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I am asked to find the set of Pareto Optimal Allocations in an economy where there are two agents namely $1$ and $2$, with the following utility functions and endowments.

$$u_1({x_1}^1,{x_2}^2)= \beta log({x_1}^1)+(1-\beta)log({x_2}^2) \ ,\ {\omega}_1 = (0,1) \ \beta \in (0,1) \ $$ $$u_2({x_2}^1,{x_2}^2)=min\{ {x_2}^1,{x_2}^2 \} \ ,\ {\omega}_2 = (1,0)$$

I know that I can show the set of PO on the Edgeworth Box. However, how am I supposed to show the set of PO in algebraically? Shall I try splitting cases for the second agent's utility function?

Thanks in advance.

Edit: Each agent $i$ has the preferences represented by the following utility function, $u_i$ and the endowment ${\omega}_i$. ${x_i}^t$ denotes the amount of good $t$ consumed by the agent $i$. Say, prices of the good 1 and good 2 are denoted by $P_1$ and $P_2$, respectively. I am asked to show the set of Pareto Optimal allocations in this setting.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Giskard
    Oct 30 at 13:54
  • $\begingroup$ Thanks, I have edited the post. $\endgroup$
    – user722271
    Oct 31 at 10:53
  • $\begingroup$ Hi! This does not really make things clearer. What do you mean by "I can show the set of PO on the Edgeworth Box"? And what is your purpose with "splitting cases for the second agent's utility function"? Also, it is not clear where you get stuck if you "can show the set of PO on the Edgeworth Box". $\endgroup$
    – Giskard
    Oct 31 at 11:09
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Probably the easiest way to get all Pareto optimal allocations is to maxmize a weighted sum of utilities (of the two agents) subject to the resource constraints: $$ \max_{x_1^1, x_1^2, x_2^1, x_2^2} \alpha u_1(x_1^1, x_1^2) + (1-\alpha) u_2(x_2^1, x_2^1) \text{ s.t. } x_1^1 + x_2^1 = 1 \text{ and } x_1^2 + x_2^2 = 1. $$ The PO allocations can be obtained by varying $\alpha$ over the interval $[0,1]$.

As the second agent has Leontief preferences any PO allocation will have $x_2^1 = x_2^2$. So substuting out we have: $$ \max_{x_2^1} \alpha u_1(1 - x_2^1, 1 - x_2^1) + (1-\alpha) u_2(x_2^1, x_2^1) \text{ s.t. } x_{2}^1 \le 1 $$

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