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^{{20}} Y^{{70}}`..
.The price of chocolate (X) is `p_X = $4`, and the price of strawberry Y is `p_Y = $10`. Alice has in her pocket `I = $90`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 5 chocolates and 7 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{4 \times 20 X^{19} Y^{70}}{4 \times 70 X^{20} Y^{69}} &= \frac{4}{10} \\
\frac{20 Y}{70 X} &= \frac{4}{10} \\
\frac{Y}{X} &= \frac{4 \times 70}{10 \times 20} \\
\frac{Y}{X} &= \frac{7}{5} \\
\frac{Y}{X} &= 1.4 \\
Y &= 1.4 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
4 X + 10 Y = 90 \\
4 X + 10 \times 1.4 X = 90 \\
\left( 4 + 10 \times 1.4 \right) X = 90 \\
18.0 X = 90 \\
X = \frac{90}{18.0} \\
X = 5
\end{align*}
$$
Step 3: Conclude
`X=5` and `Y = 1.4 X = 7`