<p>Given an Edgeworth box environment, with two individuals and two goods <span class="math-container">$(x,y)$</span> with maximum quantities <span class="math-container">$Q_x$</span> and <span class="math-container">$Q_y$</span>. Suppose that the utility functions for each individual are
<span class="math-container">$$ U_1(x,y) = x^\alpha y^{1-\alpha} \qquad U_2(x,y) = (Q_x - x)^\beta (Q_y - y)^{1-\beta} $$</span>
How I can compute the curve, in the space, where the two funtions are equal ?
Then, it's correct to say that the equilibrium point, without prices, is the maximum of the locus of point in wich the utilities functions are equal ?</p>
<p>To clarify the question, consider the following image; where pink and light blue surface are the utility functions of the agent, <span class="math-container">$Q_x = Q_y = 100$</span>, the orange dashed line represent the points where the functions have the same derivative and the dashed green line is the points where the two utility functions are equal.<a href="https://i.sstatic.net/A7zkP.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/A7zkP.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-05-24T16:49:46.973Z
Can you please explain the question to me in a different manner? Are you trying to find the Walrasian (competitive) equilibrium or you are trying to do something with equating the utility functions as well?
solow supremacy
CC BY-SA 4.0
2023-05-25T09:11:13.990Z
I had an intuition plotting the two functions on GeoGebra and noticed that the point in which the utilities are maxima for both is the maximum of the point where the two function are euqual, but I'm not able to find an explicit function for the equality points.
TeX Scholar
CC BY-SA 4.0
2023-05-25T14:14:38.583Z
If you're trying to find the competitive equilibrium, as it seems from your last comment, then it's going to be a point and not a curve (unless you would like to vary the individual or total endowment(s)).
solow supremacy
CC BY-SA 4.0
2023-05-25T16:36:30.980Z
Did you mean that $x,y$ are bads for agent $2$? Or are $Q_x,Q_y$ the total endowments and all the $x,y$ are in terms of agent $1$?
Nicolas Torres
CC BY-SA 4.0
2023-05-27T09:44:28.060Z
@TeXScholar Are you trying to find the set of Pareto Efficient outcomes for the above economy?
mynameparv
CC BY-SA 4.0
2023-05-29T14:26:36.443Z
@NicolasTorres yes, $Q_x$ and $Q_y$ are the total endowments of the two goods.
TeX Scholar
<mathematical-economics><cobb-douglas><edgeworth-box><equilibrium>
Equilibrium in Edgeworth box with Cobb-Douglas utility function
TeX Scholar
# Equilibrium in Edgeworth box with Cobb-Douglas utility function

(Asked by: username_0 on 2023-05-24)

Given an Edgeworth box environment, with two individuals and two goods $(x,y)$ with maximum quantities $Q\_x$ and $Q\_y$. Suppose that the utility functions for each individual are
$$ U\_1(x,y) = x^\alpha y^{1-\alpha} \qquad U\_2(x,y) = (Q\_x - x)^\beta (Q\_y - y)^{1-\beta} $$
How I can compute the curve, in the space, where the two funtions are equal ?
Then, it's correct to say that the equilibrium point, without prices, is the maximum of the locus of point in wich the utilities functions are equal ?

To clarify the question, consider the following image; where pink and light blue surface are the utility functions of the agent, $Q\_x = Q\_y = 100$, the orange dashed line represent the points where the functions have the same derivative and the dashed green line is the points where the two utility functions are equal.[![enter image description here](https://i.sstatic.net/A7zkP.jpg)](https://i.sstatic.net/A7zkP.jpg)

**Tags:** mathematical-economics, cobb-douglas, edgeworth-box, equilibrium
> **Comments:**
> **username_1**: Can you please explain the question to me in a different manner? Are you trying to find the Walrasian (competitive) equilibrium or you are trying to do something with equating the utility functions as well?  
> **username_0**: I had an intuition plotting the two functions on GeoGebra and noticed that the point in which the utilities are maxima for both is the maximum of the point where the two function are euqual, but I'm not able to find an explicit function for the equality points.  
> **username_1**: If you're trying to find the competitive equilibrium, as it seems from your last comment, then it's going to be a point and not a curve (unless you would like to vary the individual or total endowment(s)).  
> **username_2**: Did you mean that $x,y$ are bads for agent $2$? Or are $Q_x,Q_y$ the total endowments and all the $x,y$ are in terms of agent $1$?  
> **username_3**: @TeXScholar Are you trying to find the set of Pareto Efficient outcomes for the above economy?  
> **username_0**: @NicolasTorres yes, $Q_x$ and $Q_y$ are the total endowments of the two goods.  

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