Demand, supply and equilibrium
A demand function p = D(x) gives the price consumers are willing to pay per unit when x units are demanded; it usually falls as x rises. A supply function p = S(x) gives the price at which producers are willing to supply x units; it usually rises as x rises. The market equilibrium (x₀, p₀) is where D(x₀) = S(x₀) = p₀.
Consumer surplus (CS)
Many consumers would have paid more than the market price p₀. The total of these savings is the consumer surplus: CS = ∫₀^x₀ D(x) dx − p₀x₀. On a graph it is the area below the demand curve and above the price line p = p₀, from x = 0 to x = x₀.
Producer surplus (PS)
Many producers would have accepted less than p₀. Their total gain is the producer surplus: PS = p₀x₀ − ∫₀^x₀ S(x) dx. On a graph it is the area above the supply curve and below the price line, from 0 to x₀.
Examples
- Demand p = 20 − 2x at x₀ = 5, so p₀ = 10. CS = ∫₀⁵ (20 − 2x) dx − 50 = 75 − 50 = 25.
- Supply p = 2 + x at x₀ = 4, so p₀ = 6. PS = 24 − ∫₀⁴ (2 + x) dx = 24 − 16 = 8.
- Demand p = 36 − x² and supply p = 4 + x². Equilibrium: 36 − x² = 4 + x² gives x₀ = 4, p₀ = 20. CS = ∫₀⁴ (36 − x²) dx − 80 = 144 − 64/3 − 80 = 128/3. PS = 80 − ∫₀⁴ (4 + x²) dx = 80 − 16 − 64/3 = 128/3.
For a linear demand curve, CS is the area of a triangle, which gives a quick check: in the first example, ½ × 5 × (20 − 10) = 25.
Key points
- CS: area between demand curve and price line.
- PS: area between price line and supply curve.
- Find the equilibrium first when both curves are given.
- p₀x₀ is the total amount actually spent by consumers (and received by producers).
Common mistakes
- Subtracting in the wrong order, giving a negative surplus.
- Using the supply curve to compute consumer surplus or vice versa.
- Forgetting to find x₀ from the equilibrium before integrating.