∗ would have to be sold at a price below
M C — which implies firm 2’s best response is to produce x2 = 0. This implies that firm 2’s best
response function reaches zero at x1 = x∗ = 2xM ; i.e. the horizontal intercept of firm 2’s best
response function lies at x∗ . (Note: This is the answer to within-chapter-exercise 25A.5.) Since the
two firms are identical, the same is true for firm 1’s vertical intercept.
25A. Competition and Collusion in Oligopolies
973
If we connect the horizontal intercept of firm 1’s best response function with the vertical intercept
of firm 2’s best response function (with the dashed blue line) in panel (c), we then get all the different
ways in which the two firms could split production and produce x∗ , the quantity that would be
sold when p = M C as happens under Bertrand competition. If we assume that, when both firms
charge the Bertrand price of p = M C, the two firms split overall output, each firm would produce
half of x∗ as indicated at point B in the graph. Thus, Bertrand competition leads to unambiguously
higher output than Cournot competition.
Exercise 25A.7 Which type of behavior under simultaneous decision making within an oligopoly results in
greater social surplus: quantity or price competition?
B
M
Exercise 25A.8 True or False: Under Bertrand competition, xB
1 = x2 = x .
As we will note again in Chapter 26, the dramatic difference between the Bertrand and Cournot
competition seems quite strange, and it is not easy to choose between the two models on intuitive
grounds: On the one hand, it seems that firms in the real world often set prices (when they are
not in perfectly competitive settings), and this seems to speak in favor of the Bertrand model.
(In Chapter 26, for instance, I give the example of Apple coming out with a new computer and
immediately setting its price long before it finds out how much it will have to produce.) On the other
hand, the Bertrand prediction of price being set equal to marginal cost even when only two firms
are competing seems a stretch, which speaks in favor of the Cournot model which not only arrives
at the intuitively reasonable prediction that price falls between the monopoly and the competitive
level when there are only two firms but also predicts (as we will show in Section B) that oligopoly
prices converge to competitive prices as the number of firms in the oligopoly becomes large. Much
work has, as a result, been done by economists to reconcile these models of oligopoly competition.
One of the most revealing results, which we already mentioned in our discussion of Bertrand
competition, is the following: Suppose that firms really do set prices (as the Bertrand model
assumes) but they set capacities for production (which sounds a lot like the quantity setting of the
Cournot model) before announcing prices. Then under plausible conditions, it has been shown that
this Bertrand equilibrium outcome of price competition results in Cournot quantities and prices.3
Economists have therefore often come to view oligopoly competition as guided in the long run by
production capacity competition (as envisioned by Cournot) equilibrated through price competition
(as envisioned by Bertrand) in the short run when capacities are fixed. Both models appear to have
their place, and both play important roles in how we think of oligopoly competition.
25A.2.3
Sequential Strategic Decisions about Quantity: The “Stackelberg” Model
Under Bertrand competition, we concluded that it does not matter whether firms determine their
price simultaneously or sequentially – in either case, firms end up charging p = M C in equilibrium.
The same is not true for quantity competition, as we will see now.
The sequential quantity competition model is known as the Stackelberg model,4 and the firm
designated to “move first” is called the Stackelberg leader while the firm that moves second is called
the Stackelberg follower. In sequential move games, we concluded in Chapter 24 that non-credible
threats are eliminated by restricting ourselves to Nash equilibria that are subgame-perfect – i.e. to
equilibria in which early movers look forward and determine the best responses by their opponents
3 This was demonstrated by Kreps, D. and J. Scheinkman (1983), “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,” Rand Journal of Economics 14, 326-37.
4 The model is named after Heinrich Freiherr von Stackelberg (1905-46), a German economist.
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Chapter 25. Oligopoly
later on in the game. When she decides how much capacity to set, the Stackelberg leader will
then take into account the entire best response function of the follower because that function tells
the leader exactly how the follower will respond once she finds out how much the leader will be
producing. Thus, rather than “guessing” about the quantity the opposing firm will set (as is the
case under simultaneous quantity competition), the leader now has the luxury of inducing how
much the follower will set by her own actions in the first stage.
Suppose, then, that you – firm 2 – are the follower and I – firm 1 – am the leader. I already
know your best response function for any quantity that I might set – we derived this in Graph 25.2a
which we now replicate in panel (a) of Graph 25.4. In deciding how much capacity to set, I then
simply have to determine my residual demand curve given your best response function. The grey
demand curve D in panel (b) is simply the market demand curve. For any output level x1 ≥ x∗ ,
we know that your best response is simply not to produce, which implies that I know I will “own”
the market demand curve if I chose to produce above x∗ . Thus, my residual demand is equal to
market demand for quantities greater than x∗ .
If I set capacity below x∗ , however, I know that you will produce along your best response
function once you find out how much capacity I set. To arrive at my residual demand, I therefore
have to subtract the quantity that I know you will produce for any x1 < x∗ . If I set my capacity
close to x∗ , you will choose to produce relatively little, but as x1 falls, your best response quantity
rises and reaches xM , the monopoly quantity, when x1 = 0. My residual demand curve Dr therefore
begins at the monopoly price pM (which would be charged by you if I set x1 = 0) and reaches the
market demand curve D when it crosses M C.
Once we have figured out firm 1’s residual demand, we can now do what we always do to identify
my firm’s optimal capacity: simply plot out the M Rr curve that corresponds to Dr and find its
intersection with M C. Because all the relationships are linear, this intersection occurs at half
the distance between x∗ and zero – which happens to be the monopoly quantity xM . Thus, the
Stackelberg leader, firm 1, will set x1 = xM , and the Stackelberg follower will produce half this
amount as read off its best response function. Given what I as the leader have done in the first
stage, you as the follower are doing the best you can, and given your predictable output decisions
in the second stage (as summarized in your best response function), I have done the best I can. We
have reached a sub-game perfect equilibrium.
Exercise 25A.9 Determine the Stackelberg price in terms of pM – the price a monopolist would charge –
and M C.
Adding this outcome to our predicted outputs for Bertrand, Cournot and monopoly settings
from Graph 25.3, we can then see that the Stackelberg quantity competition results in greater
overall output than simultaneous Cournot competition but less overall output than Bertrand price
competition.
Exercise 25A.10 Where is the predicted Stackelberg outcome in Graph 25.3c?
25A.2.4
The Difference between Sequential and Simultaneous Quantity Competition
We can now step back a little and ask why the Stackelberg model differs fundamentally from the
Cournot model. Why, for instance, don’t I threaten to act like a Stackelberg leader when you and
I are competing simultaneously?
Suppose you and I set quantity simultaneously before we arrive at the AEA meetings, but I
call you ahead of time and tell you that I will produce the Stackelberg leader quantity. Would you
25A. Competition and Collusion in Oligopolies
975
Graph 25.4: Stackelberg Equilibrium
have any reason to believe me when I threaten to do this? The answer is that you should not take
my threat seriously. After all, if you thought that I thought you would produce xM /2, my best
response (according to my best response function in Graph 24.5) would be to produce less than
xM ! (You can see this in panel (c) of the graph where the horizontal (dashed) grey line that passes
through M at an output level of xM /2 for you crosses my best response function to the left of xM .)
Your best response to me producing less than xM would then be to produce more than xM /2. My
threat to produce xM is therefore simply not credible when I try to bully you over the phone.
When the game assumes a sequential structure, however, the threat becomes real because you
know how much I have produced by the time that you have to decide how much to produce. It’s
no longer an idle threat for me to say I will produce the Stackelberg leader quantity – I have just
done so. Now it is indeed a best response for you to produce the Stackelberg follower quantity,
and given that you will do so it is best for me to have produced the Stackelberg leader quantity. It
is the sequential structure of the game that results in the difference in equilibrium behavior, and
without that sequential structure, there is no way for me to credibly threaten to do anything other
than produce the Cournot quantity.
976
25A.3
Chapter 25. Oligopoly
Incumbent Firms, Fixed Entry Costs and Entry Deterrence
The insight that the sequential structure of the oligopoly quantity competition changes the outcome
of that competition can then get us to think of other ways in which sequential decision-making might
matter. An important case is the case in which one firm is the incumbent firm that currently has
the whole market but is threatened by a second firm that might potentially enter the market and
turn its structure from a monopoly to an oligopoly. Is there anything (aside from sending someone
with a baseball bat) the incumbent firm can do to prevent the potential entrant from coming into
the market? The answer depends on two factors: (1) how costly is it for the potential entrant to
actually enter the market and begin production, and (2) to what extent can the incumbent firm
credibly threaten the potential entrant.
25A.3.1
Case 1: Incumbent Quantity Choice follows Entrant Choice
Suppose the potential entrant has to pay a one-time fixed entry cost F C in order to be able to begin
production. Now consider the case in which the potential entrant makes her decision on whether
to enter the market before either firm makes a choice about how much to produce. Panels (a) and
(b) in Graph 25.5 picture two such scenarios. In both panels, firm 2 first decides whether or not
to enter, and if she does not enter, firm 1 sets its quantity x1 . If firm 2 does enter, the firms are
assumed to choose their production quantities simultaneously in panel (a) and sequentially in panel
(b).
Graph 25.5: Possible Sequences of Entry and Quantity Choices
25A. Competition and Collusion in Oligopolies
977
Recall that we solve games of this kind from the bottom up in order to find subgame perfect
equilibria. If firm 2 does not enter, we know that firm 1 will optimize by simply producing the
monopoly quantity and thus will make the monopoly profit π M while firm 2 will make zero profit. If
firm 2 enters, on the other hand, the two firms will engage in simultaneous Cournot competition in
panel (a), with each firm making the Cournot profit π C but with firm 2 paying the fixed entry cost
F C. Firm 2 therefore looks ahead and makes its entry decision based on whether or not (π C − F C)
is greater than zero. Put differently, so long as the profit from producing the Cournot quantity at
the Cournot price is greater than the fixed cost of entering, firm 2 will enter the market. Similarly,
in panel (b), firm 2 knows that she will be a Stackelberg follower if she enters, and so she will enter
so long as the profit π SF from producing the Stackelberg follower quantity at the Stackelberg price
is greater than the fixed cost of entering.
Exercise 25A.11 True or False: Once the entrant has paid the fixed entry cost, this cost becomes a sunk
cost and is therefore irrelevant to the choice of how much to produce.
Exercise 25A.12 Is the smallest fixed cost of entering that will prevent firm 2 from coming into the market
greater in panel (a) or in panel (b)?
Notice that in neither of these cases can the incumbent firm (firm 1) do anything to affect firm
2’s entry decision because the entry decision happens before quantities are set. This implies that
firm 2’s entry decision is entirely dependent on the size of the fixed entry cost F C. The problem
(from firm 1’s perspective) is once again that there is no way it can credibly threaten firm 2, a
problem that can disappear if firm 1 gets to commit to an output quantity before firm 2 makes its
entry decision (as we will see next).
25A.3.2
Case 2: Entry Choice follows Incumbent Quantity Choice
Now consider the sequence pictured in panel (c) of Graph 25.5 where the incumbent (firm 1)
chooses its quantity x1 before the potential entrant (firm 2) makes its decision on whether to enter
the market and produce. Again, we can solve the resulting game from the bottom up, beginning
with the case in which firm 2 has decided to enter the market. Firm 2’s optimal quantity is then
simply given by its best response function (derived in Graph 25.2) to the quantity set by firm 1
(which is known to firm 2 at the time it makes its quantity decision). Firm 1 knows firm 2’s best
response function – which implies that if firm 2 enters the market, firm 1 is simply a Stackelberg
leader. Thus, if firm 2 enters, the equilibrium payoffs are the Stackelberg profits, π SL and π SF ,
minus the fixed entry cost for firm 2.
