https://pedestrianobservations.com/2020/02/13/metcalfes-law-for-high-speed-rail/ February 13, 2020 7:27 AM (EST)
<aside> ⚠️ NB Call-outs are commentary on the article bigger than would fit in an [in-line comment] Ignore numbering for [citation] marks as they are generally ripped from the relative Wikipedia content and do not correlate here.
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I wrote a Twitter thread about high-speed rail in the United States that I’d like to expand to a full post, because it illustrates a key network design principle. It comes from Metcalfe’s law: the value of a network is proportional to the square of the number of nodes. [ ie $value = nodeValue*nodes^2$ or $v \propto nodes^2$ ] The upshot is that once you start a high-speed rail network, the benefits to extending it in every direction are large even if the subsequent cities connected are not nearly so large as on the initial segment. Conversely, isolated networks from the initial segments are of lower value.
The implication for the United States is that, first of all, it should invest in high-speed rail on the entire Northeast Corridor from Boston to Washington, aiming for 3-3.5 hour end-to-end trip times. And as the Corridor is completed, the priority should be extensions in all directions: south to Atlanta, north to Springfield and (by legacy rail) Portland, west to Pittsburgh and Cleveland, northwest to Upstate New York and Toronto.
To quantify the benefits, I’m going to look purely at railroad finances: construction costs go out, annual profits go in. Intercity high-speed rail pretty much universally turns an operating profit, the question is just how it compares with interest on capital construction. For this, in turn, we need to estimate ridership.
The theoretical model for ridership is called a gravity model: ridership between two cities of populations Pop_A and Pop_B at distance d is proportional to
$$ \frac{Pop_A^{Ga}*Pop_B^{Gb}} {d^2} $$
However, two complications arise. First of all, there are some diseconomies of scale: the trip time from the train station to one’s ultimate destination is likely to be much higher if the city is as huge as Tokyo or New York than if it is smaller. Empirically, this can be resolved by raising the populations of both cities to an exponent slightly less than 1; on the data I have, which is Japanese (east and west of Tokyo), Spanish (Madrid-Barcelona, Madrid-Seville), and French (see post here – all its sources link-rotted), the best exponent looks like 0.8.
$$ \frac{Pop_A^{0.8}*Pop_B^{0.8}} {d^2} $$
And second, at short distance, the gravity model fails for two reasons: first, access time dominates so in-vehicle time is less important, and second, passengers drive more and take fast trains less. In fact, on the data I’m most certain of the quality of – that from Japan – ridership seems insensitive to distance up to and beyond the distance of Tokyo-Osaka, which is 515 km by Shinkansen. Tokyo-Hiroshima, 821 km and 3:55 by Shinkansen, under-performs Tokyo-Osaka by a factor of about 1.6 if the model is
$$ Pop_A^{0.8}*Pop_B^{0.8} $$
if we lump in air with rail traffic; of course, air travel time is incredibly insensitive to distance over this range, so it may not be fair to do so. French data taken about 3 hours out of Paris overperforms the mid-distance Shinkansen, although that’s partly an artifact of lower fares on the TGV.
To square this circle, I’m going to make the following assumption: the model is,
$$ \frac{Pop_A^{0.8}*Pop_B^{0.8}}{min\:\{500 km, d\}^2} $$
If the populations of the two metro areas so connected are in millions then the best constant for the model is 75,000: that is, take out the number the formula spits, multiply by
$$ 500^2 = 250,000 $$
to get rid of the denominator at low d, multiply by 0.3, and make that your annual number of passengers in millions.
Finally, operating costs are set at $0.05/seat-km or $0.07/passenger-km, which is somewhat lower than on the TGV but realistic given how overstaffed and peaky the TGV is. This is inclusive of the capital costs of rolling stock, but not of fixed infrastructure. Fares are set at $0.135/passenger-km, a figure chosen to make New York-Boston and New York-Washington exactly $49 each, but on trips longer than 770 km, the fares rise more slowly so that profit is capped at $50/trip. Of note, Shinkansen fares are about $0.23/p-km on average, so training data on Shinkansen fares for a network that’s supposed to charge lower fares yields conservative ridership estimates; I try to be conservative since my model is not the most reliable.