<p>I have a leader who has to optimise quantity, x, inputting the quantity of follower, y.</p>
<p>Problem is quantity equation of leader and follower are not separable.</p>
<p>Leader's objective function is -
<span class="math-container">$$ \sqrt{x} (y^3 - y^2) + \sqrt{1-x} (y^2 - 2y)$$</span></p>
<p>Follower's best response function is -
<span class="math-container">$$ y = \frac{\sqrt{x} y^3 - \sqrt{1-x} ~ 3y}{\sqrt{x} (y-1)^4 - \sqrt{1-x} y^2} $$</span></p>
<p>If I substitute y into leader's objective function and differentiate wrt x. I'll get differential equation (with dy/dx) and setting it equal to 0</p>
<p>Is this the right way to approach this problem? Is there a better way? Please help.</p>
<p>For reference, Attached is the differentiation of leader's objective function, after substituting x, wrt y</p>
<p><a href="https://i.sstatic.net/ezocHZvI.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ezocHZvI.jpg" alt="Differential equation" /></a></p>

CC BY-SA 4.0
2024-06-18T06:41:48.070Z
Hi Elina, is it possible to post the quation in Tex and not as pic? Thank you
T123
CC BY-SA 4.0
2024-06-18T07:44:14.943Z
I have just posted the differential equation as Pic, because it is not directly related to my question. I want to know the idea with which I am doing, is it correct or is there a better way to approach this. Thank you for looking at my problem. Can you please give your insights @T123
Elina Gilbert
CC BY-SA 4.0
2024-06-18T11:59:43.257Z
"Follower's best response function" should be "objective function", right?
VARulle
CC BY-SA 4.0
2024-06-18T15:33:10.097Z
Sorry for confusion @VARulle. Follower's best response function is solution of y after optimizing its objective function, taking x as given
Elina Gilbert
CC BY-SA 4.0
2024-06-20T05:59:39.497Z
There is simply no reason to include the image. Can you explain what use you think any reader will have for it?
Giskard
CC BY-SA 4.0
2024-06-21T06:22:38.230Z
Sequential equilibrium is not really defined for games with a continuum of actions.
Michael Greinecker
CC BY-SA 4.0
2024-06-21T07:09:53.083Z
@MichaelGreinecker No, sequential equilibrium is defined for continuum of action. Stackelberg game is a perfect example of it
Elina Gilbert
CC BY-SA 4.0
2024-06-21T07:12:59.847Z
@Giskard I included because the image shows that solving for x requires solving differential equation. If there is a better approach than this, I am asking for guidance on that
Elina Gilbert
CC BY-SA 4.0
2024-06-21T07:52:45.830Z
@ElinaGilbert Sure: https://www.jstor.org/stable/1912767
Michael Greinecker
<microeconomics><mathematical-economics><oligopoly><difference-equations><solution-concept>
Stackelberg Model with inseparable variables: How to find sequential equilibrium?
Elina Gilbert
# Stackelberg Model with inseparable variables: How to find sequential equilibrium?

(Asked by: username_0 on 2024-06-18)

I have a leader who has to optimise quantity, x, inputting the quantity of follower, y.

Problem is quantity equation of leader and follower are not separable.

Leader's objective function is -
$$ \sqrt{x} (y^3 - y^2) + \sqrt{1-x} (y^2 - 2y)$$

Follower's best response function is -
$$ y = \frac{\sqrt{x} y^3 - \sqrt{1-x} ~ 3y}{\sqrt{x} (y-1)^4 - \sqrt{1-x} y^2} $$

If I substitute y into leader's objective function and differentiate wrt x. I'll get differential equation (with dy/dx) and setting it equal to 0

Is this the right way to approach this problem? Is there a better way? Please help.

For reference, Attached is the differentiation of leader's objective function, after substituting x, wrt y

[![Differential equation](https://i.sstatic.net/ezocHZvI.jpg)](https://i.sstatic.net/ezocHZvI.jpg)

**Tags:** microeconomics, mathematical-economics, oligopoly, difference-equations, solution-concept

> **Comments:**
> **username_1**: Hi username_0, is it possible to post the quation in Tex and not as pic? Thank you  
> **username_0**: I have just posted the differential equation as Pic, because it is not directly related to my question. I want to know the idea with which I am doing, is it correct or is there a better way to approach this. Thank you for looking at my problem. Can you please give your insights @username_1  
> **username_2**: "Follower's best response function" should be "objective function", right?  
> **username_0**: Sorry for confusion @username_2. Follower's best response function is solution of y after optimizing its objective function, taking x as given  
> **username_3**: There is simply no reason to include the image. Can you explain what use you think any reader will have for it?  
> **username_4**: Sequential equilibrium is not really defined for games with a continuum of actions.  
> **username_0**: @username_4 No, sequential equilibrium is defined for continuum of action. Stackelberg game is a perfect example of it  
> **username_0**: @username_3 I included because the image shows that solving for x requires solving differential equation. If there is a better approach than this, I am asking for guidance on that  
> **username_4**: @username_0 Sure: https://www.jstor.org/stable/1912767  

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## 0 answers