The incumbent firm, however, would very much like to remain the only firm in the market.
Short of sending in big guys with baseball bats to beat up firm 2, the only way to persuade firm 2
to stay out of the incumbent’s (monopoly) market is for the incumbent to insure that firm 2 cannot
make a positive profit by entering. And the only way to do that is to commit to producing a larger
quantity in order to drive the price down sufficiently to keep firm 2 from wanting to come into the
market. Whether it is possible for firm 1 to do this and thereby to make a profit higher than that
of a Stackelberg leader depends on just how big the fixed entry cost F C is for firm 2.
This is illustrated in the two panels of Graph 25.6. In panel (a), we plot the profit that the
incumbent can expect from different output levels if it remains the only firm in the market. The
highest possible profit occurs at the monopoly quantity xM (which, as we have seen, is also the
Stackelberg leader quantity xSL ). If the fixed entry cost is very high, the incumbent can simply
produce xM and rest assured in its monopoly given that it is simply too costly for any potential
978
Chapter 25. Oligopoly
entrant to enter the market. This is illustrated in panel (b) where, for F C ≥ F C, firm 1 produces
xM while firm 2 stays out of the market (and thus produces zero). If the fixed entry cost is very
low, on the other hand, there is little that firm 1 can do to keep the entrant out of the market –
and so firm 1 simply produces the Stackelberg leader quantity xSL and accepts firm 2’s production
of the Stackelberg follower quantity xSF . This is illustrated in panel (b) for F C ≤ F C.
Graph 25.6: Setting Quantity to Deter Entry
The interesting case of entry deterrence arises for fixed entry costs between F C and F C. Suppose, for instance, that F C is just below F C – i.e. suppose that firm 2 would make a slightly
positive profit by entering if firm 1 behaved like a Stackelberg leader and produced xSL . If firm 1
then produces just a little more than xSL , this will insure that firm 2 can no longer make a positive
profit by entering. The incumbent firm can therefore deter entry by producing above xSL . While
this will mean that firm 1’s profit falls below the monopoly profit, it is preferable to engaging in
Stackelberg competition with firm 2 (in which case firm 1 would only get π SL ). As the fixed entry
cost falls, it becomes harder and harder for firm 1 to do this – necessitating higher and higher
levels of output to deter entry. But it’s worth it as long as the incumbent’s profit remains above
the Stackelberg leader profit π SL . Thus, the highest quantity that firm 1 would ever be willing
SL
to produce to deter entry, xED
. When fixed entry costs fall
max , is the quantity that will insure π
below F C, it is too costly for the incumbent to deter entry – and firm 1 reverts back to producing
simply the Stackelberg leader quantity.
This is, then, a more rigorous treatment of an idea that we raised in Chapter 23 when we
discussed the possibility that a monopoly might be restrained in its behavior (and might produce
more than the monopoly quantity) if it feels threatened by potential competitors. Notice that, if it
could, the incumbent firm would like to reduce its output back to the monopoly quantity xM once
it has successfully deterred an entrant, but the only way that deterrence could succeed is if the
incumbent was able to commit to not doing so by setting output prior to firm 2’s entry decision. It
is this commitment that made the threat to the entrant credible – were it possible to then go back
on the commitment, the threat would not be credible and entry could not be deterred.
It is a little like the general that would like to strike fear into the opposing army on the battlefield
by telling them that his army will fight to the death. Of course just saying “We will fight to the
25A. Competition and Collusion in Oligopolies
979
death!” is not credible – anyone can say it. So the general might cross a bridge into the battlefield
and then burn the bridge down – thus cutting off any possibility of retreat. This would certainly
make the threat to fight to the death more credible – just as the incumbent firm’s threat to increase
production to prevent entry becomes credible when the firm actually does it and thus cuts off any
possibility of retreat.
25A.4
Collusion, Cartels and Prisoner’s Dilemmas
So far, we have assumed that you and I will act as competitors within the oligopoly – strategically
competing on either price or quantity decisions. Now suppose instead, however, that I call you
before the AEA meetings and say: “Why don’t we stop competing with each other and instead
combine forces to see if we can’t do better by coordinating what we do?”
Logically, we should be able to do better if we don’t compete. After all, if we could act like
one firm that has a monopoly, we would be able to do at least as well as we can do if we compete
by simply producing the same quantity as we do under oligopoly competition. But we know from
Graph 25.3c that as a monopoly we would produce less than we do under Cournot, Stackelberg or
Bertrand competition. Our joint profit would therefore be higher if we could find a way of splitting
monopoly production and charging a higher price than it would be under any competitive outcome
that results in a price below the monopoly price. We therefore have an incentive to find a way to
collude instead of compete.
25A.4.1
Collusion and Cartels
A cartel is a collusive agreement (between firms in an oligopoly) to restrict output in order to raise
price above what it would be under oligopoly competition. The most famous cartel in the world is
OPEC – the Organization of Petroleum Exporting Countries – which is composed of countries that
produce a large portion of the world’s oil supply. Oil ministers from OPEC countries routinely meet
to set production quotas for each of the countries. Their claim is to aim for a stable world price of
oil, but what they really aim for is a high price for oil. There are many other examples of attempts
by producers of certain goods to form cartels, some of which we will analyze in end-of-chapter
exercises.
Suppose our two little firms are currently engaged in Cournot competition, with each of us
producing xC as depicted in Graph 25.3b. It’s then easy to see how we can do better – all we have
to do is figure out what the monopoly output level xM would be and agree to each limit our own
production to half of that. This would allow us to sell our economist cards at the AEA meetings
at the monopoly price pM , with each of us making half the profit we would if our individual firm
was the sole monopoly. The same cartel agreement would make each of us better off if we currently
engaged in Bertrand competition.
Exercise 25A.13 * How might the cartel agreement have to differ if we were currently engaged in Stackelberg competition? (Hint: Think about how the cartel profit compares to the Stackelberg profits for both
firms, and use the Stackelberg price you determined in exercise 25A.9 along the way.)
25A.4.2
A Prisoner’s Dilemma: The Incentive of Cartel Members to Cheat
Suppose, then, that you and I enter a collusive cartel agreement and decide to each produce half of
xM in order to maximize our joint profit. It is certainly in our interest to sign such an agreement.
980
Chapter 25. Oligopoly
But is it optimal for us to stick by our agreement as we prepare to come to the AEA meetings with
our economist cards?
Suppose I believe you will stick by the agreement. We can then ask what I would have to
gain from producing one additional set of economist cards above the quota we set in our cartel.
In panel (a) of Graph 25.7, we assume that we we have agreed to behave as a single monopolist,
jointly producing xM which allows us to sell all our cards at price pM . Were we, as a monopoly, to
produce one more set of cards, we would have to drop the price in order to sell the larger quantity.
This would result in a loss of profit equal to the magenta area since we can no longer sell the initial
xM goods at the price pM . It would also result in an increase in profit equal to the blue area since
we get to sell one more set of cards. For a monopoly, the quantity xM is profit maximizing because
the magenta area is slightly larger than the blue area – i.e. our monopoly profit would fall if we
produced one more set of cards.
Graph 25.7: The Incentive to Cheat on a Cartel Agreement
But now think of the question of whether to produce one more set of cards from the perspective
of one of the members of the cartel that has agreed to behave as a single monopolist. In our cartel
agreement, we agreed that I would produce half of the monopoly output level xM and you would
produce the other half. If you produce one more set of cards, you will therefore lose only half the
magenta area in profit from having to accept a price slightly lower than pM for the half of xM you
are producing under the cartel agreement, but you would get all of the blue area in additional profit
from the additional unit you produce. Since the magenta area is only slightly larger than the blue
area, half of the magenta area is certainly smaller than all of the blue area in the graph — which
means your profit will increase if you cheat and produce one more set of cards than you agreed to
in the cartel.
Panel (b) looks at this another way and asks not only whether it would be in your best interest
to produce one unit of output beyond the cartel agreement but how much more you would in fact
want to produce assuming you believe that I will be a sucker and stick by the agreement to produce
only half of xM . The residual demand Dr that you would face given that I produce x1 = 0.5xM
is equal to the market demand D minus 0.5xM which intersects M C at the quantity 1.5xM . The
corresponding residual marginal revenue curve M Rr has twice the slope and therefore intersects
M C at 0.75xM – implying that it would be optimal for you to produce 0.75xM rather than 0.5xM
25A. Competition and Collusion in Oligopolies
981
as called for in your cartel agreement. Put differently, if you believe I will produce 0.5xM , your
best response is to produce 0.75xM .
Exercise 25A.14 Can you verify the last sentence by just looking at the best response functions we derived
earlier in Graph 25.2?
Now, if you are smart enough to figure out that it is in your best interest to cheat on the cartel
agreement, chances are that I am smart enough to figure this out as well. But that means that,
unless we can find a way to enforce the cartel agreement, the cartel will unravel as each of us cheats.
And if each of us knows that the other will cheat, we are right back to Cournot competition and
will end up behaving as if there was no cartel agreement at all.
Put in terms of the game theory language we developed earlier, we face a classic Prisoner’s
Dilemma: We would both be better off colluding and producing in accordance with the agreement
than we would be by competing with one another (either in Bertrand or Cournot competition), but
we also both have a strong incentive to cheat on the agreement (whether the other party cheats or
not) and bring more economist cards to the AEA meetings than we had promised. As we noted in
our discussion of Prisoner’s Dilemmas, these types of games do not result in the optimal outcome
for the two players unless the players can find a way to enforce the agreement. Inconveniently for
us, cartel agreements are usually illegal. (Usually, but not always – as we will see shortly.)
Exercise 25A.15 The Prisoners’ Dilemma you and I face as we try to maintain a cartel agreement works
toward making us worse off. How does it look from the perspective of society at large?
While the incentives of cartel members therefore contain seeds that undermine cartel agreements,
there are real world examples of cartel agreements that have lasted for long periods. They may
not always be successful at maintaining exactly monopoly output, but they often do restrict output
beyond what Cournot competition would predict. This raises the question of how firms can overcome
the Prisoners’ Dilemma incentives that would, if unchecked, lead to a full unraveling of a cartel.
We can think of two possible ways of accomplishing this: First, firms might find ways of hiring
an outside party to enforce the cartel, just as our two prisoners in the classic Prisoners’ Dilemma
might do by joining a “mafia” that enforces silence when the prisoners are interrogated by the
prosecutor. Second, in our discussion of repeated Prisoners’ Dilemmas in Chapter 24, we found
that, if the game is repeated an infinite number of times or, more realistically, if the players know
that there is a decent chance that they will meet again each time that they meet, cooperation in
the Prisoners’ Dilemma can emerge as part of a subgame perfect equilibrium strategy. We will now
briefly discuss each of these paths that can lead to successful cartel cooperation among oligopolists.
