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

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?

• There is no difference if in the objective function there is a constant, proceed as usual, treating the constant as a number, and you'll have the result given by @Dank trader in the answer below. Jan 27, 2023 at 17:25

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