Constrained optimization to find utility maximizing allocation

Question

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?

Answer 1

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.