25A.4.3
Enforcing Cartel Agreements through Government Protection
In 1933, in the midst of the Great Depression, Congress passed the National Industrial Recovery Act
(NIRA) at the urging of the newly inaugurated President Franklin D. Roosevelt who proclaimed it
“the most important and far-reaching” legislation “ever enacted by the American Congress.” The
act represented a stark departure from laissez faire attitudes toward industry, envisioning a more
planned economy in which industrial leaders would coordinate production and prices to “foster fair
competition”, with compliance enforced by the newly created National Recovery Administration
(NRA). In essence, the act legalized cartels in major manufacturing sectors – thus putting the
force of law behind oligopolists’ efforts to set price and quantity within particular markets. It
982
Chapter 25. Oligopoly
generally received strong support from large corporations but was opposed by smaller firms.5 The
NIRA has become the clearest example in the U.S. of how oligopolists can employ the government
as an enforcer of cartel agreements to limit quantity and raise price. Less than two years after its
enactment, the U.S. Supreme Court unanimously declared the portion of the NIRA that established
cartels as unconstitutional.
Exercise 25A.16 Why would oligopolists who cannot voluntarily sustain cartel agreements want to have
such agreements enforced?
While this large-scale establishment of cartels vanished in the U.S. with the demise of the
NIRA, similar legislation often governs industry in other countries. And, there continue to be more
modest attempts to establish cartels through government action, typically with the stated purpose
of benefitting the “general welfare” but the actual consequence of restricting quantity and raising
price. In the 1990’s, for instance, Congress authorized the Northeast Interstate Dairy Compact that
permitted the setting of minimum wholesale prices of milk across six New England states (amending
extensive federal price regulation of milk that predated the establishment of the Compact) and
restrictions of competition from milk producers in other regions. Other regional milk cartels were
similarly authorized in other regions. The stated intent of such legislation was to “assure the
continued viability of dairy farming in the Northeast and to assure consumers of an adequate, local
supply of pure and wholesome milk” at “a fair and equitable price”. The cooperative suggested
that “dramatic price fluctuations, with a pronounced downward trend, threaten the viability of the
Northeast dairy region” and that “cooperative, rather than individual state action, may address
more effectively the market disarray.” But the ultimate aim of the cartel was the same as that of
all cartels: to curtail competition and raise price. Predictably, such legislation tends to be fought
vigorously by consumer groups and is advocated by firms producing the cartel good.6
In some cases, it is generally recognized that the purpose of government sponsored cartels is to
limit competition in order to raise price. Few, for instance, would argue that this is not the prime
mission of OPEC – the Organization of Petroleum Producing Countries that meets frequently to set
production quotas for each of its 13 member countries. Yet one would not be able to tell this from
the official mission statement by OPEC which states: “OPEC’s mission is to coordinate and unify
the petroleum policies of Member Countries and to ensure the stabilization of oil prices in order
to secure an efficient, economic and regular supply of petroleum to consumers, a steady income to
producers and a fair return on capital to those investing in the petroleum industry.” The words
sound similar to those used to advocate for the NIRA in 1933 and continue to be similar to those
articulated whenever government enforcement for cartel agreements is sought by firms.
25A.4.4
Self-Enforcing Cartel Agreements in Repeated Oligopoly Interactions
Alternatively, we can turn to the case where oligopolists who seek to establish a cartel agreement
know that they will meet repeatedly. From our game theory chapter, we know that this is not
sufficient for cooperation to emerge: If the firms know they will interact repeatedly but that this
interaction will end at some definitive point in the future, subgame perfection leads to an unraveling
of cooperation from the bottom of the repeated game tree upwards. The firms know that, in their
5 The act also encouraged collective bargaining through unions, set maximum work hours and minimum wages
and forbid child labor.
6 To the extent to which milk cartels are intended to support the viability of small, family-owned dairy farmers,
they appear not to be very successful. Most of the economic benefits accrue to larger corporate dairy farms, with
little evidence that cartels slow the disappearance of smaller, less efficient farms.
25B. The Mathematics of Oligopoly
983
final interaction, neither will have an incentive to stick by the cartel agreement. But that means
that in the second to last period, there will also be no incentive to cooperate since there is no
credible way to punish non-cooperation in the final interaction. But that then means that there
is no way to enforce cooperation in the third-to-last interaction given that both firms know that
non-cooperation will take place in the last two periods. And by the same logic, cooperation cannot
emerge in any period.
But the real world is rarely quite as definitive as setting up a finitely repeated set of interactions
with a clear end-point. Rather, firms will know that they are likely to interact again each time that
they meet, and for our purposes, we can therefore treat such interactions as infinitely repeated.
Again, as we saw in our discussion of repeated Prisoners’ Dilemmas in Chapter 24, this removes
the “unraveling” feature of finitely repeated games because there is no definitive final interaction.
And it opens the possibility of simple “trigger strategies” under which firms begin by complying
with the cartel agreement, continue to do so as long as everyone complied in previous interactions,
and revert to oligopoly competition if someone deviates from the agreement. Such strategies can
sustain cartel cooperation so long as the immediate payoff from violating the cartel agreement is
not sufficiently large to overcome the long-run loss from the disappearance of the cartel and the
reversion to oligopoly competition.
Real world strategies of this type are complicated by the fact that firms might not in fact be
able to tell for sure whether another firm has violated the agreement. For instance, suppose that
oil producers cannot observe how much oil is produced by any given company but they can only
see the price that oil sells for in the market. Suppose further that oil price in any given period
depends on both the overall quantity of oil supplied by the oligopoly firms and unpredictable (and
unobservable) demand shocks to the oil market. If a firm then observes an unexpectedly low price
in a given period, it might be because a member of the cartel has cheated and has produced more
oil than the agreement specified, but it might also be because of an adverse demand shock in the
oil market. Firms in such markets may then find it difficult to be certain about whether cartel
members are cheating and run the risk of mis-interpreting an unexpectedly low price as a sign of
cheating. Economists have introduced such complicating factors into economic models of oligopolies
and cartels, and it becomes plausible to observe equilibria in which cartel agreements break down
and re-emerge in repeated oligopoly interactions. This corresponds well to observed cartel behaviors
in some industries.
Exercise 25A.17 In circumstances where firms are not certain about demand conditions in any given
period, why might a more forgiving trigger strategy (like Tit-for-Tat) that allows for the re-emergence of
cooperation be better than the extreme trigger strategy that forever punishes perceived non-cooperation in
one period?
25B
The Mathematics of Oligopoly
Throughout most of this section, we will assume for simplicity that firms face a constant marginal
cost M C = c (with no recurring fixed costs) and that the market demand for the oligopoly good x
is linear and of the form
x = A − αp.
(25.1)
In some of our end-of-chapter exercises, we will explore how the various oligopoly models are
affected by different assumptions, including different marginal costs and the presence of recurring
984
Chapter 25. Oligopoly
fixed costs for the firms. For now, note that, under our current assumptions, were the oligopoly
to function as a single monopoly, we know from our work in Chapter 23 that, assuming no price
discrimination, the firm would produce the monopoly quantity xM and sell it at the monopoly price
pM where
xM =
A + αc
A − αc
and pM =
.
2
2α
(25.2)
Exercise 25B.1 Verify xM and pM in equation (25.2).
25B.1
Bertrand Competition
From our work in part A, we know that Bertrand competition, whether simultaneous or sequential,
will result in both firms setting price equal to marginal cost. It is therefore quite easy to determine
the overall Bertrand oligopoly output level by simply substituting M C = c for price in the market
demand function to get the joint output level x = A − αc. Assuming that the consumers will come
to our two firms in equal numbers when we charge the same price, this implies Bertrand output
levels for our two firms of
A − αc
(25.3)
2
sold at the Bertrand price of pB = c. Thus, for the linear demand and constant M C model we
are using, the Bertrand model predicts that each of the two firms will produce the quantity that a
single monopolist would choose to produce on her own, because the “competitive” quantity is twice
the monopoly quantity.
The Bertrand model becomes more interesting, as we will see in Chapter 26, when firms can
differentiate their products, i.e. when firms are not producing identical products but are still part of
an oligopoly. We will also demonstrate in end-of-chapter exercise 25.1 how the inclusion of recurring
fixed costs and differences in marginal costs across firms can alter the stark Bertrand predictions.
B
xB
1 = x2 =
25B.2
Quantity Competition: Cournot and Stackelberg
Next we briefly describe the mathematics behind Cournot and Stackeberg competition as treated
in Section A before covering some other aspects of quantity competition in Section 25B.3.
25B.2.1
Cournot Competition
In order to calculate the best response functions for our two firms in the economist card oligopoly
described in part A, we begin (as we did in Graph 25.2c) by calculating my residual demand given I
assume you produce x2 . If the market demand is given by equation (25.1), then my residual demand
if you produce x2 is simply
xr1 = A − αp − x2 .
(25.4)
To make this analogous to the residual demand curve graphed in Graph 25.2c, we need to put
it in the form of an inverse demand function; i.e.
A − x2
1
pr1 =
−
x1 .
(25.5)
α
α
25B. The Mathematics of Oligopoly
985
Exercise 25B.2 Verify that pr1 is in fact the correct inverse demand function.
We know from our work in Chapter 23 that the marginal revenue curve for any linear inverse
demand function is itself a linear function with the same intercept as the inverse demand function
but twice the slope; i.e. the relevant marginal revenue function for my firm given that I assume you
will produce x2 is
M R1r =
A − x2
α
−
2
x1 .
α
(25.6)
Exercise 25B.3 Derive this M R function using calculus.
Given this residual marginal revenue for my firm, I can now determine the optimal quantity to
produce (assuming I think you are producing x2 ) by setting equation (25.6) equal to marginal cost
M C = c. Solving this for x1 , I get
x1 =
A − x2 − αc
.
2
(25.7)
Since our two firms are identical, your best response to thinking that I produce some quantity
x1 is symmetric. Put differently, for any quantity x1 that I am producing, we can now write down
the best response for you in terms of x1 , and for any quantity of x2 that you are producing, we can
write down my best response in terms of x2 . This gives us the best response functions x1 (x2 ) and
x2 (x1 ) as
x1 (x2 ) =
A − x2 − αc
A − x1 − αc
and x2 (x1 ) =
.
2
2
(25.8)
In a Nash equilibrium, the quantity x2 that I predict you will be producing has to be your best
response to what I am producing; i.e. x2 = x2 (x1 ). We can therefore substitute x2 (x1 ) into our
expression for x1 (x2 ) and solve for x1 , which then gives us the Cournot output level for me as
xC
1 =
A − αc
.
3
(25.9)
Since our two firms are identical, your Nash equilibrium quantity should then be the same.
Exercise 25B.4 Verify that this is correct.
Exercise 25B.5 Verify that these quantities are in fact the Nash equilibrium quantities; i.e. show that,
given you produce this amount, it is best for me to do the same, and given that I produce this amount, it is
best for you to do the same.
Note that this implies that together we will produce 2(A − αc)/3 which is larger than the
monopoly quantity (A − αc)/2 we derived in equation (25.2) and smaller than the competitive and
Bertrand quantities (A − αc).
Exercise 25B.6 How does the monopoly price pM (derived in equation (25.2)) compare to the price that
will emerge in the Cournot equilibrium? How does it compare to the Bertrand price?
