**A homogeneous function is a function in which the sum of the exponents of the variables in each term is the same.**

**For example:**

**f(x,y,z) = 4xyz + 1/2x**

^{2}y - .72z^{3}

^{ }**is a homogeneous function of order 3, because it is equivalent to**

**4xyz + 1/2x**

^{2}yz^{0}– .72x^{0}y^{0}z^{3 }

^{ }**And**

**(1 + 1 + 1) = (2 + 1 + 0) = (0 + 0 + 3) = 3**

**A homogeneous function in which the sum of the exponents of the variables is 1 is called a linear homogeneous function.**

**Euler proved a theorem about the first partial derivatives of homogeneous functions. For linear homogeneous functions, and in particular for the example above, the theorem states that:**

**the value of the function f**

^{ }at a point f^{0}(x^{0},y^{0},z^{0}) is equal to the sum of each first partial derivative multiplied by the value of the variable at that point. In symbols:

**f(x**

^{0},y^{0},z^{0}) =**δ**

**f/**

**δ**

**x(x**

^{0}) +**δ**

**f/δy(y**

^{0}) + δf/δz(z^{0})**Now for the payoff.**

**Suppose there is a production function for an economy in which the two variables are Capital and Labor [measured how? Ah, that is a very big question and another story.] Let us represent the function as**

**f = f(K,L)**

**where K stands for capital and L stands for Labor.**

**The partial derivative of f with respect to Capital, or K, can be interpreted as the increase in the value of the production function of the society if Labor is held constant and one unit of Capital is added. In short, it can be interpreted as the marginal product of Capital. Similarly for Labor.**

**Now, suppose that each unit of Capital is paid a profit equal to its Marginal Product and each unit of Labor [one hour, one employee, whatever] is paid a wage equal to its [his/her] Marginal Product.**

**EULER’S THEOREM ASSERTS THAT IF THE PRODUCTION FUNCTION , f, IS LINEAR HOMOGENEOUS THEN THE TOTAL OUTPUT OF THE SOCIETY IS EXACTLY EXHAUSTED BY PAYING EACH FACTOR OF PRODUCTION, LABOR OR CAPITAL, ITS MARGINAL PRODUCT.**

**AND THIS IS INTERPRETED AS MEANING THAT CAPITAL AND LABOR EACH RECEIVES A REWARD PROPORTIONAL TO WHAT IT CONTRIBUTES TO THE SOCIETY. SO, CAPITAL AND LABOR COOPERATE, AND FAIR IS FAIR IN A CAPITALIST ECONOMY.**

**TA DA!!**

**Never mind the various definitional problems, which are huge. The question is: Does the US economy have a linear homogeneous production function?**

**Well, an economy with a linear homogeneous production function can be shown to have three properties that follow mathematically from that assumption:**

**1. The economy exhibits constant returns to scale**

**2. The economy is in long run equilibrium**

**3. The economy has a zero rate of profit.**

**Hmm. Does that describe the US economy? Does it describe any capitalist economy? No.**

**SO, NOT SO TA DA.**

## 7 comments:

First, I have zero background with economics. But it seems like you've shown that a linear homogeneous production function is sufficient for the society to be "fair". And then you say our society doesn't have such a production function. But that doesn't mean our society isn't fair.

Can you show that every function which is not linear homogeneous is not "fair"? From what I can tell a homogenous function of order greater than 1 can be fair (in the way you define fair: capital and labor can be paid proportionally to what they contribute, with order > 1 there's still money left over).

Just based on what you have given it doesn't seem like we can draw a conclusions one way or the other on how fair our society is.

(By the way, I'd love to see how those points 1-3 are represented mathematically and how they follow from the assumption of a linear homogenous function. Do they follow from a broader class of functions too?)

The professor is illustrating how linear homogeneous functions can implicate ideology. He's not trying to prove whether or not our economy is fair.

I'm going to enjoy chewing on this for some time. I studied this level of math a very long time ago, and I look forward to revisiting it. Thank you.

Sorta, kinda relevant recent obituary in the NYT: https://www.nytimes.com/2018/09/26/obituaries/voltairine-de-cleyre-overlooked.html

@Alex

As Prof. Wolff did not reply, I'll take the liberty of replying in his stead. It should go without saying, but I'll say it nevertheless: if I'm mistaken, Prof. Wolff is free to correct me.

But it seems like you've shown that a linear homogeneous production function is sufficient for the society to be "fair".You recognise that Prof. Wolff did show explicitly that an aggregate linear homogeneous production function is a sufficient condition for society to be fair. Your claim is that he did not show it to be a necessary condition.

You are right on that account. However, a cursory background on economics and on the debate that argument is a part of would have helped. It should also help understand why it is also a necessary condition.

Let me start with the debate.

