Chapter 14
Traditional Models of Imperfect Competition
Now we will relax some assumptions of perfect competition; the
assumptions about the number (and size) of firms in the industry,
homogeneity of products, and freedom of entry. Models of imperfect
competition include a mixture of perfect competition and monopoly.
A. Homogeneous Oligopoly (Assumptions)
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•
•
•
•
•
•
Few (large) firms selling a single homogeneous product.
Perfect competition on the buyer side.
The product obeys the law of one price.
No transaction or information costs.
Barriers to entry are prohibitive.
All firms face the same cost structure (identical firms).
Industry demand is P = f(Q) (inverse demand function), where Q =
q1+q2+…+qn=qi =nqi and qi = output of firm i. The price depends on output of
all firms.
The firm wants to max i = Pqi-C(qi) = f(Q)qi-C(qi) = f(q1+q2+…+qn)qi -C(qi).
Quasi-Competitive Model
Each firm assumes that its decisions do not affect price. It acts
like a price taker (MR=AR=D=P).
To max i,
C(q i )
π i
P
0
q i
q i
 P  MCi (q i )  0
MR=P*
Solve for
Assume constant
C
MC

MR
qi*.
Q*=qi*
D
=
nqi*
Also, MCi = MCj = MC for all i  j.
• Thus, the industry will reach the competitive solution
(P=MC).
• The “price-taking assumption” leads to behavior as if
MR=AR=D=P, that is, that the demand curve is horizontal
when in fact it is not.
• How long could this last before firms realize that they face
downward sloping demand curves and restrict output to
maximize profit?
Cartel Model
Firms realize their effect on price and collude to take advantage of their
market power. Collusion will lead to a monopoly solution because the firms
coordinate their outputs explicitly to achieve maximum  for the industry.
They realize that they are interdependent and their fortunes are tied together.
They cooperate for the good of all. The cartel attempts to maximize:
π  PQ  C1 (q1 )  C2 (q2 )  ...  Cn (qn )
 f(q1  q 2  ...  q n )(q1  q 2  ...  q n )   Ci (qi )
Product Rule
P Ci (qi )
π
FOC is
 P  (q1  q 2  ...  q n )

 0 for all i.
q i
q i
q i
P
 PQ
 MC i (q i )  0
q i
This equals MR(Q) because by
definition it does not matter which
firm’s quantity changes.
P*


M
MR
D
MC(Q)=MCi(qi)
Q*=nqi*
 MR(Q)  MCi (q i )  0
All firms produce the same qi with the
Also, MCi (q i )  MC(Q)
same cost. This is the monopoly solution.
because all firms face the same costs.
Problems with Cartels
•Collusion is illegal in most countries.
•Collusion requires substantial information.
•Each member has an incentive to increase qi because P > MCi.
•The cartel is unstable because each firm has an incentive to
cheat. It would require a strong policeman to direct the cartel.
OPEC oil cartel is a special problem because Saudi Arabia has
lower costs.
•Those outside the cartel have incentive to increase production
because price is high (interlopers).
Cournot Model
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•
•
•
•
Each firm recognizes that its output affects price, but thinks that changes
in its output will not cause other firms to change their output.
P
q j
So
 0 but
 0 for i  j. “Conjectural variation” assumed = 0.
q i
q i
 i
P
 P  qi
 MC i ( q i )  0
q i
q i
Max πi = Pqi – Ci(qi) ;
FOC
For the Cournot model, qi < Q for the cartel model.
MR for Cournot > MR for Cartel.
Cournot firm’s output will exceed the cartel firm’s output because each
firm’s MR(qi) > MR(Q) for the cartel.
But, because qi(P/qi) will be negative, P>MCi(qi). Thus, output will
occur between M and C at a point such as A.
M
The Cournot model followed to its conclusion
A
MCi = MC
C
indicates that each firm will produce the same qi and
n
C.
that total industry output will be Q 
D
MR
Q*
n 1
where n = number of firms and C = the competitive
industry output level (Footnote 3, page 423 of text).
• As n increases, Q approaches C.
• If n = 1, Q = 1/2C, which is the
monopoly output for a horizontal
MC curve.
• If n = 2, Q = 2/3C.
$
Monopoly or Cartel
M
C
MR
Q=½C
C
• LESSON: Output expands toward competitive output as n
increases. This result also follows from the fact that as n
increases q i P q i approaches 0. This gets smaller and
smaller as n gets larger because, with more firms in the
industry, an individual firm has less market power (P/qi
gets smaller). P approaches MCi(qi) as n gets larger.
MCi = MC
D
Q
Conjectural Variations Model
So far we have not allowed for each firm to realize that its qi actions will elicit a
qj reaction from others in the industry. In reality, each firm will have
“conjectures” about how other firms will react to its qi actions. No interaction
would result in the Cournot solution:
 q j  q i  0 , i  j.
If we relax this assumption, the FOCs for each firm to maximize i = Pqi-Ci(qi):
 P
π i
P q j 


