We now turn to the situation when there are a small number of firms
in the industry and these firms have the option of colluding with
or competing with each other. To begin with, we assume that there are
only two firms---a situation called duopoly. Then in the next Topic
we will consider a larger number of firms---first four and then ten.

**When there are only two firms in the industry, it is in their advantage
to collude and set the price and their individual outputs at levels that
will maximize their joint profits. This situation is shown in Figure 1
where the demand curve, given by DD, is the individual firm's share of
the market demand under circumstances where the two firms are identical
with respect to size and costs of production.
**

**
The results, as should be clear from the discussion in the previous Topic,
are that each firm is a monopoly supplier of half the industry output and,
given agreement that the price charged should be
P_{{0}} , chooses the output that creates equality between
its marginal revenue (and the industry marginal revenue) with its own marginal
costs**, at point

1. P = 70 − 0.0325 Q .

Under complete collusion, with the firms of equal size so that Q = 2 q , each individual firm's demand curve is

2. P = 70 − 0.065 q .

The marginal revenue curve of the two firms combined is obtained by calculating the change in the total revenue of the industry for each successive one-unit change in industry output---that is,

which simplifies to

3. MR = 70 − 0.065 Q

where MR is the marginal revenue of the industry. The above equation turns out to be identical to the demand curves of the individual firms. The individual firms' marginal revenue relations, assuming that both produce the same output so that Q = 2 q , is given by

4. MR = 70 − 0.13 q .

The individual firms' total, average and marginal cost curves are calculated in a fashion that makes the average cost curves U-shaped and the marginal cost curves upward sloping beyond some appropriate output level as shown in the Figures here presented.

**There are two problems with collusion. First, it is illegal in most advanced
industrial countries, so that any collusive arrangements can not be written
down and legally enforced. Second, given this illegality, any collusive
arrangement can be be profitably broken by one of the parties to the
detriment of the other. Suppose that one of our two firms decides to
break their collusive arrangement and to act independently, while the other
firm chooses to follow that arrangement. The situation with respect to the
deviant firm is presented in Figure 2 below.
**

**
The demand for the deviant firm's output is much more elastic than the
industry demand, given the constant output of the other firm, and the deviant
firm's marginal revenue, denoted by MR , is also much flatter
and closer to the firm's demand curve when it increases its output beyond
that agreed to in the collusive arrangement. The collusive demand and marginal
revenue curves are given by the dotted lines in the figure extending downward to
the right from point h . The deviant firm's demand curve
must pass through the collusive demand curve at the collusive price and quantity
equilibrium. When the deviant firm increases
its output** by dq ,

**
At this point, it becomes reasonable for the colluding partner firm to also
break with the collusive arrangement and produce its most profitable output,
given the output of its break-away partner. At that point, the break-away
partner can increase its profits by adjusting its output to the most
profitable level given the new level of output of the other firm. The two
firms will continue to adjust their outputs in this fashion until neither
firm can gain by further adjusting its output. The resulting equilibrium
is called the Cournot equilibrium, after Antoine Augustin Cournot (1801-1877),
and is presented in Figure 3 below which, given our assumption that the two
firms are identical, represents the equilibrium of each of them.
**

To obtain this equilibrium we assume that each firm adjusts its output to maximize its profits, which are equal to

5. π = P(Q) q − C(q) ,

where π is the individual firm's profit, Q is the level of industry output, q is the level of output of the individual firm (where the two firms are assumed to be identical), P is the price of that output, P(Q) is the function presented in Equation 1, giving the level of P associated with each level of Q, and C(q) is a function giving the firm's total costs associated with each level of its output. To maximize its profit, each firm adjusts q until π is at its maximum, at which point