986
25B.2.2
Chapter 25. Oligopoly
Cournot Competition with more than 2 Firms
We can also demonstrate how Cournot competition changes as the number of firms increases. To
be a bit more general, suppose that the inverse market demand function is p(x) and that all firms
have the same cost function c(xi ) that gives the total cost of production as a function of the firm’s
production level xi . Suppose there are N firms in the oligopoly, and let’s denote the output levels
of all firms other than firm i as x−i = (x1 , x2 , ..., xi−1 , xi+1 , ..., xN ). Firm i’s profit maximization
problem given x−i is then
max πi = p(xi , x−i )xi − c(xi ) = p(x1 + x2 + ... + xi−1 + xi + xi+1 + ... + xN )xi − c(xi ). (25.10)
xi
The first order condition
dp(xi , x−i )
dc(xi )
=0
xi + p(xi , x−i ) −
dx
dxi
(25.11)
can then be written as
M Ri =
dc(xi )
dp(xi , x−i )
= M Ci .
xi + p(xi , x−i ) =
dx
dxi
(25.12)
As we did in our work on monopoly in equation (23.9), we can express the M Ri above as
dp xi
M Ri = p 1 +
.
(25.13)
dx p
Since we are assuming all firms are identical, in equilibrium they will produce the same quantity.
This means that N xi = x, and this in turn means we can write the M Ri equation as
dp x 1
1
dp xi N
=p 1+
= p 1+
,
(25.14)
M Ri = p 1 +
dx p N
dx p N
N ǫD
where ǫD = (dx/dp)(p/x) is the price elasticity of market demand. Using this as the expression
for M Ri , and recognizing that in equilibrium marginal costs will be the same for all our firms (even
though we are allowing M C to be non-constant by expressing costs as c(x)), we can write equation
(25.12) as
1
M Ri = p 1 +
= M C.
(25.15)
N ǫD
Note that, as N becomes large, this implies that price approaches M C just as it does under
perfect competition. Thus, as oligopolies with identical firms become large, Cournot competition
approaches perfect competition (as well as Bertrand competition.)
Exercise 25B.7 Compare this equation to equation (23.15) in our chapter on monopolies. How are they
related?
Exercise 25B.8 Can you make a case for why the Cournot model gives intuitively more plausible predictions
than the Bertrand model for oligopolies in which identical firms produce identical goods?
25B. The Mathematics of Oligopoly
25B.2.3
987
Stackelberg Competition
Now suppose we return to our linear demand and constant M C example and suppose that we
set quantity sequentially, with me (firm 1) being the Stackelberg leader and you (firm 2) being the
Stackelberg follower. Subgame perfection requires that I first figure out what your optimal response
will be for any x1 I might set in the first stage of the game. But this is simply your best response
function which we already calculated to be
A − x1 − αc
.
(25.16)
2
I can then determine the residual demand for my goods by subtracting what I know you will
produce from the market demand; i.e.
x2 (x1 ) =
A − x1 − αc
.
(25.17)
2
To get the inverse residual demand curve Dr that we graphed in Graph 25.4b, we solve this for
p to get
xr1 = A − αp − x2 (x1 ) = A − αp −
pr1 =
A + αc
1
−
x1 .
2α
2α
(25.18)
Exercise 25B.9 Verify that this is the correct inverse residual demand function for me.
Exercise 25B.10 In Graph 25.4b, the residual demand curve has a kink at the level of M C. Verify that
the function we derived above in fact meets the market demand curve at p = M C. How would you fully
characterize the residual demand curve mathematically (taking into account the fact that it is kinked)?
From pr1 we can now derive my residual marginal revenue curve by once again recognizing that
it will have the same intercept but twice the slope; i.e.
1
A + αc
− x1 .
(25.19)
2α
α
We can then set this equal to M C = c and solve for my optimal Stackelberg leader (SL) quantity
M R1r =
A − αc
.
(25.20)
2
Given this output level for firm 1, firm 2’s best response function implies the optimal Stackelberg
follower (SF ) quantity of
xSL
1 =
xSF
=
2
A − αc
.
4
(25.21)
Exercise 25B.11 How does the overall level of Stackelberg output relate to the monopoly quantity and
the Cournot quantity? What is more efficient in this setting (from society’s vantage point): Cournot or
Stackelberg competition?
Exercise 25B.12 What will be the output price under Stackelberg competition, and how does this relate to
the Cournot and monopoly prices?
Exercise 25B.13 Can you draw a graph analogous to Graph 25.3c, indicating the monopoly outcome
(assuming the two firms would split the monopoly output level), the Cournot outcome, the Stackelberg
outcome and the Bertrand outcome? Carefully label all the points.
988
25B.3
Chapter 25. Oligopoly
Oligopoly Competition with Asymmetric Information
So far we have assumed that firms always know the costs of other firms, but this is not generally
true in the real world. Suppose, for instance, we have a relatively new oligopoly, with firm 1
having lost its monopoly status given the successful entry of firm 2 into the industry. It might
then be reasonable to assume that firm 1’s costs are well-known (given it’s history as a monopolist)
but firm 2’s costs might not be known. Or suppose that it is known that firm 2 invented a new
manufacturing process but it is not yet known how costly that process is. Either of these scenarios
results in an oligopoly in which firm 2 knows firm 1’s costs but firm 1 does not know firm 2’s
costs. Put differently, we now have asymmetrically informed firms – and thus one player (firm
1) with incomplete information. The resulting oligopoly quantity setting game is an example of
a simultaneous Bayesian game. (If you have not done Section B of Chapter 24, you can skip to
Section 25B.4.)
To be more concrete, suppose that the oligopoly once again faces the same market demand
x = A − αp, with inverse market demand of p = (A/α) − x/α. In a 2-firm oligopoly, this inverse
demand can then again be written as p = (A − x1 − x2 )/α, with xi simply indicating firm i’s
production level. Firm 1 is assumed to have marginal cost of c as before, but firm 2 might have
either “high” marginal costs of cH or “low” marginal costs of cL , with cH > cL . The high cost
“type” in firm 2 occurs with probability ρ while the low cost “type” occurs with probability (1 − ρ).
Firm 2 knows its type but firm 1 only has beliefs about firm 1’s type (based on the probability
with which each type occurs). We will consider Cournot competition in this setting (and explore
Bertrand competition briefly in two within-chapter exercises at the end of the section).
It seems intuitive that firm 2 will produce a different level of output depending on whether its
costs are high or low. A “strategy” for firm 2 therefore involves settling on a quantity depending
on whether the firm is a high or a low cost type.7 But firm 1 does not have the luxury of setting its
quantity with the knowledge of firm 2’s cost structure – it has to settle on a single quantity given
its beliefs about the likelihood of firm 2 being a high cost rather than low cost type. Put differently,
firm 1 needs to solve the optimization problem
A − x1 − xL
A − x1 − xH
2
2
− c x1 + (1 − ρ)
− c x1
max ρ
x1
α
α
(25.22)
L
where xH
2 and x2 are the firm 2 production levels of high and low cost types. Depending on
which type i firm 2 is assigned (by “Nature”), it solves the optimization problem
max
xi2
A − x1 − xi2
i
− c xi2 .
α
(25.23)
The first order condition of the optimization problem in (25.22) solves to
x1 =
L
A − αc − ρxH
2 − (1 − ρ)x2
2
(25.24)
for firm 1, and the first order conditions for the optimization problems (for the two types) in
(25.23) solve to
7 Remember from Chapter 24 that a simultaneous Bayesian game involves Nature assigning types first, and a
strategy for each player therefore involves a plan of action for each possible type that might be assigned.
25B. The Mathematics of Oligopoly
xH
2 =
A − x1 − αcL
A − x1 − αcH
and xL
2 =
2
2
989
(25.25)
for firm 2.
Exercise 25B.14 Show that the first order condition for firm 1 approaches an expression similar to the
first order condition for each of the firm 2 types as firm 1’s uncertainty diminishes; i.e. as ρ approaches
zero or 1.
Substituting the first order conditions for firm 2 into equation (25.24) and solving for x1 , we get
firm 1’s optimal quantity x∗1 as
A − 2αc + α(ρcH + (1 − ρ)cL )
.
(25.26)
3
Now suppose that firm 1 actually knew firm 2’s type. This would imply that it would produce
(A − 2αc + αcH )/3 if it knew it was facing a high cost firm and (A − 2αc + αcL )/3 if it knew it
was facing a low cost firm. But since it does not know what type it is facing, firm 1 produces a
quantity in between these – thus producing less than it would under complete information when it
faces a high cost opponent and more when it faces a low cost opponent.
Firm 2 has an informational advantage and will, we we will see shortly, try to use that to its
advantage. Suppose, for instance, it has high marginal costs cH . Substituting firm 1’s output level
from equation (25.26) into xH
2 in expression (25.25), we can solve for the output level of firm 2
when it has high costs. This gives us
x∗1 =
xH∗
2 =
2A + 2αc − α(3 + ρ)cH − α(1 − ρ)cL
6
(25.27)
which, by adding and subtracting αρcH can be written as
A + αc − 2αcH
α(1 − ρ) H
+
(c − cL ).
(25.28)
3
6
In the absence of informational asymmetries, the high cost firm 2 would produce only the first
term in this expression – which implies that it will produce more than it would under complete
information when it knows it has high costs but its opponent does not. We just saw that firm 1 will
produce less than it would under complete information when it faces a high cost opponent. Firm 2
is therefore using its informational advantage to its advantage.
We can similarly solve for xL∗
2 to get
xH∗
2 =
αρ H
A + αc − 2αcL
−
(c − cL ),
(25.29)
3
6
and we can now see that firm 2 will produce less than it would under complete information
when it knows it is a low cost type – allowing firm 1 to produce more.
xL∗
2 =
Exercise 25B.15
**
Verify the last equation.
Exercise 25B.16 * Can you tell whether the Cournot price will be higher or lower under this type of
asymmetric information than it would be under complete information? (Hint: For both the case of a high
cost and a low cost type, can you see if overall production is higher or lower in the absence of asymmetric
information?)
990
Chapter 25. Oligopoly
Exercise 25B.17 * Suppose the two firms engage in price (Bertrand) competition, and suppose c > cH .
What price do you expect will emerge?
Exercise 25B.18 * Suppose again the two firms engage in price (Bertrand) rather than quantity competition, and suppose cL < c < cH . This case is easier to analyze if we assume sequential Bertrand competition
– with firm 1 setting its price first and firm 2 setting it after it observes p1 (and after it finds out its cost
type). What equilibrium prices would you expect? Does your answer change with ρ?
25B.4
Fixed Entry Costs and Entry Deterrence
We showed in Section 25A.3 that, for particular fixed costs of entry, it is possible for an incumbent
firm to deter entry by a new firm if the incumbent firm is able to set quantity prior to the potential
entrant’s entry decision. Given our work above, we can now show exactly the range of fixed costs
for which the intuition we developed in part A is correct. Recall that the sequence of moves required
for entry deterrence has the incumbent firm setting quantity first, followed by an entry and quantity
decision by the potential entrant. (That sequence is pictured in panel (c) of Graph 25.5).
First, we can begin by asking how high fixed entry costs F C would have to be in order for the
incumbent firm to not have to worry about challenges from an entrant. Suppose firm 1 produces the
monopoly quantity xM (in equation (25.2)) which we have shown is exactly equal to the Stackelberg
leader quantity xSL (in equation (25.20)) under our linear assumptions about demand and costs.
The best firm 2 could then do if it did enter is to produce the Stackelberg follower quantity xSF
(in equation (25.21)) and to sell that quantity at the Stackelberg price which you should have
calculated in exercise 25B.12 to be
A + 3αc
.
(25.30)
4α
The profit π2 for firm 2 from entering is then equal to revenue minus the cost of production
minus the fixed cost of entry F C, i.e.
pS =
π2 = pS xSF − cxSF − F C =
(A − αc)2
− F C.
16α
(25.31)
Exercise 25B.19 Verify that this equation is correct.
