<p>I think you are trying to find a feasible allocation that maximises the sum of the utilities of the two individuals. So we can write the objective function as:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{x,y} &amp; \ xy^5 + 10 (8-x)(8-y) \\ \text{s.t.} &amp; \ 0\leq x \leq 8, 0 \leq y \leq 8\end{eqnarray*}</span>
We can set up the Lagrangian in the following way:
<span class="math-container">\begin{eqnarray*} \mathcal{L}(x, y) = xy^5 + 10 (8-x)(8-y) + \mu_xx + \mu_yy - \lambda_x(x-8) - \lambda_y(y-8)\end{eqnarray*}</span>
K-T FONCs are</p>
<p><span class="math-container">\begin{eqnarray*} \dfrac{\partial\mathcal{L}}{\partial x} = y^5 - 10(8-y)+\mu_x-\lambda_x = 0 \\ \dfrac{\partial\mathcal{L}}{\partial y} = 5xy^4 - 10(8-x)+\mu_y-\lambda_y = 0 \\ x\geq 0, \ \mu_x \geq 0, \ \mu_xx = 0 \\ y\geq 0, \ \mu_y \geq 0, \ \mu_yy = 0 \\ x\leq 8, \ \lambda_x \geq 0, \ \lambda_x(x-8) = 0 \\ y\leq 8, \ \lambda_y \geq 0, \ \lambda_y(y-8) = 0 \end{eqnarray*}</span></p>
<p>One of the solutions to the above set of K-T FONCs is <span class="math-container">$(x, y) = (8,8)$</span> and the corresponding <span class="math-container">$(\mu_x, \mu_y, \lambda_x, \lambda_y) = (0,0,8^5, 5(8^5))$</span>, which is also the solution to the above optimization problem.</p>

CC BY-SA 4.0
2022-05-11T18:08:59.637Z
2022-05-11T18:08:59.637Z
Amit
<p>I am trying to find the allocation of goods X and Y in order to maximize utility between two consumers.</p>
<p>The two utility functions are:
<span class="math-container">$$
U1 = xy^5
$$</span>
<span class="math-container">$$
U2 = 10xy
$$</span></p>
<p>There are 8 of good X and 8 of good Y.</p>
<p>Intuitively, I can tell that the utility maximizing allocation will be (8,8) for consumer 1 and (0,0) for consumer 2 based on the social welfare function
<span class="math-container">$$
U = xy^5 + 10(8-x)(8-y)
$$</span>
However, I am trying to prove this mathematically and this is what I have done so far:
<span class="math-container">$$
dU/dx = y^5 -10(8-y)=0
$$</span>
<span class="math-container">$$
dU/dy = 5xy^4 -10(8-x)=0
$$</span>
This is where I feel stuck because solving the first order condition gives me
<span class="math-container">$$
y^5 = 10(8-y)
$$</span>
Y cannot equal 8 here since then the equality doesn't make sense... where am I going wrong?</p>

CC BY-SA 4.0
2022-05-11T14:14:56.510Z
First of all the the objective that you are maximizing is not concave. Secondly, the solution is at the corner i.e. where $x_2 = y_2 = 0$. It'll not satisfy the FOCs you have mentioned. You have to solve this problem: $\max_{x, y} \ xy^5 + 10(8-x)(8-y)$ subject to these constraints $0 \leq x \leq 8, 0 \leq y \leq 8$. To solve this, you can use Kuhn-Tucker conditions to find the solution.
Amit
CC BY-SA 4.0
2022-05-11T17:04:16.437Z
Is it given that social welfare $U = U1+U2$?  As worded the question seems not entirely explicit on this point.
Adam Bailey
<microeconomics><mathematical-economics><utility>
Constrained optimization to find utility maximizing allocation
Econ2hard
# Constrained optimization to find utility maximizing allocation

(Asked by: username_0 on 2022-05-11)

I am trying to find the allocation of goods X and Y in order to maximize utility between two consumers.

The two utility functions are:
$$
U1 = xy^5
$$
$$
U2 = 10xy
$$

There are 8 of good X and 8 of good Y.

Intuitively, I can tell that the utility maximizing allocation will be (8,8) for consumer 1 and (0,0) for consumer 2 based on the social welfare function
$$
U = xy^5 + 10(8-x)(8-y)
$$
However, I am trying to prove this mathematically and this is what I have done so far:
$$
dU/dx = y^5 -10(8-y)=0
$$
$$
dU/dy = 5xy^4 -10(8-x)=0
$$
This is where I feel stuck because solving the first order condition gives me
$$
y^5 = 10(8-y)
$$
Y cannot equal 8 here since then the equality doesn't make sense... where am I going wrong?

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

> **Comments:**
> **username_1**: First of all the the objective that you are maximizing is not concave. Secondly, the solution is at the corner i.e. where $x_2 = y_2 = 0$. It'll not satisfy the FOCs you have mentioned. You have to solve this problem: $\max_{x, y} \ xy^5 + 10(8-x)(8-y)$ subject to these constraints $0 \leq x \leq 8, 0 \leq y \leq 8$. To solve this, you can use Kuhn-Tucker conditions to find the solution.  
> **username_2**: Is it given that social welfare $U = U1+U2$?  As worded the question seems not entirely explicit on this point.  

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## 1 answers

### Answer by username_1 on 2022-05-11. Score: 3 | Accepted answer

I think you are trying to find a feasible allocation that maximises the sum of the utilities of the two individuals. So we can write the objective function as:

\begin{eqnarray\*} \max\_{x,y} & \ xy^5 + 10 (8-x)(8-y) \\ \text{s.t.} & \ 0\leq x \leq 8, 0 \leq y \leq 8\end{eqnarray\*}
We can set up the Lagrangian in the following way:
\begin{eqnarray\*} \mathcal{L}(x, y) = xy^5 + 10 (8-x)(8-y) + \mu\_xx + \mu\_yy - \lambda\_x(x-8) - \lambda\_y(y-8)\end{eqnarray\*}
K-T FONCs are

\begin{eqnarray\*} \dfrac{\partial\mathcal{L}}{\partial x} = y^5 - 10(8-y)+\mu\_x-\lambda\_x = 0 \\ \dfrac{\partial\mathcal{L}}{\partial y} = 5xy^4 - 10(8-x)+\mu\_y-\lambda\_y = 0 \\ x\geq 0, \ \mu\_x \geq 0, \ \mu\_xx = 0 \\ y\geq 0, \ \mu\_y \geq 0, \ \mu\_yy = 0 \\ x\leq 8, \ \lambda\_x \geq 0, \ \lambda\_x(x-8) = 0 \\ y\leq 8, \ \lambda\_y \geq 0, \ \lambda\_y(y-8) = 0 \end{eqnarray\*}

One of the solutions to the above set of K-T FONCs is $(x, y) = (8,8)$ and the corresponding $(\mu\_x, \mu\_y, \lambda\_x, \lambda\_y) = (0,0,8^5, 5(8^5))$, which is also the solution to the above optimization problem.


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