Utility maximization consumer

Introduction: In consumer behavior it is assumed that a consumer wants to maximize his utility.

Model:

• $p_1$ is the price of good $x$
• $p_2$ is the price of good $y$
• $I$ is the available income

Consider the utility maximization problem

$\begin{array}{ll}
\mbox{maximize}&U(x,y)\\
\mbox{subject to}&p_1x+p_2y=I,\\
\mbox{where} &  x \in D_1 \ \mbox{and} \ y \in D_2.
\end{array}
$

An extremum location $(x,y)=(c,d)$, where $c \in D_1$ and $d \in D_2$ that is not a boundary point, satisfies the following system of equations:

$\left\{
\begin{array}{lcl}
MRS(x,y)&=&{\dfrac{p_1}{p_2}}\\
p_1x + p_2 y &=&I.
\end{array}
\right.
$