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) = 8 X^{{32}} Y^{{18}}`..
.The price of chocolate (X) is `p_X = $8`, and the price of strawberry Y is `p_Y = $9`. Alice has in her pocket `I = $100`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 8 chocolates and 4 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{8 \times 32 X^{31} Y^{18}}{8 \times 18 X^{32} Y^{17}} &= \frac{8}{9} \\
\frac{32 Y}{18 X} &= \frac{8}{9} \\
\frac{Y}{X} &= \frac{8 \times 18}{9 \times 32} \\
\frac{Y}{X} &= \frac{1}{2} \\
\frac{Y}{X} &= 0.5 \\
Y &= 0.5 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
8 X + 9 Y = 100 \\
8 X + 9 \times 0.5 X = 100 \\
\left( 8 + 9 \times 0.5 \right) X = 100 \\
12.5 X = 100 \\
X = \frac{100}{12.5} \\
X = 8
\end{align*}
$$
Step 3: Conclude
`X=8` and `Y = 0.5 X = 4`