Price Consumption Curve and Derivation of Demand

Core Idea

The price consumption curve (or price expansion path) shows how a consumer's optimal bundle changes as the price of one good varies, holding income and the other good's price constant. By plotting the optimal quantity at each price, we derive the demand curve. This shows that demand curves come from utility maximization under constraints.

How It's Best Learned

Rotate the budget line by changing the price of one good; find new optimum each time; plot the resulting price-quantity points to see the demand curve emerge.

Explainer

You already know two things: a consumer's budget line shows all affordable combinations of two goods, and an indifference curve shows all combinations that provide equal utility. The consumer's optimum is where the budget line is tangent to the highest reachable indifference curve. The price consumption curve (PCC) is what you get when you ask: if the price of one good changes, how does this optimal bundle change?

Start with a concrete setup: a consumer choosing between coffee (good X) and other goods (good Y), with fixed income. Lower the price of coffee. The budget line *rotates outward* on the coffee axis—coffee is cheaper, so the endpoint on the X-axis moves right while the maximum Y stays fixed. This rotation produces a new budget line tangent to a new, higher indifference curve at a new optimal bundle. Reduce the price again, find the new optimum, and again. Each optimal bundle is a point in (X, Y) space. Connecting all these points traces the price consumption curve—the path of optimal choices as the price of coffee varies continuously.

Deriving the demand curve is a direct translation. Each point on the PCC tells you the price of coffee and the optimal quantity of coffee at that price. Plot these (price, quantity) pairs on a separate graph with price on the vertical axis and quantity on the horizontal. The resulting curve *is* the demand curve. This derivation matters because it reveals that demand curves are not arbitrary—they are the observable consequence of utility maximization under a budget constraint. The shape of the demand curve reflects the shape of the underlying indifference curves.

The PCC also reveals how preference structure drives demand elasticity. If indifference curves are L-shaped (perfect complements, like left and right shoes), the consumer always buys the goods in fixed proportions regardless of price—the demand curve is inelastic. If the consumer readily substitutes coffee for other goods when its price rises (indifference curves with high curvature), the demand curve will be more elastic. The connection between the geometry of preferences and the slope of the demand curve is one of the deeper insights this construction provides.