 P  qi 
  MCi (q i )  0
q i
 q i ji q j q i 
Effect on P of a change in
the firm’s own output.
Effect on P of the other firms’ output changes that
result from my firm’s output change.
There is no general theory about the form and size of the qj/qi. One could
imagine many possibilities. Thus, without some assumptions about firms’
conjectures, we do not know much about where this model will lead. The
solution is indeterminate and may be unstable. “There is no theory of
oligopoly”??
Price Leadership - One possibility is to assume
that the industry has a price leader, maybe a low-cost
firm or maybe the largest firm, and a residual group of
followers who are quasi-competitors (fringe firms).
The leader sees its market as the excess demand below
price P1 and above price P2 (QL = QT - QSC). It
equates its MCL with its MRL and sells QL at PL.
Fringe firms sell QSC at PL (dictated by leader and
“taken” by the fringe). Total Q is QT = QL + QSC.
$ DT
Competitive firms supply
the quantity indicated by
point A and the leader
supplies 0 at P1.
MC’L
Competitive firms
supply 0.
DL
 MC
A
P1
PL
i
(fringe)  Q SC
i
D
P2
L
MRL
MCL
0
QSC QL
QT
D
T
Example: Cournot’s Duopoly
Assume two firms producing spring water at zero marginal cost and that the
demand for spring water is: Q = q1 + q2 = 120-P P = 120 - Q.
$
120
Demand
Quasi-Competitive Solution: The firm thinks its decisions do
not affect P. The firm acts as a Price Taker, MR = P. Optimal
M
P = MC = 0. Optimal q1 = 60 =q2, =0, Q = 120 at point C.
60
R
40
S
30
Cartel Solution: D = AR = P = 120 – Q. All firms act
MR
2 =R
C
together
as
one.

=
PQ
=
(120
Q)Q
=
120Q
–
Q
MC = AC = 0
60 80 90 120 Q
(C = 0). FOC  Q  120  2Q  MR  MC  0.
Optimal Q = 60, q1 = 30 =q2, P = 60,  = 3600 at point M, 1 = 1800 = 2.
Cournot Solution:  q j  q i  0 , i  j but  P  q i  0
2



1  Pq1  (120  q1  q 2 )q1  120q1  q  q1q 2
2  Pq 2  (120  q1  q 2 )q 2  120q 2  q1q 2  q 22
  1  q 1  120  2 q 1  q 2  0 because q 2 q1  0
FOC
  2  q 2  120  q 1  2 q 2  0 because q1 q 2  0
2
1
Q
C  80
3
2
Q  120  80
3
These FOCs are called reaction functions. They show how each firm reacts to
the other firm’s decisions. For equilibrium to exist, each firm must do what the
other firm expects. Thus, solving simultaneously gives point R where optimal
Q = 80, q1 = 40 =q2, P = 120 – (q1 + q2) = 40, 1 = 1600 = 2, and  = 3200.
120
$
Demand
MC = AC = 0


