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?
1 Answer
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As you can see in this post, 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.
