Example
Alice buys chocolate (X) for `p_X = $4` and strawberries (Y) for `p_Y = $2`. Her budget is `I = $12`. Her utility is measured by `u \left( X, Y \right) = X Y^2`.
Step 1: MRS = MRT
$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ -\frac{Y^2}{2XY} &= -\frac{2}{4} \\ - \frac{Y}{2X} &= - \frac{2}{4} \\ Y &= X \end{align*} $$Step 2: Plug back in the budget
$$ \begin{align*} Xp_X + Yp_Y &= I \\ 4X + 2Y &= 12 \\ 4X + 2X &= 12 \\ 6X &= 12 \\ X &= 2 \end{align*} $$Step 3: Conclude
Alice will buy 2 chocolates and 2 strawberries.
Question
Alice's utility function is `u \left( X, Y \right) = 7 X^{{6}} Y^{{30}}`..
.The price of chocolate (X) is `p_X = $6`, and the price of strawberry Y is `p_Y = $10`. Alice has in her pocket `I = $216`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 6 chocolates and 18 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{7 \times 6 X^{5} Y^{30}}{7 \times 30 X^{6} Y^{29}} &= \frac{6}{10} \\
\frac{6 Y}{30 X} &= \frac{6}{10} \\
\frac{Y}{X} &= \frac{6 \times 30}{10 \times 6} \\
\frac{Y}{X} &= \frac{3}{1} \\
\frac{Y}{X} &= 3.0 \\
Y &= 3.0 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
6 X + 10 Y = 216 \\
6 X + 10 \times 3.0 X = 216 \\
\left( 6 + 10 \times 3.0 \right) X = 216 \\
36.0 X = 216 \\
X = \frac{216}{36.0} \\
X = 6
\end{align*}
$$
Step 3: Conclude
`X=6` and `Y = 3.0 X = 18`