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 $6270 on chocolate (X) and strawberries (Y). Her utility is `u \left( X, Y \right) = 8 X^{95} Y^{950}`.

What is her demand for X?

Alice's demand for chocolate (X) is `X = \frac{570}{p_X}`.

Step 1: Equalize MRS and MRT

$$ \begin{align*} MRS &= MRT \\ - \frac{MU_X}{MU_Y} &= -\frac{p_X}{p_Y} \\ - \frac{8 \times 95 X^{94} Y^{950}}{8 \times 950 X^{95} Y^{949}} &= - \frac{p_X}{p_Y} \\ \frac{Y}{X} &= \frac{950}{95} \frac{p_X}{p_Y} \\ Y &= 10 \frac{p_X}{p_Y} X \end{align*} $$

Step 2: Plug into the budget

$$ \begin{align*} X p_X + Y p_Y &= 6270 \\ X p_X + 10 \frac{p_X}{p_Y} X p_Y &= 6270 \\ X p_X + 10 X p_X &= 6270 \\ \left( 1 + 10 \right) X p_X &= 6270 \\ 11 X &= 6270 \\ X &= \frac{6270}{11 p_X} \\ X &= \frac{570}{p_X} \end{align*} $$