# The Calculus of Prostitution

By Mark Trodden | March 17, 2007 8:11 pm

Here’s a question for our highly mathematically talented readership: what does the following condition describe?

[(Î´U/Î´L) / (Î´U/Î´C) | Sp=0] â‰¤ w – [(Î´U/Î´r) / (Î´U/Î´C) | S = 0]

If you said

“An individual will start to sell prostitution if the price for selling the first amount of prostitution, minus the costs of a worsened reputation for doing so, exceeds the shadow price of leisure evaluated at zero prostitution sold.”

you were spot on. That’s right; according to Marc Abrahams’ Improbable Research column in The Guardian, this is the equation to describe when a prostitute finds it worthwhile to sell (typically) her services.

The story is a little unclear, but the very least you’ll need to make sense of the equation is a definition of the variables:

• U is the “utility”
• L is the amount of leisure you have.
• C is the amount of goods and services you, as a consumer, consume.
• S is the amount of prostitution you, as a prostitute, sell to your customers.
• W is the going price for prostitutes.
• R is a measure of your reputation.

And here I am working on particle cosmology when there are these huge open problems in other fields!

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• hueoblue

I’m declared in a BSc(it’s doubtful whether there should actually be such a thing) Econ program and I wish that someone would make it clear to economists that our use of mathematics outside of statistics is almost always illegitamate. I am pretty sure that the poor use of mathematics and the subsequent hand waving is a manifestation of our latent physics envy.

• Fermi-Walker Public Transport

Agreed, this does look like an illegitame use of mathematics. For a start, it is assumed that the functions can be differentiated. There is no reason for this assumption, let alone whether there may be other relevent variables, differentiable or not.

• John Beamer

I agree that the math isn’t great … but I am not sure it is implying that the variables are differentiable, it is short hand for rate of change, which is a small but subtle difference.l

• hueoblue

Actually, we do assume that it is continuosly differentiable for all positive real numbers, although we treat the numbers as ordinal. All kinds of assumptions are made to ensure differntiability as well as convexity of the function. None of these assumptions are justified by observation and are made because it makes the math manageable. A terrible reason to make an assumption.

• spyder

R is a measure of your reputation.
Discarding this particular, near-impossible, variable measurement, such math assessments might be verifiable if applied to the brothel industries in Nevada and Alaska (Thailand equations, for example, would require determining the original sales values of the girl, into the slavery of prostitution, by her own family). I notice that the proposed equation does not take into account the medical costs (mostly preventive), nor the long-term depreciation of the value of the service, and so forth. The brothel industry is having a resurgence in the US, and as such must be evaluated as one would calibrate other service investments.

Perhaps this equation might be better reconfigured to analyze the economic decisions of those going into the porn business in LA??

• Isaac

Economists use math differently than physicists. Physicists use it to generate quantifiable predictions; economists use it to generate qualitative predictions.

The relevant comparison of the math in economics is not with physics but with the prose in social theory: the math is often a much clearer way to communicate a complex social process than words. Additionally, it forces an internal consistency that social theorists often lack.

Economists make assumptions to make the model tractable so that others can see the core of the idea about how something in the world is supposed to work and not because these assumptions hold in all their strictness. A lot of yeoman’s work is done seeing if the resuts of a model hold when assumptions are relaxed in order to see how robust the idea is.

• Fermi-Walker Public Transport

Also, what are the units for all the variables ? Some have implied units
such as time or money, but what unit does one use for reputation ?
Since none is given, I propose that the unit for reputation be called
the “Franklin”, after the singer Aretha Franklin who sang that great song about respect. Since it is hard to determine whether the unit is MKS or CGS, lets use natural units and set everything equal to one.

• http://www.blumensacha.wordpress.com Sacha

This is a bit harsh on economists! I doubt that the equation is taken to be literally true, but moreso an idealisation.

• http://www.blumensacha.wordpress.com Sacha

It’s probably just about succintly communicating an idea.

• Ike

Most economics is all idealization, zero realization.

Time for a new discipline: physical economics. Necessary first step: getting economists to accept conservation of energy. There’s actually a wiki on this: http://en.wikipedia.org/wiki/Physical_economics

Physical economics addresses physical reality:

1) the “total physical output of the economy”: agricultural and industrial output – T.

2) the “physical cost of sustaining the human population”: the direct and indirect consumption of households and necessary capital investments into housing, educational, and health services – V.

3) the “physical cost of sustaining existing levels of production and consumption into the future”, or cost of maintaining capital investments in machinery, infrastructure, agricultural land, and the natural resource base – C.

Growing prosperous economies are characterized by a sustained rate of increase in the ratio T/(V+C)…In contrast, collapsing economies are characterized by a sustained rate of decrease in the ratio T/(V+C).

In collapsing economies (1990s Russia, for example), prostitution would be more likely to be an act of economic desperation. Physical economics offers better explanations for the real world. Thus, physicists should consider taking over the econ departments… only good would come of it.

• http://www.jyotirmoy.net Jyotirmoy

This seems to be a problem with a lot of economics–what are essentially parables are presented as if they were testable quantitative theories. But this does not mean that the parables themselves are worthless. For example I think this story about prostitution which takes social stigma into account is more interesting that another story (which has also been told in economic journals) of the earnings of female prostitutes being explained by their reduced chances of marriage.

