<p>As you can see in <a href="https://economics.stackexchange.com/a/58316/11824">this post</a>, that there are also &quot;corner&quot; solutions to this problem under some conditions. These are solutions where <span class="math-container">$x_1=0$</span> or <span class="math-container">$x_2=0$</span>.
For this reason, you may use Kuhn-Tucker (KT) conditions or any other method to determine the demands. Knowing quasi-concavity of <span class="math-container">$u$</span> can be useful in getting the sufficiency of KT conditions to deliver the solution of the optimization problem. To see that <span class="math-container">$u$</span> is quasi-concave, observe that <span class="math-container">$u(x_1,x_2)=2x_1x_2+x_1+2x_2$</span> is an increasing transformation of a concave function <span class="math-container">$v(x_1,x_2)=\ln(x_1+1)+\ln(2x_2+1)$</span> (which is a sum of concave functions) and <span class="math-container">$u=e^v-1$</span>, therefore, it is quasi-concave.</p>

CC BY-SA 4.0
2024-05-06T13:29:54.807Z
2024-05-06T13:29:54.807Z
Amit
<p>A consumer has the following utility function
<span class="math-container">$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$</span>
I have maximized his utility function, and found its demand functions, for <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>, using Lagrange. However, is it necessary to check beforehand if the function is quasi-concave? And how can I establish that?</p>

<microeconomics><mathematical-economics><optimization>
Using lagrange on a quasi-concave utility function
Noah
# Using lagrange on a quasi-concave utility function

(Asked by: username_0 on 2024-05-06)

A consumer has the following utility function
$$u(x\_1,x\_2)=2x\_1x\_2+x\_1+2x\_2$$
I have maximized his utility function, and found its demand functions, for $x\_1$ and $x\_2$, using Lagrange. However, is it necessary to check beforehand if the function is quasi-concave? And how can I establish that?

**Tags:** microeconomics, mathematical-economics, optimization

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## 1 answers

### Answer by username_1 on 2024-05-06. Score: 2 | Accepted answer

As you can see in [this post](https://economics.stackexchange.com/a/58316/11824), that there are also "corner" solutions to this problem under some conditions. These are solutions where $x\_1=0$ or $x\_2=0$.
For this reason, you may use Kuhn-Tucker (KT) conditions or any other method to determine the demands. Knowing quasi-concavity of $u$ can be useful in getting the sufficiency of KT conditions to deliver the solution of the optimization problem. To see that $u$ is quasi-concave, observe that $u(x\_1,x\_2)=2x\_1x\_2+x\_1+2x\_2$ is an increasing transformation of a concave function $v(x\_1,x\_2)=\ln(x\_1+1)+\ln(2x\_2+1)$ (which is a sum of concave functions) and $u=e^v-1$, therefore, it is quasi-concave.


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