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) = 6 X^{{70}} Y^{{24}}`..

.

The price of chocolate (X) is `p_X = $7`, and the price of strawberry Y is `p_Y = $6`. Alice has in her pocket `I = $846`.

What quantities X and Y maximizes Alice's utility?

Alice will buy 90 chocolates and 36 strawberries.

Step 1: MRS = MRT

$$ \begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{6 \times 70 X^{69} Y^{24}}{6 \times 24 X^{70} Y^{23}} &= \frac{7}{6} \\ \frac{70 Y}{24 X} &= \frac{7}{6} \\ \frac{Y}{X} &= \frac{7 \times 24}{6 \times 70} \\ \frac{Y}{X} &= \frac{2}{5} \\ \frac{Y}{X} &= 0.4 \\ Y &= 0.4 X \end{align*} $$

Step 2: Plug back in the budget

$$ \begin{align*} 7 X + 6 Y = 846 \\ 7 X + 6 \times 0.4 X = 846 \\ \left( 7 + 6 \times 0.4 \right) X = 846 \\ 9.4 X = 846 \\ X = \frac{846}{9.4} \\ X = 90 \end{align*} $$

Step 3: Conclude

`X=90` and `Y = 0.4 X = 36`