@ike: I don’t know what you mean by making economists accepting the law of the conservation of energy. I’m yet to meet an economist who refuses to believe in that basic physical fact. If you are talking about taking material flows into account, that was the theme of a lot of post-WW2 economics, and even the great von Neumann himself pitched in. Even before that the circulation of goods in a closed economy was studied by the Physiocrats in the eighteenth century and Marx (among others) in the nineteenth.

• http://magicdragon.com Jonathan Vos Post

So, you folks are commenting what might be summarized as:

“this math sucks!”

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• Ike

Well, a typical economic argument is that demand creates supply; for example, since consumer demand for energy and water will always be high, we will never run out of energy and water. This kind of disconnection from physical reality is somthing only an economist could be comfortable with, and it’s probably due to a lack of education in the physical sciences.

Another example is perpetual economic growth as measured by the GDP. GDP isnt’ measured in terms of energy or mass balance; but rather by some artificial ‘index’: consumption + investment + (government spending) + (exports âˆ’ imports) = GDP. Where’s the physical reality? Where’s the predictive utility? This is a mathematical formulation that calls for endless growth in a closed system; it’s arbitrary and idealized and has little if anything to do with reality – which is why economic predictions are of little value.

• dave tweed

If you’re going after areas where economics really has a disconnet with reality, I’d say it’s in the sentence “when a prostitute finds it worthwhile to sell (typically) her services” (which to be fair might be a quick blog phrasing rather than intentional). What that really ought to say (and economists have been slowly cottoning on to) is “when a prostitute ought to find it worthwhile to sell (typically) her services”. Economics in the past often seemed termined to figure out what “perfectly analysing” (obviously wrt a given value system) individuals would do, ignoring the fact that in large swathes of their behaviour people aren’t and are swayed by inbuilt prejudices, etc. It’d be like physicists coming up with ever more precisely stated and proved theorems in classical mechanics, completely ignoring the fact physics (appears) inherently quantum. The best mathematics can’t be useful when your modelling assumptions are wrong.

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• http://predelusional.blogspot.com/ Stephen Uitti

Yes. This is the equation i use. (I’m a consultant).

Heard on a podcast:

BA: I’m just a whore for astronomy.
PG: You get paid for this?
BA: No.
PG: Then you’re not even up to being a whore!

• http://www.blumensacha.wordpress.com Sacha

“Well, a typical economic argument is that demand creates supply; for example, since consumer demand for energy and water will always be high, we will never run out of energy and water.”

I don’t think that any economist could think that it is impossible to run out of energy and water. Rather, they *might* think that, if there is a demand for a good, it is possible that someone will come up with a way to supply it, but of course this is not guaranteed. If there is an incentive for someone to act in a certain way, then they might act in that way. Where is the reference for the idea that “a typical economic argument is that demand creates supply” ?

• http://noteforent.typepad.com Note

The equation also misses another key variable, which would modify the dubious quantity of reputation: how well the person can keep a secret.

• Gypsy

The basic assumption of supply and demand only works if no outside influence is interjected. The demand for the prostitutes services are increased by her actions just like the demand for SUV’s are increased by the marketing done by the car makers. That would effect the “W” in the equation changing the value of U.
Just rambling for fun.

• hueoblue

Accusing economists of not recognizing limited resources is silly, There are many problems in economics but that isn’t one of them. The most obvious limited resource is land. Of course land and natural resource economists recognize that there is only so much land to till, or ore in the mine. It would be absurd not to.

• http://blumensacha.wordpress.com/ Sacha

On the topic of the post, it often seems to be true that people working in one field wonder about the things that people work at in other fields.

• Chris Tunnell

Talk about nit-picky.

1) This is a fitted model
2) Determining a good model to fit involves understanding symmetries of the problem
3) Extrapolating predictions from the model yields something interesting

The thoroughness of a model only becomes important when faced with wrong or boring predictions from the model. Unless you’re in maths, hand-waving is fine until your wrong or boring.

Maybe I’m missing something…

• Bodhidharma

When will people learn that mathematics is just an exact *language*? It is a formal language (for lazy people who like shortcuts ;)) and can be used to express many things – what is expressed doesn’t have to be true. Just because something is formulated in mathematical terms doesn’t make it true!

• mandt

“it is short hand for rate of change,” Absolutely RU/RF=5. Who needs C

• http://hanson.gmu.edu Robin Hanson

Let me get this straight: The mere fact that someone would use math to model a theory of prostitution seems ludicrous to you? And what exactly is your preferred way to express precise theories about prostitution? Is it the very idea of having theories on this topic that offends you? Or the idea of being precise about them?

• http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

Robin Hanson – what on earth makes you think I find the idea ludicrous?

• http://hanson.gmu.edu Robin Hanson

Mark, I took your post as critical from the apparent sarcasm of your last sentence, and since most comments here seems to have interpreted your post as critical, none of which you corrected. If your position is otherwise, feel free to correct us all.

• http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

Robin Hanson – The last line is only sarcastic in the sense that I’m obviously not going to switch fields. I was merely being lightly humorous in suggesting that studying hookers sounds a little more fun to a layperson than studying the fundamental properties of matter.

As a scientist, I generally think that mathematics is the only way to get specific predictions from a theory.

Many commenters seem to have taken the post as ignition to light a waiting fuse of discontent with the way economists and others use mathematics. I’m afraid the blame for this general perception doesn’t lie with me. Many others have taken issue with the details of how this particular equation is constructed. If you disagree with them, feel free to correct them.

• http://hanson.gmu.edu Robin Hanson

Mark, fair enough, thanks for the clarification.

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### Cosmic Variance

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### About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.

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