### Lecture 7: Oligopoly

Advanced Microeconomics II

Yosuke YASUDA

National Graduate Institute for Policy Studies

January 7, 2014

### Oligopoly Models

We have so far looked at the extreme market structures of monopoly and perfect competition. However, most real-world markets are somewhere between these extremes:

◮ The number of the firms is more than one but less than the

“very large number.”

◮ The situation in which there are a few competitors is called oligopoly (duopoly if the number is two).

One thing the monopoly and perfect competition have in common is that each firm does not have to worry about its rivals’ reactions.

◮ In the case of monopoly, this is trivial as there are no rivals.

◮ In the case of perfect competition, the idea is that each firm is so small that its actions have no significant impact on rivals. By contrast, an important characteristic of oligopolies is the strategic interdependence between competitors, which is appropriately analyzed by game theory.

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### Hotelling Model (1)

Two convenience stores are going to open new shops on the street.

◮ Each store has to decide the location between 0 and 1.

◮ Customers are located uniformly on the street, and each customer goes to the nearest shop.

◮ If both shops choose the same location, each receives half of the customers.

◮ Each store is going to maximize the number of customers. The game is defined as follows:

◮ Players: Two stores, 1, 2.

◮ Strategies: Shop location along the street, s_{i}∈[0, 1] for
i= 1, 2.

◮ Payoffs: The number of customers described by
u_{i} =

( _{s}

i+sj

2

1 −^{s}^{i}^{+s}_{2} ^{j}

if si ^{≤ s}j

if s_{i} > s_{j} for i = 1, 2, i 6= j.

### Hotelling Model (2)

There is a unique Nash equilibrium where both shops open at the middle, (0.5, 0.5), which is shown by the following three steps:

1. Choosing separate locations never becomes a NE.

2. Choosing the same location other than the middle point also fails to be a NE.

3. If both shops choose the middle, then no one has an incentive to change the location.

✞

✝

☎

Rm The equilibrium is often referred as the principle of✆

minimum differentiation to explain little product differentiation, agglomeration of shops, similar target policies set by two political parties (the median voter theorem), and so on.

✞

✝

☎

Q What does happen if there are more than two players?✆

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### Bertrand Model of Duopoly (1)

Two firms producing perfectly substitutable goods (no product differentiation) compete in their prices.

◮ A downward demand function is given, D(p).

◮ The firms have a common marginal cost c.

◮ The firm with lower price will serve the entire market demand: if the price is the same, each firm serves the half of it.

The game is defined as follows:

◮ Players: Two firms, 1, 2.

◮ Strategies: Prices they will charge, p_{i} ∈[0, ∞) for i = 1, 2.

◮ Payoffs: Profits described by

πi =

(p_{i}− c)D(p_{i})

(pi−c)D(pi)

02

if p_{i} < p_{j}
if pi = sj

if p_{i} > p_{j}

for i = 1, 2, i 6= j.

### Bertrand Model of Duopoly (2)

There is a unique Nash equilibrium in which both firms charge the
price equal to their common marginal cost, i.e., p1 = p2 = c. (If
there are n firms, p_{1} = ... = pn= c will be the unique equilibrium.)
This is shown by the following three steps:

1. Charging different prices (by firms) never becomes a NE. 2. Charging the same price other than the marginal cost also

fails to be an equilibrium.

3. If both firms choose p_{i} = c, then no firm has an (strict)
incentive to change the price.

Bertrand Paradox Even if there are only two competitors, prices will be set at the level of marginal cost. In reality, there are many industries that look suitable for the Bertrand model but prices are (much) higher than marginal cost.

There are at least three explanations which can reasonably resolve this Bertrand paradox: 1) product differentiation, 2) capacity constraints, and 3) dynamic interaction, i.e., collusion (or cartel).

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### Cournot Model of Duopoly (1)

Two firms producing perfectly substitutable goods (no product differentiation) compete in their quantities.

◮ A (inverse) linear demand function is given, p = a − q.

◮ The firms have a common marginal cost c.

◮ The market price is set equal to the highest price that clears the market. That is,

p= a − (q_{1}+ q_{2}).
The game is defined as follows:

◮ Players: Two firms, 1, 2.

◮ Strategies: Quantities they will produce, q_{i}∈[0, ∞) for
i= 1, 2.

◮ Payoffs: Profits described by
π_{i} = [a − (q_{1}+ q_{2}) − c]q_{i}
for i = 1, 2, i 6= j.

### Cournot model of duopoly (2)

_{✞}

✝

☎

Q How can we derive a Nash equilibrium of this game?✆

In Nash equilibrium, each firm tries to maximize her profit given other firm’s (equilibrium) strategy:

maxqi

[a − (q_{i}+ q_{j}) − c]q_{i}.

This is just a unconstrained optimization problem (assuming q_{j} is
given). The first order condition gives best reply (BR) function:

dπ_{i}

dq_{i} ^{= a − 2q}^{i}^{− q}^{j}^{− c}^{= 0}

⇒ q_{i}= r_{i}(q_{j}) = ^{a}^{− c}

2 ^{−}

q_{j}

2 for i = 1, 2, i 6= j.

