We now demonstrate that if a good rises in price, the quantity demanded of it is likely to fall. A geometric analysis of maximising utility subject to a budget constraint is used. Begin with a situation in which a consumer is currently in a utility maximising position, at the point indicated by “a” in Figure 3.10. Money income is m, prices are P_y1 and P_x1, and maximised utility is U₁.

Now suppose that the price of good x falls from P_x1 to P_x2, but money income and the price of good y remain unchanged. This rotates the individual’s budget constraint anti-clockwise; its position changes from the line connecting the points (m/ P_y1) and (m/ P_x1) to the line (m/ P_y1) to (m/ P_x2). The set of feasible consumption choices is now enlarged, and the consumer attains a new utility maximum at the point “b” on the indifference curve U₂.

What has happened to the quantity of good x that the consumer demands as the price of good x falls? It has increased, from x₁ to x₂ . The individual’s demand curve for x is, therefore, negatively sloped (at least in this region) as a lower price has resulted in a greater quantity demanded.