# Finding Pareto Optimality

Consider an exchange economy with two individuals and 2 commodities. The endowments of the commodities are 1 each. Which one of the following are Pareto Optimum.

a. {$\frac{1}{2},\frac{1}{2},(\frac{1}{2}\frac{1}{2})$} b. {$(\frac{1}{4},\frac{1}{4}),(\frac{1}{2}\frac{1}{2})$} c. {$(\frac{3}{4},\frac{3}{4}),(\frac{1}{2},\frac{1}{2})$} d. None of the above.

I have a hunch that it is d. I am assuming that preferences are perfect complements and hence all the options from a to c will give the two individuals lesser utility. Hence d. Is this the right way to go about it. If not, please help?

• It seems there´s a much simpler idea to focus on rather than the complementarity. What is the definition of Pareto Optimality that you have been given? – Fix.B. May 9 '16 at 19:50
• Yes. The conditions for Pareto Optimality suffice in this case. – Abhinav Dutta May 10 '16 at 4:25

To be a Pareto optimum, there must not exist another feasible allocation that makes every agent at least as well off and one or more agents strictly better off.

So, let us consider the options here.

Answer B - look at the individual allocations here and consider them relative to the economy's total endowment. Since the allocations given here do not exhaust the economy's total endowment, we could make one of the agents better off by simply giving him some of what has not been allocated. Doing this would not make the other agent worse off. So B cannot be P.O.

Answer C - Consider the allocations here relative to the total endowment. Is such an allocation feasible?

Answer A - Has the total endowment been allocated? If so, is there any way to increase what one agent has without decreasing what the other has?

• Thanks for the intuition. So I'll try to arrive at the right answer. We have a total endowment of 1 for each good. For option A. The consumers exhaust the endowments and there is no scope for further reallocation without making anybody worse-off. For option c, the individuals consume more than the endowments, so I guess option c is an unfeasible point in the Edgeworth box. So option a is a PO allocation. – Abhinav Dutta May 9 '16 at 18:12
• Correct. So you must either choose A or D. So you have to ask yourself whether or not the allocations in A satisfy the definition of P.O. – 123 May 9 '16 at 18:16
• I think it does so I will settle for a. – Abhinav Dutta May 9 '16 at 18:24
• That would be my choice also. – 123 May 9 '16 at 18:27