Metcalfe’s Law for High-Speed Rail, Redux
Americans are in big infrastructure spending mood, and my post from February using Metcalfe’s law to argue in favor of expansive high-speed rail in the eastern half of the United States has been attracting renewed attention. That post looked at how Metcalfe’s law that the value of a network rises in proportion to the square of the number of nodes implied that once a strong HSR corridor existed, for example the Northeast Corridor, extensions would be strong as well even if they connected much smaller cities. People have been asking me to extend that analysis to more lines that do not touch the Northeast Corridor, so here goes.
As a reminder, I’m using a simple gravity model, of the following level of sophistication:

The model is that the annual ridership in millions between two metropolitan areas A and B, with populations in the millions, is,
The theoretical reason for the 0.8 exponents is diseconomies of scale: the average person in Tokyo is farther from Tokyo Station than the average person in a small city is from their respective intercity rail station. Empirically, the best fit exponent for observed data in Japan and Europe is 0.8 – see sources in my previous post and in this post (sourced to since-rotted links) for France. The 500 km minimum is an artifact of the impact of station access time and the option of driving instead of taking the train.
Fares are set at typical Continental European levels rather than Japanese ones. As in the previous post, this means $0.135 per passenger-km, which breaks down as $0.07/p-km in operating expenses including rolling stock but excluding infrastructure and $0.065/p-km in profit, up to a total profit of $50/passenger. Beyond $50 in profit, which normally occurs at 770 km, fares only rise with operating expenses, to be more competitive with airlines. The goal is to find lines that have annual profits of more than 2-3% of construction costs.
A note of caution on the model
There are arguments to be made to refine the gravity model above in either direction. Ridership estimates in Britain are well above what the model predicts. High Speed 2 projects 3 trains per hour between London and Birmingham, running nonstop between the two cities so that no other city pairs can be added. The model gives an annual ridership equal to,
which fills around 1 train per hour in each direction to 50% of seated capacity. It’s possible the model does give higher ridership figures for very close-by cities – London and Birmingham are only 180 km apart – or it’s possible some unknown factor exists. Or HS2’s traffic estimates could be completely off.
In the case of the US, it’s likely any HSR will run faster than legacy 1960s Shinkansen. However, there’s a serious malus coming from higher car ownership, lower car traffic levels, and much weaker city centers. This is unlikely to be a problem for traffic to New York, but the last post dealt with that, whereas today we’re looking mostly at lines that aren’t about New York. Even Chicago is extremely auto-oriented by the standards of London or Paris, let alone Tokyo.
Metcalfe’s law for HSR in the Midwest: the initial line
The Midwest benefits from two things: it is flat, which reduces construction costs to $20-25 million per km if European norms are followed, and it has near-megacity Chicago in the middle. Unfortunately, Chicago is big but not big enough, and while the secondary cities are pretty big, there aren’t additional medium-size cities nearby the way Lyon has Saint-Etienne, Marseille has Toulon, etc. HSR can succeed, but the return on investment is for the most part marginal. The one exception is lines that can leverage the Northeastern network, including eventually not just the Northeast Corridor but also tie-ins to Pittsburgh and Cleveland, both of which are at reasonable HSR distance from New York.
By itself, the core Midwestern network would connect Chicago (10 million people) with Toledo (0.8, a distance of 370 km) and thence split toward Detroit (5 million, 100 km from Toledo) and Cleveland (3 million, 180 km). This leads to the following O&D ridership matrix, in millions:
| City W\City E | Toledo | Detroit | Cleveland |
| Chicago | 1.58 | 6.86 | 3.77 |
| Toledo | — | 0.91 | 0.6 |
| Detroit | — | — | 2.62 |
And in annual operating profits, in millions of dollars:
| City W\City E | Toledo | Detroit | Cleveland |
| Chicago | 38.08 | 209.56 | 134.68 |
| Toledo | — | 5.91 | 7.07 |
| Detroit | — | — | 47.65 |
This is not a lot of ROI. It’s $443 million a year, for a 650 km system, which should cost maybe $15 billion. It’s 3% by itself, which isn’t horrible, but compares poorly with Northeastern lines even though it connects the Midwest’s numbers 1, 2, and 4 metro regions.