We can therefore say that, so long as F C > (A − αc)2 /16α, the profit from entering if the
incumbent firm is producing the monopoly output level is negative and firm 2 would choose to not
enter while firm 1 would produce xM without feeling the threat of competition from the potential
entrant. In terms of the notation in Graph 25.6b, this implies
F C = (A − αc)2 /16α.
(25.32)
Next, we can ask at what fixed entry cost the incumbent firm would be better off accepting the
Stackelberg outcome rather than attempting to raise quantity in order to keep the entrant from
coming into the market. To answer this, we first have to determine, for any given F C, how much
firm 1 would have to produce in order to keep firm 2 from entering. Whatever x1 is produced,
firm 2 will respond (if it enters) by producing according to its best response function x2 (x1 ) (in
equation (25.8)). This allows us to calculate the price that firm 1 can expect to emerge for any
quantity x1 conditional on firm 2 entering the market
25B. The Mathematics of Oligopoly
p(x1 ) =
991
A − x1 + αc
A x1 + x2 (x1 )
−
=
.
α
α
2α
(25.33)
Exercise 25B.20 Verify that this derivation of p(x1 ) is correct.
Firm 2 will enter if (p(x1 )x2 (x1 ) − cx2 (x1 )) > F C. Substituting in for x2 (x1 ) and p(x1 ), this
implies firm 2 will enter so long as
(A − x1 − αc)2
> F C.
4α
(25.34)
Exercise 25B.21 Again, verify that this derivation is correct.
Firm 1 is in full control of what x1 will be when firm 2 has to make its entry decision, which
implies that firm 1 has to make sure that the inequality in (25.34) goes in the other direction (if it
wants to keep firm 2 out). Firm 1 therefore has to solve
(A − x1 − αc)2
≤ FC
(25.35)
4α
for x1 . Doing so, we get the minimum output for firm 1 to deter firm 2 from entering as
xED
= A − αc − 2(αF C)1/2 .
1
(25.36)
2
When fixed entry costs are below F C = (A − αc) /16α, the incumbent firm now has a choice:
it can either produce the entrance deterrent quantity xED
and keep firm 2 from entering, or it can
1
produce the Stackelberg leader quantity and accept firm 2’s competition. If the incumbent settles
into Stackelberg leadership and accepts firm 2 entry, its profit π1SL will be
π1SL =
(A − αc)2
.
8α
(25.37)
Exercise 25B.22 Verify that this is correct. Does it make sense that profit for the Stackelberg leader is
exactly twice the profit of the Stackelberg follower (which we calculated in equation (25.31)) when F C = 0?
The profit from producing a quantity x as the sole producer in the market (graphed in panel
(a) of Graph 25.6) is
A−x
A − x − αc
π = (p(x) − c)x =
x
(25.38)
−c x=
α
α
Since the incumbent can always just decide to be Stackelberg leader, the most she is ever willing
to produce to deter entry is an amount that sets equations (25.37) and (25.38) equal. Doing so and
solving for x (using the quadratic formula) we get the highest quantity that would ever be produced
to deter entry as8
xED
max =
(2 + 21/2 )(A − αc)
.
4
(25.39)
8 The quadratic formula gives two solutions for x. However, one of these is less than the Stackelberg leader quantity
and we can therefore discard that solution as economically irrelevant.
992
Chapter 25. Oligopoly
Exercise 25B.23 As noted in the footnote, the quadratic formula also gives a second solution, namely
x = (2 − 21/2 )(A − αc)/4. Can you locate this solution in panel (a) of Graph 25.6?
Setting this equal to equation (25.36), we can calculate the lowest fixed cost F C at which entry
deterrence is still optimal for firm 1 as
FC =
(2 − 21/2 )(A − αc)
8
2
.
(25.40)
Thus, if the fixed entry cost falls below F C, the incumbent firm will make no effort to deter
firm 2 from entering, and the two firms simply play the Stackelberg game. If the fixed entry cost
falls between F C and F C (from equation (25.34)), the incumbent firm will raise its output to xED
1
(from equation (25.36)) and will thereby successfully deter firm 2 from entering the market. Finally,
if the fixed entry cost is higher than F C, the incumbent can safely produce the monopoly quantity
xM without worrying about firm 2 entering.
25B.5
Dynamic Collusion and Cartels
The mathematics behind our Section A discussion of cartels and collusion is relatively straightforward. We will briefly illustrate mathematically the temptation by members of cartels to cheat on
cartel agreements before illustrating how dynamic collusion can nevertheless emerge under the right
conditions.
25B.5.1
The Temptation to Cheat on a One-Period Cartel Agreement
Continuing with the assumption that market demand is given by x = A − αp, we already calculated
that a monopolist facing this market demand will produce xM = (A − αc)/2 and sell at pM =
(A + αc)/2α. Two identical firms in an oligopoly facing the same market demand would therefore
maximize their joint profit if they agree to each produce half the monopoly quantity; i.e. xCartel
=
i
xM /2 = (A − αc)/4.9 If both parties to a cartel agreement abide by the agreement, this implies
that profit for each cartel member i would be
xM
(A − αc)2
A + αc
A − αc
=
−c
=
.
(25.41)
πiCartel = pM − c
2
2α
4
8α
Now suppose that firms i and j have entered such a cartel agreement but firm i, rather than
blindly following the agreement, asks itself if it could produce a different quantity and do better.
If firm j sticks by the agreement to produce xM /2, this means firm i would choose xi to solve
3(A − αc) − 4xi
A − (xM /2) − xi
xi .
(25.42)
− c xi =
max πi =
xi
α
4α
Solving the first order condition, we can then calculate the optimal quantity for firm i conditional
on firm j sticking by the cartel agreement. Denoting this quantity as xD
i ,
xD
i =
3(A − αc)
,
8
(25.43)
9 Of course other production quotas for the two firms can also maximize joint profits so long as the quotas add up
to the monopoly quantity. End-of-chapter exercise 25.9 explores how unequal the quotas could be in principle.
25B. The Mathematics of Oligopoly
993
which is 50% greater than half the monopoly quantity assigned to firm i in the cartel agreement.
The profit from deviating, πiD , conditional on firm j not deviating from the cartel agreement can
then be calculated to be
πiD =
9(A − αc)2
.
64α
(25.44)
Exercise 25B.24 Verify πiD . Is it unambiguously larger than πiCartel ?
25B.5.2
Collusion in Finitely Repeated Oligopoly Quantity Setting
It is clear from what we just derived that, unless there is some outside enforcement mechanism that
can get the two firms to abide by the cartel agreement, it is not possible to sustain the agreement in
equilibrium once the firms meet. As we pointed out in Section A, the two firms are caught in a classic
prisoners’ dilemma – they both know that an enforced cartel agreement makes both of them better
off, but without enforcement, it is rational for both of them to cheat. The equilibrium continues
to be the Cournot equilibrium despite the cartel agreement. And, as explained in Section A, this
does not change when the firms interact repeatedly a finite number of times (since cooperation of
repeated Prisoners’ Dilemma games unravels from the bottom up under subgame perfection).
As we noted in Section A, however, there are many real world instances of collusion in oligopolies –
which casts doubt on the real-world relevance of the result that collusion cannot arise under subgame
perfection in finitely repeated oligopoly interactions. We already discussed in Section A some of the
real world considerations that might in fact be responsible for instances of firm collusion despite this
theoretical result. It may, for instance, be that firms found a way to enforce their cartel agreement,
perhaps by employing government in some fashion. Or it may, as we discussed in Chapter 24, be
the case that there is a Bayesian dimension to the game that we have not considered – that, for
instance, there are firms that will always play Tit-for-Tat even if it is not in their best interest to
do so, and that firm 1 might be uncertain about whether it is in fact playing such an opponent.
We have shown that, even if the probability of encountering an “irrational” Tit-for-Tat opponent
is small, the mere possibility that one of the players might be such an opponent may be enough
for “rational” players to want to establish a reputation for cooperating. Or it may be the case
that firms are uncertain about whether they will interact again, which in essence turns the finitely
repeated game into one that can, in some sense, be modeled like a game of infinitely repeated
interactions.
25B.5.3
Infinitely Repeated Oligopoly Interactions
As we saw in Chapter 24, the unraveling of cooperation in finitely repeated prisoners’ dilemmas is
due to the fact that there is a definitive end to the interactions of the players. In the real world, we
rarely know when the last time is that we interact with someone, and so it might be with firms in
an oligopoly. We could model this directly as a probability that firms will interact again when they
find themselves interacting. Or we can model the game as an infinitely repeated game in which the
firms discount the future. We will do the latter here, assuming that $1 next period is worth $δ this
period, where δ < 1. Recall that this means that a stream of income of y per period starting this
period is worth y/(1 − δ), and a stream of income of y per period starting next period is worth
δy/(1 − δ).
We will now show that, assuming firms do not discount the future too much, collusion between
firms in an oligopoly can emerge in infinitely repeated settings. One possibility that we previously
994
Chapter 25. Oligopoly
raised in Chapter 24 is that players employ “trigger strategies” – strategies that presume cooperation
initially but that “trigger” eternal non-cooperation if non-cooperation ever enters the game. In the
context of oligopolies in cartel agreements that assign to each of two identical firms half of the
monopoly output in each period, such a strategy would be: “Produce (xM /2) in the first period;
every period thereafter, produce (xM /2) if everyone in previous periods has stuck by the cartel
agreement but produce the Cournot quantity xC otherwise.” One instance of non-cooperation
therefore “triggers” the Cournot equilibrium from then on.
Such a trigger strategy, if adopted by both players, is a subgame perfect equilibrium of the
infinitely repeated oligopoly game so long as one of the firms cannot make enough additional profit
immediately by deviating this period to compensate for the loss of cartel profits in the future. Put
differently, when firm i considers whether to deviate, it knows that it can get πiD from equation
(25.44) this period at the cost of settling for the Cournot profit πiC for every period thereafter – i.e.
deviating results in profit of πiD + δπiC /(1 − δ). Not deviating, on the other hand, implies a profit
of π Cartel every period starting now – or, in present value terms, πiCartel /(1 − δ). Deviating from
the trigger strategy therefore does not pay so long as
δπiC
πiCartel
> πiD +
.
(1 − δ)
(1 − δ)
(25.45)
We previously calculated (in equation (25.9)) the Cournot quantity to be xC = (A − αc)/3, and
in exercise 25B.6 you should have derived the Cournot price as pC = (A + 2αc)/3α. This implies a
Cournot profit for each firm of πiC = (A − αc)2 /9α.
Exercise 25B.25 Verify that this is the correct per period profit in the Cournot equilibrium.
Plugging the relevant quantities into the inequality (25.45), we get
9(A − αc)2
δ(A − αc)2
(A − αc)2
>
+
.
8α(1 − δ)
64α
9α(1 − δ)
(25.46)
Solving for δ, we then get that
9
≈ 0.53.
(25.47)
17
Thus, so long as $1 next period is worth more than $0.53 this period, neither firm will want to
deviate from the proposed trigger strategy – which implies the two firms will collude in accordance
with their cartel agreement.
