Utility Possibility Frontier with two consumers and 3 commodities

Question

I am trying to solve the following problem:

Consider a pure exchange economy with three commodities and two households with individual endowments:

$$e_1=(1,2,3), e_2=(3,2,1),$$

respectively, and utility functions

$$u_1(x_{11},x_{12},x_{13})=x_{11}+2x_{12}+3x_{13}$$ and $$u_2(x_{21},x_{22},x_{23})=3x_{21}+2x_{22}+x_{23}$$ respectively.

Which of the following is the Utility Possibility Frontier? Options:

A. $\displaystyle \max\left\{ u_1+\frac{u_2}{2}, u_1 + u_2, \frac{u_1}{2} +u_2 \right\} = 32$

B. $ \displaystyle \max\left\{ u_1+\frac{u_2}{3}, \frac{3}{4} u_1 + \frac{3}{4} u_2, \frac{u_1}{3} +u_2 \right\} = 24$

C. $\displaystyle \max\left\{ u_1+\frac{u_2}{3}, u_1 + u_2, \frac{u_1}{3} +u_2 \right\} = 24$

D. None of the Above

My attempt:

I tried calculating the Pareto optimal allocations by taking two commodities at a time. I found out that between good $1$ and good $2$, the Pareto efficient allocations are where: $$x_{21}= 4 \quad \text{or} \quad x_{12} = 4$$

Similarly, between good $2$ and good $3$, I find that:

$$x_{22}= 4 \quad \text{or} \quad x_{13} = 4$$ are Pareto optimal.

Similarly, between good $1$ and good $3$, I find that:

$$x_{21}= 4 \quad \text{or} \quad x_{13} = 4$$ are Pareto optimal.

Also I observe that the preferences among good $1$, good $2$ and good $3$ of individual 1 are:

$$ \text{good } 3 > \text{good } 2 > \text{good } 1 $$

And those of individual $2$ are:

$$ \text{good } 1 > \text{good } 2 > \text{good } 3 $$

Now I consider the following allocations:

$$((x_{11},x_{12},4), (x_{21},x_{22},0))$$

And the fact that $$x_{11} + x_{21} = 4$$ $$x_{12} + x_{22} = 4$$

I find that:

$$u_1 = x_{11}+2x_{12}+3x_{13} = x_{11}+2x_{12} + 12$$

$$u_2 = 3x_{21}+2x_{22}+x_{23} = 3x_{21}+2x_{22}$$

$$\implies u_1 + u_2 /3 = 16 + 2x_{12} + 2/3 x_{22}$$ $$\implies u_1 + u_2 /3 = 16 + 2/3(3x_{12} + x_{22})$$ $$\implies u_1 + u_2 /3 = 16 + 2/3(2x_{12} + 4)$$

$$\implies u_1 + u_2 /3 \le 24 $$ because $x_{12} \le 4$.

Similarly I consider the allocation:

$$((0,x_{12},x_{13}), (4,x_{22}, x_{23}))$$

And find that:

$$ u_1 /3 + u_2 \le 24 $$

I am stuck here. I am not able to see how I can take it from to the options given. Please drop hints as to how I can.

Thank you for reading this.

Answer 1

The utility possibility frontier (UPF) plots the maximum total combination of utilities that can be achieved, given the preferences and total resources. To fix ideas, let's suppose we are plotting $u_1$ in the $y$ coordinate and $u_2$ in the $x$ coordinate. The easiest way to find the UPF is to start with a single agent, say agent 1 and give all the resources to him, so allocation $(4,4,4)$ his utility will be 24, so that point, $(0,24)$, in the plane is definitely part of the UPF (by the way this calculation already precludes option $a$). Now, think that you are going to decrease agent $1$'s utility in order to increase agent $2$'s utility. Since we are characterizing the frontier, you want to think what is the best way to move some resources from player $1$ to player $2$. After some thought, it is obvious that the best is to take away some of good $1$ from player $1$ since he only loses $1$ util while the other player gains $3$ utils. Therefore, the point $(3,23)$ is also part of the $UPF$ note that the line connecting this two points is $u_1+\frac{u_2}{3}=24$, (unfortunately, this realization does not narrow the possible answers, since $b$ and $c$ share this equation as its first argument).

Now, this point was found by moving of the good 1 from agent 1 to agent 2, however, there are only 4 such units, so after agent 2 has all units of good 1, in order to keep increasing her utility, the most efficient transfer is to take away from agent 1 some of the units of good 2 he enjoys. Here the loses from agent 1 equal the gains from agent 2, so we should have this defined by a line with equal coefficients: $\alpha u_1+\alpha u_2=C$ both options have this format, so we only need to make sure which one is correct given that the constant is 24.

Note that the point where $x_1=(0,4,4)$ and $x_2=(4,0,0)$ is part of this line and if we compute $u_1(0,4,4)+u_2(4,0,0)=32\neq 24$, but $\frac34 u_1(0,4,4)+\frac 34u_2(4,0,0)=24$, as desired. You can conclude that the answer is $b$.

The $\max$ operator ensures that the graph has the correct "kink" points when the type of good you are transferring from one agent to the other changes.