# Finding demand function in Walrasian equilibrium

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 $$A, I get $$(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.

• 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. Dec 14, 2022 at 18:44
• 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? Dec 15, 2022 at 6:50
• 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. Dec 15, 2022 at 7:25
• Thank you sir I understand it now. Dec 15, 2022 at 8:39