This is, of course, as our discussion of the Folk Theorem in the appendix to Chapter 24 illustrated, not the only way to sustain collusion in infinitely repeated oligopoly games. Furthermore,
in a world where there is less certainty than what we have assumed here, the trigger strategy we
proposed here seems far too severe since it eternally punishes deviations. Consider, for instance, a
world in which firms in an oligopoly cannot observe the output of other firms but only see what the
equilibrium price turned out to be in every period. In a 2-firm oligopoly, this is enough to infer the
other firm’s output – but only if firms know market demand perfectly. If there is some uncertainty
in each period about what exactly market demand looks like – if there are, as we put it in Section
A, unobservable market demand “shocks” – then it becomes more difficult to know whether an
unexpectedly low price was due to unexpectedly low market demand in a given period or whether
it was due to the other firm cheating on its cartel agreement. A number of economists have investigated such settings closely and have concluded that more forgiving trigger strategies are likely to
δ>
25B. The Mathematics of Oligopoly
995
be optimal – strategies where a price below some level “triggers” punishment (i.e. deviations from
the cartel agreement) for some period but eventually collusion is restored. Our only point here is
that, when firms interact without knowing that their interactions will end at some point, collusion
may well be sustainable despite the incentives to deviate from cartel agreements in finitely repeated
games.
Conclusion
We have now moved from a model of perfect competition in which firms could behave nonstrategically (since their actions had no influence on price) to models of perfect monopoly in Chapter 23 to the intermediate case of oligopoly. Any deviation from perfect competition introduces
strategic considerations and eliminates the possibility of modeling firms as price-takers. In the
monopoly setting, we illustrated different types of pricing policies that monopolists might employ
to strategically shape their economic environment, and in the oligopoly case we have illustrated
how less-than-perfect competition results in pricing and output in between the extremes of perfect
competition and monopoly so long as oligopolists do not form cartels and are not perfect Bertrand
competitors. In the process, we have also illustrated that the potential threat of competition can,
assuming sufficiently low entry costs, alter the quantities produced by monopolists (or “incumbent”
firms) in a socially desirable direction.
The welfare implications of different forms of oligopoly competition are relatively straightforward, but the policy question of how to deal with oligopoly markets to enhance efficiency runs into
complications similar to some of those we discussed in our chapter on monopolies. There are often
very good underlying economic reasons for the existence of oligopolies, reasons that mirror those
for the existence of natural monopolies. For instance, a firm has to pay a relatively large fixed
cost before it can begin producing cars, which results in U-shaped average cost curves for which
the bottom of the “U” occurs at large quantities relative to market demand. In such instances,
the nature of production does not permit the existence of many small firms that can all act as
price-taking competitors, nor would such a market arrangement be efficient if it could be forced
(since it would result in high average costs for cars as each firm needs to recoup its fixed costs). The
loss of efficiency from pricing above marginal cost by Cournot competitors can therefore easily be
outweighed by the gain in efficiency from having a small number of firms produce at lower points
of their average cost curves.
As a result, the thrust of anti-trust policy in oligopoly markets is focused on attempts to detect
and deter collusion by oligopoly firms that seek to escape oligopoly competition by forming cartels
that behave more like monopolies. Without knowing the cost functions of firms in an oligopoly
(as well as demand conditions on the consumer side), however, it is not always easy for regulators
charged with fostering competitive behavior in oligopolistic markets to detect collusion, and firms
in an oligopoly (just as natural monopolists) have no particular incentive to reveal their true cost
functions to regulators. Suspected colluders are then often taken to court for alleged violations
of anti-trust laws (that make such anti-competitive collusion illegal), and courts are then charged
with investigating the underlying economics of the relevant market to determine the extent to
which collusion has in fact taken place and what damages have resulted from such collusion.10
To the extent to which colluding firms can be shown to have had explicit interactions in which
10 Such court cases may arise from federal regulators initiating law suits, or they often arise from firms who charge
competitors with collusive behavior. Damages to both consumers and competitors who did not participate in the
collusion are then assessed.
996
Chapter 25. Oligopoly
they discussed and coordinated pricing and production decisions, evidence of collusion can be
found in records that do not require explicit knowledge of cost functions, but the assessment of
damages requires such information in order to determine the extent to which the observed prices
and production levels deviated from what one would have expected under oligopoly competition.
But one can easily envision instances where firms are quite clever in how they engineer their collusive
relationship without making explicit cartel agreements that can be entered as evidence in court.
Again, these complications lead to quite interesting ways in which courts have successfully or
mistakenly dealt with allegations of collusion, and if this is interesting to you, a course in anti-trust
economics (or law and economics) should be fascinating.
Oligopoly market structures are not, however, the only market structures that fall in between the
extremes of perfect competition and perfect monopoly. Perfect competition involves the assumption
of no barriers to entry, while monopoly and oligopoly markets require significant barriers to such
entry of new firms. In Chapter 26, we will therefore introduce a final type of market structure known
as “monopolistic competition” in which barriers to entry are low (unlike for oligopoly and monopoly
market structures) but firms can engage in innovation that differentiates their product (unlike in
the case of perfect competition where we have assumed all firms produce identical products.) The
potential for product differentiation through innovation also exists in oligopoly markets (or, for that
matter, for monopolists who fear innovative potential competitors), and we will treat this explicitly
in Chapter 26 as well.
End of Chapter Exercises
25.1 * In the text, we demonstrated the equilibrium that emerges when two oligopolists compete on price when
there are no fixed costs and marginal costs are constant. In this exercise, continue to assume that firms compete
solely on price and can produce whatever quantity they want.
A: We now explore what happens as we change some of these assumptions. Maintain the assumptions we
made in the text and change only those referred to in each part of the exercise. Assume throughout that costs
are never so high that no production will take place in equilibrium, and suppose throughout that price is the
strategic variable.
(a) First, suppose both firms paid a fixed cost to get into the market. Does this change the prediction that
firms will set p = M C?
(b) Suppose instead that there is a recurring fixed cost F C for each firm. Consider first the sequential case
where firm 1 sets its price first and then firm 2 follows (assuming that one of the options for both firms is
to not produce and not pay the recurring fixed cost). What is the subgame perfect equilibirum? (If you
get stuck, there is a hint in part (f).)
(c) Consider the same costs as in (b). Can both firms produce in equilibrium when they move simultaneously?
(d) What is the simultaneous move Nash Equilibrium? (There are actually 2.)
(e) True or False: The introduction of a recurring fixed cost into the Bertrand model results in p = AC
instead of p = M C.
(f) You should have concluded above that the recurring fixed cost version of the Bertrand model leads to a
single firm in the oligopoly producing. Given how this firm prices the output, is this outcome efficient –
or would it be more efficient for both firms to produce?
(g) Suppose next that, in addition to a recurring fixed cost, the marginal cost curve for each firm is upward
sloping. Assume that the recurring fixed cost is sufficiently high to cause AC to cross M C to the right
of the demand curve. Using logic similar to what you have used thus far in this exercise, can you again
identify the subgame perfect equilibrium of the sequential Bertrand game as well as the simultaneous
move pure strategy Nash equilibria?
B: Suppose that demand is given by x(p) = 100 − 0.1p and firm costs are given by c(x) = F C + 5x2 .
(a) Assume that F C = 11, 985. Derive the equilibrium output xB and price pB in this industry under
Bertrand competition.
25B. The Mathematics of Oligopoly
997
(b) What is the highest recurring fixed cost F C that would sustain at least one firm producing in this industry?
(Hint: When you get to a point where you have to apply the quadratic formula, you can simply infer the
answer from the term in the square root.)
25.2 In exercise 25.1, we checked how the Bertrand conclusions (that flow from viewing price as the strategic variable)
hold up when we change some of our assumptions about fixed and marginal costs. We now do the same for the case
where we view quantity as the strategic variable in the simultaneous move Cournot model.
A: Again, maintain all the assumptions in the text unless you are asked to specifically change some of them.
(a) First, suppose both firms paid a fixed cost to get into the market. Does this change the predictions of the
Cournot model?
(b) Let xC denote the Cournot equilibrium quantities produced by each of two firms in the oligopoly as
derived under the assumptions in the text. Then suppose that there is a recurring fixed cost F C for each
firm (and F C does not have to be paid if the firm does not produce). Assuming that both firms would
still make non-negative profit by each producing xC , will the presence of F C make this no longer a Nash
equilibrium?
(c) Can you illustrate your conclusion from (c) in a graph with best response functions that give rise to a
single pure strategy Nash equilibrium with both firms producing xC ? (Hint: You should convince yourself
that the best response functions are the same as before for low quantities of the opponent’s production
but then, at some output level for the opponent, jump to 0 output as a best response.)
(d) Can you illustrate a case where F C is such that both firms producing xC is one of 3 different pure strategy
Nash equilibria?
(e) Can you illustrate a case where F C is sufficiently high such that both firms producing xC is no longer a
Nash equilibrium? What are the two Nash equilibria in this case?
(f) True or False: With sufficiently high recurring fixed costs, the Cournot model suggests that only a single
firm will produce and act as a monopoly.
(g) Suppose that, instead of a recurring fixed cost, the marginal cost for each firm was linear and upward
sloping – with the marginal cost of the first unit the same as the constant marginal cost assumed in the
text. Without working this out in detail, what do you think happens to the best response functions – and
how will this affect the output quantities in the Cournot equilibrium?
B: Suppose that the cost function for both firms in the oligopoly have the cost function c(x) = F C + (cx2 /2),
with demand given by x(p) = A − αp (as in the text).
(a) Derive the best response function x1 (x2 ) (of firm 1’s output given firm 2’s output) as well as x2 (x1 ).
(b) Assuming that both firms producing is a pure strategy Nash equilibrium, derive the Cournot equilibrium
output levels.
(c) What is the equilibrium price?
(d) Suppose that A = 100, c = 10 and α = 0.1. What is the equilibrium output and price in this industry
assuming F C = 0?
(e) How high can F C go with this remaining as the unique equilibrium?
(f) How high can F C go without altering the fact that this is at least one of the Nash equilibria?
(g) For what range of F C is there no pure strategy equilibrium in which both firms produce but two equilibria
in which only one firm produces?
(h) What happens if F C lies above the range you calculated in (g)?
25.3 In exercise 25.2, we considered quantity competition in the simultaneous Cournot setting. We now turn the
sequential Stackelberg version of the same problem.
A: Suppose that firm 1 decides its quantity first and firm 2 follows after observing x1 . Assume initially that
there are no recurring fixed costs and that marginal cost is constant as in the text.
(a) Suppose that both firms have a recurring F C (that does not have to be paid if the firm chooses not to
produce). Will the Stackelberg equilibrium derived in the text change for low levels of F C?
(b) Is there a range of F C under which firm 1 can strategically produce in a way that keeps firm 2 from
producing?
(c) At what F C does firm 1 not have to worry about firm 2?
(d) Could F C be so high that no one produces?
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Chapter 25. Oligopoly
(e) Suppose instead (i.e. suppose again F C = 0) that the firms have linear upward sloping M C curves, with
M C for the first output unit equal to what the constant M C was in the text. Can you guess how the
Stackelberg equilibrium will change?
(f) Will firm 1 be able to engage in entry deterrence to keep firm 2 from producing?
B: * Consider again the demand function x(p) = 100 − 0.1p and the cost function c(x) = F C + 5x2 (as you did
in exercise 25.1 and implicitly in the latter portion of exercise 25.2).
(a) Suppose first that F C = 0. Derive firm 2’s best response function to observing firm 1’s output level x1 .
(b) What output level will firm 1 choose?
(c) What output level does that imply firm 2 will choose?
(d) What is the equilibrium Stackelberg price?
(e)
*
Now suppose there is a recurring fixed cost F C > 0. Given that firm 1 has an incentive to keep firm 2
out of the market, what is the highest F C that will keep firm 2 producing a positive output level?
