How Pricing Tolls Right Eliminates Congestion

Chris Bradford over at Austin Contrarian has been making some solid points in favor of congestion pricing. (here, here, here and here)  Chris’s core argument in favor of congestion tolling is that:

congestion pricing does more than relieve congestion.  Congestion pricing tells us when a road needs more capacity.  Additional capacity costs money, and drivers are willing to pay only so much for it.  That “so much” is exactly equal to the price they are willing to pay to avoid congestion.

The idea that toll profits send a signal to road operators to produce additional capacity is often neglected in discussions of the benefits of congestion pricing.  Without pricing, the only signal is the manifestation of congestion itself.  This is problematic, as the only solution is to build more roads when congestion is observed.  Actually if done right, years before congestion occurs with the help of foresight and luck on the part of transportation planners and agencies.  This problem feeds the dangerous new highway –> sprawl –> congestion –> highway expansion –> sprawl, etc., etc. positive feedback loop.  This feedback loop is quite a powerful mechanism that helps drive the unhealthy types of sprawl.

Chris is on the right track, but sets a sub-ideal objective (in my opinion) when he says:

The optimal congestion toll should be set just high enough to achieve free-flow (45 mph) traffic.

Since the goal should not only be to avoid congestion, but to get the highest number of commuters through the system as possible, I would restate that as:

The optimal congestion toll should be set at exactly the price that maximizes traffic flow.

As Chris said, “Congestion pricing is hard.”  Although it seems complicated, you might be shocked at how easy it is, in concept, to price roads optimally.  That’s because it’s somewhat counter-intuitive: flow is maximized if revenue is maximized.  (it’s approximately true, the variance is negligible enough that it’s not significant in practice.)

I’ll say it again, flow is maximized if revenue is maximized.  Don’t believe me?  OK, I’ll have to convince you rationally.

First, and most importantly, you have to understand some basic traffic engineering concepts.  (as a structural engineering student, I always assumed I’d never use anything I learned in the required traffic engineering course…)  The simplest explanation is using the fundamental diagram of traffic flow:


Looking at the third diagram, we see that traffic flow peaks (Qmax) at a particular density of traffic, D (cars/km) before reducing due to congestion.  This makes sense if you consider that in gridlock, despite a high density of cars on the road, not many cars are actually passing though a particular point.  Thus, a toll is optimal if it is priced to achieve maximum possible traffic flow, Qmax and maximum velocity Vc from the second diagram.  When a toll is introduced to a congested road, a certain number of drivers are incentivized away by the toll, which decreases traffic density (D) to a point where those who are left, travel more quickly, than if those drivers had simply added to congestion.

It is much simpler to understand the revenue concept:

Revenue ($/hour) = Toll ($/car) x Traffic Flow Q (car/hour)

Thus, the revenue is (approximately) maximized when maximum traffic flow is achieved.  (I say approximately because, the optimal toll is slightly higher because of the elasticity of toll pricing, but I think there there are more upsides than down of charging a tinsy bit more than for Qmax, even if it turns out to be significant)  One can maximize revenue by carefully selecting the optimal toll for the road at a given time.  We can derive with calculus, but I’ll steer us clear of calculus in this blog unless a reader really, really wants to challenge me on this.

So, we can conclude that traffic flow can only be maximized by carefully pricing the roads.  If the toll road operator charged a toll one dollar more than optimal, we can see that traffic density (D) will move to the left of the dashed line going through Qmax (into the free flow area), traffic flow would be drastically cut, and revenue would be reduced accordingly.  If the toll road operator charged a dollar less than optimal, traffic density (D) will move to the right of the dashed line into the congestion area, which will also reduce flow and revenue.  The trick is to be able to price the tolls correctly and dynamically, while maintaining price predictability to keep commuters loyal.

Next, I plan to discuss why the private sector is better equipped to get this tricky optimal pricing mechanism right.  I hope to follow that up with a discussion of the other economic, social and environmental benefits of congestion pricing, as well as dispelling some of the Urban[ism] Legends surrounding congestion pricing and private roadways.  (as time permits between feedings of the little guy)  Stay tuned…

The Nature of the Living City

Sandy Ikeda posted an abstract for a short essay he is contributing to a Festschrift honoring Jane Jacobs.  He quite eloquently describes the nature of the living city:

A city is not a man-made thing.  Rather, it emerges from the actions of its inhabitants, who interact in unpredictable yet orderly ways.  Under the right conditions – the right “rules of the game” – what arises is vital, creative, radically unpredictable, and profitable:  the living city.

Neither can it be inefficient, because that too presupposes a system-wide plan.  Both efficiency and inefficiency presume that we know how things ought to be, what success and failure look like, and that’s impossible in the urban dynamic.  Instead, borrowing from ecology (and certain heterodox schools of economic thought), we might say that a living city is a “dynamically stable” process, in which the forces of positive and negative feedback, as well as sudden mutation and diversity, combine under the right conditions to generate order through time.  It embodies trial and error, surpluses and shortages, apparently useless duplication, conflict and disappointment, trust and opportunism, and discovery and radical change.  These are in the nature of the living city.

Another piece to look forward to!  Sounds like Sandy touches on some similar themes to Mathieu Helie’s upcoming piece on Emergent Urbanism