In contrast, suppose a Northeastern system preexists, or perhaps is built at the same time, including a Pittsburgh-Cleveland connection. What then? Well, the question is really what the ROI is on connections from west of Cleveland to east of Cleveland. There are four metro areas east of Cleveland on the way to New York: Pittsburgh (2.5 million, 200 km from Cleveland), Harrisburg (0.7, 280 km from Pittsburgh), Philadelphia (7 million, 170 km from Harrisburg), New York (22 million, 140 km from Philadelphia). Washington has 10 million people and is 220 km from Philadelphia, but because a Washington-Philadelphia-Harrisburg route is circuitous, trains can only charge for 220 km, which is $29.70, and then earn the usual rate of $0.135/km farther west up to a maximum of $50 in profit, which is reached 730 km west of Harrisburg, or somewhat west of Toledo. With this in mind, we use the same pair of tables as above for the new city pairs, first ridership and then operating income:
| City W\City E | Pittsburgh | Harrisburg | Philadelphia | New York | Washington |
| Chicago | 1.75 | 0.34 | 1.56 | 3.12 | 1.48 |
| Toledo | 0.52 | 0.11 | 0.43 | 0.79 | 0.36 |
| Detroit | 2.26 | 0.35 | 1.49 | 2.81 | 1.3 |
| City W\City E | Pittsburgh | Harrisburg | Philadelphia | New York | Washington |
| Chicago | 85.36 | 16.77 | 77.94 | 156.23 | 74.03 |
| Toledo | 12.9 | 4.64 | 21.6 | 39.53 | 17.95 |
| Detroit | 70.6 | 17.47 | 74.53 | 140.73 | 64.84 |
The total operating income is $875 million a year, which combines with our internal $443 million to produce an 8.8% ROI. This relies on estimating HSR ridership at hefty distances – New York-Chicago is 1,340 km and around 5 hours, New York-Detroit is 1,070 km and around 4 hours. But we do have ridership estimates for city pairs of that magnitude in both Europe and Japan and they’re fine, except for airline-dominated Tokyo-Fukuoka. If anything, this is more robust than making assumptions on how many people are willing to travel by train between two cities without public transportation like Cleveland and Detroit.
Metcalfe’s law for the Midwest: further lines
Past plans for a Chicago-centered Midwestern HSR network called for four spokes: east toward Cleveland and Detroit, northwest toward Milwaukee and Minneapolis, southeast toward Indianapolis and Cincinnati, and southwest toward St. Louis and perhaps Kansas City. These spokes do pan out financially, but the ROI is not great. Even a line that doesn’t touch Chicago can work, namely HSR between Cleveland, Columbus, and Cincinnati – those three cities are too small and weak-centered to produce internal ridership, but New York-Columbus is in similar shape to New York-Detroit.
Milwaukee (2 million, 140 km from Chicago)
Milwaukee’s metro area touches Chicago’s. HSR between the two cities alone is not worth it, since at this distance, top speed isn’t as relevant as station access time. However, the addition of other cities makes this worthwhile. Since Milwaukee is just on city, we put ridership and operating income in the same table:
| City | Ridership | Operating income |
| Chicago | 3.3 | 29.99 |
| Toledo | 0.42 | 13.92 |
| Detroit | 1.27 | 50.42 |
| Cleveland | 0.66 | 29.62 |
| Pittsburgh | 0.34 | 17.16 |
| Harrisburg | 0.07 | 3.59 |
| Philadelphia | 0.34 | 17.25 |
| New York | 0.71 | 35.34 |
| Washington | 0.33 | 16.3 |
The ROI within the Midwest alone on what should be about $3 billion in construction is around 4% – higher than the bare Chicago-Cleveland/Detroit system. With Northeastern tie-ins, this rises to 7%, if one is confident in second-order but noticeable extra revenue from trains from New York, which would necessarily be a two-seat ride and take almost 6 hours with transfer time.
St. Louis (3 million, 460 km from Chicago) and Kansas City (2.5 million, 400 km from St. Louis)
The Chicago-St. Louis line has received some investments in the last 10 years that the state of Illinois pretends are high-speed rail. Those are expensive – there’s extensive surplus extraction by actors including politicians and the freight railroads – and perform exactly as one should expect trains that are slower than the legacy trains that the TGV replaced 40 years ago. However, this says nothing about whether trains that Europeans and East Asians would recognize as fast could succeed on that corridor. Could they?
