# Finding the set of Pareto Optimal Allocations

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?

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.
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: $$\max_{x_2^1} \alpha u_1(1 - x_2^1, 1 - x_2^1) + (1-\alpha) u_2(x_2^1, x_2^1) \text{ s.t. } x_{2}^1 \le 1$$