# Utility Possibility Frontier with two consumers and 3 commodities

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.

## 1 Answer

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.

• Hi Gabriel. Thanks for taking the time to write the answer. While the rest of your logic in solving this is sound, my doubt here is regarding the max operator. How do you justify that in the UPF? Apr 22, 2019 at 20:34
• So hopefully by now it is clear that the UPF is piece-wise linear. You could write it’s equation as a function with three different rules for three different parts of the domain, and be correct. The max operator is a shortcut that allows you to write it down more compact. To see it, draw all three lines in a graph and note that the UPF traces the lower envelope. The max operator captures this fact (it might be counter-intuitive and you might think it should capture the upper-envelope, but try to figure out on your own that that is not the case). Apr 23, 2019 at 21:25