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) = 4 X^{{10}} Y^{{7}}`..

.

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 = $34`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 4 chocolates and 2 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{4 \times 10 X^{9} Y^{7}}{4 \times 7 X^{10} Y^{6}} &= \frac{5}{7} \\ \frac{10 Y}{7 X} &= \frac{5}{7} \\ \frac{Y}{X} &= \frac{5 \times 7}{7 \times 10} \\ \frac{Y}{X} &= \frac{1}{2} \\ \frac{Y}{X} &= 0.5 \\ Y &= 0.5 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 5 X + 7 Y = 34 \\ 5 X + 7 \times 0.5 X = 34 \\ \left( 5 + 7 \times 0.5 \right) X = 34 \\ 8.5 X = 34 \\ X = \frac{34}{8.5} \\ X = 4 \end{align*} $$

Step 3: Conclude

`X=4` and `Y = 0.5 X = 2`