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) = 3 X^{{6}} Y^{{80}}`..

.

The price of chocolate (X) is `p_X = $3`, and the price of strawberry Y is `p_Y = $8`. Alice has in her pocket `I = $860`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 20 chocolates and 100 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{3 \times 6 X^{5} Y^{80}}{3 \times 80 X^{6} Y^{79}} &= \frac{3}{8} \\ \frac{6 Y}{80 X} &= \frac{3}{8} \\ \frac{Y}{X} &= \frac{3 \times 80}{8 \times 6} \\ \frac{Y}{X} &= \frac{10}{2} \\ \frac{Y}{X} &= 5.0 \\ Y &= 5.0 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 3 X + 8 Y = 860 \\ 3 X + 8 \times 5.0 X = 860 \\ \left( 3 + 8 \times 5.0 \right) X = 860 \\ 43.0 X = 860 \\ X = \frac{860}{43.0} \\ X = 20 \end{align*} $$

Step 3: Conclude

`X=20` and `Y = 5.0 X = 100`