The Bridge

The engineer was Jules Dupuit, a Frenchman occupied in the eighteen-forties with the thoroughly practical business of public works: bridges, roads, canals, the infrastructure a state must decide whether and where to build. The decision turned on a comparison of the cost of the work against its benefit. A bridge required stone, timber, land, labour, and the continuing expense of keeping a crossing safe; its accounts could identify those demands on resources. The benefit was harder to enter. How does one put a number on the value a bridge confers upon the people who use it? A merchant who would have given ten francs to get his goods across, and is charged a toll of two, carries away eight francs of benefit that the toll-box never records. A traveller crossing free may gain just as surely. The preceding plate drew the surplus as an area. Dupuit's question gives that area a public purpose: to decide whether an undertaking is worth the resources it consumes, including the worth that its proprietor never collects.

The amount paid for a service cannot by itself measure the advantage of having it. Dupuit begins his 1844 inquiry by challenging a calculation that valued roads through the expenditure on using them. Suppose an improvement lets the same goods be carried at half the former expense. The recorded outlay falls, although the community can now obtain the service with resources left for something else. To infer that the roads have become half as useful would reverse the meaning of the improvement. Nor does a toll tell us all that a crossing is worth: it establishes that those who pay are willing to give at least that much, leaving the larger valuations unmeasured. Dupuit's method is to consider what would happen as the charge rose, and to count the journeys abandoned at each step. A person who would cross at two francs but cease at three puts the incremental advantage somewhere within that interval. Add the valuations of the journeys, with their alternatives held in view, and the account begins to include the benefit that a single posted charge leaves behind.¹

A small numerical bridge makes the distinction visible. Suppose that, over a given period, one hundred crossings would be made at a zero toll, and that each additional pound of toll reduces crossings by ten, until a toll of ten stops them altogether. Assume that each crossing uses no additional resources, creates no congestion, and confers no benefit or harm on outsiders; the bridge's fixed cost will be counted separately. At zero toll the crossings have a combined willingness to pay of five hundred pounds: the area of a triangle one hundred crossings wide and ten pounds high. A toll of four leaves sixty crossings. The collector receives two hundred and forty pounds, and those who still cross retain a surplus of one hundred and eighty. Together these are four hundred and twenty pounds. The remaining eighty belongs to nobody. It is the value of the forty journeys deterred by the toll, journeys worth something to their users and costing nothing further to accommodate. The cash account records two hundred and forty; the benefit account must distinguish what users keep, what the collector receives, and what the charge prevents from arising.²

A bridge can pass the benefit test and fail the toll-box test. In this example the largest revenue from a single uniform toll is two hundred and fifty pounds, obtained by charging five and admitting fifty crossings. Suppose the construction and upkeep have an equivalent cost of three hundred pounds over the period being considered. Free use yields five hundred of benefit against three hundred of cost, yet no uniform toll can cover that cost. The undertaking therefore presents a financing problem even though its use could justify its resources. A grant or tax does not make the construction free; it moves the obligation to people whose money and alternatives must also be counted. The cost of collection, the changes in behaviour induced by financing, and the distribution of the burden can alter the decision. Nor does zero toll remain the right charge when another crossing wears out the bridge or delays other users. Dupuit himself declines to infer from the lost journeys that every toll should disappear. The worth of building, the terms of use, and the means of payment are connected decisions, each requiring its own account.³

The comparison also has to begin from the journey people would otherwise make. A new canal may take trade from several roads, open a cheaper quarry, or offer a slower but less expensive route. Its benefit cannot be found by pricing every ton as though it would otherwise have travelled the whole canal distance by the most costly imaginable road. Some goods would have come from elsewhere; some would never have been carried. Suppose a new route reduces the full cost of a journey from six pounds to two, and journeys rise from twenty to sixty. The original twenty gain four pounds each. The forty newly undertaken journeys do not all gain four: their willingness to pay lay between the old and new costs, which is why they were previously forgone. If demand is linear over that interval, their average gain is two pounds. The total gain is therefore one hundred and sixty, rather than the two hundred and forty obtained by multiplying every new journey by the full saving. Time, reliability, and damage in transit belong in the comparison too. Counting passages is a beginning; weighing what has improved for each passage is the work.⁴

The measure remains a measure in money, and the purse accompanies the preference into every valuation. A wealthy traveller can offer more for a small convenience than a poor traveller can offer for an urgent need. Dupuit recognizes the distinction: willingness to sacrifice money is not a direct instrument for comparing the intensity of different people's wants. Adding pounds makes a money account, while the distribution of those pounds requires further judgment. Even for one person, a sufficiently large change in circumstances can change the demand being used to value the improvement. The estimates must therefore say whose opportunities and resources they describe. They must also acknowledge that a whole schedule of possible use cannot be read from one observed toll and one count of crossings. Dupuit's answer to imperfect information is to seek useful limits: the benefit need not be known to the last franc if a defensible range lies wholly above or below the cost. Where the range straddles that cost, the uncertainty is part of the decision, not something that the precision of a drawn triangle has removed.⁵

Dupuit's bridge carries the margin back to the whole. The willingness to pay for one further crossing becomes, when the crossings are added, an estimate of the service the work supplies; the toll is then one means of dividing and financing that service. A busy bridge with modest receipts may confer a large benefit. A handsome account of receipts may conceal an unnecessarily severe exclusion. Neither observation dispenses with the cost of construction or proves that this is the best undertaking available. The same labour, materials, and public money could have served another purpose, and that purpose has to stand beside the visible bridge in the reckoning. The engineer has taught us to count the advantage that the toll-box misses. The next plate asks us to follow what an expenditure displaces, including the work that will not be built and the benefit that will consequently remain unseen.