Consider a pure exchange economy with two consumers and two commodities. Each consumer has consumption set $X_i=\mathbf{R}_{+}^2$, and endowments are given by
$$
w_i= \begin{cases}(1,3) & \text { for } i=1 \\ ,(3,1) & \text { for } i=2 .\end{cases}
$$
Utility functions are of "Leontief type" (i.e., indifference sets are L-shaped), described by utility functions of the form:
$$
u_i\left(x_i\right)=\min \left\{x_{i 1} / \alpha_i, x_{i 2} / \beta_i\right\}
$$
where $\alpha_i$ and $\beta_i$ are positive constants. For each of the following two cases, determine the set of Pareto optima, Walrasian, and core allocations for the economy. Illustrate each of the sets in an Edgeworth box and in a net trade diagram.
$$
\left(\alpha_i, \beta_i\right)= \begin{cases}(2,1) & \text { if } i=1 \\ (1,2) & \text { if } i=2 .\end{cases}
$$
$$
\left(\alpha_i, \beta_i\right)=(2,1) \text { for } i=1,2 .
$$
1.15 Repeat the preceding exercise for a two-person exchange economy in which endowments are given by
$$
w_i= \begin{cases}(0,4) & \text { if } i=1, \\ (4,0) & \text { if } i=2\end{cases}
$$
and preferences are represented by the utility function
$$
u_i\left(x_i\right)=x_{i 1}+2 x_{i 2} \text { for } i=1,2 .
$$