In Memphis, the road to riches is constanly under construction. Recently completed or ongoing projects include a multi-year, $4.5 million widening of Poplar in Germantown, a $3.9 million widening of Germantown Road, and a much anticipated segment of Wolf River Boulevard between Farmington and Kimbrough. Will these roads lead Memphis to a headache-free rush hour? Maybe not. A simple economic model leads to a surprising conclusion: building new capacity may not reduce congestion at all.
The key to the paradox lies in the idea of latent demand: Memphians try to avoid roads that will be congested. If congestion is severe enough, many commuters will switch to a different route or even resort to carpooling. So for every vehicle on a congested road, there are other vehicles that would like to use the road but have avoided it. When congestion is reduced by increased capacity, many commuters switch back to the formerly congested road, offsetting the reduction in congestion. In other words, capacity generates its own demand rather than just increasing space for the current drivers on the road.
Another force driving the paradox is the self-interest of drivers. There is a certainly a personal cost to driving on Germantown Parkway during rush hour: you sit bored in traffic listening to the same three songs on the radio and lose time. But there is also a social cost: you add one more vehicle to the congestion, and cause other drivers to sit bored in traffic listening to the same three songs on the radio and lose time. Drivers base their decisions on their personal cost while ignoring the social cost.
The combination of these two forces results in the Pigou-Knight-Downs Paradox (side note: a more apt model for the Wolf River connection is Braess' Paradox).
To illustrate the paradox, let's simplify the city. Imagine 5,000 workers in Memphis have only two routes home to Germantown or Collierville: Poplar, or Walnut Grove and Wolf River. Assume Poplar is the more direct route, but is narrower and more congested than the less direct but wider Wolf River route. Suppose Poplar takes 20 minutes when there's no traffic but the time it takes rises linearly with the level of congestion (the ratio of the flow of traffic to capacity). Suppose Walnut Grove and Wolf River takes 30 minutes, regardless of traffic. Some notation: define full capacity as the flow of traffic at which the speed falls to half the speed with no traffic. Call the traffic flow on Poplar F_P and the traffic flow on the Wolf River route F_W. Call the capacity of Poplar C_P. Finally, let the time it takes to travel home on Poplar be T_P and the time it takes to travel home on Wolf River T_W.
If the capacity of Poplar, C_P, is at least 10,000, twice the total number of drivers in our model (5,000), then there is no paradox. Poplar is always the quickest route, and no one will take Wolf River. But Poplar doesn't have this capacity. Suppose Poplar's capacity is less than 10,000. Then the drivers will choose the quickest route, which will change depending on the number of drivers that choose each route. The result will be that both routes will always take 30 minutes. Wolf River takes 30 minutes regardless of traffic, so if Poplar will take more than 30 minutes, commuters will switch to Wolf River until traffic on Poplar has fallen to the point where both routes take 30 minutes. If Poplar will take 25 minutes, drivers will switch from Wolf River to Poplar until both routes take 30 minutes.
Given that drivers will change their behavior to equalize the travel time on the two routes, what happens when we widen Poplar? Assume that Poplar currently has just enough capacity to carry all 5,000 commuters, but traffic will be tight. If all 5,000 take Poplar, the drive will take 40 minutes. But if drivers choose the shortest route, 2,500 will take Poplar and 2,500 Wolf River. Both routes will take 30 minutes. Suppose a widening project increases the capacity of Poplar to 7,000. Here's the paradox: Poplar will still take 30 minutes! If drivers choose the quickest route, 3,500 will take Poplar and 1,500 will take Wolf River. Capacity generates demand. Only if Poplar's capacity is more than doubled to over 10,000 will travel time on Poplar decrease.
Can increases in capacity reduce congestion? When drivers minimize only private cost (their own driving time), capacity does not reduce congestion. If by law or agreement, the drivers acted to minimize total social cost (the sum of the time it takes everyone to get to their destination), extra capacity does reduce congestion.
Our model suggests that there are more efficient ways to deal with congestion, such as congestion pricing that forces drivers to pay for the negative externalities they generate.
Is road work wasting taxpayer dollars? Our model has some problems: the two routes don't lead to exactly the same place, for instance, and travel time on Wolf River wouldn't remain constant at all levels of traffic. Therefore, it's best to take these results with a grain of salt. But regardless of any model predictions, the distance from Cameron Brown Park to Taco Bell will be cut roughly in half. So I say it's money well spent.
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