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Unformatted text preview: CHAPTER 2 7 OLIGOPOLY We have now investigated two important forms of market structure: pure
competition, where there are typically many small competitors, and pure
monopoly, where there is only one large ﬁrm in the market. However,
much of the world lies between these two extremes. Often there are a
number of competitors in the market, but not so many as to regard each
of them as having a negligible effect on price. This is the situation known
as oligopoly. The model of monopolistic competition described in Chapter 24 is a
special form of oligopoiy that emphasizes issues of product differentiation
and entry. However, the models of oligopoly that we will study in this
chapter are more concerned with the strategic interactions that arise in an
industry with a small number of ﬁrms. There are several models that are relevant since there are several different
ways for ﬁrms to behave in an oligopolistic environment. It is unreason
able to expect one grand model since many different behavior patterns can
be observed in the real world. What we want is a guide to some of the
possible patterns of behavior and some indication of what factors might be
important in deciding when the various models are applicable. QUANTlTY LEADERSHIP 481 For simplicity, we will usually restrict ourselves to the case of two ﬁrms;
this is called a situation of duopoly. The duopoly case allows us to cap
ture many of the important features of ﬁrms engaged in strategic interaction
without the notational complications involved in models with a larger num
ber of ﬁrms. Also, we will limit ourselves to investigation of cases in which
each ﬁrm is producing an identical product. This allows us to avoid the
problems of product differentiation and focus only on strategic interactions. 27.1 Choosing a Strategy If there are two ﬁrms in the market and they are producing a homogeneous
product, then there are four variables of interest: the price that each ﬁrm
charges and the quantities that each ﬁrm produces. When one ﬁrm decides about its choices for prices and quantities it may
already know the choices made by the other ﬁrm. If one ﬁrm gets to set its
price before the other ﬁrm, we call it the price leader and the other ﬁrm
the price follower. Similarly, one ﬁrm may get to choose its quantity ﬁrst,
in which case it is a quantity leader and the other is a quantity follower.
The strategic interactions in these cases form a sequential game.1 On the other hand, it may be that when one ﬁrm makes its choices it
doesn’t know the choices made by the other ﬁrm. In this case, it has to
guess about the other ﬁrm’s choice in order to make a sensible decision
itself. This is a simultaneous game. Again there are two possibilities:
the ﬁrms could each simultaneously choose prices or each simultaneously
choose quantities. This classiﬁcation scheme gives us four possibilities: quantity leadership,
price leadership, simultaneous quantity setting, and simultaneous price set—
ting. Each of these types of interaction gives rise to a different set of
strategic issues. There is also another possible form of interaction that we will examine.
Instead of the ﬁrms competing against each other in one form or another
they may be able to collude. In this case the two ﬁrms can jointly agree
to set prices and quantities that maximize the sum of their proﬁts. This
sort of collusion is called a cooperative game. 27.2 Quantity Leadership In the case of quantity leadership, one ﬁrm makes a choice before the other
ﬁrm. This is sometimes called the Stackelberg model in honor of the 1 We will examine game theory in more detail in the next chapter, but it seems appro
priate to introduce these speciﬁc examples here. 482 OLIGOPOLY (Ch. 27) ﬁrst economist who systematically studied leaderfollower interactions.2 The Stackelberg model is often used to describe industries in which there
is a dominant ﬁrm, or a natural leader. For example, IBM is often con—
sidered to be a dominant ﬁrm in the computer industry. A commonly
observed pattern of behavior is for smaller ﬁrms in the computer industry
to wait for IBM’s announcements of new products and then adjust their
own product decisions accordingly. In this case we might want to model
the computer industry with IBM playing the role of a Stackelberg leader,
and the other ﬁrms in the industry being Stackelberg followers. Let us turn now to the details of the theoretical model. Suppose that
ﬁrm 1 is the leader and that it chooses to produce a quantity yl. Firm 2
responds by choosing a quantity yg. Each ﬁrm knows that the equilibrium
price in the market depends on the total output produced. We use the
inverse demand function p(Y) to indicate the equilibrium price as a function
of industry output, Y = y1 + yg. What output should the leader choose to maximize its proﬁts? The
answer depends on how the leader thinks that the follower will react to its
choice. Presumably the leader should expect that the follower will attempt
to maximize proﬁts as well, given the choice made by the leader. In order
for the leader to make a sensible decision about its own production, it has
to consider the follower’s proﬁtmaximization problem. The Follower’s Problem We assume that the follower wants to maximize its proﬁts 115% 19041 + 92ly2 — 62(92)
2 The follower’s proﬁt depends on the output choice of the leader, but from
the viewpoint of the follower the leader’s output is predeterminedthe
production by the leader has already been made, and the follower simply
views it as a constant. The follower wants to choose an output level such that marginal revenue
equals marginal cost: A
MR2 = p(y1+ 1/2) + Aim = M02.
