Demand

Where a demand curve comes from

A demand curve shows how the quantity a consumer buys depends on price. It falls out of utility maximization. For a Cobb–Douglas consumer (utility \(u = x_1^{a}x_2^{1-a}\)) with income \(m\) and prices \(p_1, p_2\), the demand for good 1 is:

\[x_1 = a\,\frac{m}{p_1}\]

• Downward-sloping: higher \(p_1\) → less \(x_1\) (that’s the demand curve).
• Income shifts it: higher \(m\) → the whole curve shifts out (\(x_1\) rises at every price) — good 1 is a normal good.
• Nice Cobb–Douglas fact: spending on good 1 is \(p_1 x_1 = a\,m\) — a constant share \(a\) of income, no matter the price.
• \(p_2\) doesn’t appear — Cobb–Douglas demand for good 1 is independent of the other price.

Try it. Raise the price to slide up the demand curve; raise income to shift it out. Watch spending stay a constant share of income.

Income m: 100 Taste for good 1, a (share): 0.50 Price of good 1, p₁: 5

What to notice

• Slide price up → you move along the demand curve (quantity falls).
• Slide income up → the whole curve shifts out (normal good).
• Spending p₁·x₁ always equals a·m — the Cobb–Douglas constant-share result — even as price changes.