Constrained maximization problem

Question

I need help with the following optimization problem

$$ \max\;\alpha\ln(x(1-y^2))+(1-\alpha)\ln(z) $$ where the maximization is with respect to $x,y,z$, subject to \begin{align} \alpha x+(1-\alpha)z&=C_1\\ \alpha y\sqrt{x(x+\gamma)}-\alpha x&=C_2 \end{align} where $0\leq\alpha\leq1$, $\gamma>0$, and $x,z\geq0$, and $|y|\leq1$.

Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

Thank you

Answer 1

You should use Lagrange multipliers, write out the Lagrangian, differentiate it w.r.t. $x,y$ and $z$ and set them to zero.

Your Lagrangian for this problem would be:

$$\mathcal{L}(x,y,z)=\alpha ln(x(1-y^2))+(1-\alpha)ln(z)+\lambda_1(\alpha x+(1-\alpha)z-C_1)+\lambda_2(\alpha y\sqrt{x(x+\gamma)}-\alpha x-C_2)$$

You need to set $\frac{\partial \mathcal{L}}{\partial x}=\frac{\partial \mathcal{L}}{\partial y}=\frac{\partial \mathcal{L}}{\partial z}=0$ and eliminate $\lambda_1$ and $\lambda_2$ to get your optimal $x^*$, $y^*$ and $z^*$.

Answer 2

Use the second constraint to eliminate $y$ and the first constraint to eliminate $z$ from the expression that you want to maximize. The result is a function of $x$, in fact a linear combination of three terms of the form $\log(A_k x+B_k)$.

The complete picture also depends on the values of $C_1$ and $C_2$, about which we have been told nothing.