# Escalating Confusion

Mises Daily: Wednesday, September 04, 2002 by

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In a recent article for Slate, Steven Landsburg explains that he and his colleagues at the University of Rochester teach in one of the top-ranked economics departments in the world. He then makes the amusing admission that "somehow last summer, we managed to spend a week in a state of collective befuddlement, obsessing over a seemingly impenetrable conundrum that came up over lunch: If people stand still on escalators, then why don't they stand still on stairs?" For the average reader who might find such confusion hard to believe, Landsburg offers the following explanation:

"For those of us who were too dense to see what all the fuss was about, one of our colleagues spelled out the paradox: 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.
"In other words, a step either is or is not worth the effort, and whatever calculation tells you to walk (or not) on the escalator should tell you to do exactly the same thing on the stairs."

Something was obviously wrong here. After entertaining various theories of why marginal analysis is inapplicable to the staircase versus the escalator situation, Landsburg and his colleagues realized they had been incorrectly calculating the marginal benefit of an additional step:

"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.…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."

Sorry, Try Again

Unfortunately, Landsburg’s resolution of the "paradox" isn’t correct.[1] A marginal step on a staircase saves just as much time as a marginal step on an escalator. When Landsburg appeals to the obvious fact that "if you stand still on the stairs, you’ll never get anywhere," he has dropped the marginal framework without realizing it. In other words, he has solved the problem as the layman would.

Austrian economists often chastise their mainstream colleagues for overlooking the important role of time in human action. However, despite Landsburg’s confession to the contrary, the confusion over staircases was not due to a focus on distance rather than time. To see this, we can solve the paradox on entirely neoclassical grounds (complete with cardinal utility), while still measuring benefits as a function of distance:

Following the mainstream technique for recursive problems, we can define a value function, indexed by the step on which a person finds himself. Thus V,s(k) gives us the utility of being on the kth step on the staircase, while V,e(k) gives us the utility of being on the kth step on the escalator.  Suppose that k=10, and that the utility of reaching the top (i.e. the destination) is 100. Finally, suppose the disutility of taking a marginal step is 10.

Within this framework, we can compute the value of being on any given step of the staircase by reasoning backward. We know the value of reaching the top is 100; i.e., we know V,s(10) = 100. We compute V,s(9) by figuring out what the optimal action would be for a person finding himself on the ninth step. Such a person could do nothing, or he could take a step (incurring disutility of 10) and reach the top (incurring utility of 100).  Thus V,s(9) = 90.  And so on.

Turning to the escalator, the situation is different. Reaching the top is still worth 100, so V,e(10) = 100.  But a person on the ninth step of the escalator doesn't need to incur disutility in order to reach the top; his optimal action is to refrain from stepping. Because he will still achieve a utility of 100 without expending any resources, V,e(9) = 100. The same reasoning shows V,e(8) = V,e(7) = … = V,e(1) = 100.

With these value functions so defined, the marginal analysis becomes trivial: Taking a step on the staircase brings a person one step closer to his goal; that is, he moves from step k to step k+1. This has a marginal benefit of 10. The marginal cost is also 10, and so the person would be indifferent; i.e., the person would be willing to take the step. But on the escalator, things are different. Because the value of being on each step is the same, moving from step k to step k+1 confers no marginal benefit, and so it is not worth the marginal cost to take a step.

An Austrian Example

Obviously, the above reasoning is not meant to persuade typical readers of Mises.org.[2] My point is that standard neoclassical tools are perfectly adequate to solve the problem,[3] even when we gauge benefits as a function of distance. The real error contained in the initial description of the "paradox" was not measuring benefits in terms of distance; it was the assumption that being one step closer "is the same whether you’re on the stairs or on the escalator."  (This will be clearer in the example below.)

Austrians work in the subjectivist tradition of Carl Menger. They know that utility is determined in the minds of acting individuals when they comprehend the world. Costs and benefits do not reside in material objects. Consequently, just because two objects may share similar characteristics does not mean different individuals will value the objects in the same way. The unique circumstances of each person affect the way he or she values a given object. Furthermore, Menger explained, objects may be valued, not because they are directly useful in themselves, but because they represent to an individual the means to attain a valued end.

For example, suppose that Joe and Sally each have a new pool. Before they can go swimming, they obviously need to fill their pools with water. Sally is fortunate enough to have a hose that she turns on and drapes over the edge of her pool. Joe, on the other hand, has no hose, and so must use a bucket to fill his pool.

Now, following Landsburg's approach, we might wonder why Joe keeps going back and forth from faucet to pool, carting heavy bucketfuls of water, while Sally does not do so. After all, the costs are roughly the same (we assume Sally is quite masculine), and in both cases a trip to the faucet will add one bucketful of water to the pool.

The answer of course is that a pool with X gallons of water in it is a different good to Joe than it is to Sally.  For Sally, a pool filled halfway with water has the same value as a pool full of water, since the former is a means to the latter.  (This is no different from observing that a winning lotto ticket assumes the subjective value of the prize to which its bearer is entitled,[4] despite the fact that people can use money but not lotto tickets to buy things.)

Of course, differences due to timing may affect the valuations; if Sally has guests coming over for a party, a pool filled halfway may not be the same as a full one.  But this is true in the staircase example too; if someone is rushing to catch a plane, he might run up an escalator.  Nevertheless the general conclusion still follows, that the valuation of water depth in a swimming pool depends on whether or not the pool is filling up without the application of labor.  Likewise, the valuation of height on an escalator depends on whether or not the escalator is moving.

Steven Landsburg and his mainstream colleagues were tripped up by calculating marginal benefits in terms of distance, and then they erroneously thought the solution was to calculate it in terms of time. But there is nothing intrinsically valuable about height or time (or depth) as such; what is important is the entire situation of the actor, and the significance he attributes to these objective quantities. Had he kept this fact in mind, Landsburg would’ve easily understood why people walk up stairs but often stand still on escalators.

Robert P. Murphy, a Rowley Fellow of the Mises Institute, is an economics graduate student at New York University. See his Mises.org Articles Archive and send him MAIL.

[1]  I emailed this solution to Landsburg to see if he agreed, but he (uncharacteristically) never replied.  We can only hope that he is on vacation, and not stuck at the bottom of some staircase.

[2]  Sam Bostaph has pointed out that a true explanation would need the ability to handle cases such as these: "On the stairs, sometimes I stand still (is that another angina pain, or just heartburn from that last taco?), sometimes I run (My God, why is my three-year old screaming upstairs?), and sometimes I walk quite deliberately (I'll be damned if I'll give her the satisfaction of seeing how mad I am!)."

[3]  Roger Garrison has likewise used indifference curve analysis to illustrate the higher benefits of taking a step on a staircase. Both of us were surprised by Landsburg’s confusion.

[4]  Strictly speaking, the value assigned to a means can't equal the full value assigned to its end; otherwise no one would ever cash in winning lotto tickets. Guido Hülsmann has related this fact to originary interest, but such concerns lie outside the scope of this article.