David has utility function U(x; y) - minf3x;3yg. He has 13 units of good>x and 9

Question

David has utility function U(x; y) - minf3x;3yg. He has 13 units of good>x and 9 units of y. Aner's utility function for the same two goods is U(x:y) - 2x 2y. Aner has 6 units of x and 15 units of y. The current allocation is (a) Pareto optimal and envy free (neither envies the allocation of the other) (b) Not Pareto optimal and not envy free. (c) Pareto optimal and not envy free. (d) Not Pareto optimal and envy free. e) Since they have di erent preferences, there is not enough information to determine the answer.

Explanation / Answer

Solution:

For David, U(x, y) = min{3x,3y} and (x, y) = (13,9)

For Aner, U(x, y) = 2x + 2y and (x, y) = (6,15)

Checking for envy-freeness :

An allocation is said to be envy-free if none of the players prefer bundle of their neighbour (or any other player). When they consume their own bundle, the respective level of satisfaction is

For David, U=min{3*13,3*9} = min{39,27} = 27

For Aner, U=2*6 + 2*15 = 12+30=42

Now, suppose the bundles were exchanged (let's see if any of David and Aner prefers another player's bundle to his own)

Then, for David U=min{3*6,3*15} = min{18,45}=18 < 27, so David will prefer his own bundle.

For Aner U = 2*13 + 2*9 = 26 + 18 = 44 > 42 implying Aner will prefer David's bundle to his own. Thus, the given allocation isn't envy free.

Checking for Pareto optimality :

An allocation is Pareto optimal if there exists no other allocation such that one person can be made better-off without making other person worse-off. We can either do this by hit and trial method (randomly checking for some other allocation say by transferring a little from one person to another) or we could solve for the pareto optimal allocation (which by seeing the preferences we can tell that optimality for David will occur at kink, so 3x=3y or x=y there, and accordingly proceed further)

Since we have to just check if given bundle is Pareto optimal or not, and not really find the allocation which is, I'll follow a simpler hit and trial method.

We have already seen above that by consuming their alloted bundles, David derives utility of 27 and Aner derives utility of 42. Say I make a transfer: 1 unit of good x from David to Aner. Now, David has bundle (12,9) and Aner has (7,15)

Their utilities are: For David U=min{3*12,3*9}= min{36,27}=27

For Aner U = 2*7 + 2*15 = 14+30=44

Clearly, David receives same level of satisfaction as before, i.e, 27 while Aner receives a higher level of satisfaction (44, earlier it was 42). So, there exists such allocation where one (Aner) can be made better-off without making other (David) worse-off. Thus, given allocation isn't Pareto optimal.

So correct answer is (b) Not pareto optimal and not envy free.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.