A consumer whose utility function is given by $U(x,y)=2x^4y$ spends his income $I=40$ on the goods $x$ and $y$ with prices $p_1=32$ and $p_2=1$, respectively. Determine the maximum utility.
$64$
$(1,8)$
$(2,2)$
$16$
Correct: The information translates in the utility maximization problem
$\begin{array}{ll}
\mbox{maximize}&2x^4y\\
\mbox{subject to}&32x+y=40,\\
\mbox{where} & x\geq 0 \ \text{ and } \ y \geq 0.
\end{array}$
$MRS(x,y)={\dfrac{p_1}{p_2}}$ then results in the equation $\dfrac{8x^3y}{2x^4}=\dfrac{32}{1}$, which gives $y=8x$.
Then we use the budget equation: $32x+8x=40$ gives $x=1$. Therefore, $y=8$.
To verify that $(x,y)=(1,8)$ is indeed the bundle that gives maximum utility we observe that $U(0,40)=U(1\frac{1}{4},0)=0<16=U(1,8)$.
Hence, $U(1,8)=16$ is the maximum utility.
Go on.
Wrong: There is no combination of $x$ and $y$ that satisfies the budget contrainst and gives a utility of $64$.
See Example.
Wrong: The question is not to find the bundle that gives the maximum utility.
Try again.
Wrong: The question is not to find the bundle that gives the maximum utility.
Try again.