II. 1 OLIGOPOLY COMPETITION II. 1. 1. The Bertrand Model II.1.2. Price Setting with Capacity Constraints II.1.3. The Cournot Model -

II. 1 OLIGOPOLY COMPETITION

II. 1. 1. The Bertrand Model

II.1.2. Price Setting with Capacity Constraints II.1.3. The Cournot Model

- Symmetric Duopoly - Symmetric Oligopoly - Asymmetric Duopoly

II.1.4. The Model of Stackelberg Competition

SYMMETRIC OLIGOPOLY________________________________

The Model:

Same as symmetric duopoly, but with n ≥ 2 symmetric firms.

Q = q ₁ + q ₂ + q ₃ +…+ q _n

Firm 1:

Π 1 = [a – b (q ₁ + q ₂ +…+ q _n )] q ₁ – c q ₁

→ FOC: a – b (q ₁ + q ₂ + q ₃ +…+ q _n ) – b q ₁ – c = 0

; Similarly:

;

…

.

n symmetric firms:

→ q ₁ = q ₂ =…= q _n

→ i = 1, 2, …, n

i = 1, 2, …, n

Solution:

decreasing in n,

increasing in n,

decreasing in n,

decreasing in n,

€

CS

= 1 2b

n

n + 1 (a − c)

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

increasing in n,

€

DWL

= 1 2b

a − c n + 1

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

decreasing in n.

→ if n = 1, E ^N ≡ E ^M ; if n = 2, E ^N ≡ E; if n → ∞, E ^N → E ^C .

ASYMMETRIC DUOPOLY_________________________________

Same model as the previous one, but with:

n = 2, D ≡ P – b Q, Q = q _A + q _B , C(q _i ) = c _i q _i , i = A, B.

Best reply functions:

Solution:

Comparative statics, with Δ c _A et Δ c _B . Graphically:

c _A decreases → q _B decreases et q _A increases.

E q

q

E’

COMPARATIVE STATICS / EXAMPLES___________________

• Example 1 : Input costs

- Suppose that the market for transatlantic flights between London and NYC is served by 2 companies, AA (American Airlines) and BA (British Airways).

- Same marginal costs for both firms:

50% labor costs, 50% fuel.

- Suppose that the oil price increases in such a way that the fuel price increases by 80%

→ MC increases by 40% → what effect on tariff ?

- Suppose competition à la Cournot.

Intuitively: Q decreases and P increases.

Analytically:

If marginal costs increase by 40%, then prices increase by = 26,6%.

• Example 2 : Exchange rate fluctuations

- Consider a duopoly with 2 producers of microships, from 2 different countries, the U.S. and Japan.

- All sales are made in the U.S., in U.S.$. The costs of inputs are paid with the national currency.

- Competition à la Cournot.

- In an initial equilibrium, both firms have the same costs and the U .S.

market is divided equally between the 2 firms . The market price is $ 16,6.

The exchange rate is 100 Yen / $. The costs structure is such that MC _US = MC _JAP = $ 10 = 1000 Yen.

- What is the impact on market shares of a 50% devaluation of the yen?

Graphically:

Analytically:

After the devaluation, the exchange rate is 150 Y / $;

MC _US = $ 10 ; MC _JAP = Y 1000 = $ 6,6 ;

q

q

E

E’

;

;

€

P

= 16, 6 = a + 2c

3 = a + 20

3 ⇔ a = 29, 8 ;

€

s

= 29, 8 − 13, 4

(2 * 29, 8) − 16, 6 = 16, 4

43 ≅ 38% ;

• Exemple 3 : New technology

- Initially:

Firm A uses an old technology to produce, with MC = 15; Firm B uses a new technology, with MC = 10; Observed price = 16,66 and b = 1.

- How much would firm A be willing to pay for the modern technology?

→ Compare benefits under both scenari.  

With the new technology:

Status quo :

Therefore,

and

Firm A adopts the new technology if:

The Bertrand / Cournot dilemma

- Depends on the industry;

- Decisions timing (ST versus LT) ;

- Banking sector, software, etc: quantities are flexible;

- Both models are likely part of a more general model, in which firms decide prices,

capacities, quantities, timing, etc.

II. 1. 4. Stackelberg Compatition

- The Model:

cf. Cournot, but with sequential decisions (with a “natural” leader).

Firm A : leader ; Firm B : follower.

- Example :

Parallel imports of drugs.

- By backwards induction:

Firm B :

Π B = P(q _A , q _B ) q _B – c q _B = (a – b q _A – b q _B ) q _B – c q _B

→ CPO : a – b q _A – 2 b q _B – c = 0

Firm A :

Π A = P(q _A , q _B ) q _A – c q _A = (a – b q _A – b – c) q _A

→ CPO : a –2b q _A – – c = 0

- Solution :