Finding distributions of wealth from utility functions

Question

Maximize $$\begin{align} U_1(x) \qquad & (1)\\ \text{s.t.} \quad U_2(a-x) = c \qquad & (2)\\ \end{align}$$

where we have bundles $x = (x_1, x_2), \quad a = (3,5)$, and want to split our resources between the two users.

$$\begin{align} U_1(x_1, x_2) & = (x_1 + 1)x_2 \\ U_2(x_1, x_2) & = 36 - (x_1 - 4)^2 - (x_2 - 6)^2 \end{align}$$

The question is

In equations $(1)–(2)$, choosing different $c$ would lead to different $x$. Find all those $x$ with $x_1 ∈ [0, 3]$ and $x_2 ∈ [0, 5]$. Those set of points will comprise those distributions of wealth which are reasonable in the sense that no person’s utility can be improved without hurting someone else.

I don't understand how one can solve $(1)-(2)$, since I think this gives

$(x_1 + 1)x_2 - 36 − (x_1 − 4)^2 − (x_2 − 6)^2 = c$

I can plot this as $c = f(x_1,x_2)$, but I feel this isn't what the question is asking for, since it doesn't define a set of points. If anyone can give me the direction in which to pursue this question, or let me know if I need to provide more information. Thanks.

Answer 1

From what it looks like, you are literally trying to subtract $(2)$ from $(1)$, incorrectly as well (You didn't distribute the minus out correctly.) It is saying to use equations 1 through 2 to solve this problem.

You have to set up a Lagrangian with the maximization problem:

$$\max_{x_1, x_2} \ (x_1 + 1)x_2 \\ \text{s.t.} \ 36 - (x_1 - 4)^2 - (x_2 - 6)^2 = c $$

So we have

$$\mathcal{L} = (x_1 + 1)x_2 - \lambda[36 - (x_1 - 4)^2 - (x_2 - 6)^2 - c]$$

and you should be able to move from there.