eco 444

Handout 12

ECO 444

Konrad Grabiszewski

Stackelberg Duopoly

Let P (Q) be the demand function with Q denoting the aggregate quantity. We assume the

following form of P (Q) (with a > 0):

P (Q) =

  a − Q if Q < a

0 otherwise

(1)

Each firm in the market determines how much to produce. Let qi denote the quantity produced

by firm i. We assume that the total cost of firm i is Ci(qi) = ciqi, where ci is marginal cost. We

impose the following “technical” assumption:

a > ci for each player i (2)

At this point, you might recognize that we are having the same setup as in the Cournot Oligopoly.

Now, it is a good time to refresh you knowledge about that model (see Handout 8).

We assume that there are two firms in the market – A and B. Firms make their decisions in

stages. In stage 1, firm A (leader) chooses qA. Then, in stage 2, firm B (follower) observes A’s

choice and chooses qB. Recall that in the Cournot model, firms make choices at the same time

(simultaneous-move game).

We assume that cost-wise A and B identical. That is, cA = cB = c. We solve this game using the

procedure of backward induction. That is, we start from the end: first, we find B’s best-response;

second, we determine A’s optimal choice given B’s best-response.

1

Optimization for B. We solve the following optimization problem of firm B: maximize uB with

respect to qB.

∂uB ∂qB

= a − 2qB − qA − c (3)

∂2uB ∂q2B

= −2 (4)

Given that SOC is satisfied (see (4)), we obtain best-response of firm B from FOC in equation

(3).

q̃B = 1

2 (a − c − qA) (5)

Optimization for A. Now, we solve the optimization problem of firm A: maximize uA with

respect to qA but with qB as given in (5). First, we need to incorporate the fact that firm A knows

how firm B will react to A’s choice of quantity qA. That is, we need derive A’s utility function

given that q̃B = 1 2 (a − c − qA) (see equation (5)).

uA = qA (a − qA − q̃B − c) (6)

= qA

( a − qA −

1

2 (a − c − qA) − c

) (7)

= qA

( 1

2 a −

1

2 c −

1

2 qA

) (8)

= 1

2 (a − c)qA −

1

2 q2A (9)

Now, we solve the optimization problem.

∂uA ∂qA

= 1

2 (a − c) − qA (10)

∂2uA ∂q2A

= −1 (11)

2

Equilibrium strategies. Given that SOC is satisfied, from FOC depicted in equation (10), we

obtain A’s equilibrium strategy.

q∗A = (a − c)

2 (12)

This allows us to derive B’s equilibrium strategy.

q∗B = 1

2 (a − c − q∗A) =

(a − c) 4

(13)

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

Q∗ = q∗A + q ∗ B =

3

4 (a − c) (14)

P ∗ = 1

4 (a + 3c) (15)

uA (q ∗ A, q

∗ B) =

1

8 (a − c)2 (16)

uB (q ∗ A, q

∗ B) =

1

16 (a − c)2 (17)

To do.

1. Compare Firms A and B. Is it beneficial to be the first-mover?

2. Conduct the comparative statics analysis.

3. Compare Stackelberg to Cournot.

3