# Question on Pareto optimality problem

In an economy with two agents whose utility functions are $$U_A(x_1,x_2) = \alpha \cdot x_1 + x_2 \hskip 20pt U_B(y_1,y_2) = y_1 \cdot y_2.$$

The given allocations are bundle (4,0) for A and bundle (1,5) for B.

Consider the following question Taking into consideration the respective utilities for the bundles, we have $U_A=4\alpha$ and $U_B=5$. For this allocation to be a No Envy allocation, it has to be $4\alpha \geq 5$, which means alpha has to be greater than or equal to $\frac{5}{4}$.

Is this the right approach to solve this problem? If its not, please find the solution and show me the steps.

• Did you include the picture only to provide the definition of "no-envy allocation"? It does not seem to have anything to do with your question. And could you clarify what the utility functions are and where $\alpha$ comes from? – Giskard Jun 8 '15 at 15:41
• Sorry for that. Well, the utility functions are U(x1,x2)= (alpha)x1+x2 and U(y1,y2)=y1.y2. The allocation is (4,0) and (1,5). – Praveen Varghese Jun 9 '15 at 3:18
• According to the definition it is not the other's utility (or happiness) that you envy. This is because that is in their head and you cannot have it anyway. What you can envy is their bundle, which you could have. So please read the definition again and apply it. – Giskard Jun 9 '15 at 6:23

## 1 Answer

As denesp mentioned, your solution is incorrect. To check whether there is envy, you have to compare the two bundles from the point of view of the same agent. So in our case we have:

• For agent A: the utility of his own bundle is $4 \alpha$ and the utility of the other bundle is $\alpha+5$. So he feels no envy as long as $4\alpha \geq \alpha+5$, which is equivalent to: $\alpha \geq 5/3$.
• For agent B: the utility of his own bundle is 5 and the utility of the other bundle is 0. So he never feels envy.

This means that the correct answer is (c).