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) = 4 X^{{6}} Y^{{40}}`..
.The price of chocolate (X) is `p_X = $3`, and the price of strawberry Y is `p_Y = $10`. Alice has in her pocket `I = $276`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 12 chocolates and 24 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{4 \times 6 X^{5} Y^{40}}{4 \times 40 X^{6} Y^{39}} &= \frac{3}{10} \\
\frac{6 Y}{40 X} &= \frac{3}{10} \\
\frac{Y}{X} &= \frac{3 \times 40}{10 \times 6} \\
\frac{Y}{X} &= \frac{4}{2} \\
\frac{Y}{X} &= 2.0 \\
Y &= 2.0 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
3 X + 10 Y = 276 \\
3 X + 10 \times 2.0 X = 276 \\
\left( 3 + 10 \times 2.0 \right) X = 276 \\
23.0 X = 276 \\
X = \frac{276}{23.0} \\
X = 12
\end{align*}
$$
Step 3: Conclude
`X=12` and `Y = 2.0 X = 24`