(f) What is the lowest F C at which firm 1 does not have to engage in strategic entry deterrence in order to
keep firm 2 out of the market?
(g) What is the lowest F C at which neither firm will produce?
(h) Characterize the equilibrium in this case for the range of F C from 0 to 20,000.
25.4 Business Application: Entrepreneural Skill and Market Conditions: We often treat all firms as if they must
inherently face the same costs – but managerial, or entrepreneural, skill in firms can sometimes lead to a decrease
in the marginal cost of production. We investigated this in the competitive setting in exercise 14.5 of Chapter 14
and now investigate the extent to which effective managers can leverage their skill in oligopolies depending on the
market conditions they face.
A: Suppose two firms in an oligopoly face a linear demand curve, constant marginal costs M C1 and M C2 and
no recurring fixed costs.
(a) Suppose first that the market conditions are such that firms compete on price and can easily produce any
quantity that is demanded at their posted prices. If the firms simultaneously choose price, what happens
in equilibrium?
(b) Does your answer change if the firms post prices sequentially, with firm 1 posting first?
(c) When firms face the same costs, we concluded that the Bertrand equilibrium is efficient. Does the same
still hold when firms face different marginal costs?
(d) Next, suppose that instead firms have to choose capacity and they therefore are engaged in quantity
competition. What happens in equilibrium compared to the situation where both firms face the same
marginal cost equal to the average of M C1 and M C2 we assume in this exercise?
(e) Could it be that firm 2 does not produce in the Cournot equilibrium? If so, how much does firm 1 produce?
(f) If firms set quantity sequentially, do you think it matters whether firm 1 or firm 2 moves first?
(g)
*
In (b) you were asked to find the subgame perfect equilibrium in a sequential Bertrand pricing market
where firm 1 moves first. How would your answer change if firm 2 moved first? Is there a subgame perfect
equilibrium in which the efficient outcome is reached? What is the subgame perfect equilibrium that
results in the least efficient outcome? (Hint: Think about firm 2’s payoffs for all its possible strategies in
stage 1 – given she predicts firm 2’s response.)
B: The two oligopoly firms operate in a market with demand x = A − αp. Neither firm faces any recurring
fixed costs, and both face a constant marginal cost. But firm 1’s marginal cost c1 is lower than firm 2’s – i.e.
c1 < c2 .
(a) In a simultaneous move Bertrand model, what price will emerge, and how much will each firm produce?
(b) Does your answer to (a) change if the Bertrand competition is sequential – with firm 1 moving first? What
if firm 2 moves first? (Assume subgame perfection.)
(c) How does your answer change if the two firms are Cournot competitors (assuming that both produce in
equilibrium)?
(d) What if the two firms are engaged in Stackelberg competition, with firm 1 as the first mover? What if
firm 2 is the first mover?
(e) How would each firm behave if it were a monopolist?
25B. The Mathematics of Oligopoly
999
(f) Suppose A = 1000, α = 10, c1 = 20 and c2 = 40. Use your results from above to calculate the equilibrium
outcome in each of the above cases. Illustrate your answer in a table with p, x1 , and x2 for each of the
cases. Do the results make intuitive sense?
(g) Add a column to your table in which you calculate profit in each case. What market conditions are most
favorable in this example for the good manager to leverage his skills?
(h) What would be the efficient outcome? Add a row to your table illustrating what would happen under the
efficient outcome.
(i) Which of the oligopoly/monopoly scenarios in your table is most efficient? Which is best for consumers?
(j) Are there any scenarios in your table that would result in the same level of overall production if the
marginal costs for each of the two firms were the same and equal to the average we have assumed for them
(i.e. c1 = c2 = 30)?
* Business Application: Quitting Time: When to Exit a Declining Industry:11 We illustrated in the text the
strategic issues that arise for a monopolist who is threatened by a potential entrant into the market – and in Chapter
26, we will investigate firm entry into an industry where demand increases. In this exercise, suppose instead that an
industry is in decline in the sense that demand for its output is decreasing over time. Suppose there are only two
firms left – a large firm L and a small firm S.
A: Since our focus is on the decision of whether or not to exit, we will assume that each firm i has fixed capacity
k i at which it produces output in any period in which it is still in business; i.e. if a firm i produces, it produces
x = ki . Since L is larger than S, we assume k L > k S . The output that is produced is produced at constant
marginal cost M C = c. (Assume throughout that, once a firm has exited the industry, it can never produce in
this industry again.)
(a) Since demand is falling over time, the price that can be charged when the two firms together produce
some output quantity x declines with time – i.e. p1 (x) > p2 (x) > p3 (x) > ... where subscripts indicate
the time periods t = 1, 2, 3, .... If firm i is the only firm remaining in period t, what is its profit πti ? What
if both firms are still producing in period t?
25.5
(b) Let ti denote the last period in which demand is sufficiently high for firm i to be profitable (i.e. to make
profit greater than or equal to zero) if it were the only firm in the market. Assuming they are in fact
different, which is greater: tL or tS ?
(c) What are the two firms’ subgame perfect strategies beginning in period (tS + 1)?
(d) What are the two firms’ subgame perfect strategies in periods (tL + 1) to tS ?
(e) Suppose both firms are still in business at the beginning of period tL before firms make their decision of
whether to exit. Could both of them producing in this period be part of a subgame perfect equilibrium?
If not, which of the two firms must exit?
(f) Suppose both firms are still in business at the beginning of period (tL −1) (before exit decisions are made).
Under what condition will both firms stay? What has to be true for one of them to exit – and if one of
them exits, which one?
(g) Let t denote the last period in which (pt (k S + k L ) − c) ≥ 0. Describe what happens in a subgame perfect
equilibrium, beginning in period t = 1, as time goes by – i.e. as t, tL and tS pass. Is there ever a time
when price rises as the industry declines?
(h)
*
Suppose that the small firm has no access to credit markets – and therefore is unable to take on any
debt. If the large firm knows this, how will this change the subgame perfect equilibrium? True or False:
Although the small firm will not need to access credit markets in order to be the last firm in the industry,
it will be forced out of the market before the large firm exits if it does not have access to credit markets.
(i) How does price now evolve differently in the declining industry (when the small firm cannot access credit
markets)?
B: Suppose c = 10, k L = 20, k S = 10 and pt (x) = 50.5 − 2t − x until price is zero.
(a) How does this example represent a declining industry?
(b) Calculate tS , tL and t as defined in part A of the exercise.
(c) Derive the evolution of output price as the industry declines.
11 This exercise is derived from Osborne, Martin J. (2004), “An Introduction to Game Theory,” New York: Oxford
University Press.
1000
Chapter 25. Oligopoly
(d) How does your answer change when firm S has the credit constraint described in A(h) – i.e. when the
small firm has no access to credit markets.
(e) How would your answer change if the large rather than the small firm had this credit constraint?
(f)
*
Suppose firm S can only go into debt for n time periods. Let n be the smallest n for which the subgame
perfect equilibrium without credit constraints holds, with n < n implying the change in equilibrium you
described in part A(h). What is n? (Assume no discounting).
(g) If n < n, how will output price evolve as the industry declines?
25.6 Business Application: Financing a Strategic Investment under Quantity Competition: Suppose you own a firm
that has invented a patented product that grants you monopoly power. Patents only last for a fixed period of time
– as does the monopoly power associated with the patent. Suppose you are nearing the end of your patent and you
have the choice of investing in research that will result in a patented technology that reduces the marginal cost of
producing your product.
A: The demand for your product is linear and downward sloping and your current constant marginal cost is
M C. There is one potential competitor who faces the same constant M C. Neither of you currently face any
fixed costs, and the competitor observes your output before he decides whether and how much to produce.
(a) If this is the state of things when the patent runs out, will you change your output level? What happens
to your profit?
(b) Suppose you can develop an improved production process that lowers your marginal cost to M C ′ < M C.
Once developed, you will have a patent on this technology – implying that your competitor cannot adopt
it. You would finance the fixed cost of this new technology with a payment plan that results in a recurring
fixed cost F C for the life of the patent. If you do this, what do you think will happen to your output?
(c) If M C ′ is relatively close to M C, will you be able to keep your competitor out? In this case, might it still
be worth it to invest in the technology?
(d) If the technology reduces marginal costs by a lot, might it be that you can keep your competitor from
producing? If so, what will happen to output price?
(e) Do you think that investments like this – intended to deter production by a competitor – are efficiency
enhancing?
(f) Suppose the potential competitor could also invest in this technonlogy. Might there be circumstances
under which your firm will invest and your competitor does not?
B: * Suppose again that demand is given by x = A − αp, that there are currently no fixed costs, that all firms
face a constant marginal cost c and that you are about to face a competitor (because your patent on the good
you produce is running out).
(a) What will happen to your output level if you simply engage in the competition by producing first. What
will happen to your profit?
(b) If you lower your marginal cost to c′ < c by taking on a recurring fixed cost F C, what will be your profit
assuming that your competitor still produces. (If you have done exercise 25.4, you can use your results
from there to answer this.)
(c) Suppose that A = 1000, c = 40 and α = 10. What is the highest F C can be for you to decide to go ahead
with the investment if the new marginal cost is c′ < c and assuming the competitor cannot get the same
technology? Denote this F C 1 (c′ ).
(d) Now consider the competitor. Suppose he sees that firm 1 has invested in the technology (and thus lowered
its marginal cost to c′ .) Firm 2 finds out that the patent on firm 1’s technology has been revoked – making
it possible for firm 2 to also adopt the technology at a recurring fixed cost F C. What is the highest F C
at which firm 2 will adopt the technology in equilibrium? Denote this F C 2 .
(e) Suppose c′ = 20. For what range of F C will firm 1 adopt and firm 2 not adopt the technology even if it
is permitted to do so?
25.7 Business Application: Financing a Strategic Investment under Price Competition: In exercise 25.6, we investigated the incentives of firms to finance technologies that lower marginal costs. We did so in a sequential setting
where firms compete by setting quantity, with the incumbent firm moving first. Can you repeat the exercise under
the assumption that firms are sequentially competing on price (with firm 1 moving first)?
25B. The Mathematics of Oligopoly
1001
25.8 Business Application: Deal or No Deal: Acquisitions of Up-Start Firms by Incumbents: Large software companies often produce a variety of different software, and sometimes a small up-start develops a competing product.
The large firm then faces a decision of whether to compete with the up-start or whether to “acquire” it. Acquiring
an up-start firm implies paying its owners to give up and join your firm. Since the two firms will jointly make less
money than the merged firm can make on this product, the two parties have to negotiate an acquisition price. What
price will emerge will depend on the market conditions the firms face as well as the way the bargaining unfolds. In
end-of-chapter exercises 24.5 and 24.9, we discussed two bargaining models that we apply here. In the first, known
also as an ultimatum game, one firm would make a take-it-or-leave-it offer, and the other either accepts or rejects.
In the second, the parties make alternating offers until an offer is accepted.12
A: Suppose that the firms face a linear downward sloping demand curve, the same constant marginal cost and
no recurring fixed costs.
(a) Let Y denote the overall gain in profit to the industry if an acquisition deal is cut. How is Y divided
between the firms under three bargaining environments: An ultimatum game in which the incumbent firm
proposed an acquisition price, an ultimatum game in which the up-start firm proposes the price and an
alternating offer game.
(b) Which of your answers in (a) might change if firm 2 is very impatient while firm 1 can afford to be patient?
