Stackelberg Model of Duopoly
Stackelberg’s Model of Duopoly also has to do with companies trying to decide how much of a
homogeneous good to produce. The principal difference between the Cournot model and the Stackelberg model is that instead of moving simultaneously (as in the Cournot model) the firms now
move sequentially. Firm 1 moves first and then firm two moves second. How does this affect the
equilibrium of this game?
We now have two firms sequentially deciding how much of a given good to produce. Let 𝑞1 and 𝑞2
denote the quantities each firm can produce. Let Q denote the sum of 𝑞1 and 𝑞2 . Thus
𝑄 = 𝑞1 + 𝑞2
(1)
We also need a price function to tell us how much each unit of quantity will sell for (P(Q)). In
this case the price function (P(Q)) is equal to 𝛼 − Q iff Q < 𝛼 and is equal to 0 if Q > 𝛼. The
parameter 𝛼 captures demand. If the firms produce more of the good than what is demanded then
they net no money. Thus our price function is the following
𝑃 (𝑄∣𝑄 < 𝛼) = 𝛼 − 𝑄
(2)
𝑃 (𝑄∣𝑄 > 𝛼) = 0
(3)
Finally we need to address the cost of each item. In this case assume that the cost to produce each
item is c. Further, assume that c is less than 𝛼. This assumption means that there is some levels
of production which are profitable. Thus we have the following cost function for firms 1 and 2.
𝐶1 (𝑞1 ) = 𝑐𝑞1
(4)
𝐶2 (𝑞2 ) = 𝑐𝑞2
(5)
The final thing we need to do is write out utility functions for each firm. Assuming that this is a
normal form game we can write out firm 1 and 2’s utility functions as the following. Essentially,
these functions represent the quantity of the good produced times the amount of money each unit
generates.
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝑃 (𝑄) − 𝐶1 (𝑞1 ))
(6)
𝑈2 (𝑞1 , 𝑞2 ) = 𝑞2 (𝑃 (𝑄) − 𝐶2 (𝑞2 ))
(7)
Since we have definitions of P(Q) and 𝐶𝑖 (𝑞𝑖 ) we can rewrite these utility functions in the following
form.
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − 𝑞2 − 𝑐)
(8)
𝑈2 (𝑞1 , 𝑞2 ) = 𝑞2 (𝛼 − 𝑞1 − 𝑞2 − 𝑐)
(9)
(10)
We use backwards induction to find the nash equilibrium of this game. This means that we start
at the bottom of the game and move up. So the first thing we must figure out is what firm two
1
will produce in any equilibrium. To do this we must take the first order condition of their utility
function1 . First we make the necessary algebraic changes.
𝑈2 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − 𝑞2 − 𝑐)
𝑈2 (𝑞1 , 𝑞2 ) = 𝑞2 𝛼 − 𝑞1 𝑞2 −
𝑞22
(11)
− 𝑞2 𝑐
(12)
𝑈2 (𝑞1 , 𝑞2 ) = 𝑞2 𝛼 − 𝑞1 𝑞2 − 𝑞22 − 𝑞2 𝑐
∂U2 (𝑞 1 ,q2 )
= 𝛼 − 𝑞1 − 2𝑞2 − 𝑐
∂q2
(13)
Now we take the first derivative of the utility function.
(14)
(15)
Now we set the first derivative equal to zero. This gives us the value of 𝑞1 which optimizes player
2’s utility function.
0 = 𝛼 − 𝑞1 − 2𝑞2 − 𝑐
(16)
2𝑞2 = 𝛼 − 𝑞1 − 𝑐
𝛼 - q1 - c
𝑞2∗ =
2
(17)
(18)
So far this process has been exactly the same as the process of finding the equilibrium of the
Cournot model. Now is which the process changes. Since the game is sequential we must now
return to player 1’s utility function.
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − 𝑞2 − 𝑐)
(19)
(20)
Since we have used backwards induction it is no longer a mystery to firm 1 as to the amount that
firm 2 will produce. So we can now insert the amount of production we know firm two will do in
equilibrium into firm 1’s utility function.
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − (
𝛼 - q1 - c
) − 𝑐)
2
(21)
(22)
The first step is to do some algebra to make this utility function comprehensible.
𝛼 - q1 - c
) − 𝑐)
2
1
1
1
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − ( 𝛼 − 𝑞1 − 𝑐) − 𝑐)
2
2
2
1
1
1
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − 𝛼 + 𝑞1 + 𝑐 − 𝑐)
2
2
2
1
1
1
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 ( 𝛼 − 𝑞1 − 𝑐)
2
2
2
1
1 2
1
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 𝛼 − 𝑞1 − 𝑞1 𝑐
2
2
2
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 (𝛼 − 𝑞1 − (
1
In other words, we take the first derivative.
2
(23)
(24)
(25)
(26)
(27)
Now we take the first derivative of firm 1’s utility function.
1
1
1
𝑈1 (𝑞1 , 𝑞2 ) = 𝑞1 𝛼 − 𝑞12 − 𝑞1 𝑐
2
2
2
∂U1 (q1 ,q2 )
1
1
= 𝛼 − 𝑞1 − 𝑐
∂q1
2
2
(28)
(29)
Now we set the first order condition equal to zero to optimize the function.
∂U1 (q1 ,q2 )
1
1
= 𝛼 − 𝑞1 − 𝑐
∂q1
2
2
1
1
0 = 𝛼 − 𝑞1 − 𝑐
2
2
1
1
𝑞1 = 𝛼 − 𝑐
2
2
𝛼−c
∗
𝑞1 =
2
(30)
(31)
(32)
(33)
Thus we have found the quantity that firm one will produce in equilibrium. The last step is to
insert this quantity into firm 2’s critical value of 𝑞2 .
𝑞2∗ =
𝛼 - q1 - c
2
1
𝑞2∗ = (𝛼 − 𝑞1 − 𝑐)
2
1
𝛼−c
∗
𝑞2 = (𝛼 −
− 𝑐)
2
2
1
1
1
𝑞2∗ = (𝛼 − 𝛼 + 𝑐 − 𝑐)
2
2
2
1 1
1
∗
𝑞2 = ( 𝛼 − 𝑐)
2 2
2
1
1
∗
𝑞2 = 𝛼 − 𝑐
4
4
𝛼−c
𝑞2∗ =
4
(34)
(35)
(36)
(37)
(38)
(39)
(40)
We have thus solved for the quantities each firm will produce in this game when the firms move
sequentially instead of simultaneously.
3