Introduction

A few months ago I debated /u/Musicotic on the subject of Marx, I didn't really finish that debate. This post takes it further. I hope that people will see some arguments that are relevant to current debates. I won't point them out clearly though, that would spoil the fun.... I'll just say one thing, does anyone remember what Keynes said about the foundations of Marxism?

In Capital III Marx presents the Transformation Problem. That leads him to an alteration of his earlier theories (one that he hinted at earlier). Marx's previous books implied that the labour-theory-of-value applies separately to each commodity. In Capital III he changes that so the LTV applies to all commodities in aggregate. So, the labour-value put into all commodities is proportional to the price of all commodities. But the labour-value put into each one is not proportional to the price of that one commodity.

Most discussions about these later theories of Marx focus on the Transformation Problem. That is, they focus on discovering a procedure to find price-of-production that are consistent with Marx's other theories. Here I'm going to take a different path and instead concentrate on the aggregate labour-theory-of-value, and ask the question: is it plausible?

Musicotic put it like this in our previous discussion.

The aggregate theory is rather that the sum of prices is equal to the sum of (the monetary expression of) labour times, not that incomes (?) are proportional to labour-values.

Mathematical form is that at time t, ∑P(t)=τ(t)⋅∑L(t) , where τ(t) is the MELT at time t, L(t) is the labour hours at time t, and P(t) are the prices at time t.

Musicotic put the last line in TeX, which is more readable if you have "TeX-All-The-Things":

Mathematical form is that at time t, [; \sum P(t) = τ(t) \cdot \sum L(t) ;], where [; τ(t) ;] is the MELT at time t, [; L(t) ;] is the labour hours at time t, and [; P(t) ;] are the prices at time t.

I find Musicotic's writing very difficult to understand, that's why I'm concentrating on this part. This is an RI of this view, of Musicotic, Marx and many Marxists. My criticisms are variations on Bohm-Bawerk's and others.

What are we talking about?

In debates with Marxists, the first thing I often read is "Marx was talking about value not price". Now, value has two different meanings in Marx. Firstly, it refers to labour-value. In this debate, Labour-value refers to Marx's system of adding up the labour put into commodities. Secondly, there's exchange-value which is just another word for price - one used by the Classical Economists too.

Marx's labour-value is reasonably simple. For Marx the labour-time put into a commodity is the average that an averagely skilled worker would require. A trainee worker may take 2 hours to make a widget that would take the average worker 1 hour. In that case the labour-time in that widget is 1 hour, not 2. Secondly, work put into a commodity must be "socially necessary". Unnecessary work doesn't count. Thirdly, this labour-time is weighted for skills. So, some work is worth more than others. A lawyer's time is worth more than that of an unskilled worker. Marx saw this difference as a unskilled labour multiplied. A lawyer may create 3x the labour-value of an unskilled labourer, for example (so for one hour of work our lawyer creates 3 labour-value units). Marx never created a way of deriving these multipliers from anything other than differences in wage rates.

Now, you can't have a labour-value theory of labour-value. What I have described above is simply a definition of Marx's labour-value. It must be related to something to give a theory that can actually predict something. That something is usually exchange-value - i.e. price.

The equation that Musicotic gives is fairly good:

∑P(t)=τ(t)⋅∑L(t)

Musicotic describes L(t) as labour hours in period t. I think it should be labour-value in period t, I expect this is just a typo. P is prices.

Marx needs a theory of price because ultimately what he's talking about is profits. Profits are the result of prices. There are the costs - the price of labour and the price of capital inputs. Then there's the revenue - the sum of the sale price of the goods. The profit is the difference between them.

This is how Hilferding (a Marxist) put it:

... we learn that, since the total price is equal to the total value, the total profit cannot be anything else than the total surplus value.

The value τ has a timebase - this is a problem. Let's say that τ(t) varies randomly across time t. If you think about it that means that there is no theory. Any two things can be summed and a random variable can be put between them. For example, instead of L(t) I could use W(t). That's the weight of all commodities sold. I could then replace τ by ω the "monetary expression of weight". My function ω(t) would vary all over the place, of course. This would not prove my aggregate weight theory of value. Similarly, a changing τ does not prove an aggregate labour theory of value. However, an unchanging τ gets closer to that. Most Marxists I've seen suggest an unchanging τ, or at least one that changes very little.

Relationship to the Transformation Problem

Many, if not most, criticisms of Marx focus on the Transformation Problem. Marx starts in Capital I with a per-commodity version of the labour-theory-of-value. The problem with that theory is that it implies different profit rates in different sectors. I describe that here and here more mathematically.

Marx brings together several ideas and suggests a way of solving this problem. I've already discussed two of those, the aggregate LTV and his definition of labour-value. He added to that the following:

Firstly, the labour-power concept. Marx recognized a problem - how could the price of labour itself be measured in labour hours? He introduced the idea of "labour power". In Marx, labour power is what Capitalists buy and labour is what workers do. So, it may be possible to buy for $10 an hour of labour-power. That could result in an hour of work that will produce goods worth $14.

Next, his theory of exploitation - the worker creates the whole product, but the Capitalist only pays him for a portion of it. Marx thought of this through working time. A labourer works for part of the day for himself and part of the day for the Capitalist employing him. That extra labour-value was called "surplus-value". So, the profit made is proportional to the degree of exploitation. That can be expressed as a ratio of hours to hours for the shares of the day I describe. Marx reasoned that because labour-value costs the same for all sectors the rate of exploitation is the same for all sectors. The rate of exploitation is also called the rate of surplus-value.

Finally, Marx needed to create a reasonable theory of profit-rate. One that didn't involve some sectors being wildly more profitable than others. So, Marx moved to what he called prices-of-production (a term used by Ricardo for roughly the same thing).

