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) = X^{{16}} Y^{{20}}`..

.

The price of chocolate (X) is `p_X = $4`, and the price of strawberry Y is `p_Y = $5`. Alice has in her pocket `I = $324`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 36 chocolates and 36 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{1 \times 16 X^{15} Y^{20}}{1 \times 20 X^{16} Y^{19}} &= \frac{4}{5} \\ \frac{16 Y}{20 X} &= \frac{4}{5} \\ \frac{Y}{X} &= \frac{4 \times 20}{5 \times 16} \\ \frac{Y}{X} &= \frac{4}{4} \\ \frac{Y}{X} &= 1.0 \\ Y &= 1.0 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 4 X + 5 Y = 324 \\ 4 X + 5 \times 1.0 X = 324 \\ \left( 4 + 5 \times 1.0 \right) X = 324 \\ 9.0 X = 324 \\ X = \frac{324}{9.0} \\ X = 36 \end{align*} $$

Step 3: Conclude

`X=36` and `Y = 1.0 X = 36`