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Price, Value, and Exploitation using Input-Output Tables

Part 4: An Economy with Workers, Means of Production, and Capitalists

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Now we are ready to add profit, and hence exploitation, to the mix. Finally, something approaching capitalism.

This will be a model of simple reproduction. There is no economic growth, and the entire net product produced by workers is entirely consumed by capitalists and workers. Nothing is set aside for growth or expanding production; the system reproduces itself at the same level of gross output.

What’s new, in contrast to the previous scheme, is the addition of an exploiting class. Capitalists do not provide labor to the economy, yet they still consume products. Surplus labor must be performed by the workers to produce both these consumption items and the means of production required to make them.

This is important: all of our talk about prices and profits can sometimes mask that underneath it all is surplus labor.

The Physical System

Before diving further into prices, let’s first discuss the physical system. This is nearly identical to the previous model, but now total consumption divides into

• The workers' consumption bundle: c^(W)^
• The capitalists' consumption bundle (a.k.a. the surplus product): c^(K)^

So total consumption is:

c = c^(W)^ + c^(K)^

and the gross product can still be written as

q = (I - A)^-1^ c

as long as we understand that c now contains the consumption for both classes.

Value, Necessary Labor, and Surplus Labor

Our previously mentioned definition of value

v = vA + 𝓁

v = 𝓁(I - A)^-1^

can allow us to invisgate how the workers’ labor divides into the necessary labor (needed to reproduce the working class) and the unnecessary or surplus labor (which goes to the capitalists).

The total labor required for production of all products is L. Value is defined such that the total value of the consumption bundle equals the total labor:

L = 𝓁 q = vc

Now separate c into its class components:

L = vc

L = vc^(W)^ + vc^(K)^

L = Necessary Labor + Surplus Labor

So for workers to produce their consumption bundle and the means of production for their bundle (as well as the means of production for the means of production, and the means of production for the means of production of the means of production, and the …. etc.) they must collectively perform vc^(W)^ hours of labor.

For the workers to produce the capitalists’ consumption bundle and its means of production (and the means of production of its means of production ad nauseum) they must collectively perform vc^(K)^ hours of labor.

Note that v includes embodied labor that 𝓁 alone doesn’t capture, i.e. v contains the labor needed to produce the means of production, the means of production of the means and production, and so on. So we don’t need to explicitly include the labor to produce the means of production (Aq), it’s already wrapped up in the labor value.

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If you want to explore the justification for this definition of value, note:

(I - A)^-1^ = I + A + A^2^ + A^3^ + …

and consider the interpretation of what the n^th^ power of the input-output matrix represents.

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Now for profit and prices.

The Price System

When sector j sells its product, the prices must cover:

• the means of production used to produce product-type j: p a~★,j~ q~j~
• the wages for the labor: w𝓁~j~q~j~
• and also a profit for the capitalists: Π~j~.

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If it helps, recall that a~★,j~ q~j~ is the vector of material inputs required for sector j to product its output at level q~j~.

We can rewrite this as

z^(j)^ = a~★,j~ q~j~

z^(j)^ = [z~1,j~ z~2,j~ … z~j,j~ … z~n,j~]^T^

So the term

p a~★,j~ q~j~

is also

pz^(j)^

which is just the following sum of input costs for sector j:

p~1~ z~1,j~ + p~2~ z~2,j~ + … + p~j~ z~j,j~ + … + p~n~ z~n,j~

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So, following the reasoning of the previous systems we have our initial price equation

p~j~q~j~ = p a~★,j~ q~j~ + w𝓁~j~q~j~ + Π~j~

We’re not done with this equation, though. Let’s rewrite profits as using a sectoral rate of profit r~j~.

The profit rate r~j~ tells us how much capitalists get back in profits relative to what they advance for industry j. Capitalists advance money-capital covering the wages and the costs of the means of production. So the profit rate is defined as the following ratio:

r~j~ = Profits / Money-Capital Advanced

r~j~ = Π~j~ / ( p a~★,j~ q~j~ + w𝓁~j~q~j~).

Rearrange this to get an expression for profits

Π~j~ = r~j~ ( p a~★,j~ q~j~ + w𝓁~j~q~j~).

Sraffa, and some others, exclude wages from the advanced capital, but we’ll stick with the above as it’s closer to Marx and the classical economists.

Substitute this definition of profit back into the our initial price price equation above

p~j~q~j~ = (1 + r~j~)(p a~★,j~ q~j~ + w𝓁~j~q~j~)

and divide by q~j~ to get:

p~j~ = (1 + r~j~)(p a~★,j~ + w𝓁~j~) .

Now, assume capital is feely mobile and competition equalizes the rate of profit across all sectors. If one sector has a lower profit rate than the average then capitalists disinvest, labor exits, supply shrinks, and prices rise. This causes the profit rate to increase and restimulates investment in that sector. This is one institutional mechanism underlying the law of value.

The a.) movement of capital from sectors with low to high rate of profit, and b.) the effects of supply and demand on actual market prices are mechanisms that equalize profits and also reallocate labor.

