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^{{70}} Y^{{24}}`..
.The price of chocolate (X) is `p_X = $7`, and the price of strawberry Y is `p_Y = $6`. Alice has in her pocket `I = $846`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 90 chocolates and 36 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{6 \times 70 X^{69} Y^{24}}{6 \times 24 X^{70} Y^{23}} &= \frac{7}{6} \\
\frac{70 Y}{24 X} &= \frac{7}{6} \\
\frac{Y}{X} &= \frac{7 \times 24}{6 \times 70} \\
\frac{Y}{X} &= \frac{2}{5} \\
\frac{Y}{X} &= 0.4 \\
Y &= 0.4 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
7 X + 6 Y = 846 \\
7 X + 6 \times 0.4 X = 846 \\
\left( 7 + 6 \times 0.4 \right) X = 846 \\
9.4 X = 846 \\
X = \frac{846}{9.4} \\
X = 90
\end{align*}
$$
Step 3: Conclude
`X=90` and `Y = 0.4 X = 36`