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.