Random Finds in Heterodox Economics, #2

If you don’t like dd’s latest post, wait a minute.

I checked out the FAQ on labor theory of value, and it didn’t answer my question. It seems that you think it is the correct theory, so maybe you can help me.

Three workers, A, B, and C. They can work for three years and produce something that is worth $315. Here is how it works.

– For 1 year, A works on the product,
– then for 1 year, B works on the product,
– then for 1 year, C works on the product and finishes it.

The workers sell the product for $315, then try to figure out how to split the money.

First idea: split equally, $105 each. Each person spent one year, so each deserves equal shares.

This idea is silly; anyone who doesn’t think so, give me $1 million and I’ll pay you $1 million plus one extra dollar thirty years from now. You should be happy! Of course you wouldn’t be, since the first idea ignores the time value of money, and time preference generally.

Second idea: A gets $110, B gets $105, C gets $100. This is because A, while he did work for a year just like the others, had to wait two years for his payoff. B only had to wait one year, and C didn’t have to wiat at all. Therefore, A should get more (and assuming a simple 5% discount rate, we get the above).

The second idea makes sense to me at least. However, it seems to make a mockery of the labor theory.

A capitalist could make profit by simply paying A and B $100 each at the end of their year of work, so they didn’t have to wait until the end of the production for their money. Then A gets $100 after one year, B gets $100 after two years, and C gets $100 after three years, and the capitalist gets $15 after three years. This is equivalent (to the workers) to the second idea above.

But wait! The capitalist did no labor to get his $15, and therefore is exploiting the workers, right?

Where have I gone wrong? Is the labor theory of value contradicted by the time value of money? If so, I’m pretty sure I know which is “provably wrong.” :-)

I think I fast-forwarded through the ending of the labor theory comment: the capitalist actually has the following cash flow:

-$100 in year 1
-$100 in year 2
+$215 in year 3

Not just +$15 in year 3.

Does this still count as exploitation, since he has earned something without laboring?

“just that economics needs a value theory of some sort and at the moment it’s the only game in town”

You mean the ‘objective’ type, don’t you? And why?

must say I’m underwhelmed by the counterexamples that form the basis of the Cambridge (UK) side of the controversy. Do they show anything more than the fact that results of any kind derived for the case N=2 rarely hold exactly for general N?

Well yes John mate. They show that aggregative measures of capital can’t be found independently of an assumption about the rate of profit, and thus can’t be used to support marginal product theories of profit. That is pretty huge, innit?

I mean seriously. If there is no non-question-begging way to define inputs of capital, then you can’t use a Cobb-Douglas production function, all of Solow is useless and you are pushed in the direction of a theory of the wage rate which exploitation figures much more highly than marginal product.

This is one point on which I disagree with Vienneau; I think that the nonaggregative nature of capital is the whole point of the CCC, although this is because I’m not a Sraffian.

Speaking of transformation and while we wait for Daniel to give us an explanation, here’s a joke (not a great one, but one):

http://eh.net/lists/archives/hes/jul-2001/0019.php

Y1 -100 = Y3 -110
Y2 -100 = Y3 -105
Y3 +215 = Y3 +215

I see no exploitation, but then, I don’t see any profit, either.

“I see no exploitation, but then, I don’t see any profit, either.”

I clearly don’t understand LTV, and need to do more research if I want to understand it.

I thought that it (and its followers) would object to a capitalist making 5% return on investment without exerting any labor. The fact that that plan would be equivalent to a fair plan for the workers to split the payout confused me.

Any reason why I should reconsider Daniel?

Not if you care about exploitation, yes if you care about theories of value. Basically, without something like a value theory, you can’t come up with something as basic as a noncircular definition of the capital stock, which I (and few others) regard as a bit of a theoretical problem.

The mathematical sense. It’s now settled ground that the approach taken by the Cambridge (US) side – that one could measure the capital stock taken as an aggregate by using the dollar prices of capital equipment – is not satisfactory given that it means that anything you then go on to say about the rate of profit is a circularity (because the price of capital equipment depends on the rate of profit). There is some room for dispute about how important this is (see above) but the central point is purely mathematical.

Comments on “All I have to say has already crossed your mind,” especially Argument II

I haven’t been able to get the Morgenstern, Clayton, or Diaconis papers, so perhaps they explain some of the following.

