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) = 6 X^{{20}} Y^{{24}}`..
.The price of chocolate (X) is `p_X = $10`, and the price of strawberry Y is `p_Y = $8`. Alice has in her pocket `I = $440`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 20 chocolates and 30 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{6 \times 20 X^{19} Y^{24}}{6 \times 24 X^{20} Y^{23}} &= \frac{10}{8} \\
\frac{20 Y}{24 X} &= \frac{10}{8} \\
\frac{Y}{X} &= \frac{10 \times 24}{8 \times 20} \\
\frac{Y}{X} &= \frac{3}{2} \\
\frac{Y}{X} &= 1.5 \\
Y &= 1.5 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
10 X + 8 Y = 440 \\
10 X + 8 \times 1.5 X = 440 \\
\left( 10 + 8 \times 1.5 \right) X = 440 \\
22.0 X = 440 \\
X = \frac{440}{22.0} \\
X = 20
\end{align*}
$$
Step 3: Conclude
`X=20` and `Y = 1.5 X = 30`