For PROF XAVIER ONLY

Handout 9

ECO 444

Konrad Grabiszewski

Cournot Oligopoly

In the market, there are N firms producing a single good. Let i denote a generic firm. Each firm

decides how much to produce (quantity competition). Let qi denote the quantity produced by

firm i. Let Q denote the aggregate quantity; Q = q1 + ... + qN .

Firms incur costs, and we assume a linear cost function for each firm; Ci(qi) = ciqi, where ci is

marginal cost. We impose the following “technical” assumption.

a > ci for each player i (1)

We assume a linear demand function.

P (Q) =

  a−Q if Q < a

0 otherwise

(2)

1 Monopolist

Our first task is to solve the problem of monopolist. Let q denote the quantity produced by

the monopolist. Since there is only one firm, the aggregate quantity Q is the same as q. The

monopolist maximizes u(q) = qP (q) − cq = q(a − q) − cq = −q2 + aq − cq with respect to q. We

compute the first-order derivative (FOD) and the second-order derivative (SOD).

du

dq = −2q + a− c (3)

d2u

dq2 = −2 (4)

From FOD we obtain the first-order condition (FOC) where q∗ is a candidate for a solution of the

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monopolist’s optimization problem.

−2q∗ + a− c = 0 (5)

Of course, it is possible that at q∗ is the minimum rather than the maximum. Since we maximize

the utility function, we need to make sure that q∗ is the maximum. Here, we rely on the second-

order condition (SOC); that is, d 2u dq2

(q∗) < 0. Note that in our case d 2u dq2

(q) < 0 for each q. Hence,

we know that SOC is satisfied and q∗ is the maximum.

We compute q∗, P ∗, and u (q∗).

q∗ = 1

2 (a− c) (6)

P ∗ = 1

2 (a + c) (7)

u (q∗) = 1

4 (a− c)2 (8)

Comparative statics. We analyze the impact of exogenous parameter, a and c, on equilibrium

values, q∗, P ∗, and u (q∗).

∂q∗

∂a > 0 (9)

∂q∗

∂c < 0 (10)

∂P ∗

∂a > 0 (11)

∂P ∗

∂c > 0 (12)

∂u (q∗)

∂a > 0 (13)

∂u (q∗)

∂c < 0 (14)

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2 Two identical firms

There are two firms – Firm A and Firm B. They have the same marginal costs; i.e., cA = cB = c.

Normal-form representation:

• N = {A, B} is the set of players;

• Si = [0,∞) is the set of strategies of player i; a strategy qi is the quantity produced;

• ui is the utility function of player i defined as

ui(qA, qB) = qiP (Q) − cqi (15)

2.1 Best-response correspondences

We solve the following optimization problem: maximize ui with respect to qi.

FOD and SOD.

∂ui ∂qi

= a− 2qi − qj − c (16)

∂2ui ∂q2i

= −2 (17)

Note that SOC is satisfied. Hence, from FOC, we obtain firm i best-response.

q̃i = 1

2 (a− c− qj) (18)

From equation (18) we obtain A and B best-responses:

q̃A = 1

2 (a− c− qB) (19)

q̃B = 1

2 (a− c− qA) (20)

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2.2 Equilibrium strategies

From (19) and (20), we derive two equations which describe equilibrium strategies of firms A and

B denoted as q∗A and q ∗ B, respectively. We know that (q

∗ A, q

∗ B) is a Nash equilibrium if and only if

(a) q∗A is a best-response of firm A to firm B choosing q ∗ B and (b) q

∗ B is a best-response of firm B

to firm A choosing q∗A.

Conditions (a) and (b) are captured below.

q∗A = 1

2 (a− c− q∗B) (21)

q∗B = 1

2 (a− c− q∗A) (22)

Hence, we solve a system of two equations with two unknowns. We obtain the equilibrium values.

q∗A = 1

3 (a− c) (23)

q∗B = 1

3 (a− c) (24)

We compute market price, total quantity, and profits at equilibrium:

Q∗ = q∗A + q ∗ B =

2

3 (a− c) (25)

P ∗ = 1

3 (a + 2c) (26)

uA (q ∗ A, q

∗ B) = uB (q

∗ A, q

∗ B) =

1

9 (a− c)2 (27)

2.3 Comparative statics

In our model, there are two exogenous parameters: c and a. Note that Q∗, q∗A, q ∗ B, uA (q

∗ A, q

∗ B),

and uB (q ∗ A, q

∗ B) increase in a and decrease in c. However, P

∗ increases in both c and a.

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3 N identical firms

There are N firms with the same marginal costs; i.e., ci = c. Note that Q = q1 + ... + qN . Let

Q−i = Q− qi.

3.1 Best-response correspondences

We solve the following optimization problem: maximize ui with respect to qi.

FOD and SOD.

∂ui ∂qi

= a− 2qi −Q−i − c (28)

∂2ui ∂q2i

= −2 (29)

Note that SOC is satisfied.

We can re-write (28) as a−2qi−Q−i−c = a−qi−Q−c. From FOC, we obtain firm i best-response.

a− q̃i − Q̃i − c = 0 (30)

3.2 Equilibrium strategies

Let Q∗ = q∗1 + ... + q ∗ N . Given (30), we obtain the system of N equations.

 a− q∗1 −Q∗ − c = 0

...

a− q∗N −Q ∗ − c = 0

We sum these N equations.

Na− (q∗1 + ... + q ∗ N ) −NQ

∗ −Nc = 0

Na−Q∗ −NQ∗ −Nc = 0

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Na− (N + 1)Q∗ −Nc = 0

We derive equilibrium aggregate quantity, price, firm i quantity and profit.

Q∗ = N

N + 1 (a− c) (31)

P ∗ = a + Nc

N + 1 (32)

q∗i = 1

N + 1 (a− c) (33)

ui (q ∗ 1, ..., q

∗ N ) =

1

(N + 1)2 (a− c)2 (34)

You might want to analyze what happens at equilibrium when N goes to infinity (i.e., perfect

competition).

3.3 Comparative statics

In our model, there are three exogenous parameters: c, a, and N. Note that Q∗, q∗i , and

ui (q ∗ 1, ..., q

∗ N ) increase in a and decrease in c. However, P

∗ increases in both a and c. Finally,

P ∗, q∗i , and ui (q ∗ 1, ..., q

∗ N ) decrease in N while Q

∗ increases in N.

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