We aren't modeling these market price fluctuations due to supply and demand, though. We’re interested in attractors, not actual trajectories. Again, think long-term prices, or regulating prices. So note that this profit rate r is the regulating center of the observed rates — not the actual momentary measurement existing at any given time due to existing market prices.

Using a single profit rate across all sectors is analogous to how we used a single wage for all workers. If you want to resist this assumption and keep each distinct sectoral profit rate r~j~, then you’ll need to introduce a diagonal matrix of profit rates R.

When, or if, there is one regulating profit rate for the economy then we can drop the index for the profit rate,

r~j~ = r,

Now, just as we’ve done before, let’s turn our price equation for one sector into a vector price equation for all sectors’ prices:

p = (1+r)(p A + w 𝓁).

This is our equation for natural prices in a system with exploitation but no growth.

Let’s solve it now for p to get

p = w 𝓁 (ϱ I - A)^-1^

where ϱ = 1/(1+r).

Note that this is close, but not identical, to the the price equation discussed last time when there was no exploitation: p = w 𝓁 (I - A)^-1^ = w v.

The Transformation Problem

As r → 0, the system with exploitation collapses to the non-exploitative case. But as r increases, p shifts while v still remains unchanged. This is the core of the transformation problem when expressed with input-output analysis. Surplus distribution affects prices but not values (as conventionally defined).

Just to say this again to be clear: as the profit rate r changes our definition of value

v = 𝓁 (I - A)^-1^

is not impacted. Each of the above quantities in the value equation are set in our example and they don’t change as more or less product goes to the capitalists. The profit rate doesn’t impact 𝓁 or A, and hence v is invariant as r changes.

But the prices

p = w 𝓁 (ϱ I - A)^-1^

do change as the profit rate (or ϱ) changes.

If v is invariant to changes in the profit rates but p isn’t, then we have a problem if we want to claim that prices can be expressed in terms of value.

Don’t panic, though. There is a way to address this problem head on without hiding from it or throwing out the use of input-output tables. Unfortunately I won’t do it justice in these posts, but in the next post I will still show the relationship between prices, profit rates and surplus labor using an example. And the fact that such a relationship can still be shown to exist hints that labor is still there, somewhere, in the prices.

Before solving the above price equation with an example, though, let’s discuss the circular flow of the economy in this exploitative case.

The Circular Flow and a Discussion of Profit

(Note: this is a reformulation of Marx’s M → C (L + MP) … P… → M’)

As before,

• Workers contribute L hours of labor

• In return, they receive wages wL.

Capitalists inject money-capital to pay for

• Wages: wL

• Inputs: pAq

This injection of money capital can be written as M using Marx’s notation, and we can also connect it to the input-output quantities via

M = mq

M = pAq + w 𝓁 q

where m = pA + w 𝓁 are the unit costs.

Don't confuse M with the total money supply. It is not a stock of total money in the economy, instead it's the flow of money-capital.

Workers make the product q, but capitalists own it and sell it in three parts:

• z = Aq back to industries as means of production

• c^(W)^ to workers as means of consumption

• c^(K)^ back to capitalists as their means of consumption.

Under simple reproduction there is no growth, so no investments to increase capacity. The same level of means of production z are invested during each round of production.

When capitalists sell the total product at natural prices, they receive a flow of revenue equaling:

M’ = pq

From this, they set aside a part to advance another round of capital. In simple reproduction, this new round of capital is identical to the previous one: M = m q = pAq + w 𝓁 q.

After advancing capital M and receiving M’ = pq from sales, they still have (they hope) a net positive flow of money coming into their pockets which is equal to

M’ - M = pq - pAq - wL

This positive net in-flow is the total profit Π of the capitalists, and using the definition of the profit rate we can write it as

Π = rM

Π = r mq

Π = r(pAq + wL)

This profit Π is also written as ΔM using the more classical Marxist notation.

Capitalists aim to acquire c^(K)^ without laboring. Hence, the system must compel workers to engage in surplus labor for the production of this surplus product that capitalists consume.

The profits that capitalists make are spent on purchasing this surplus product.

Let’s discuss what each class does with the money distributed to them.

First, let’s discuss the industries - not really a class, just a section of our model. In the aggregate, industries (or the capitalists that own industries) sell pAq to other industries. This quantity of money flows out of industry but then right back into it. It’s like a closed loop.

Now the workers. Capitalists hire workers for their labor-power. The workers can’t check out whenever they want - capitalists make them work for some specific time. The length of the working day isn't a natural law, though, it's part of the class struggle. Organized labor can fight for a reduction in the working hours, and fight for increases in pay as well. This isn’t mechanical, it’s determined by our struggle.

Now, as long as there are capitalists with power and self-preservation, the wages they pay to workers are only enough for them (in the aggregate) to purchase their means of consumption

pc^(W)^ = wL.

The workers can not use their wage to afford the entire consumption bundle c that they've created: c^(W)^ < c. Workers (collectively) are forced to produce the entire final consumption bundle when employed, but are never able to buy it back. Workers must labor for more than they could ever equivalently receive back in value via consumption.