“The essence of the result we derive below has been expressed by Clayton 1986 (p. 38). Imagine

“A and B are witness to some coin tossing. Bayesian A is firmly committed to the belief that all coins are fair, and so uses d1/2 for q, the probability of heads. B is firmly committed to the belief that coins are never fair, and uses a uniform prior on [0,1/3]U[2/3,1]. Both A and B will use Bayes theorem to coherently update their priors as they see data, but they will never agree, nor should they”

Probabilities should represent information, not mere belief. Before I start calculating, I would want to know what information A has that committed A to A’s belief, and the same for B. Note that all it takes to destroy the result is the minimal claim “I might be wrong,” putting a small density on all points between 0 and 1.

It doesn’t seem that this is the essence of Rosser’s argument II. Let me try to summarize it.

You are a single decision-maker, trying to decide what to do. The problem is, there is another person who is trying to outthink you; your opponent will anticipate your action, or anticipate your anticipation of his anticipation of your action, etc. Let n be the number of your opponent’s anticipations (i.e. the two examples in the previous sentence are n=1 and n=2).

Now, if you knew n for sure, then you could choose the optimal decision (which in this case is a particular mixed strategy). However, as n gets large, the optimal decision might not converge, but instead oscillate between two choices.

Rosser then (erroneously, I think) concludes that there is no solution to the problem, so all such decisions must be made, not by careful rational analysis, but by animal spirits.

Two problems I see with this argument.

First, the solution might converge; in his example it doesn’t, but in others it does. Then, some decisions could be made using rational analysis. The example given involved the words “Cauchy” and “posterior mean” which set off alarm bells in my head (Cauchy distributions are famous for not having a mean, so often it can cause hidden problems when you try to use means as summary measures). It also uses continuous, infinite-ranged probability distributions, which often cause problems when people don’t remember that they are just limits of discrete, finite-ranged distributions (this is quite common in the probability literature, and causes all sorts of nonsense to be published; people forget that infinity isn’t a real number, either for engineers or mathematicians. I’m not sure if Diaconis or Rosser are particularly susceptible to this sort of thing).

Second, if the mathematical limit does not converge, you can’t make any other conclusion other that “my model (or the problem) is ill-posed.”

The model above says “We don’t know much about n, other than it could get large. Let’s not think about it too much, and just see what happens when n gets really large. Maybe it will give us a sensible answer.” When the limit doesn’t converge, you should then refine your model; the divergence tells you, “Look, this particular “n gets large” shortcut, although it works a lot of the time, isn’t open to you in this case. Go back and do some more work.”

In this case, the “more work” that is needed is a distribution on n. After all, no one actually anticipates infinitely often. So figure out what you know about your opponent’s anticipation strategy. My friends often win games this way (“Bill will most likely think either three or four moves ahead”).

Perhaps we could put a uniform distribtuion on all possible n from 1 to N, then see what happens when N goes to infinity. In the example in the paper, where for all even n the posterior mean was y and for all odd n the posterior mean was -y, such a limit would give a posterior mean of zero (which, it seems, is the solution Rosser wants it to give).

So, instead of non-convergence = no answer, non-convergence = bad model that needs to be fixed. Here, instead of assuming n is large, actually model what you know about it.

Now there may be a little problem with my idea. This strategy I’ve suggested might just be a subset of the strategies already considered. In other words, adding this “put a distribution on n” step in the argument might lead to a whole other anticipation game, as he anticipate what n-distribution I am usign. I don’t think this is the case; here, the n-distribution describes my information, whereas before, the mixed distribution described my actions. However, the information about n depends on what he thinks about my distribution on n, so there still may be a problem.

It’d be good to be clear on simple things, like time value of money, before worrying about things like labor theory of value, or cauchy distributions*.

In the example given, as far as I can tell, the capitalist is simply breaking even; we’ll chain the other direction this time:

Y1 -100 = Y1 -100
Y2 -100 = Y1 -95
Y3 +215 = Y1 +195

* a) it doesn’t take a cauchy distribution for means to not be very useful. The mean is simply the first coefficient in an approximation, and trying to treat highly skewed distributions with only the mean is somewhat like approximating addresses with only latitude.
b) the cauchy distribution is well known to those of us who have ever worked someplace where “Dilbert” is popular: theoretically, projects are supposed to behave like they have expected completion dates, and naively one might expect that the longer one has gone without observing completion, the more likely completion will be in the near future; practically, as Schroeder points out, the longer many projects have gone on without success, the farther in the future their expected success will be. CS people know about Real Soon Now, and folk wisdom knows about Mañana.

