Thread: Second Order Difference Equation- Cournot

Hi , i hope someone can help me with this. I am aware of how to solve diffrence equations however unsure about how to approach this. any help greatly appreciated.
I have part of the answer showing partial differentiation but i dont see how thats applicable to this....


2 firms supplying the market with demand function
P(x+y)= alpha - beta(x + y)

x and y are output levels of Firm1 and Firm2 respectively. assume each firm makes Cournot conjecture and find second order difference equation

(cournot conjecture implies each firm chooses its output level fot <t+1> to max profit in belief that the other firm will maintain output at period <t> value)


This is the solution im given, there are however many steps missing and im wondering if anyone could explain the solution to me especially obtaining the 1st line and reason for partial differentiation.

Ot+1 = <alpha -beta(Xt+1 + Yt)> Xt+1
dOt+1 = alpha- 2betaXt - betaYt
dXt

Xt+1 = alpha/2beta - 0.5Yt

Ot+1= <alpha - beta(Xt + Yt+1)> Yt+1
dOt+1 = alpha - 2betaYt+1 - betaXt = 0
dYt


So Yt= alpha/2beta - 0.5 Xt-1
So Xt - 0.25Xt-2 = 0

Originally Posted by molimoli

Hi , i hope someone can help me with this. I am aware of how to solve diffrence equations however unsure about how to approach this. any help greatly appreciated.
I have part of the answer showing partial differentiation but i dont see how thats applicable to this....


2 firms supplying the market with demand function
P(x+y)= alpha - beta(x + y)

x and y are output levels of Firm1 and Firm2 respectively. assume each firm makes Cournot conjecture and find second order difference equation

(cournot conjecture implies each firm chooses its output level fot <t+1> to max profit in belief that the other firm will maintain output at period <t> value)


This is the solution im given, there are however many steps missing and im wondering if anyone could explain the solution to me especially obtaining the 1st line and reason for partial differentiation.

Ot+1 = <alpha -beta(Xt+1 + Yt)> Xt+1
dOt+1 = alpha- 2betaXt - betaYt
dXt

Xt+1 = alpha/2beta - 0.5Yt

Ot+1= <alpha - beta(Xt + Yt+1)> Yt+1
dOt+1 = alpha - 2betaYt+1 - betaXt = 0
dYt


So Yt= alpha/2beta - 0.5 Xt-1
So Xt - 0.25Xt-2 = 0

Isn't there something missing in this question, like the costs of production?

CB

I dont think so, ive checked it again.

This is the actual question pasted:, i reworded it slightly originally just to simplify

8) Suppose there are only two firms supplying the market which has a demand function:
P(x+y)= alpha - beta(x + y)
where x and y are the output levels of Firm 1 and Firm 2 respectively. Assume each firm makes a Cournot conjecture about the other firm’s output level. Find and solve the second order difference equation for x.
[Cournot conjecture implies that each firm chooses its output level for t+1 to maximize profit in the belief that the other firm will maintain its output level at the period t value].