Utility Maximization

A consumer maximizes utility when:
  1. `MRS = MRT`
  2. `Xp_X + Yp_Y = I`

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) = 8 X^{{20}} Y^{{10}}`..

.

The price of chocolate (X) is `p_X = $2`, and the price of strawberry Y is `p_Y = $10`. Alice has in her pocket `I = $60`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 20 chocolates and 2 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{8 \times 20 X^{19} Y^{10}}{8 \times 10 X^{20} Y^{9}} &= \frac{2}{10} \\ \frac{20 Y}{10 X} &= \frac{2}{10} \\ \frac{Y}{X} &= \frac{2 \times 10}{10 \times 20} \\ \frac{Y}{X} &= \frac{1}{10} \\ \frac{Y}{X} &= 0.1 \\ Y &= 0.1 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 2 X + 10 Y = 60 \\ 2 X + 10 \times 0.1 X = 60 \\ \left( 2 + 10 \times 0.1 \right) X = 60 \\ 3.0 X = 60 \\ X = \frac{60}{3.0} \\ X = 20 \end{align*} $$

Step 3: Conclude

`X=20` and `Y = 0.1 X = 2`