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) = 5 X^{{50}} Y^{{35}}`..
.The price of chocolate (X) is `p_X = $5`, and the price of strawberry Y is `p_Y = $7`. Alice has in her pocket `I = $850`.
What quantities X and Y maximizes Alice's utility?
Alice will buy 100 chocolates and 50 strawberries.
Step 1: MRS = MRT
$$
\begin{align*}
MRS &= MRT \\
-\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\
\frac{5 \times 50 X^{49} Y^{35}}{5 \times 35 X^{50} Y^{34}} &= \frac{5}{7} \\
\frac{50 Y}{35 X} &= \frac{5}{7} \\
\frac{Y}{X} &= \frac{5 \times 35}{7 \times 50} \\
\frac{Y}{X} &= \frac{1}{2} \\
\frac{Y}{X} &= 0.5 \\
Y &= 0.5 X
\end{align*}
$$
Step 2: Plug back in the budget
$$
\begin{align*}
5 X + 7 Y = 850 \\
5 X + 7 \times 0.5 X = 850 \\
\left( 5 + 7 \times 0.5 \right) X = 850 \\
8.5 X = 850 \\
X = \frac{850}{8.5} \\
X = 100
\end{align*}
$$
Step 3: Conclude
`X=100` and `Y = 0.5 X = 50`