Dave, thanks for responding.

“It’d be good to be clear on simple things, like time value of money, before worrying about things like labor theory of value, or cauchy distributions”

That’s fair. Let’s see now.

“In the example given, as far as I can tell, the capitalist is simply breaking even; we’ll chain the other direction this time:”

Yes, if the capitalist has a 5% time preference as well (like the workers), then he breaks even. If he has a smaller time preference, he makes a profit (is it commone for capitalists to have smaller or larger discout rates than workers? Workers need the money, but capitalists have more investment opportunities?) However, that was not my point.

My point was that he made _any monetary return at all_ on his original investment at all without putting in any labor.

I was confused by the fact that the capitalist could make a 5% return on investment without putting in any socially useful labor. I thought that adherents of the labor theory of value (LTV) would object to that. I then showed that it was equivalent to another distribution only to the workers that a LTV adherent would, I thought, accept.

D^2 then, I believe, put me right when he told me that the LTV doesn’t take a position on distribution, but I wanted to hear what other people had to say as well.

So, would you object to the capitalist making 5% on his investment without putting any labor into it?

“it doesn’t take a cauchy distribution for means to not be very useful”

True; I only meant that Cauchy distributions _frequently_ give strange results when dealing with means and variances, rather than some other summary measures. It is usually the catch-all counterexample to anything involving means and variances (e.g. law of large numbers, minimum variance unbiased estimators, etc.).

“theoretically, projects are supposed to behave like they have expected completion dates, and naively one might expect that the longer one has gone without observing completion, the more likely completion will be in the near future”

I’m not sure this is true, even theoretically. For example, if a project will be completed in an exponential amount of time, then the expected time (and the distribution of the time) until completion never changes (Dilbert would be proud :-)

This is a standard fallacy when means are involved.

“the longer many projects have gone on without success, the farther in the future their expected success will be”

Again, I don’t think this is poses any theoretical problem; I teach this in first-quarter probability. Many lifetime distributions have similar qualities. For example, when I buy a computer game, the expected length of time until it breaks down starts out at X, say. Once I actually get it installed and start playing, the expected time until it breaks down increases dramatically; it “wears-in.” Isn’t that what you experience?

Another example; my wife will have a child soon; when it is born. I will assign X as its expected life. A year later, the expected life will have increased, no?

“When you assign a uniform distribution over outcomes because you have no information (as you explicitly suggest above), that’s a version of the principle of insufficient reason.”

Well, there are lots of rationales for it: transformation groups and maximum entropy are other ones that lead to the same sorts of conclusions. See Jaynes ; examples of transformation groups leading to uniform distributions are in the section beginning on page 17.

“Most of the time it will work. Sometimes (particularly when dealing with complex systems), it could be very bad indeed.”

I would be eternally in your debt if you would answer two questions about this (ever since your post, they are driving me crazy):

1) Can you give an example where the principle of insufficient reason doesn’t work when assigning probabilities to a finite set of possibilities? I’m pretty sure none exist, but I’m willing to learn.

This usually clinches it for me; probability theory (and I believe, with far less confidence, mathematics) are about finite sets and well-behaved limits of finite sets.

2) Can you give an example of one of the complex systems when it is “very bad indeed” to assign equal probabilities if you have no information. I thought you said I would find an example in one of the Rosser papers, but I couldn’t; if you could point me to one in there, or another you know about, I would really appreciate it.

Dsquared,

First, a sincere thanks for the response.

“For the first, the PIR is going to give you wrong answers when you’re trying to make guesses about a set of outcomes which doesn’t have a probability measure.”

Can you give me an example of a finite set without a probability measure?

In the Pruss paper you linked, he starts with infinite sets:

“Suppose I know with complete certainty that X could have any value in the interval [0,1], and there is no reason to favor any value over any other.”

“Suppose, however, that (a) I believe the universe is infinite in extent,”

When you start with a finite set, then take a well-behaved limit to an infinite set, then probability calculations always work; they give the correct (usually intuitive) answer when such a limit exists, and refuses to answer when it doesn’t. For example, in Pruss:

“For instance, a quick calculation shows that the likelihood that the blast will occur within a thousand miles of Earth is less than 10^-44 of the likelihood that the blast will occur somewhere in the Andromeda Galaxy. Any likelihood that is that much smaller than some other likelihood is negligible.”

