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) = X^{{10}} Y^{{9}}`..
.The price of chocolate (X) is `p_X = $1`, and the price of strawberry Y is `p_Y = $9`. Alice has in her pocket `I = $38`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 20 chocolates and 2 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{1 \times 10 X^{9} Y^{9}}{1 \times 9 X^{10} Y^{8}} &= \frac{1}{9} \\
\frac{10 Y}{9 X} &= \frac{1}{9} \\
\frac{Y}{X} &= \frac{1 \times 9}{9 \times 10} \\
\frac{Y}{X} &= \frac{1}{10} \\
\frac{Y}{X} &= 0.1 \\
Y &= 0.1 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
1 X + 9 Y = 38 \\
1 X + 9 \times 0.1 X = 38 \\
\left( 1 + 9 \times 0.1 \right) X = 38 \\
1.9 X = 38 \\
X = \frac{38}{1.9} \\
X = 20
\end{align*}
$$
Step 3: Conclude
`X=20` and `Y = 0.1 X = 2`