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^{{100}} Y^{{54}}`..
.The price of chocolate (X) is `p_X = $10`, and the price of strawberry Y is `p_Y = $9`. Alice has in her pocket `I = $1078`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 70 chocolates and 42 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{6 \times 100 X^{99} Y^{54}}{6 \times 54 X^{100} Y^{53}} &= \frac{10}{9} \\
\frac{100 Y}{54 X} &= \frac{10}{9} \\
\frac{Y}{X} &= \frac{10 \times 54}{9 \times 100} \\
\frac{Y}{X} &= \frac{3}{5} \\
\frac{Y}{X} &= 0.6 \\
Y &= 0.6 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
10 X + 9 Y = 1078 \\
10 X + 9 \times 0.6 X = 1078 \\
\left( 10 + 9 \times 0.6 \right) X = 1078 \\
15.399999999999999 X = 1078 \\
X = \frac{1078}{15.399999999999999} \\
X = 70
\end{align*}
$$
Step 3: Conclude
`X=70` and `Y = 0.6 X = 42`