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 to $84 on chocolate (X) and strawberries (Y). Her utility is `u \left( X, Y \right) = 10 X^{28} Y^{56}`.

What is her demand for X?

Alice's demand for chocolate (X) is `X = \frac{28}{p_X}`.

Step 1: Equalize MRS and MRT

$$ \begin{align*} MRS &= MRT \\ - \frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ - \frac{10 \times 28 X^{27} Y^{56}}{10 \times 56 X^{28} Y^{55}} &= - \frac{p_X}{p_Y} \\ \frac{Y}{X} &= \frac{56}{28} \frac{p_X}{p_Y} \\ Y &= 2 \frac{p_X}{p_Y} X \end{align*} $$

Step 2: Plug into the budget

$$ \begin{align*} X p_X + Y p_Y &= 84 \\ X p_X + 2 \frac{p_X}{p_Y} X p_Y &= 84 \\ X p_X + 2 X p_X &= 84 \\ \left( 1 + 2 \right) X p_X &= 84 \\ 3 X &= 84 \\ X &= \frac{84}{3 p_X} \\ X &= \frac{28}{p_X} \end{align*} $$