A Capitalist starts with money K. That money is used to buy capital goods and to pay workers. That produces products that are collectively sold to gather revenue Q. Profit is then Q - K. The profit rate is (Q - K) / K. Often this is turned into a profit rate per year or per period.

The "Price of Production" theory suggests that all of these per period profit rates are equalized over time.

Kx(1 r) = Qx

Where Kx is capital invested in any particular sector and Qx is the corresponding revenue. The profit rate per period is r.

To bring all these things together Marx suggested that all profit comes from surplus-value. As a result, profit is directly proportional to surplus-value by the same proportion that total labour-value is proportional to the total prices. So, profit rate is proportional to surplus-value divided by other labour-value.

r = S / (C V)

Where r is the profit rate. S is total surplus-value. C is total capital input called "constant capital" by Marxists. V is "variable-capital" this is the portion of labour-value where the labourer works for themselves.

Years after Marx died Bortkiewicz showed that this process doesn't work in long-term equilibrium. Bortkiewicz created another process that does work in equilibrium. But, that process relies nearly entirely on prices not labour-values. Also, it doesn't guarantee the same relationship that Hilferding summarized above. The relationship between total labour-value and total prices turns out to be different to the relationship between total surplus-value and total profit. One can fall while the other rises, I described all that here.

This triggered a century of work on fixing the problem. Some decided to abandon the idea of equilibrium. They claim that Marx never meant the theory to work in that sense. Other's created complicated vector algebra intending to prove that small changes to the structure of the problem rendered it solvable.

This whole Transformation Problem debate is about consistency- how consistent are Marx's ideas with each other? If the problem were solved then it would be solved for all similar objective value theories. In other words, it would be consistent with my weight theory-of-value too. As long as is were structured in the corresponding way (i.e. a surplus-weight and a weight theory of exploitation). Whether it's correct is quite a different matter.

Problems with the Aggregate LTV

Here I'm going to talk about correctness not consistency. Is Marx's view plausible given what we know about the economy? There are several issue here, but I'll concentrate on only two.

Is Money Special?

The equation we're discussing refers to price:

∑P(t)=τ⋅∑L(t)

How is this price counted? It could be in money, but it could be in anything else. In Marx money is not special, it's just another commodity.

Think about using different commodities in this equation. As the rate of profit changes the price of different commodities varies in different ways. As a result, it's important what price is measured in. If it's measured in dollars then that's different to if it's measured in, say, bricks. There is a different aggregate LTV for each commodity that we could potentially use for pricing, and each one gives different results. If we were to measure in dollars and bricks then, clearly, the factor τ would not be the same for both. Let's call those factors Δ and β. If the rate of profit changed then the factor Δ could remain a constant across time, but it would change over time for β. Or vice-versa, if β remained constant then Δ would change. Why will become more clear later.

We could ask - how plausible is this in a world of fiat money? But, I think we should give Marx his due and consider commodity money only since that was his world. Perhaps Marx meant P to be a measure of real prices - i.e. he meant it to be adjusted for inflation and deflation. I've never seen anyone suggest this.

How Do Prices End Up Working?

To explain this problem I'm going to use some tables. Bohm-Bawerk presented tables to explain this in his book criticising Marx. But, I'm going to use the ones given by Hilferding in his counter-criticism. We can more-or-less forget about equilibrium here.

So, capital advanced is what capitalists spend to make the commodities. Constant capital is labour-value spent on capital goods which are assumed to be used up in one period. Together, variable capital and surplus value are the labour-value created by the worker. That is split between the worker's part (variable capital) and the capitalists part (surplus-value). Then there's profit. Total labour-value is the total in the output after the period. Finally there's the production price of the output.

We assume 1:1 correspondence between labour-value and price at the start. The columns Capital Advanced, Profit and Production Price are money quantities, everything else is labour-value.

Here, the exploitation rate is 100% that means that variable capital and surplus-value are always the same. Out of an hour each worker is spending half creating his own wage and half creating the profit of the capitalist. Marx tells us that total profit is equal to total surplus value. That allows total profit to the calculated. Then total profit is spread across the three commodities proportional to the amount of capital advanced. As a result, the profit rate is the same. Here it's 10% (50/500 = 70/700 = 30/300 = 0.1). We then get the production price by adding the profit to the cost, for example for C that's 300 30 = 330.

Now, let's change the exploitation rate to 66.7%. This gives us the following table:

The total price-of-production is the same and so is total labour-value - the aggregate LTV is obeyed. The profit rate was calculated by S/(C V) as a result, it is 7.8% this time, not 10%.

We can think of these as two successive periods, that's how Bohm-Bawerk and Hilferding do it. I prefer to look at it differently, I see them as two parallel worlds. In one parallel world the exploitation rate is different. Notice that in both worlds all the labour-value totals are the same. The constant capital figures are all the same. If we add together variable capital and surplus-value the sum is always the same (e.g. for B it's 30 30 = 60 then 36 24 = 60. So, in labour-value terms there is no difference between the two scenarios. There is no reason to imagine any difference between the production processes.

But the prices are different! For example, commodity B is 770 in the first table and 761 in the second. The difference is opposite for commodity C which is 330 in the first table and 339 in the second. (I could have made these differences larger if I'd changed the numbers a bit).

Let's say that commodities B and C are (imperfect) substitutes. If the price of B is high then why don't people use more C? Or if the price of C is high then why don't people use more B? The short answer is - that can't happen in this system. The theory I've described determines everything, leaving no room for decisions to be made between goods on price. Here we get to the implausible weirdness - profits affect relative prices, but not relative consumption. This is even stranger when we realize that shifts in distribution between profit and wages will undoubtedly affect consumption in reality, but can't here.