Staircases, Escalators and Non-Convexities
Published by Michael Greinecker on Sunday, October 08, 2006 at 05:31Steven Landsburg thinks that marginal analysis explains why people stand still on the escalators but not on staircases.
The problem is the following:
Taking a step has a certain cost, in terms of energy expended. That cost is the same whether you're on the stairs or on the escalator. And taking a step has a certain benefit—it gets you one foot closer to where you're going. That benefit is the same whether you're on the stairs or on the escalator. If the costs are the same in each place and the benefits are the same in each place, then the decision to step or not to step should be the same in each place.For Landsburg, the role of marginal analysis was in question. But he's happy again:
Regarding escalators, the solution came in a blinding flash. Marginal analysis does work. It is right to compare the costs and benefits of each individual step. (And thank God it's right; otherwise I'd have to retract everything I've told my students since the day I started teaching.) But before you can weigh costs against benefits, you've got to measure the benefits correctly. And in this case, "getting one foot closer to where you're going" is the wrong way to measure benefit. Who cares how close you are to where you're going? What matters is how long it takes to get there. Benefits should be measured in time, not distance. And a step on the stairs saves you more time than a step on the escalator because—well, because if you stand still on the stairs, you'll never get anywhere. So walking on the stairs makes sense even when walking on the escalator doesn't.I strongly disagree, the correct solution is different. Landsburg and his colleagues actually rejected the correct solution:
It was observed early on that if you stand still on stairs, you'll never get anywhere. But for reasons I can no longer entirely reconstruct, that explanation was dismissed as overly simplistic.Let´s make a very simple model. There are n stairs leading to something giving you utility U. The escalator has a speed of 1/step and our individual too, but with a small cost c with nc>U (it pays to walk up there). It takes n long to walk up the stairs and, with some modification, n-m long to use the escalator, where m are the steps done by our individual in question. Only what is up the stairs gets you utility. Time is of no interest to our individual. Since the escalator brings our individual up with no cost, he/she gets utility U. Walking up the stair gives utility U-cn>0, which is better than keeping standing. What is the worth of the nth step? It is U-c, for there is no other way to get all the way up. Now what is the worth of the (n-1)th step? That depends. It is U-c if the individual takes the nth step afterwards and -c otherwise. So the marginal worth of a step is not well-defined. The marginal analysis might well suggest not to take the (n-1)th step, although that is clearly the only rational thing to do.
The problem should be familiar to everyone knowing some economics. The marginal utility of walking up the stairs is eventually increasing. We are faced with non-convexities where marginal conditions don´t give us sufficient conditions for optimality.
Landsburg should consider teaching that to his students.
4 comments:
Subscribe to:
Post Comments (Atom)
Benefit is extent (costly) step was a necessary element to achieving valued goal. That varies in case of stairs and escalator. If valued result is “being on second floor,” benefit of step on escalator is zero - so stand still there - while each step (assuming ability to make them all) on stairs is obvious. Actually, however, that is not the benefit. Rather, benefit is being on second floor at particular time – and usually, the earlier the better. That is, it has both result and time dimension. There are three variables - benefit of getting to second floor, benefit of getting there sooner (some people run up stairs), and cost of steps (greater if running). People vary. Among those who find being on second floor of some benefit, these variables predict (and reality confirms) that some still do not walk up stairs (effort/time too great); some stand on escalator, and some walk up escalator; (Actually, it is more complicated - take into account running up - or taking the stairs quickly when escalator is clogged so taking steps is impossible.)
That wouldn't really change the problem of increasing marginal benefit.
Hi, just wanted to know whether you were the same michael greinecker that had written an opinion about Paul Krugman's book on amazon...
Posted here because no e-mail adress was sorrily available on your blog
Thanx in advance
Pancho Villa
I once wrote a review of a German translation of "The Accidental Theorist" on the German Amazon site, but there's no Krugman review, among my Amazon.com reviews.