A consumer whose utility function is given by $U(x,y)=x^2y^3$ spends his income $I=64$ on the goods $x$ and $y$ with prices $p_1=4$ and $p_2=16$, respectively. We determine the bundle of goods $(x,y)$ with maximum utilty.
This information translates in the utility maximization problem
$\begin{array}{ll}
\mbox{maximize}&x^2y^3\\
\mbox{subject to}&4x+16y=64,\\
\mbox{where} & x\geq 0 \ \mbox{and} \ y \geq 0.
\end{array}
$
$MRS(x,y)={\dfrac{p_1}{p_2}}$ then results in the equation $\dfrac{2xy^3}{3x^2y^2}=\dfrac{4}{16}$, which gives $y=\frac{3}{8}x$.
Then we use the budget equation: $4x+16(\frac{3}{8}x)=64$ gives $x=6\frac{2}{5}$. Therefore, $y=2\frac{2}{5}$.
To verify that $(x,y)=(6\frac{2}{5},2\frac{2}{5})$ is indeed the bundle that gives maximum utility we observe that $U(0,4)=U(16,0)=0<566\frac{722}{3125}=U(6\frac{2}{5},2\frac{2}{5})$.
Hence, $(x,y)=(6\frac{2}{5},2\frac{2}{5})$ is indeed the bundle that gives maximum utility.
This information translates in the utility maximization problem
$\begin{array}{ll}
\mbox{maximize}&x^2y^3\\
\mbox{subject to}&4x+16y=64,\\
\mbox{where} & x\geq 0 \ \mbox{and} \ y \geq 0.
\end{array}
$
$MRS(x,y)={\dfrac{p_1}{p_2}}$ then results in the equation $\dfrac{2xy^3}{3x^2y^2}=\dfrac{4}{16}$, which gives $y=\frac{3}{8}x$.
Then we use the budget equation: $4x+16(\frac{3}{8}x)=64$ gives $x=6\frac{2}{5}$. Therefore, $y=2\frac{2}{5}$.
To verify that $(x,y)=(6\frac{2}{5},2\frac{2}{5})$ is indeed the bundle that gives maximum utility we observe that $U(0,4)=U(16,0)=0<566\frac{722}{3125}=U(6\frac{2}{5},2\frac{2}{5})$.
Hence, $(x,y)=(6\frac{2}{5},2\frac{2}{5})$ is indeed the bundle that gives maximum utility.