Finding the set of Pareto Optimal Allocations

Question

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.

Answer 1

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