# Consumer Demand

The demand depicts the quantity of products the consumer purchases for different prices.

## Example

Alice has 12 to buy chocolate (X) and strawberries (Y). Her utility is measured by u \left( X, Y \right) = X Y^2. Depending on the prices p_X and p_Y, what is Alice's demand for chocolate? ### Step 1: Equalize MRS and MRT \begin{align*} MRS &= MRT \\ - \frac{Y}{2X} &= - \frac{p_X}{p_Y} \\ Y &= 2X \frac{p_X}{p_Y} \end{align*} ### Step 2: Plug in the budget constraint \begin{align*} X p_X + Y p_Y = 12 \\ X p_X + 2X \frac{p_X}{p_Y} p_Y = 12 \\ 3 X p_X = 12 \\ X = \frac{4}{p_X} \\ \end{align*} Alice's demand for chocolate is X = \frac{4}{p_X}. ### Question Now Alice can spend up to159 on chocolate (X) and strawberries (Y). Her utility is u \left( X, Y \right) = 7 X^{53} Y^{106}.

What is her demand for X?

Alice's demand for chocolate (X) is X = \frac{53}{p_X}.

### Step 1: Equalize MRS and MRT

\begin{align*} MRS &= MRT \\ - \frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ - \frac{7 \times 53 X^{52} Y^{106}}{7 \times 106 X^{53} Y^{105}} &= - \frac{p_X}{p_Y} \\ \frac{Y}{X} &= \frac{106}{53} \frac{p_X}{p_Y} \\ Y &= 2 \frac{p_X}{p_Y} X \end{align*}

### Step 2: Plug into the budget

\begin{align*} X p_X + Y p_Y &= 159 \\ X p_X + 2 \frac{p_X}{p_Y} X p_Y &= 159 \\ X p_X + 2 X p_X &= 159 \\ \left( 1 + 2 \right) X p_X &= 159 \\ 3 X &= 159 \\ X &= \frac{159}{3 p_X} \\ X &= \frac{53}{p_X} \end{align*}