# 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 = \$8. Her utility is measured by u \left( X, Y \right) = X \times Y.

### Step 1: MRS = MRT

\begin{align*} MRS &= MRT \\ -\frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ -\frac{Y}{X} &= -\frac{4}{2} \\ - \frac{Y}{X} &= - 2 \\ Y &= 2X \end{align*}

### Step 2: Plug back in the budget

\begin{align*} Xp_X + Yp_Y &= I \\ 4X + 2Y &= 8 \\ 4X + 2\left( 2X \right) &= 8 \\ 4X + 4X &= 8 \\ 8X &= 8 \\ X &= 1 \end{align*}

## Step 3: Conclude

Alice will buy 1 chocolate bar (X) and 2 strawberries (Y).

### Question

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