<p>If <span class="math-container">$a&gt;0$</span> then <span class="math-container">$u(x,y)$</span> is a concave function by sum and domain extension theorem. </p>
<p>The optimal bundle will satisfy the following conditions:
<span class="math-container">$$\begin{eqnarray*}i)&amp; \quad \ \frac{MU_x}{MU_y}= \frac{p_X}{p_Y} \\ ii)&amp; \quad \ xp_X+yp_Y=M \end{eqnarray*}. $$</span></p>
<p>Solving <span class="math-container">$i)$</span> gives: <span class="math-container">$ \frac{\sqrt{y}}{a \sqrt{x}} = 2 \implies y=4a^2x $</span>.</p>
<p>By solving <span class="math-container">$ii)$</span> we get: <span class="math-container">$2x+ 4a^2x = 80 \implies  x=\frac{40}{1+2a^2} $</span>.</p>
<p>And therefore, <span class="math-container">$y = 4a^2x =\frac{160a^2}{1+2a^2}$</span>.</p>

CC BY-SA 4.0
2023-01-27T06:20:26.653Z
2023-01-27T17:21:50.040Z
2023-01-27T17:21:50.040Z
Dank trader
<p>The utility function is given by: <span class="math-container">$$u(x, y) = 2x^{\frac{1}{2}} + 2ay^{\frac{1}{2}}.$$</span></p>
<p>The optimal bundle should be expressed as a function of <span class="math-container">$a$</span>. Other variables are given by:</p>
<p><span class="math-container">$$\begin{eqnarray*}\text{Income} &amp;&amp;= 80\\
p_X &amp;&amp;= 2\\
p_Y &amp;&amp;= 1 \end{eqnarray*}$$</span></p>
<p>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?</p>

CC BY-SA 4.0
2023-01-27T17:25:11.247Z
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.
BakerStreet
<microeconomics><utility><lagrangian>
How can I solve a Utility Maximization problem using the Lagrangian method where the Utility formula has an exogenous constant $a$?
mrtavsh
# How can I solve a Utility Maximization problem using the Lagrangian method where the Utility formula has an exogenous constant $a$?

(Asked by: username_0 on 2023-01-26)

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?

**Tags:** microeconomics, utility, lagrangian
> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2023-01-27. Score: 3

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


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