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Maybe the title doesn't reflect what I mean perfectly but basically, I wanna derive demand functions from those two utility functions: where $x_{11}$ is the consumption of good 1 by agent 1 and $x_{12}$ is the consumption of good 2 by agent 1 and so on. $$ u_1(x_{11},x_{12}) = A \cdot x_{11}+x_{12} $$

and endowments of agent 1 are $e_1 = (e,0)$

$$ u_2(x_{21},x_{22}) = x_{21}\cdot x_{22} $$

and endowments of agent 2 are $e_2 = (0,e)$

When $A>p_1/p_2$ or $A<p_1/p_2$, I get $(e,0)$ and $ (0,e \cdot p_1/p_2)$ pairs, but I don't know how to solve the case where $A=p_1/p_2$.

For the second agent, I got this from the maximization problem: $x_{21} = p_2 \cdot e /2 p_1$ and $x_{22} = e/2$.

So the first question is how to find demand in case of $A=p_1/p_2$ and how to construct Edgeworth Box in that case.

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    $\begingroup$ There is no demand "function" because a function must have a unique value. In the case you mentioned, everything on the budget line is a utility maximizing bundle. $\endgroup$ Dec 14, 2022 at 18:44
  • $\begingroup$ Thanks but when $A = p_1/ p_2$ what we get for $x_{11}$ and $x_{12}$ pair ? also, may I ask what do you mean by the uniqueness of the function? $\endgroup$
    – Tatanik501
    Dec 15, 2022 at 6:50
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    $\begingroup$ A function from a set $A$ to a set $B$ associates with every point in the set $A$ exactly one point in the set $B$. Here, you don't get one point, you get many pairs. Namely, the whole budget line. $\endgroup$ Dec 15, 2022 at 7:25
  • $\begingroup$ Thank you sir I understand it now. $\endgroup$
    – Tatanik501
    Dec 15, 2022 at 8:39

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