How to derive utility possibility frontier?

Question

I'm trying to derive the utility possibility frontier of a economy whose consumption contract curve is $$y_A = \frac {y} {x} x_A$$ and $$y_B = \frac {y} {x} x_B$$where $x_A + x_B = x$ and $y_A + y_B= y$.

$$U_i = \sqrt {x_iy_i}$$

While deriving the UPF. they have added up $U_A$ and $U_B$, and I don't understand why. Why do we add up utilities while deriving UPF?

(I gave all the details of the question because I'm not sure if they added the utilities only because of the peculiarities of the question or if its an algorithm. )

Answer 1

Given that $u_A(x_A, y_A) = \sqrt{x_Ay_A}$, $u_B(x_B, y_B) = \sqrt{x_By_B}$ are the utility functions of A and B, and total endowment of X and Y in this pure exchange economy is $\omega_X$ and $\omega_Y$, utility possibility frontier (UPF) is the set of all utility pairs $(\mu_A, \mu_B)$ such that the there exist a Pareto optimal allocation $((x_A, y_A), (x_B, y_B))$ with the property: $\mu_A = u_A(x_A, y_A)$ and $\mu_B = u_B(x_B, y_B)$. So to find UPF, we will use the following conditions:

• Optimality:

$\displaystyle \frac{y_A}{x_A} =\frac{y_B}{x_B}$

• Feasibility:

$x_A + x_B = \omega_X$ and $y_A + y_B = \omega_Y$

• Utility:

$\mu_A = \sqrt{x_Ay_A}$ and $\mu_B = \sqrt{x_By_B}$

Solving the above system of equations we get the UPF as: $\mu_A + \mu_B = \sqrt{\omega_X\omega_Y}$.

Answer 2

In general, one could use the following procedure to find UPF (in a two-person case):

1. fix one person's utility at some arbitrary level, say $U_A=\bar u$
2. maximize the other person's (i.e. person $B$) utility subject to the resource constraints, as well as $U_A=\bar u$
3. $B$'s maximum utility, call it $U_B^*$, found in step 2 should be a function of $\bar u$, and other parameters of the problem: $U_B^*(\bar u,\dots)$
4. lastly, you can vary the value of $\bar u$ (in the range of acceptable values for $\bar u$) to trace out the UPF.

There's no a priori reason that points on the UPF should equal to the sum of individual utilities, since utility is ordinal (unless assumed otherwise). It may be that the square root utility chosen for this question is what's causing the result. But I haven't done the math to verify it.