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) = 3 X^{{80}} Y^{{21}}`..
.The price of chocolate (X) is `p_X = $8`, and the price of strawberry Y is `p_Y = $3`. Alice has in her pocket `I = $404`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 40 chocolates and 28 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{3 \times 80 X^{79} Y^{21}}{3 \times 21 X^{80} Y^{20}} &= \frac{8}{3} \\
\frac{80 Y}{21 X} &= \frac{8}{3} \\
\frac{Y}{X} &= \frac{8 \times 21}{3 \times 80} \\
\frac{Y}{X} &= \frac{7}{10} \\
\frac{Y}{X} &= 0.7 \\
Y &= 0.7 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
8 X + 3 Y = 404 \\
8 X + 3 \times 0.7 X = 404 \\
\left( 8 + 3 \times 0.7 \right) X = 404 \\
10.1 X = 404 \\
X = \frac{404}{10.1} \\
X = 40
\end{align*}
$$
Step 3: Conclude
`X=40` and `Y = 0.7 X = 28`