(c) Let Y B represent the overall gain in profit when the alternative to a deal is Bertrand competition; let Y C
represent the same when the alternative is Cournot competition and let Y S represent the same when the
alternative is Stackelberg competition. Which is biggest? Which is smallest?
(d) Let π M denote monopoly profit; let π C denote one firm’s Cournot profit; and let π SL and π SF denote
the Stackelberg leader and follower profits. In terms of these, what will be the acquisition price under
the three bargaining settings if the alternative is Bertrand competition? What about if the alternative is
Cournot competition or Stackelberg competition?
(e) Which of these acquisition prices is largest? Which is smallest?
(f) Do you think acquisition prices for a given bargaining setting will be larger under Cournot competition
than under Stackelberg competition? Does your answer depend on which bargaining setting we are using?
(g) If part of the negotiations involves laying the groundwork to set expectations about what kind of economic
environment will prevail in the absence of a deal, what would you advise the up-start firm to say at the
first meeting with the incumbent? Does your answer depend on what kind of bargaining environment you
expect?
(h) Would your advice be any different for the incumbent?
B: Let firm 1 be the large incumbent firm and firm 2 the up-start firm. Assume they have no recurring fixed
costs and both face the same constant marginal cost c. The demand for the product is given by x(p) = A − αp.
(a) Suppose the firms expect to be Bertrand competitors if they cannot agree on an acquisition price. If firm
1 is the proposer in the ultimatum bargaining game, what is the subgame perfect acquisition price? What
if firm 2 is the proposer?
(b) What is the acquisition price if the two firms engage in the alternating offer game?
(c) Repeat (a) for the case where the two firms expect to be Cournot competitors.
(d) Repeat (b) if the two firms expect to be Cournot competitors. How does it compare to the answer you
arrived at in (b)?
(e) Repeat (a) if the two firms expect firm 1 to be a Stackelberg leader?
(f) Repeat (b) if the two firms expect firm 1 to be the Stackelberg leader?
(g) Suppose A = 1000, c = 20 and α = 40. What is the acquisition price in each of the cases you analyzed
above? Can you make intuitive sense of these?
25.9 Business and Policy Application: Production Quotas under Cartel Agreements: In exercise 25.8, we investigated the acquisition price that an incumbent firm might pay to acquire a competitor under different bargaining and
economic settings. Instead of one firm acquiring or merging with another, two firms in an oligopoly might choose to
enter a cartel agreement in which they commit to each producing a quota of output (and no more).
A: Suppose again that both firms face a linear downward sloping demand curve, the same constant marginal
cost, and no recurring fixed costs.
12 In exercise 24.5 you should have concluded that the proposing party gets all the gains in a subgame perfect
equilibrium, and in exercise 24.9 you should have concluded that they will split the gains equally. Assume these
bargaining outcomes throughout this exercise.
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Chapter 25. Oligopoly
(a) Under the different bargaining settings and economic environments described in exercise 25.8,13 what are
the profits that the two firms in the cartel will make in terms of π M , π C , π SL and π SF (as these were
defined in A(d) of exercise 25.8)? (If you have already done this in A(d), skip to (b).)
(b) It turns out that π C = (4/9)π M , π SL = (1/2)π M and π SF = (1/4)π M for examples like this. Using this
information, can you determine the relative share of profit that each firm in the cartel will get for each of
the bargaining and economic settings from (a)?
(c) Assuming the cartel agreement sets xM – the monopoly output level – as the combined output quota
across both firms, what fraction of xM will be produced by firm 1 and what fraction by firm 2 under the
different bargaining and economic settings we are analyzing?
(d) Assume that any cartel agreement results in xM being produced, with each firm producing a share
depending on what was negotiated. True or False: For any such cartel agreement, the payoffs for firms
could also have been achieved by one firm acquiring the other at some price.
(e) Explain why the firms might seek government regulation to force them to produce the prescribed quantities
in the cartel agreement.
(f) In the early years of the Reagan administration, there was a strong push by the US auto industry to have
Congress impose protective tariffs on Japanese car imports. Instead, the administration negotiated with
Japanese car companies directly – and got them to agree to “voluntary export quotas” to the US, with
the US government insuring that companies complied. How can you explain why Japanese car companies
might have agreed to this?
(g) Suppose the firms cannot get the government to enforce their cartel agreement. Explain how such cartel
agreements might be sustained as a subgame perfect equilibrium if, each time the firms produce, they
expect there is a high probability that they will again each produce as the only firms in the industry in
the future?
(h) If you are a a lawyer with the antitrust division of the Justice Department and were charged with detecting
collusion among firms that have entered a cartel agreement – and if you thought that these agreements
were typically sustained by trigger strategies, in which market setting (Bertrand, Cournot or Stackelberg)
would you expect this to happen most frequently?
B: Suppose again that firms face the demand function x(p) = A − αp, that they both face marginal cost c and
neither faces a recurring fixed cost.
(a) For each of the bargaining and economic settings discussed in exercise 25.8, determine the output quotas
x1 and x2 for the two firms.
(b) Verify that the fraction of the overall cartel production undertaken by each firm under the different
scenarios is what you concluded in A(c).
(c) Suppose A = 1000, c = 20 and α = 40. What is the cartel quota for each of the two firms under each of
the economic and bargaining settings you have analyzed?
(d) In terms of payoffs for the firms, is the outcome from the cartel agreement any different than the outcome
resulting from the negotiated acquisition price in exercise 25.8?
(e)
*
(f)
*
Suppose the two firms enter a cartel agreement with a view toward an infinite number of interactions.
Suppose further that $1 one period from now is worth $δ < $1 now. What is the lowest level of δ for each
of the bargaining settings such that the cartel agreement will be respected by both firms if they would
otherwise be Cournot competitors?
Repeat (e) for the case of Bertrand and Stackelberg competitors.
(g) Assuming that cartel quotas are assigned using alternating offer bargaining, which cartels are most likely
to hold: Those that revert to Bertrand, Cournot or Stackelberg? Can you explain this intuitively? Which
is second most likely to hold?
25.10 Policy Application: Mergers, Cartels and Antitrust Enforcement: In exercises 25.8 and 25.9, we illustrated
how firms in an oligopoly can collude through mergers or through the formation of cartel agreements. We did this for
different bargaining and economic environments and concluded that payoffs for the firms might differ dramatically
depending on the environments in which the negotiations between firms take place. Suppose now that you are a
lawyer in the anti-trust division of the Justice Department – and you are charged with limiting the efficiency costs
from collusive activities by oligopolists.
13 There is a total of 9 such cases: 3 market settings (Bertrand, Cournot, Stackelberg) and three bargaining settings
(ultimatum game with firm 1 proposing, ultimatum game with firm 2 proposing, and the alternating offer game).
25B. The Mathematics of Oligopoly
1003
A: Suppose that cartel agreements are always negotiated through alternating offers – i.e. suppose the firms
always split the gains from forming a cartel 50-50. Suppose further, unless otherwise stated, that demand curves
are linear, firms face the same constant marginal costs and no recurring fixed costs.
(a) Suppose you have limited resources to employ in pursuing antitrust investigations. Given that breaking
up some forms of collusion leads to greater efficiency gains than breaking up others, which firms would
you focus on – those that would revert to Bertrand, Cournot or Stackelberg environments?
(b) Given that some cartels are more likely than others to last, which would you pursue if you wanted to
catch as many as possible?
(c) Given the likelihood that one form of collusion is more likely to last than the other, would you focus more
on collusion through mergers and acquisitions or on collusion through cartel agreements?
(d) Suppose that you were asked to focus on collusion through mergers and acquisitions. In what way would
the size of recurring fixed costs figure into your determination of whether or not to pursue an antitrust
case against firms that have merged? What tradeoff do you have to consider?
B: Suppose that demand is given by x(p) = 1000 − 10p and is equal to marginal willingness to pay. Firms face
identical marginal costs c = 40 and identical recurring fixed cost F C.
(a) Suppose two Cournot oligopolists have merged. For what range of F C would you decide that there is no
efficiency case for breaking up the merger?
(b) Repeat (a) for the case of Stackelberg oligopolists.
(c)
*
Repeat (a) for the case of Bertrand oligopolists.
(d) It is often argued that antitrust policy is intended to maximize consumer welfare, not efficiency. Would
your conclusions change if you cared only about consumer welfare and not efficiency?
25.11 Policy Application: Subsidizing an Oligopoly: It is common in many countries that governments subsidize
the production of goods in certain large oligopolistic industries. Common examples include aircraft industries and
car industries.
A: Suppose that a 2-firm oligopoly faces a linear, downward sloping demand curve, with each firm facing the
same constant marginal cost and no recurring fixed cost.
(a) If the intent of the subsidy is to get the industry to produce the efficient output level, what should be the
subsidy for Bertrand competitors?
(b)
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How would your answer to (a) change if each firm faced a recurring fixed cost?
(c) What happens (as a result of the subsidy) to best response functions for firms who are setting quantity
(rather than price)? How does this impact the Cournot equilibrium?
(d) How would you expect this to impact the Stackelberg equilibrium?
(e) Suppose policy-makers can either subsidize quantity-setting oligopoly firms in order to get them to produce
the efficient quantity, or they can invest in lowering barriers to entry into the industry so that the industry
becomes competitive. Discuss how you would approach the trade-offs involved in choosing one policy over
the other.
(f) How would your answer be affected if you knew that it was difficult for the government to gather information on firm costs?
(g) Suppose there are recurring fixed costs that are sufficiently high for only one firm to produce under
quantity competition. Might the subsidy result in the entry of a second firm?
B: Suppose demand is given by x(p) = A − αp, that all firms face constant marginal cost c and there are no
recurring fixed costs.
(a) If the government introduces a per-unit subsidy s < c, what happens to the marginal costs for each firm?
(b) How do the Monopoly, Bertrand, Cournot and Stackelberg equilibria change as a result of the subsidy?
(c) Suppose A = 1000, c = 40 and s = 15. What is the economic incidence of the subsidy in each economic
environment – i.e. what fraction of the subsidy is passed onto consumers and what fraction is retained by
producers?
(d) How would your answer to (c) change if the government instead imposed a per unit tax t = 15?
(e) How much of a tax or subsidy has to be set in order to get the efficient level of output under each of the
four market conditions?
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Chapter 25. Oligopoly
(f) Suppose you are advising the government on policy and you have two choices: Either you subsidize the
firms in the oligopoly, or you lower the barriers to entry that keep the industry from being perfectly
competitive. For each of the four market conditions, determine what cost you would be willing to have
the government incur to make the industry competitive rather than subsidize it?
(g) Suppose that pollution was produced in this industry – emitting a constant level of pollution per unit of
output, with a cost of b per unit of output imposed on individuals outside the market. How large would
b have to be under each of the market conditions in order for the outcome to be efficient (without any
government intervention)?
25.12 Policy Application: Government Grants and Cities as Cartels: In exercise 19.6, we explored the idea of city
wage taxes and noted that these were exceedingly rare and occurred primarily in very large cities. We explained this
by noting that labor demand and supply are more wage elastic locally than they are nationally – because firms and
workers can move from one city to another more easily than they can move from one country to another. We then
suggested that it would make sense for a mayor of a city (that wants to raise revenues by taxing wages) to ask the
national government to increase wage taxes nationally and pass back the revenues to cities and other communities
in the form of grants. Review the logic behind this. If cities persuaded the national government to do this, in what
way are they overcoming a prisoners’ dilemma? Have they found a way to successfully collude (in a way similar to
cartels)?