Deriving the Demand Curve

Where does a demand curve come from? It’s not drawn from thin air — it’s derived from the consumer’s optimization problem. When the price of good $X$ changes (holding income, preferences, and the price of $Y$ constant), the consumer re-optimises and picks a new bundle. Each new optimum becomes one point on the demand curve: a price-quantity pair.

The trick is to trace out what happens as $P_X$ falls step by step. Start with a high $P_X$: the budget line is steep and the consumer buys relatively little $X$. Now lower $P_X$: the budget line rotates outward (pivoting at the $Y$-intercept), and the tangency point shifts. Connect all these tangency points and you get the price-consumption curve (PCC) — the path of optimal bundles as $P_X$ varies. Project the $X$-coordinates of each tangency onto a Price vs. Quantity diagram, and you have the demand curve.

Each step is a separate optimisation problem at a different price. The demand curve isn’t drawn from intuition — it’s the trail of consumer choices left behind as price falls.

This is the micro-foundation of demand. Every point on a demand curve is an optimal choice — it reflects the consumer’s preferences (indifference curves) interacting with their constraint (the budget line). When you see a downward-sloping demand curve, you’re seeing the trace of utility maximization across different prices. This downward slope — the everyday observation that buyers buy less when prices rise — is what economists call the law of demand.