6. dπ/dq = P(Q) + q dP(Q)/dq − dC(q)/dq = P(Q) + q dP(Q)/dQ dQ/dq − dC(q)/dq = 0

which, upon rearrangement, becomes

7. P(Q) + q dP(Q)/dQ dQ/dq = dC(q)/dq .

The term dC(q)/dq is simply the marginal cost calculated in cournot.R and denoted as MC in the three Figures above. The term to the left of the equality in Equation 3 is the firm's marginal revenue, denoted below as MR, which reduces to

and becomes

8. MR = 70 − 0.0975 q

Essentially, each firm's demand curve has the same slope as in Figure 2, with
its intersection with the vertical asis being lower than the intersection of
the industry demand curve by an amount equal to slope of the firm demand curve
multiplied by the other firm's output,
under the assumption that the other firm is pricing the same way as it
is. **The difference between this Cournot equilibrium and the collusive one is that each
firm adjusts its output independently of the other firm's output to maximize its
profit, whereas under collusion it adjusts its output in conjunction with an
agreed-upon equivalent adjustment of the other firm's output.**

Under this equilibrium, both firms produce outputs of 506 thousand units, selling them at a price of $37.11 and earning profits of 4474.58 thousand dollars. This is 1168.96 thousand dollars less that either firm could earn if it were the only one to break the collusive arrangement and 713.27 thousand dollars less than they could each earn under complete collusion.

Of course, we can not take very seriously the magnitudes of the numbers in the example above. That example is based solely on the assumptions that the average total cost curve is U-shaped, that the industry demand curve is negatively sloped, and that the two firms are identical. These assumptions are consistent with a wide variety of possible numerical results---all that is important is the direction of the effects that arise from one or both firms breaking the collusive arrangement.

**An important tool used to analyse the interaction of firms under conditions
where collusion is possible but such arrangements can be easily broken is
game theory analysis. The classic example for the duopoly analysis here is
the prisoner's-dilemma game** which can be described as follows. Suppose that
there are two criminals jointly guilty of a serious crime who have been
arrested by the police and are being interviewed separately and simultaneously.
Each prisoner has the option of either confessing or not confessing to the
crime. If neither confess a conviction will be hard to obtain and both will
be convicted of a smaller crime and have to serve 1 year in jail. If one
prisoner confesses but the other does not, the one who confesses will go
free while the other will serve 20 years in prison. If both confess, they
will each have to serve 10 years in prison. The situation is outlined in
the Table below:

Prisoner #2 | |||

Confess | Do Not | ||

Prisoner #1 | Confess | (-10 , -10) | (0 , -20) |

Do Not | (-20 , 0) | (-1 , -1) |

The numbers in brackets give the cost to Prisoner #1 on the left and the cost to Prisoner #2 on the right.

Consider the situation from the point of Prisoner #1. If he assumes that
Prisoner #2 will confess, he spends 10 years in jail if he also confesses and
20 years in jail if he does not confess. If, on the other hand, he assumes
that Prisoner #2 will not confess, he spends no time in jail if he confesses
and 1 year in jail if he does not confess. He is better off confessing,
regardless of which decision Prisoner #2 makes. The situation is
exactly the same for Prisoner #2. By confessing, he serves 10 years in jail as
opposed to 20 years if Prisoner #1 confesses and zero time in jail as opposed
to to 1 year if Prisoner #1 does not confess. Both prisoners will thus choose
to confess---neither will later wish they had done the opposite after
observing what the other prisoner has decided to do. This is called a **Nash
equilibrium after the the famous game theorist John Nash (1929, ). Each
participant adopts the strategy that is best for him regardless of which
strategy the other participant chooses. Of course, in many games there is
no such strategy possible, in which case a Nash equilibrium cannot occur.**

**Now consider the two firms in the duopoly case analyzed above.** As can be seen
in the Table below, **the results are exactly comparable to the prisoner's
dilemma game except that the Nash equilibrium is for both firms to not abide
by any collusive agreement.**

Firm #2 | |||

Collude | Do Not | ||

Firm #1 | Collude | (5187.85 , 5187.85) | (4014.43 , 5643.54) |

Do Not | (5643.54 , 4014.43) | (4474.58 , 4474.58) |

Firm #1 will be better off by not colluding both in the case where Firm #2
abides by the collusive agreement and in the case where it does not, earning
5643.54 thousand dollars verses 5187.85 thousand dollars in the former case
and 4474.57 verses 4014.43 in the latter case. The situation is exactly the
same for Firm #2. If Firm #1 adheres to a collusive agreement Firm #2 will
earn 5643.54 thousand dollars by breaking the agreement verses 5187.85 by also
colluding. And if Firm #1 breaks the collusive agreement, firm #2 will earn
4474.58 thousand dollars as opposed to 4014.43 thousand dollars by also
breaking it. As noted above, **this equilibrium was established by Cournot,
using what became a Nash equilibrium as a result of Nash's game-theory work
many years later.
**