The Nash equilibrium (q_{1}^{∗}, q^{∗}_{2}) must satisfy these equations. Solving
the simultaneous equations, we obtain

q^{∗}_{1} = q_{2}^{∗} = ^{a}^{− c}
3 ^{.}

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### Extension: Demand and Cost (1)

Consider the slightly general version of the Cournot duopoly model in which the market demand is given by

p= a − b(q_{1}+ q_{2}),

and the firms’ marginal costs are c_{1} and c_{2}, respectively.

In Nash equilibrium, each firm tries to maximize her profit given other firm’s (equilibrium) strategy:

maxqi

[a − b(q_{i}+ q_{j}) − c_{i}]q_{i}.

The first order condition provides the best reply (BR) function: dπi

dq_{i} ^{= a − 2bq}^{i}^{− bq}^{j}^{− c}^{i}^{= 0.}

⇒ q_{i}= ri(qj) = ^{a}^{− c}^{i}
2b ^{−}

qj

2 for i = 1, 2, i 6= j.

### Extension: Demand and Cost (2)

Def When the best reply function is downward sloping, we call it strategic substitution. On the other hand, if BR is upward sloping, then it is strategic complementarity.

The Nash equilibrium (q_{1}^{∗}, q^{∗}_{2}) becomes as follows:
q^{∗}_{1} = ^{a}^{−}^{2c}^{1}^{+ c}^{2}

3b ^{and q}

∗
2 ^{=}

a−2c_{2}+ c_{1}

3b ^{.}

The equilibrium market price is

p^{∗} = a − b(q^{∗}_{1}+ q^{∗}_{2}) = ^{a}^{+ (c}^{1}^{+ c}^{2}^{)}

3 ^{.}

The equilibrium profit for each firm is
π^{∗}_{i} = (q^{∗}_{i})^{2}= ^{1}

3b^{(a − 2c}^{i}^{+ c}^{j}^{)}

2_{.}

Note that q^{∗}_{i} and π^{∗}_{i} are increasing in c_{j} while decreasing in c_{i}.

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### Extension: Number of Firms

Suppose there are n firms, and the marginal costs are identical (= c) across them. Then, the best reply becomes

dπ_{i}

dq_{i} ^{= a − 2bq}^{i}^{− bq}^{−i}^{− c}^{= 0}

⇒ q_{i}= ^{a}^{− c}
2b ^{−}

q−i

2 for i = 1, ..., n,

where q−i =^{P}_{j6=i}q_{j}. Solving the linear equations (you can assume
the equilibrium being symmetric, i.e., q_{1}^{∗} = · · · = q_{n}^{∗}), we obtain

q^{∗}_{i} = ^{a}^{− c}
b(n + 1)^{.}

The total quantity and the market price are equal to
q^{∗} = ^{n(a − c)}

b(n + 1) ^{and p}

∗_{=} ^{a}^{+ nc}

n+ 1^{.}

It follows that the markup at the equilibrium becomes
m^{∗}= ^{p}

∗_{− c}

p^{∗} ^{=}
a− c
a+ nc^{.}

=

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### Bertrand or Cournot?

One may want to ask “Which model should we use?”

◮ Both the Bertrand and Cournot models can be seen as particular cases of a more general model of oligopolistic competition where firms choose prices and quantities.

◮ Bertrand is more reasonable when firms can adjust capacities faster than prices, e.g., software.

◮ Cournot is more appropriate when prices can vary faster than capacities, e.g., wheat, cement.

These models are different games, i.e., price vs. quality competition, but we do not need different solution concepts.

◮ The single solution concept (Nash equilibrium) can explain different market outcomes depending on the situations.

◮ In other words, we do not need different assumptions about firms’ behaviors. Once a model is specified, then Nash equilibrium gives us the result of the game.

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### Bertrand Model with Product Differentiation (1)

Consider the Bertrand duopoly model of differentiated products in which the demand for each firm is given as

qi= a − pi+ bpj for i = 1, 2, i 6= j, where 0 < b < 2.

◮ That is, the demand increases as its own price decreases while the rival’s price increases.

◮ The firms have different marginal costs c1 and c2, respectively. In Nash equilibrium, each firm tries to maximize her profit given other firm’s (equilibrium) strategy:

maxpi

(pi^{− c}i)(a − pi+ bqj).

### Bertrand Model with Product Differentiation (2)

By the first order condition, we obtain the BR:
dπ_{i}

dp_{i} ^{= a − 2p}^{i}^{+ bp}^{j}^{+ c}^{i}^{= 0.}

⇒ p_{i} = ri(pj) = ^{a}^{+ c}^{i}

2 ^{+}

bpj

2 for i = 1, 2, i 6= j.

✞

✝

☎

Rm Since the best reply is upward sloping, it shows strategic✆ complementarity.

Solving the pair of equations yields
p^{∗}_{1} = ^{a}

2 − b ^{+}

2c1+ bc2

(2 − b)(2 + b)^{.}
p^{∗}_{2} = ^{a}

2 − b ^{+}

2c_{2}+ bc_{1}
(2 − b)(2 + b)^{.}

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