A reasonable estimate for the Chicago-St. Louis construction cost is $10 billion; St. Louis-Kansas City would be another $10 billion, perhaps slightly costlier per km because Missouri’s terrain isn’t quite so flat as Illinois’s. Ignoring transfer penalties, we get the following ridership and operating income out of it:
| City N\City SW | St. Louis | Kansas City |
| Cleveland | 0.43 | 0.19 |
| Toledo | 0.22 | 0.09 |
| Detroit | 0.76 | 0.32 |
| Chicago | 4.56 | 1.33 |
| Milwaukee | 0.87 | 0.27 |
| St. Louis | — | 1.5 |
| City N\City SW | St. Louis | Kansas City |
| Cleveland | 21.32 | 9.45 |
| Toledo | 10.97 | 4.32 |
| Detroit | 37.84 | 15.99 |
| Chicago | 136.3 | 66.59 |
| Milwaukee | 34.07 | 13.59 |
| St. Louis | — | 39.1 |
St. Louis generates 6.84 million riders and $240 million in operating profit, which is above our 2% minimum but not by much. Moreover, 6.84 million riders means a train every hour, at which point there are real frequency artifacts for a service that shouldn’t take much longer than an hour and a half to Chicago. So it’s marginal, though still plausible. But if this is plausible, Kansas City isn’t: it generates $150 million. There are small intermediate stop locations like Springfield and Columbia, but they’re too small to make a difference.
The Ohio Hub
The four largest metro areas of Ohio are roughly collinear. Going southwest of Cleveland, one has Columbus (2.5 million, 220 km), then Dayton (1 million, 110 km), and finally Cincinnati (2.3 million, 90 km). Construction costs are likely to be low because of the terrain – only around Cincinnati are there significant hills. 420 km for $10 billion is plausible. What is the ridership, and what is the revenue?
| City E\City W | Columbus | Dayton | Cincinnati |
| New York | 1.81 | 0.71 | 1.18 |
| Philadelphia | 0.98 | 0.37 | 0.6 |
| Washington | 0.83 | 0.1 | 0.55 |
| Harrisburg | 0.24 | 0.09 | 0.14 |
| Pittsburgh | 1.3 | 0.56 | 0.79 |
| Cleveland | 1.5 | 0.72 | 1.41 |
| Columbus | — | 0.62 | 1.22 |
| Dayton | — | — | 0.58 |
| City E\City W | Columbus | Dayton | Cincinnati |
| New York | 90.71 | 35.44 | 59.13 |
| Philadelphia | 48.91 | 18.52 | 30.25 |
| Washington | 39.79 | 4.93 | 27.68 |
| Harrisburg | 10.9 | 4.3 | 6.78 |
| Pittsburgh | 35.48 | 19.14 | 31.87 |
| Cleveland | 21.5 | 15.5 | 38.4 |
| Columbus | — | 4.46 | 15.81 |
| Dayton | — | — | 3.42 |
The total operating income is $563 million a year, which is 5.6% ROI. The biggest cells – New York-Columbus, New York-Cincinnati, Philadelphia-Columbus, Washington-Columbus – are reasonably certain. The internal Midwestern numbers are more suspect, as are the numbers involving Pittsburgh – these are cities where car ownership approaches 100% and the remainder are carless out of poverty, and the destinations are fairly decentralized.
Indianapolis and points south
Indianapolis (2.5 million, 280 km from Chicago) is an attractive-looking target. By itself it’s not much, just like slightly-bigger St. Louis, but unlike St. Louis, it has cities behind it in Cincinnati (170 km) and Louisville (1.5 million, 180 km) that are not as far from everything as Kansas City is. Moreover, Indianapolis-Cincinnati also unlocks travel to Columbus, probably with a transfer because the bluffs around Cincinnati force trains from both Indianapolis and Columbus to enter from the north, without through-service.
