Exchange economy find core

Question

Find Pareto optimal allocations and the core for the following economies

Consider an exchange economy with two agents and two goods. Let consumption sets are the nonnegative orthant. Agent 1’s utility is represented by $x+y$ and agent 2’s utility is $xy$. Endowments are (e,0) for agent 1, and (0,e) for agent 2 with 𝑒 > 0.

When I find MRS and equalize them, I am going to find $x_2=y_2$ This is interior point (contract curve. ). By feasibility constraint, $x_1=y_1$.

Its core is given as follows. Why?

How can we find (determine) core? What it’s appropriate calculation to find core? For example, why we assume $e/2$?

Answer 1

Core Allocations are Pareto efficient allocations that must satisfy individual rationality i.e. these allocations must yield at least as much satisfaction to the individuals as their respective endowments. In the given question, core allocations are represented by line connecting $(e/2, e/2)$ to $(e,e)$.

Dashed line on the left graph is representing the set of efficient allocations and on the right graph is representing the Core allocations.