Now hold income fixed and change one price. In Cobb-Douglas Demand the formula $x^* = \frac{a}{a+b}\frac{m}{p_1}$ already told us how $x^*$ responds. Here we see the same thing as a picture, and learn which properties every demand curve has and which it doesn't.
Key idea
As $p_1$ changes the budget line rotates around its $y$-intercept, and the optimum traces the price offer curve. Plotting $x^*$ against $p_1$ gives the demand curve. Every demand function is homogeneous of degree zero: multiplying all prices and income by the same $k > 0$ leaves the choice unchanged.
Why it works
Homogeneity: the budget $kp_1x + kp_2y \le km$ is the same set as $p_1x + p_2y \le m$, and preferences didn't change, so the best bundle didn't either. Only relative prices and real income matter, which is why pure inflation (all prices and incomes up together) doesn't change what people buy. Downward slope is not guaranteed by the theory alone. A price fall makes the good relatively cheaper, which always pulls quantity up, and also makes the consumer richer, which pulls quantity up for a normal good but down for an inferior one. Only if that second pull is stronger than the first does demand slope up: a Giffen good.
Worked example: Points on a demand curve
$U = xy$, $m = 120$. Find $x^*$ at $p_1 = 2, 3, 6$.
- $x^* = \dfrac{1}{2}\cdot\dfrac{120}{p_1} = \dfrac{60}{p_1}$.
- $p_1 = 2$: 30. $p_1 = 3$: 20. $p_1 = 6$: 10.
- Spending $p_1x^*$ is \$60 every time: a unit-elastic demand curve.
Answer: 30, 20 and 10.
Worked example: Doubling everything
With $U = xy$, $m = 120$, $p_1 = 3$, $p_2 = 2$, the consumer buys (20, 30). Prices and income all double. What does she buy?
- New budget: $6x + 4y = 240$. Divide by 2: $3x + 2y = 120$, the old budget.
- Same budget set, same preferences: she still buys (20, 30).
Answer: (20, 30).
Common mistake
Treating every inferior good as a Giffen good. Inferior means the income effect works against the substitution effect; Giffen means it also wins. Most inferior goods (store brands, bus rides for some people) still have down-sloping demand, because their income effects are small. Check: a Giffen good must be inferior and must take a large share of the budget, so that a price change has a big income effect.
Giffen goods are rare, and for a long time no clean example was known. Robert Jensen and Nolan Miller (2008) found Giffen behavior for rice among very poor households in Hunan, China: when a subsidy made their rice cheaper, some households bought less rice and more of other foods. Economists agree that Giffen goods are possible in theory and unusual in practice.