W The marginal revenue has the usual interpretation. When the follower
increases its output, it increases its revenue by selling more output at the 2 Heinrich von Stackelberg was a German economist who published his inﬂuential work
on market organization, Marktform and Gleichgewz‘cht, in 1934. QUANTITY LEADERSHIP 483 market price. But it also pushes the price down by A39, and this lowers its
proﬁts on all the units that were previously sold at the higher price. The important thing to observe is that the proﬁt—maximizing choice of
the follower will depend on the choice made by the leader. We write this
relationship as 92 = ﬁlm) The function f2(y1) tells us the proﬁtmaximizing output of the follower
as a function of the leader’s choice. This function is called the reaction
function since it tells us how the follower will react to the leader’s choice
of output. Let’s derive a reaction curve in the simple case of linear demand. In this
case the (inverse) demand function takes the form p(y1+112) = a—b(y1 +y2).
For convenience we’ll take costs to be zero. Then the proﬁt function for ﬁrm 2 is 7T2(y1,y2) = [a — 5(91 + y2lly2 OI'
W2(y1,y2)= ay2  b14192 — 531% We can use this expression to draw the isoproﬁt lines in Figure 27.1.
These are lines depicting those combinations of y1 and y2 that yield a
constant level of proﬁt to ﬁrm 2. That is, the isoproﬁt lines are comprised
of all points (y1,y2) that satisfy equations of the form (192 — ll3911/2 — by% = W2. Note that proﬁts to ﬁrm 2 will increase as we move to isoproﬁt lines that
are further to the left. This is true since if we ﬁx the output of ﬁrm 2 at
some level, ﬁrm 2’s proﬁts will increase as ﬁrm 1’s output decreases. Firm 2
will make its maximum possible proﬁts when it is a monopolist; that is,
when ﬁrm 1 chooses to produce zero units of output. For each possible choice of ﬁrm 1’s output, ﬁrm 2 wants to choose its own
output to make its proﬁts as large as possible. This means that for each
choice of y1, ﬁrm 2 will pick the value of 3,12 that puts it on the isoproﬁt
line furthest to the left, as illustrated in Figure 27.1. This point will satisfy
the usual sort of tangency condition: the slope of the isoproﬁt line must
be vertical at the Optimal choice. The locus of these tangencies describes
ﬁrm 2’s reaction curve, f2(y1).
To see this result algebraically, we need an expression for the marginal
revenue associated with the proﬁt function for ﬁrm 2. It turns out that
this expression is given by MR2(y1,yzl = a  byi — 2by2. 484 OLIGOPOLY (Ch. 27) y2 = OUTPUT
OF FIRM 2 lsoproﬁt lines
for firm 2 f2(Y1) Reaction
curve l‘2(y1 ) y1 y1 = OUTPUT or: FIRM I Derivation of a reaction curve. This reaction curve gives
the proﬁt—maximizing output for the follower, ﬁrm 2, for each
output choice of the leader, ﬁrm 1. For each choice of y1 the
follower chooses the output level f2(y1) associated with the iso
proﬁt line farthest to the left. (This is easy to derive using calculus. If you don’t know calculus, you’ll
just have to take this statement on faith.) Setting the marginal revenue
equal to marginal cost, which is zero in this example, we have a— byl  2by2 = O, which we can solve to derive ﬁrm 2’s reaction curve: anbyl
2b ' 92: This reaction curve is the straight line depicted in Figure 27.1. The Leader’s Problem We have now examined how the follower will choose its output given the
choice of the leader. We turn now to the leader’s proﬁt—maximization
problem. Presumably, the leader is also aware that its actions inﬂuence the output
choice of the follower. This relationship is summarized by the reaction QUANTITY LEADERSHIP 485 function f2(y1). Hence when making its output choice it should recognize
the inﬂuence that it exerts on the follower.
The proﬁt—maximization problem for the leader therefore becomes Iggx p(y1 + y2)y1 — 01(y1) such that 3/2 = f2(y1) Substituting the second equation into the ﬁrst gives us
Iggx Plyl + f2(y1)ly1 — 01(91) Note that the leader recognizes that when it chooses output yl, the to
tal output produced will be yl + f2(y1): its own output plus the output
produced by the follower. When the leader contemplates changing its output it has to recognize
the inﬂuence it exerts on the follower. Let’s examine this in the context of
the linear demand curve described above. There we saw that the reaction
function was given by (1 —~ byl
2b f2(y1) = y2 = . (27.1) Since we’ve assumed that marginal costs are zero, the leader’s proﬁts are 7r1 ($11,312) = plyi + y2)yl = ayl — by? — byiyz (272) But the output of the follower, yg, will depend on the leader’s choice via
the reaction function y2 = f2(y1).