Marxists do not believe that profit can be consistently made in exchange. I will not go directly into that, for it would be long and unnecessary, as I hope will be evident. The Marxist claim is that workers are

. By that it is meant that workers contribute more to output than they are paid as wages. It is precisely the difference between what workers get from their labour and what they put in, measured in money, Marxists argue, that explains profit, which, through accumulation, explains the fortunes of capitalists. (Note that "exploitation" does not mean being paid a physiological subsistence wage or less).exploited in productionCapitalists, in other words, do not contribute to production.

The second generation marginalists (chiefly John Bates Clark and Philip Wicksteed) set out to disprove the Marxist claim (I'll provide evidence of that in a companion comment). The argument Prof. Wolff outlined above is their argument, not Prof. Wolff's. Any departure from that argument, as the one you made (

From what I can tell a homogenous function of order greater than 1) automatically forfeits that conclusion. I am not sure this is what you want.(continuation)

But let's study your objection by comparing what results from it and what results from the Bates/Wicksteed argument.

Although Prof. Wolff did not show it explicitly, a property of the Bates/Wicksteed marginalist argument is that

p.f(K*, L*) = r.K* + w.L*

where the asterisk denotes the values of K and L that maximise the profit function p.q(K, L) - r.K - w.L (although I'm trying to employ the same notation Prof. Wolff employed, I use the asterisk because I don't know how to write super/subscript in Blogger). p is the price of the output, w is the wage rate, and r is the return on capital invested.

(While I suppose you, Alex, may not need it, it's important to clarify what that equation says, for among some rather obtuse critics of mainstream economics there's a tendency to misunderstand it: it means there are no

profits, it says nothing abouteconomicprofits.)accountingBy abandoning homogeneous linearity you forfeit that property, which means that if g is your

homogenous function of order greater than 1then eitherp.g(K*, L*) < r.K* + w.L*

or

p.g(K*, L*) > r.K* + w.L*

In the first case the economy is experiencing long run economic loses. That doesn't sound like a sustainable situation, yes? It doesn't sound "fair".

In the second case the economy is experiencing long run economic gains and nobody can ascertain how much of those economic gains go to capitalists and how much go to workers. If one

they are divided equitably between capitalists and workers, then Marxists may have nothing to object (i.e. workers are not exploited), is what one could argue.supposesBut in that case one must explain where those gains come from. I don't know about you, but the only answer I can see in marginalist theory is those gains come from consumers, for consumers together with producers, are the two essential economic agents. Consumers, that is, are paying more for those goods than is justified by their production cost. That doesn't sound "fair" either, does it?

I mentioned John Bates Clark above. Unfortunately, the history of economic thought is a subject neglected by economists, but one only needs to have read him to see where his argument fit in the debate between Marxists and their opponents:

The welfare of the laboring classes depends on whether they get much or little; but their attitude toward other classes-and, therefore, the stability of the social state-depends chiefly on the question, whether the amount that they get, be it large or small, is what they produce. If they create a small amount of wealth and get the whole of it, they may not seek to revolutionize society; but if it were to appear that they produce an ample amount and get only a part of it, many of them would become revolutionists, and all would have the right to do so. The indictment that hangs over society is that of 'exploiting labor.' 'Workmen' it is said, 'are regularly robbed of what they produce. This is done within the forms of law, and by the natural working of competition.' If this charge were proved, every right-minded man should become a socialist; and his zeal in transforming the industrial system would then measure and express his sense of justice.http://www.econlib.org/library/Clark/clkDW1.html

Incidentally, although I have no problem with Prof. Wolff's empirical claims against the production function (namely, modern economy does not look like a linearly homogeneous production function) I think that, by adopting the "aggregation" thing, he may be giving too much weight to secondary theoretical claims.

I believe there are problems more fundamental with the marginalist approach.

The first problem, affecting any kind of production function regardless of "aggregation", whether linearly homogeneous or not, whether in the long run or short, perfect competition or not, is that the marginal product of

is not the same as the marginal product ofcapital. Capital (machines, factories, offices, vehicles) may have marginal products, capitalists definitely have not.capitalistsThe second problem is that well-behaved production functions violate not only the laws of thermodynamics but also a principle of chemistry: the conservation of mass. Believe it or not. For all their alleged "physics envy", it seems to me economists do little to emulate physicists or chemists.

I've noticed that the Library of Economics and Liberty changed the addresses of its resources, so that the link provided above leads nowhere. To make things worse, Chapter I (Issues that depend on Distribution), where JBC explained his motivation for writing that book, was replaced by a Preface from the editor of the Library of Economics and Liberty (!).

Never mind that. One can still find that quote:

http://oll.libertyfund.org/titles/clark-the-distribution-of-wealth-a-theory-of-wages-interest-and-profits

It's the seventh paragraph from the beginning of the chapter.

Apologies for that.

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