# 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) = 7 X^{{36}} Y^{{80}}.. . The price of chocolate (X) is p_X =9, and the price of strawberry Y is p_Y = $8. Alice has in her pocket I =$464.

What quantities X and Y maximizes Alice's utility?

Alice will buy 16 chocolates and 40 strawberries.

### Step 1: MRS = MRT

\begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ \frac{7 \times 36 X^{35} Y^{80}}{7 \times 80 X^{36} Y^{79}} &= \frac{9}{8} \\ \frac{36 Y}{80 X} &= \frac{9}{8} \\ \frac{Y}{X} &= \frac{9 \times 80}{8 \times 36} \\ \frac{Y}{X} &= \frac{10}{4} \\ \frac{Y}{X} &= 2.5 \\ Y &= 2.5 X \end{align*}

### Step 2: Plug back in the budget

\begin{align*} 9 X + 8 Y = 464 \\ 9 X + 8 \times 2.5 X = 464 \\ \left( 9 + 8 \times 2.5 \right) X = 464 \\ 29.0 X = 464 \\ X = \frac{464}{29.0} \\ X = 16 \end{align*}

### Step 3: Conclude

X=16 and Y = 2.5 X = 40