**
The question arises as to the socially efficient levels of output that the two
firms should produce and the price at which that output should be sold. One
way of thinking about this is to imagine the price that a government regulator
should impose on the two firms to be them to produce the most efficient level
of output. This can be seen with reference to Figure 4 below.
**

**
First note that the marginal cost to the firm represents the social marginal
cost to the extent that it incorporates all the social costs of producing the
product---that is, there are no production externalities. And the firm' demand
curve, which maps the price charged against one-half of the aggregate output
of the industry and is given by the dotted line DD in the Figure, gives the
amount that the public is willing to pay (assuming no consumption externalities)
for additional units of output as they are received. Keep in mind that this
amount equals the value of other goods the public is willing to give up in order
to obtain the last unit of the good consumed. So the DD curve is the marginal
social return. Equating marginal social cost with marginal social return leads
to the optimum output at point e in the Figure at the
government imposed price of OP.** That price, which in the model we numerically
calculate equals $32.105, exceeds the firm's average total cost, which equals
$28.003, resulting in excess profits of
2390.986 thousand dollars.

**
The simple example produced here vastly under-rates the problem faced by a
government agency whose function is to regulate a duopoly, or even a monopoly,
by imposing on it the socially efficient price. To do this, the authorities
have to be able to estimate the marginal costs of the firms and the industry
along with consumers' demand curve for the product. To obtain useful information
about costs, they most surely have to examine evidence obtained directly
from the firms, which have an obvious incentive to misrepresent their
situation. Moreover, any equilibrium price, be it the monopoly price, the
Cournot equilibrium price or the socially efficient price, will tend to vary
through time with economic conditions---failure of the authorities to recognize
the correct level and path of prices and adjust the regulated prices accordingly
may result in a greater waste of resources through regulation than would be lost
if the Cournot equilibrium, or even a collusive monopoly equilibrium, were
allowed to rule.
**

**
In the simple example above, it would seem reasonable for the two firms to find
a way to collude even when such collusion is illegal and unenforceable in the
courts. All it would take is a phone call followed by a movement of the
initiating firm to the collusive equilibrium. The other firm will face an
obvious gain in long-run profits by also adopting that equilibrium, knowing
that if it does not follow suit the initiating firm will go back to the
Cournot equilibrium. The practical problem, of course, is that the range of
products and services provided by such firms in the real world are not
identical and demand and costs are changing through time, with new profit
opportunities appearing on occasion and some current activities occasionally
becoming less profitable and needing to be abandoned. Thus, even in a
situation of Cournot equilibrium the firms' prices will be constantly adjusting
through time and in relation to each other. To maximize its profits under
Cournot equilibrium the individual firm has to make the correct pricing and
production decisions, given the assumption that the other firm will also do
the same. Even if the firms have informally agreed to collude, one of them
can always argue that its recent reduction in the price of its output or,
say, the free provision of associated services, was the result of factors
specific to its situation and not a violation of the collusive agreement, even
though it may in fact be deliberately violating that agreement. So the best
understanding we can arrive at is that the prices charged by the duopolists are
likely to be at the Cournot equilibrium level or somewhere above, yet probably
below the level possible with complete collusion.
**

**
One way in which the two firms above could maximize their value is to merge
into a single firm, either by one firm purchasing the other or by both firms
selling out to a third party who would be willing to pay them more than
the present value of future Cournot-equilibrium profits in order to create a
single monopoly firm. The problem is that these actions would probably
encounter legal prohibition or subsequent government regulation in most
advanced industrial countries.**

It is now time for a test. For your own intellectual enlightenment, think up complete answers to the questions before looking at the ones provided.

Choose Another Topic in the Lesson