South of Louisville, it’s attractive to go south to Nashville (2 million, 270 km from Louisville), Chattanooga (1 million, 200 km), and finally Atlanta (7 million, 180 km). But unlike the New York-Atlanta and the New York-Chicago legs of the triangle, the Chicago-Atlanta leg is decent but not amazing, since it omits the largest city.
| City S\City N | Milwaukee | Chicago | Indianapolis | Louisville | Nashville | Chattanooga |
| Indianapolis | 1.09 | 3.94 | — | — | — | — |
| Cincinnati | 0.73 | 3.69 | 1.22 | — | — | — |
| Dayton | 0.28 | 1.62 | 0.62 | — | — | — |
| Columbus | 0.44 | 2.33 | 1.3 | — | — | — |
| Louisville | 0.5 | 2.62 | 0.86 | — | — | — |
| Nashville | 0.3 | 1.55 | 1.09 | 0.72 | — | — |
| Chattanooga | 0.11 | 0.55 | 0.37 | 0.41 | 0.52 | — |
| Atlanta | 0.4 | 1.82 | 1.07 | 1.16 | 2.48 | 1.42 |
| City S\City N | Milwaukee | Chicago | Indianapolis | Louisville | Nashville | Chattanooga |
| Indianapolis | 29.68 | 71.7 | — | — | — | — |
| Cincinnati | 28.01 | 107.8 | 13.43 | — | — | — |
| Dayton | 12.48 | 56.96 | 10.55 | — | — | — |
| Columbus | 21.77 | 98.49 | 32.1 | — | — | — |
| Louisville | 19.57 | 78.28 | 10.1 | — | — | — |
| Nashville | 15 | 73.36 | 31.8 | 12.68 | — | — |
| Chattanooga | 5.7 | 27.36 | 15.61 | 12.68 | 6.79 | — |
| Atlanta | 19.82 | 91.1 | 53.74 | 49.21 | 61.2 | 16.65 |
Reasonable construction costs are $6 billion to Indianapolis, $4 billion to each of Cincinnati and Louisville, $7 billion to Nashville, $6 billion to Chattanooga, and $5 billion to Atlanta. Indianapolis itself doesn’t generate sufficient ROI, but with the addition of Cincinnati it is pretty strong, the combined system generating $483 million, or 4.8% ROI. Then Louisville generates $108 million, or 2.7%; Nashville generates $133 million, or 1.9%; and Chattanooga and Atlanta together generate $360 million, or 3.3%. Note that the last segment generates the highest ROI, and moreover it is not really possible to start from Atlanta and move north, since Chattanooga alone doesn’t generate significant ridership to cities northeast of Atlanta, as those cities are either small (Greenville, Charlotte) or far (Washington).
Update 12-21: Madison (0.9 million, 120 km from Milwaukee) and Minneapolis (4 million, 400 km from Madison)
The above calculations are for expansions from the first Midwestern core line connecting metro regions #1, 2, and 4 to on another. But what about the #3 region, Minneapolis? Minneapolis has a metro area of 4 million, and is by far the largest Midwestern region with population growth, having grown 9% between 2010 and 2019, whereas Chicago, Detroit, and St. Louis were flat and Cleveland declined.
It should not surprise that Chicago-Minneapolis traffic alone is insufficient to justify HSR, given that Chicago-Detroit alone is not and that line requires service to Cleveland as well as points east. Fortunately, Minneapolis’s location is such that through-service from much of the rest of the Midwest is plausible. Distances are long – this isn’t the Northeast or Western Europe – but trips between Minneapolis and secondary cities like Cleveland, Detroit, Cincinnati, and Indianapolis become much faster by rail than by car. Even St. Louis-Minneapolis is feasible, even though nowadays there’s a mostly direct all-freeway route that’s 900 km long vs. 1,120 by HSR. Midwestern travel today is dominated by the car and not the plane, since car ownership is universal and flying between two secondary cities is not necessarily convenient or cheap.