 
M
60
40
30
MR
R
S
60 80 90
C
120 Q
Von Stackelberg’s Solution: If one firm can predict
accurately how the other will react, but the reverse is
not true, then the first firm has an advantage.
Say firm 1 knows that q2  (120 q1) 2. This is the
reaction function of firm 2 solved for q2. Firm 1can
calculate:  q 2  q 1   1 2 . This means that
whatever firm 1 does, firm 2 will move in the
opposite direction by ½ and firm 1 knows it.
π1  Pq1  (120  q1  q 2 )q1  120q1  q1q 2  q12
2  Pq 2  (120  q1  q 2 )q 2  120q 2  q1q 2  q 22
π1
q 2 = -1/2
FOC
 120  q1
 q 2  2q1  0 because q 2 q1  0.
q1
q1
 120  1.5q1  q 2  0 Firm 1’s reaction function has changed.
 2
 120  q 1  2 q 2  0 As before because q1 q 2  0.
q 2
Solve the reaction functions simultaneously to get optimal Q = 90 at
point S, q1 = 60, q2 = 30, P = 30, 1 = 1800, 2 = 900,  = 2700.
B. Differentiated Oligopoly
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Assume there is a small number of firms in a product group (a group of goods
whose cross-price elasticities are very high within the group, but are relatively
low with goods outside the group). A product group is a group of goods that
substitute well for one another.
Barriers to entry are high.
Each firm may spend money to differentiate its product. Methods of
differentiation may involve advertising, options, quality, location, etc.
Use i to represent the expenditures of firm i on product differentiation.
Ci  Ci (q i , z i ).
Each firm’s price will probably be somewhat different, so that the inverse
demand function for the ith firm will be: pi  gi (qi , p j , zi , z j ),
where pj and zj are prices and differentiation expenditures by all other firms.
We expect:  p i  q i  0,  p i  p j  0,  p i  z i  0,  p i  z j  0.
The pi=gi(qi|pj, zi, zj) demand curve is shifted by pj, zi, and zj.
Profits are i = piqi –Ci(qi, zi), assuming z j  z j  p j  p j  0 (Other
q i z i q i z i
firms will not change pj or zj
when firm i changes qi and zi).
Given these assumption, the FOCs are:
C i
π i
p
 pi  q i i 
0
q i
q i q i
π i
p
C i
0
 qi i 
z i z i
z i
The first says MR = MC for qi (typical).
The second says MRP = MIC for zi. This
means that the firm should increase
differentiation expenditures (zi) until the
marginal contribution to revenues equals the
marginal contribution to costs (i.e., MRP
input = MIC input).
Relaxing the zero derivative assumptions
means that the demand curve for firm i shifts
frequently and, perhaps, unpredictably. Thus,
the differentiated model is even more
complex than the homogeneous model
because there are more variables and other
firms react to changes in qi and zi. Therefore,
few definitive conclusions are possible
because of the unpredictable interdependence
among firms!
Advertising rotates and expands the
demand curve. It 1) promotes product
differentiation (rotation), 2) may
expand the market for a firm, 3) may
expand the total product group market,
4) should be undertaken to point where
MRPAdv = MICAdv (Whether products
are really different or not doesn’t
matter much; just perception matters),
5) may be subject to economies of
scale, which gives advantage to large
firms and creates barriers to entry, 6)
may convey useful, truthful
information and, thus, improve market
efficiency, and 7) may be false and
confusing, thus, leading to market
inefficiency (When not informative, it
wastes resources. When only
combative or defensive in a fixed total
market, it is inefficient). Judgments on
advertising are mixed.
Allow freedom of entry now:
Freedom of entry means that long-run profits will be zero under conditions of competition. This is
true for oligopoly, also. Economic profit attracts entry!
=(P*-AC)Q*
Even under free entry in oligopoly, there is no assurance that production will occur where AC is
minimized (as with perfect competition). It would be an accident if a firm with a downward sloping
demand curve produced where AC is minimized. If it did, it would have to make excess profits.
$
MC
p*
AC
AC
0
Q*
Q
MC
$
p*
0
D
MR
MR
Q*
AC
D
Q
Minimizing AC - Economic profit >0, Q restricted, price
high, transfer of consumer surplus to firm! In this case,
no excess capacity (Minimum AC); but economic profit
(firms have incentive to enter).
Non-minimizing AC - Zero economic profit, Q still restricted
because of negatively sloped demand curve and the resulting
separation of MR=MC from p! In this case, excess capacity
(not at minimum AC), but zero economic profit.
When demand slopes downward, the firm must either have excess  or excess
capacity or both. In both cases, Q is restricted and p is elevated relative to the
perfectly competitive solution (P=MC). The downward sloping demand curve
for the firm may result from differentiated products (  p i  q i  0) or in the case
of homogeneous oligopoly few firms in industry (  P  q i  0) . Free entry will
result in zero profits, but oligopoly normally does not have free entry.
Monopolistic Competition
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Large n; many small firms.
Free entry.
Product differentiation means firm demand curve slopes downward slightly
(fairly elastic)
Economic profits are zero (free entry)
The location of the demand curve depends on advertising and the number of
competitors and their actions.
Prices may differ somewhat among firms.
Because demand slopes downward, p > MC.
The zero profit condition determines the number of firms. A firm’s demand
curve will shift because of entry and exit until zero profit is earned.
Excess capacity exists in each firm (not at minimum AC), which is the
primary problem with monopolistic competition.
q is restricted and p is high relative to perfect competition (p=MC).
$
p1
MC
AC
p2
d1
d2
0
q2 q1
q
Start with d1, with q1 demanded at p1.
Determined by MR1 (not shown in graph) =
MC. Excess profit is earned by firms in the
industry, so other firms enter, eventually
reducing demand for this firm to d2 where zero
profit is earned at q2 and p2 where MR2=MC.
Each firm in the industry has excess capacity
(producing at higher than minimum AC).
Contestable Markets
Some economists have suggested that it is not the actual entry of firms that
constrains profits. Rather, it is the threat of entry that may constrain profits. If a
market is contestable, firms already in the industry may lower profits to zero as a
barrier to entry. Monopolies and oligopolies may be constrained to lower prices if
they are in contestable markets (constrained by threat of entry). For example,
OPEC may want oil prices to decline so the United States, Briton, and others do
not develop alternative sources of energy or increase oil exploration. Thus, perfect
competition may not be the only avenue to low or zero economic profit. Firms in
contestable markets still have excess capacity and restrict Q so that P > MR = MC.
Contestability may mean that firms may “raid” an industry by entering even where
P = AC ( = 0) and attempt to garner a large market share quickly by under pricing
existing firms, moving down their downward sloping AC curves (hoping to make
a profit eventually). This activity requires low fixed costs, low start-up cost, and
low shut-down costs. These conditions are very restrictive. They might occur
where capital equipment could be leased, unskilled labor could be hired easily, and
brands are not very important. Thus, equilibria for firms may not be stable if the
market is highly contestable. Existing firms may have to lower and raise prices
frequently to forestall entry.
The “entry forestalling price” may be the key profit constraint to existing firms.