Further Comparative Static Analysis of Demand

Inverse Demand Function

• Typically, demand curves have the quantity on the x-axis and price on the y-axis; these are the graphs of inverse demand functions
  • Especially useful if trying to find the price of a good given a quantity
  • Parameter of inverse demand function is quantity

$$\text{Using the Cobb-Douglas utility function: } u(x_1, x_2) = x_1^a x_2^b \\ \text{A Cobb-Douglas demand function looks like } (x_1^* , x_2^* ) = (\frac{am}{(a + b)p_1}, \frac{bm}{(a + b) p_2}) \\ \text{The inverse demand function would be } (p_1 , p_2 ) = (\frac{am}{(a + b)x_1^*}, \frac{bm}{(a + b) x_2^*})$$ $$\text{With perfect complements: } u(x_1, x_2) = min\{x_1, x_2\} \\ \text{The demand function looks like } (x_1^* , x_2^* ) = (\frac{m}{p_1 + p_2}, \frac{m}{p_1 + p_2}) \\ \text{The inverse demand function would be } (p_1 , p_2 ) = (\frac{m}{x_1^*} - p_2, \frac{m}{x_2^*} - p_1)$$

Income Changes

$$\text{How does } x_1^*(p_1, p_2, m) \text{ change as y changes, holding other variables constant?}$$

• Note that changes in income do not change indifference curves; it only changes the budget constraint, changing the highest curve obtainable
  • The curve containing the bundles that are demanded at different levels of income is known as the income offer curve (also known as the income expansion path)
• Two graphs can then be drawn to find the optimal amounts of both good 1 and good 2
  • The graphs mapping income (on y-axis) to quantity (on x-axis) is known as the Engel curve

$$\text{With Cobb-Douglas utility: } u(x_1, x_2) = x_1^a x_2^b \\ \text{The demand equations are } (x_1^* , x_2^* ) = (\frac{am}{(a + b)p_1}, \frac{bm}{(a + b) p_2}) \\ \text{Isolating for m: } m = (\frac{(a + b)p_1}{a}x_1^*, \frac{(a + b)p_2}{b}x_2^*) \\ \text{Note that m and the optimal amounts of either good are directly proportional to each other, } \\ \text{so in a Cobb-Douglas scenario, the Engel curves are straight lines.}$$

$$\text{With perfect complements: } u(x_1, x_2) = min\{x_1, x_2\} \\ \text{The demand function looks like } (x_1^* , x_2^* ) = (\frac{m}{p_1 + p_2}, \frac{m}{p_1 + p_2}) \\ \text{Isolating for m: } m = ((p_1 + p_2)x_1^*, (p_1 + p_2)x_2^*) \\ \text{Note that m and the optimal amounts of either good are directly proportional to each other, } \\ \text{so in a Cobb-Douglas scenario, the Engel curves are straight lines.}$$

$$\text{With perfect substitutes: } u(x_1, x_2) = x_1 + x_2 \\ \text{The demand function looks like: } \\ x_1^* (p_1, p_2, y) = \left\{ \begin{array}{lr} 0, & \text{if } p_1 > p_2\\ y / p_1, & \text{if } p_1 < p_2 \end{array} \right\} \\ x_2^* (p_1, p_2, y) = \left\{ \begin{array}{lr} 0, & \text{if } p_1 < p_2\\ y / p_2, & \text{if } p_1 > p_2 \end{array} \right\} \\ \\ \text{Suppose } p_1 < p_2 \text{. Then, } x_1^* = \frac{m}{p_1}, x_2^* = 0 \rightarrow m = p_1 x_1^*, x_2^* = 0 \\ \text{The opposite happens if } p_2 > p_1$$

• In all of the examples so far, the Engel curves have been straight lines, but this isn’t always the case
  • Engel curves are only straight lines if the consumer’s preferences are homothetic
  • Homothetic preferences are preferences in which the demand for any good goes up by the same proportion as income
  • Luxury goods: demand for the good goes up by a greater proportion than income
  • Necessary good: demand for the good goes up by a smaller proportion than income