# A question about the exchange economy

The question I am given is the following: Consider an economy that has only three goods, mineral water, orange juice, and wine available in ﬁxed amounts, and three agents, A, B and C. So in this economy agents have no money initially. There are perfectly competitive markets for these three goods. The agents are not allowed to hold any negative amount of goods. Answer the following questions.

(a) Suppose that A owns all the mineral water, B owns all the orange juice, and C owns all the wine. A only likes orange juice, B only likes wine, and C only likes mineral water. What are the equilibrium prices for mineral water, orange juice and wine and the equilibrium quantities consumed by each individual?

(b) Now suppose that A owns everything and A only likes drinking orange juice, both B and C own nothing, both B and C like drinking wine only. Discuss the equilibrium quantities consumed by each individual.

(c) Now suppose that A equally likes these three goods, B equally likes the mineral water and orange juice, but does not like wine at all, and C only likes wine. Without considering who owns mineral water, orange juice, or wine initially, what allocations are Pareto optimal?

My question is about sub-questions (b) and (c). For (b), if A owns everything and B and C own nothing, apparently there will be no trade. So what is the point for asking this question? For (c), if there is no utility function, how could I tell what is Pareto optimal? I really wish someone could teach me how to answer this kind of question and show me some examples. Thank you, and I wish you a happy new year.

• In the future, it is encouraged to format your questions nicely so that it is easier to read, and we ask that you show some of your work or what you have tried to do to solve your question. – Kitsune Cavalry Jan 2 '19 at 19:17

Hint:

These exercises seem to be encouraging you to think about the First Welfare Theorem, and whether it will hold. Recall the theorem which states that if markets are competitive (price takers), markets are complete (full information), and that consumers are locally non-satiated, then every competitive equilibrium should be Pareto optimal.

For each sub-question, there is at least one agent whose preferences are not locally non-satiated. For part (b), you are correct that there will be no trade. That is, you are at a competitive equilibrium. You should discuss if the allocation is Pareto optimal. If the allocation is PO, prove it, and if not, show a counterexample.

For part (c), you do not need to know the utility functions of each agent to do this question. Since we only care about utility up to a monotonic transformation, you can in theory make up a utility function that fits agent A's description. In this case, all three drinks are perfect substitutes, so you can express agent A's utility as

$$u_A(m, j, w) = m + j +w$$

Or any monotonic transformation of that. But this is not necessary. You can solve (c) with just the definition of a Pareto optimal allocation and some intuition. Hope this helps.

• Hi sir, thanks for your detailed answer and kind comment. It is very helpful. I have never considered this question is about the first welfare theorem and just consider how I could achieve a market clearing for this market. – Haode Wang Jan 5 '19 at 1:41
• Now I think I get some idea about how to answer Part (B). But what I still feel confused is that, in Part (C), any allocation which satisfies that B doesn't consume wine and C only consumes wine is pareto efficient. There are countless "correct" answers for Part (C). So how could I answer this part? – Haode Wang Jan 5 '19 at 1:59
• @HaodeWang Give the answer just like that. "Any set of allocations such that so and so gets...is PE" Be careful your answer is comprehensive. The allocation where A gets everything is also still Pareto efficient, etc. – Kitsune Cavalry Jan 5 '19 at 16:26
• Now I completely understand how to answer this question. Thanks for your help! – Haode Wang Jan 8 '19 at 23:55