“Admittedly, all the reasoning here cannot be reconstructed within a standard Bayesian epistemology based on the classical probability calculus.”

It can; all you do is start with a finite universe and take a limit to an infinite universe; in the limit the first _probability_ is less than 10^-44 times the second _probability_. You get the same intuitive answer as Pruss, using standard probability theory.

The limit method always works, and it satisfies our intuition about “infinite” cases when the limit converges.

When the limit isn’t well-behaved, the mathematics is telling you “You can’t use this shortcut on this problem; you can’t just say N is very, very large, you actually have to figure out how big you think N is.”

The St. Petersburg case is an example; the limit diverges (increases without bound). This means you can’t conclude anything about the answer; you can’t just say “Imagine we could play forever,” you need to be more specific. Once you put a limit on the amount of money or the number of plays, intuitive answers pop out.

You can solve most (all?) “infinite” paradoxes this way: Do all the analysis with finite quantities, then see what happens when those quantities increase indefinitely. This is the method Gauss used:

“I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit.”

2. So it isn’t chaos theory I should be looking into, it is just game theory, right? Game theory often upsets me for similar reasons to the stuff above: it assumes an infinity already exists (an infinity of anticipations). But I understand it much more than I do chaos theory.

“probabilistic reasoning in the sense of assuming well-defined and objective probability measures”

I don’t know what you mean by “objective” here. See below.

“2. I throw a dice behind a screen, which you’re never allowed to see. About 1 time in 6, so far in plays of this game to date, I say “Hey, you rolled a 6” and pay you 20x your stake, and about 5 times in 6 so far I have said “Sorry, better luck next time” and kept your stake.”

So I don’t know if you are honest or not here?

The “five times out of six” is relevant, but not a complete description of what I know. Probability is about _all_ of my information.

My judgment of your honesty would enter into the probability calculation. For example, if you were my friend and the money would go to Bill Gates if I lose, I would assign a different probability than if you were a stranger and the money went to you if I lose.

It would also be different if you didn’t know the bet was for $1 million this time (or if all the bets in the past were for $1 million).

I’m missing the point you are making here:
– I don’t see a problem with my assigning a probability representing my information, then using standard decision theory.
– I also am missing any game-theoretic issues (“he thinks that I think that he thinks …”) as in Rosser’s paper.

“Pascal’s wager, if I understand it correctly, poses a question where there’s a binary choice without a probability measure.”

Do you have a source on that? (I can’t find a source on Google) It doesn’t match my understanding of a probability measure. I’m no expert, though.

This is the wager where either an afterlife exists or not, and you choose to be religious or not?

I thought the problem was an infinite utility (or multiple infinite utilities), not about probabilities.

“In general, I think we’re talking at cross purposes.”

Sorry, I am clearly misunderstanding; first I thought it was about chaos theory, then game theory, and now it is probability theory.

“For example, game 3, which is the same as game 2 but it’s my brother, about whom you know as little as you know about me, doing the rolling. What I think Keynes and I would say would be that there is no principled way in which you can say either that gamble 2 is preferred to gamble 3, or that you are indifferent between them.”

I don’t see why either of you would think that.

Here is the principle: If you have the same information about two possibilities, then you should assign the same probabilities to them. In other words, if every piece of information you have points equally to A as to B, then p(A)=p(B).

This is what I mean by the “principle of insufficient reason”; why would you not use it in this case?

My reasoning goes as follows:
– my probability of winning game 2 equals my probability of winning game 3 (whatever it is). Why? You have specifically told me that I have no information that distinguishes your action and your brother’s action.
– Same probabilities, same outcomes implies indifference.

Intuitively, if I had to play one, I’d be indifferent between the two; I have no information one way or the other about which is more likely to win.

This also is the intuitively correct result, no? How could it be otherwise?

“What we’d question is whether probability in this sense is something that should be put on the same logical footing as frequentist probability”

These probabilities work exactly the same way as frequentist probabilities, as shown by Cox; see Jaynes. Briefly, if you represent your information as real numbers, and you want to reason consistently, then he shows that those real numbers must obey two rules:

p(A|C) + p(not-A|C) = 1
p(A and B|C) = p(A|C)p(B|AC)

These are the same rules that frequentists use for finite, discrete sets of possibilities.

So probabilities representing your information combine in the same way as frequencies do.