Question:
. Provide a complete characterization of all of the allocations, which are in the core of this economy for all possible values of the initial endowments.(由留学作业帮www.homeworkhelp.cc整理编辑)
(Recall that an allocation is in the core if it is i) Pareto optimal and ii) Individually rational (weakly preferred to the endowment)).
Perfect Allocation and Complement Utility Functions:
In a two-agent exchange economy, an endowment is pareto-optimal (PO) if it's impossible to make one agent better off (increasing utility) without hurting the other agent (decreasing utility). For a perfect complement utility function, and assuming we have a complete exchange economy with two agents, for a given level of endowment, both agents consume all of the goods, so there's no leftovers in the economy. Let's designate the consumer agents as A and B and designate the two goods as X1 and X2. Mathematically, since we're given 
Graphically, each indifference curve is in the form of an L-shape inside the Edgeworth box. Recall that on the same indifference curve, every allocation will yield the same level of utility. Referring to figure X, the red arrow indicates the direction of increasing utility for agent A, while the blue arrow indicates that of agent B. The yellow dotted lines denote all of the possible kink points for both agents, where an agent consumes an equal amount of both goods (optimally).
Answer and Explanation:
We first need to find the set of all of the possible PO allocations (refer to figure 1). Considering an endowment at point N and drawing two L-shaped utility curves that cross this point and each agent's kink point, we can easily find another allocation at B or C that both agents strictly prefer, since they're at higher utility levels (recall the direction of increasing utility for both agents). Therefore, N is not a PO allocation as it's possible to find better allocations.
Consider point P1 on the same figure. Again, try to draw two L-shaped curves that cross point P1 and the agents' respective kink points. At P1, no one prefers point N because of its lower utility level. Similarly, agent A will be worse off when moving from P1 to P2, while agent B will be better off when moving from P1 to P2.
Since it's impossible to make one agent better off without hurting the other agent at point P1, it is PO. Apply the same steps to see that point P2 is also a PO.
Extending the steps we can find a region of POs, highlighted in yellow in figure 2, including the outer edges:
Figure 2
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To find the allocations in the core, let's look into the different type of initial endowments.
Case 1: The given endowments fall within the yellow PO region.
Referring to figure 3, the first region is inside the yellow region (including the edges). Considering an initial endowment point C1, we can find another point C2 that's less preferable to the endowments. Note that both agents are indifferent when it comes to choosing between C1 and C2; therefore, we can say that C1 is less preferable to C2 and vice versa. So for any initial endowment, we'll find a corresponding line segment of core allocations where the two agents' indifference curves intersect inside the yellow PO region.
Figure 3
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Case 2: The given endowments fall outside of the yellow PO region.
Refer to figure 4, which includes the orange areas outside of the yellow PO region (excluding the lines). Considering an initial endowment at point E, it's still possible to find other allocations that are less preferable.
For example, let's consider all of the green points. If we move from E to G1, it's clear that all of the agents will be at higher utility levels. If moving from E to G2, agent A is strictly better off, while agent B is indifferent. Neither agent is better or worse off (since agent B remains on the same indifference curve). Therefore, agent B has a weak preference for the allocation G2 to E.
Hence, all of the green points are core allocations. If we apply the same logic to other similar points, we can find a region of core allocations (the green area in the figure), where any point satisfies both conditions: i) it is PO and ii) it is weakly (or strictly) preferred to the original endowment.
Figure 4
So for any given endowment, to find the core allocation, we first draw the two indifference curves that cross that point and the agents' respective kink points. We then find an area enclosed by the two curves inside the yellow PO region.
