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) = 2 X^{{50}} Y^{{12}}`..

.

The price of chocolate (X) is `p_X = $5`, and the price of strawberry Y is `p_Y = $3`. Alice has in her pocket `I = $434`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 70 chocolates and 28 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{2 \times 50 X^{49} Y^{12}}{2 \times 12 X^{50} Y^{11}} &= \frac{5}{3} \\ \frac{50 Y}{12 X} &= \frac{5}{3} \\ \frac{Y}{X} &= \frac{5 \times 12}{3 \times 50} \\ \frac{Y}{X} &= \frac{2}{5} \\ \frac{Y}{X} &= 0.4 \\ Y &= 0.4 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 5 X + 3 Y = 434 \\ 5 X + 3 \times 0.4 X = 434 \\ \left( 5 + 3 \times 0.4 \right) X = 434 \\ 6.2 X = 434 \\ X = \frac{434}{6.2} \\ X = 70 \end{align*} $$

Step 3: Conclude

`X=70` and `Y = 0.4 X = 28`