Substituting from equation (27.1) into equation (27.2) we have «1(91, y2) = ayi * by? — by1f2(y1) a — byl
=— ayl  by? — byl 2b .
Simplifying this expression gives us
a b 2
7T1(y17y2) — 51/1 “ 291 The marginal revenue for this function is O, M =
R 2 byl . Setting this equal to marginal cost, which is zero in this example, and
solving for yl gives us yl 2 2b' 486 OLIGOPOLY (Ch. 27) In order to ﬁnd the follower’s output, we simply substitute yi‘ into the
reaction function, ._ a—byi‘
y2__ 2b
*1
“411' These two equations give a total industry output of yf + yﬁ = 3a / 4b. The Stackelberg solution can also be illustrated graphically using the
isoproﬁt curves depicted in Figure 27.2. (This ﬁgure also illustrates the
Cournot equilibrium which will be described in section 27.5.) Here we
have illustrated the reaction curves for both ﬁrms and the isoproﬁt curves
for ﬁrm 1. The isoproﬁt curves for ﬁrm 1 have the same general shape as
the isoproﬁt curves for ﬁrm 2; they are simply rotated 90 degrees. Higher
proﬁts for ﬁrm 1 are associated with isoproﬁt curves that are lower down
since ﬁrm 1’s proﬁts will increase as ﬁrm 2’s output decreases. .)’2 Reaction
curve. for
firm 1 Reaction
curve fer
firm 2 Cournot
equilibrium Stackelberg
equilibrium lsoprofit
. curves for
. firml Y1 Stackelberg equilibrium. Firm 1, the leader, chooses the
point on ﬁrm 2’s reaction curve that touches firm 1’s lowest
possible isoproﬁt line, thus yielding the highest possible proﬁts
for ﬁrm 1.; _  Firm 2 is behaving as a follower, which means that it will choose an
output along its reaction curve, f2(y1). Thus ﬁrm 1 wants to choose an PRICE LEADERSHIP 487 output combination on the reaction curve that gives it the highest possible
proﬁts. But the highest possible proﬁts means picking that point on the
reaction curve that touches the lowest isoprofit line, as illustrated in Figure
27.2. It follows by the usual logic of maximization that the reaction curve
must be tangent to the isoproﬁt curve at this point. 27.3 Price Leadership Instead of setting quantity, the leader may instead set price. In order to
make a sensible decision about how to set its price, the leader must forecast
how the follower will behave. Accordingly, we must ﬁrst investigate the
proﬁtmaximization problem facing the follower. The ﬁrst thing we observe is that in equilibrium the follower must always
set the same price as the leader. This follows from our assumption that the
two ﬁrms are selling identical products. If one charged a different price from
the other, all of the consumers would prefer the producer with the lower
price, and we couldn’t have an equilibrium with both ﬁrms producing. Suppose that the leader has set a price 19. We will suppose that the
follower takes this price as given and chooses its proﬁtmaximizing output.
This is essentially the same as the competitive behavior we investigated
earlier. In the competitive model, each ﬁrm takes the price as being outside
of its control because it is such a small part of the market; in the price
leadership model, the follower takes the price as being outside of its control
since it has already been set by the leader. The follower wants to maximize proﬁts: max pyz — 02(y2).
112 This leads to the familiar condition that the follower will want to choose
an output level where price equals marginal cost. This determines a supply
curve for the follower, S (p), which we have illustrated in Figure 27.3. Turn now to the problem facing the leader. It realizes that if it sets
a price p, the follower will supply S (p) That means that the amount of
output the leader will sell will be R(p) = D(p) — S (p) This is called the
residual demand curve facing the leader. Suppose that the leader has a constant marginal cost of production c.
Then the proﬁts that it achieves for any price p are given by: mp) = (p — 0) [13(10) — 500)] = (p — C)R(p) In order to maximize proﬁts the leader wants to choose a price and output
combination where marginal revenue equals marginal cost. However, the
marginal revenue should be the marginal revenue for the residual demand
curveithe curve that actually measures how much output it will be able to 488 OUGOPOLY (Ch. 27) Demand curve
facing leader
(residual demand) MR facing leader yZ‘ y: QUANTITY Price leader. The demand curve facing the leader is the
market demand curve minus the follower’s supply curve. The
leader equates marginal revenue and marginal cost to find the
optimal quantity to supply, ’92. The total amount supplied to
the market is y; and the equilibrium price is p*. sell at each given price. In Figure 27.3 the residual demand curve is linear;
therefore the marginal revenue curve associated with it will have the same
vertical intercept and be twice as steep. Let’s look at a simple algebraic example. Suppose that the inverse de
mand curve is D(p) : aebp. The follower has a cost function 02(y2) : 36/2,
and the leader has a cost function c1(y1) 2 cyl. For any price p the follower wants to Operate where price equals marginal
cost. If the cost function is 62(y2) : 313/2, it can be shown that the marginal
cost curve is M Cg (y2) = y2. Setting price equal to marginal cost gives us 17:92 Solving for the follower’s supply curve gives yg 2 5(1)) : p.
