Consumer Surplus (Linear Demand)

Jules Dupuit, a French civil engineer, invented this in 1844 while trying to decide whether a bridge was worth building. His problem was concrete: the tolls collected obviously understate what a bridge is worth, because most people crossing would have paid more than the toll. The difference — summed over everybody — is the benefit the toll receipts miss, and it is what a cost-benefit study of a public work needs. Marshall took the idea up in 1890, gave it the name, and it has been in every textbook since.

The construction is simple. A demand curve, read horizontally, is a queue of buyers sorted by how much they would pay. The first buyer would have paid the choke price; the last would barely pay the market price. Everyone pays the same market price, so everyone except the last one gets something for nothing. Add up all those somethings and you get the area between the demand curve and the price — a triangle if the demand curve is straight, which is where $\tfrac{1}{2}(P_{\max} - P)Q$ comes from.

Real demand curves are not straight, and this is not a caveat — it is the main fact about this page. The triangle is a teaching device with a number attached to it. Actual demand curves are typically convex, bending away from the origin, which means the true area is LARGER than the triangle near the top and the straight-line estimate understates the surplus. Worse, the choke price is usually pure extrapolation: you have observations in the range where the market actually trades, and the intercept where quantity would fall to zero is a guess made by extending a line through territory nobody has measured. Since the whole triangle depends on that intercept, an estimate of consumer surplus is often more sensitive to the extrapolation than to any of the data.

Three further assumptions ride along quietly. The marginal utility of money is constant across buyers, so that a dollar of surplus to a rich buyer counts the same as a dollar to a poor one. This is what makes the areas addable, and it is a value judgement dressed as arithmetic. There is no income effect, which is why the honest theoretical objects are the compensating and equivalent variations rather than this triangle; for small changes the three nearly coincide, and for large ones they do not. And willingness to pay measures welfare, which quietly makes what people can afford part of what things are worth to them.

None of that makes the number useless. It makes it a well-defined number under stated assumptions, which is a different and more modest thing. Every cost-benefit analysis of a road, a vaccine programme or a spectrum auction rests on some version of this triangle, and the honest ones report how much the answer moves when the assumptions change. Where the calculation earns its keep is in comparing two policies under the same assumptions, where the shared errors partly cancel — not in stating an absolute value for what something is worth to society.