How can I solve a Utility Maximization problem using the Lagrangian method where the Utility formula has an exogenous constant $a$?

Question

The utility function is given by: $$u(x, y) = 2x^{\frac{1}{2}} + 2ay^{\frac{1}{2}}.$$

The optimal bundle should be expressed as a function of $a$. Other variables are given by:

$$\begin{eqnarray*}\text{Income} &&= 80\\ p_X &&= 2\\ p_Y &&= 1 \end{eqnarray*}$$

I'm able to solve similar problems using the Lagrangian Method, however it is the constant that is throwing me off. I'm lost after taking FOC's of the Lagrangian function. What should I be doing differently to account for this constant?

Answer 1

If $a>0$ then $u(x,y)$ is a concave function by sum and domain extension theorem. 

The optimal bundle will satisfy the following conditions: $$\begin{eqnarray*}i)& \quad \ \frac{MU_x}{MU_y}= \frac{p_X}{p_Y} \\ ii)& \quad \ xp_X+yp_Y=M \end{eqnarray*}. $$

Solving $i)$ gives: $ \frac{\sqrt{y}}{a \sqrt{x}} = 2 \implies y=4a^2x $.

By solving $ii)$ we get: $2x+ 4a^2x = 80 \implies x=\frac{40}{1+2a^2} $.

And therefore, $y = 4a^2x =\frac{160a^2}{1+2a^2}$.