The demand curve facing the leader—the residual demand curve—is R(p)IBM5(1))=a—bp—p=a~(b+1)p From now on this is just like an ordinary monopoly problem. Solving for
p as a function of the leader’s output yl, we have 91 (27.3) SIMULTANEOUS QUANTITY SETTING 489 This is the inverse demand function facing the leader. The associated
marginal revenue curve has the same intercept and is twice as steep. This
means that it is given by a 2 MR1=b+1—b+1 91 Setting marginal revenue equal to marginal cost gives us the equation a _ 2
b+1 bil MR1: y1=C=M01. Solving for the leader’s proﬁtmaximizing output, we have a—c(b+1) 111: 2 We could go on and substitute this into equation (27.3) to get the equilib
rium price, but the equation is not particularly interesting. 27.4 Comparing Price Leadership and Quantity Leadership We’ve seen how to calculate the equilibrium price and output in the case of
quantity leadership and price leadership. Each model determines a different
equilibrium price and output combination; each model is appropriate in
different circumstances. One way to think about quantity setting is to think of the ﬁrm as making
a capacity choice. When a ﬁrm sets a quantity it is in effect determining
how much it is able to supply to the market. If one ﬁrm is able to make
an investment in capacity ﬁrst, then it is naturally modeled as a quantity
leader. On the other hand, suppose that we look at a market where capacity
choices are not important but one of the ﬁrms distributes a catalog of
prices. It is natural to think of this ﬁrm as a price setter. It’s rivals may
then take the catalog price as given and make their own pricing and supply
decision accordingly. Whether the price—leadership or the quantityleadership model is appro—
priate is not a question that can be answered on the basis of pure theory.
We have to look at how the ﬁrms actually make their decisions in order to
choose the most appropriate model. 27.5 Simultaneous Quantity Setting One difficulty with the leader—follower model is that it is necessarily asym
metric: one ﬁrm is able to make its decision before the other ﬁrm. In some 490 OLIGOPOLY (Ch. 27) situations this is unreasonable. For example, suppose that two ﬁrms are
simultaneously trying to decide what quantity to produce. Here each ﬁrm
has to forecast what the other ﬁrm’s output will be in order to make a
sensible decision itself. In this section we will examine a oneperiod model in which each ﬁrm
has to forecast the other ﬁrm’s output choice Given its forecast, each ﬁrm
then chooses a proﬁt~maximizing output for itself. We then seek an equi
librium in forecasts#a situation where each ﬁrm ﬁnds its beliefs about the
other ﬁrm to be conﬁrmed. This model is known as the Cournot model,
after the nineteenthcentury French mathematician who ﬁrst examined its
implications.3 We begin by assuming that ﬁrm 1 expects that ﬁrm 2 will produce 3;;
units of output. (The e stands for ewpected output.) If ﬁrm 1 decides to
produce yl units of output, it expects that the total output produced will
be Y = y1 + 11%, and output will yield a market price of p(Y) : p(y1 + y§).
The proﬁt—maximization problem of ﬁrm 1 is then Himmler + 31391  C(91) For any given belief about the output of ﬁrm 2, pg, there will be some
optimal choice of output for ﬁrm 1, pl. Let us write this functional rela
tionship between the empected output of ﬁrm 2 and the optimal choice of
ﬁrm 1 as 91 = 161(95) This function is simply the reaction function that we investigated earlier in this chapter. In our original treatment the reaction function gave the follower’s output as a function of the leader’s choice. Here the reaction function gives one ﬁrm’s optimal choice as a function of its beliefs about the other ﬁrm’s choice. Although the interpretation of the reaction function is different in the two cases, the mathematical deﬁnition is exactly the same.
Similarly, we can derive ﬁrm 2’s reaction curve: 1’2 : f2(yl3)i which gives ﬁrm 2’s optimal choice of output for a given expectation about
ﬁrm 1’s output, yf. Now, recall that each ﬁrm is choosing its output level assuming that the
other ﬁrm’s output will be at y? or y§. For arbitrary values of pi and pg
this won’t happen—in general ﬁrm 1’s optimal level of output, yl, will be
different from what ﬁrm 2 expects the output to be, yf. Let us seek an output combination (yf, 143) such that the optimal output
level for ﬁrm 1, assuming ﬁrm 2 produces 213...
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