We get the following matrix of ridership:
| City S\City N | Madison | Minneapolis |
| Cleveland | 0.25 | 0.37 |
| Toledo | 0.15 | 0.18 |
| Detroit | 0.47 | 0.65 |
| Columbus | 0.17 | 0.28 |
| Dayton | 0.11 | 0.16 |
| Cincinnati | 0.27 | 0.36 |
| Louisville | 0.18 | 0.25 |
| Indianapolis | 0.49 | 0.54 |
| St. Louis | 0.32 | 0.44 |
| Chicago | 1.74 | 3.29 |
| Milwaukee | 0.48 | 1.46 |
| Madison | — | 0.84 |
And here is the matrix of operating income:
| City S\City N | Madison | Minneapolis |
| Cleveland | 12.65 | 18.7 |
| Toledo | 5.95 | 8.96 |
| Detroit | 22.24 | 32.26 |
| Columbus | 8.66 | 13.79 |
| Dayton | 5.39 | 7.89 |
| Cincinnati | 12.29 | 17.96 |
| Louisville | 8.61 | 12.53 |
| Indianapolis | 17.27 | 26.78 |
| St. Louis | 14.99 | 21.82 |
| Chicago | 29.4 | 141.28 |
| Milwaukee | 3.74 | 49.48 |
| Madison | — | 21.73 |
A reasonable construction cost for Milwaukee-Minneapolis is around $13 billion. Overall operating income is $514 million a year, so 4% ROI; one can even scratch a few fractions of 1% by including extra ridership from connections from points east of Cleveland, but I’m comfortable rounding New York-Minneapolis ridership over 2,000 km and a probably-untimed transfer in Chicago from 0.67 million to zero. At most, including East Coast-Minneapolis rail ridership provides cushion against unresolved questions such as whether people would take a 4.5-hour train between Detroit and Minneapolis or continue driving for 10 hours plus rest stops.
Metcalfe’s law in California
In California, the definition of a metro area is dicey. The combined statistical area for the Bay Area has 9.7 million people, but that includes Merced, Modesto, and Stockton, all of which are geographically in the Central Valley and would get dedicated HSR stations, some on a different branch from that going toward the Bay proper. In fact, we have 9.7^0.8 = 6.16, but if we sum each individual MSA component and raise its population to the 0.8th power, even omitting ones without planned HSR stations like Santa Cruz and Napa, we get a total of about 7. So we should use the higher figure. Likewise, in Los Angeles, taking the CSA population yields 18.7^0.8 = 10.41 whereas summing the constituent metro areas separately yields 11.3, and summing the counties, all of which are supposed to have stations, yields 12.8. We use the higher figure, 12.
Together we get 25 million intercity riders, before applying the distance penalty. The distance depends on which pair of stations we look at, since we’re summing over many different stations; it also depends on alignment choices, which don’t all have the same average speed, which means that trip time, whence the distance malus, is not perfectly congruent to distance. To simplify, we assume that LA-SF is 2:45, which at Shinkansen speed is 650 km; this is shorter than the actual LA-SF distance under most alignments, though not by much, and it’s longer than actual distances to subsidiary Northern California destinations.
With this in mind, our formula spits out 14.79 million intercity rail trips. This is a lot lower than California HSR estimates. Those estimates also include San Diego (3 million, 190 km from LA), Bakersfield (0.9 million, 180 km), Fresno (1.3 million, 170 km from Bakersfield), and Sacramento (2.6 million, 270 km from Fresno, 230 km from SF). None of these adds a lot, though. The reasons for the discrepancy include,
- California HSR assumed heavy HSR commuter traffic – Palmdale-Los Angeles was one of the top city pairs.
- California HSR assumed somewhat lower fares than the European norm, standing at $79 for LA-SF.
- California was projecting population into the future, and may have assumed less NIMBYism than the state presently has.
- The California HSR model may have had flaws; one such flaw was overestimating the impact of frequency at the LA-SF range, to the point that pruning branches such as to San Jose was said to increase ridership by improving frequency to the remaining destinations.
Not that the numbers coming out of my model are bad. The LA-SF numbers alone are worth $625 million in operating profits a year, and with Bakersfield and Fresno this grows to $875 million. The cost of the project without San Diego and Sacramento tie-ins should be on the order of $25-30 billion, in today’s money. Sacramento is maybe 90 extra km and $2 billion depending on alignment, and generates another $260 million or so; Metcalfe’s law is practically a free gift when you have a 90 km spur in flat geography. San Diego is probably something like $6 billion, the higher cost coming from the constrained urban environment and the need for some viaducts and one short tunnel, and adds around $240 million in operating profits.
I am of course aware that at no point was the cost of California HSR $25 billion in 2020 terms. In 2008 the state promised $33 billion in 2008 dollars. The discrepancy comes from some catastrophically bad decisions regarding scope at every stage of the planning phase and bad procurement. But if one looks at what the project needed rather than what has been built in the Central Valley and plugs in standardized costs, the answer is around $25 billion.







