<p>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.</p>

<p>You have to set up a Lagrangian with the maximization problem:</p>

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

<p>So we have</p>

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

<p>and you should be able to move from there.</p>

CC BY-SA 3.0
2016-09-20T17:02:39.483Z
2016-09-20T17:02:39.483Z
Kitsune Cavalry
<p>Maximize
$$\begin{align}
U_1(x) \qquad &amp; (1)\\
\text{s.t.} \quad U_2(a-x) = c \qquad &amp; (2)\\
\end{align}$$</p>

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

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

<p>The question is </p>

<p>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.</p>

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

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

<p>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.</p>

<microeconomics><mathematical-economics><utility>
Finding distributions of wealth from utility functions
Pablo
# Finding distributions of wealth from utility functions

(Asked by: username_0 on 2016-09-20)

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.

**Tags:** microeconomics, mathematical-economics, utility

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## 1 answers

### Answer by username_1 on 2016-09-20. Score: 2

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.


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