I.e., the workers must give L to produce a final product of c

vc = L

But the value of what they receive through consumption is less than the labor they originally worked!

vc^(W)^ < L

And so a surplus of the final consumption still remains:

surplus product = q - Aq - c^(W)^

As mentioned, the surplus is produced by workers but can’t be afforded by them. And it isn’t needed by industries as means of production. Where does it go?

This surplus product

c^(K)^ = q - Aq - c^(W)^

is purchased as means of consumption by capitalists with their profits. So

pc^(K)^ = Π

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At at the risk of sounding pedantic, let’s just summarize the various ways we can write out profit

Π = ΔM = rM = r mq = r( wL + pAq)

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In our simplified case of simple reproduction all of the capitalists’ profit goes toward purchasing their consumption items.

Note that the profit rate r we are using is also Marx’s profit rate, but expressed with monetary quantities and not as values (as standardly defined). We can’t yet equate prices with value so r is in terms of money. Again, the transformation problem between values and prices (and its potential solutions) really deserves its own post.

Similar to how Marx wrote the profit rate as surplus value over constant and variable capital

r = S/(C + V),

we are writing it as

r = pc^(K)^ / (pAq + wL)

It’s slightly different, but still note

• pc^(K)^ is the profit Π which is the monetary expression of the surplus product

• pAq is the monetary cost of means of production, or constant capital

• wL is the monetary cost of labor-power, or the variable capital

The above has just been another way of expressing Marx’s circuit

M → C (L, MP) → …P… → C’ → M’ = M+ΔM

Summary of the Circuit of Capital

Here’s that connection summarized one last time:

• Capitalists advance money-capital:

• M = wL + pAq

• And use it to purchase C which consists of labor L and inputs MP = Aq

• Laborers transform this into output

• C’ = q = Aq + c^(W)^ + c^(K)^

• And capitalist sell this output to receive

• M’ = pq = M + ΔM = M + rM

• They then reinvest another round of M and take home ΔM = r( wL + pAq) to purchase their consumption items, i.e.

• ΔM = pc^(K)^.

We have our familiar circuit of capital

M → C (L, MP) → …P… → C’ → M’ = M+ΔM

This process rests on capitalists owning the means of production and hence the production process. Capitalists can compel workers to provide surplus labor beyond what is needed for their own reproduction.

Summary for Value and Surplus Labor

Capitalists force workers to provide L total hours of labor.

L = vc

L = L~necessary~ + L~surplus~

Part of this total labor is L~necessary~ which is used to produce the workers’ own consumption items c^(W)^

L~necessary~ = vc^(W)^

And another part of the total labor is used to produce the capitalist’ consumption items c^(K)^

L~surplus~ = vc^(K)^

Workers are paid only enough to reproduce themselves with c^(W)^, and capitalists appropriate the surplus product c^(K)^.

A Final Note on Natural Prices

Now that we have a system of class exploitation, this class will want to preserve itself. Natural prices are now those that allow the capitalist class, and the capitalist system as a whole, to reproduce itself. Earlier, without the capitalist class, the natural prices were such that allowed for reproduction of only workers and the means of production. But here the natural prices of capitalism are those that allow for capitalists to sustain themselves at some level of profit.

This profit, and hence the distribution of labor into necessary and surplus, is determined by the class struggle. It isn’t mechanistically determined. Our model allows us to see how prices and profits are related, but by itself it doesn’t tell us what these profits are set to.

Also, it isn’t as if capitalists know what these natural prices are and set market prices to them. Instead an actual institutional mechanism must exist to “discover” these natural prices. Under capitalism it is a market mechanism of some sort that drives market prices toward natural prices.

In early capitalism there was much more competition between firms, and competing capitalists setting prices in the market would lead to the emergence of market prices settling around some natural price. Again, it is a somewhat objective process, the emergence of natural prices is beyond the will of any one individual capitalist. When monopolies form, then there is more control that one giant firm can have on market prices - and hence on the natural prices. But even then, monopolies do not mean that there is no competition (until the world is totally dominated by the Weyland-Yutani I suppose?)

If a capitalist sells their goods at market prices way below the natural price, then that results in less profit for them (and any shareholders) unless they also severely undercut their workers. That creates a pressure for them to raise their prices closer to natural prices unless they can sustain the lower prices through technological improvements.

Now, if they do make a technological breakthrough then these firms can sustain their lower prices and may tend to dominate the market. But their technological improvements and new dominating presence lowers the labor coefficients, lowers the socially necessary labor time, and has a downward pressure on natural prices.

In the opposite case, if a capitalist sells their goods at market prices above the tending natural price, then sure they make more profit, but they aren't competitive. So fewer sales. Smaller marker share. Bankruptcy is likely to follow unless they can lower prices to stay competitive.

The inner mechanisms and dynamics of market prices, how they converge to natural prices, and how the law of value also leads to technological improvement and the tendency of profit rate to fall isn’t discussed by this model. But maybe you can see ways of including them. But I also don't think it's this framework's intention to erase this dynamic side of the economy. In the last post I'll briefly discuss some work that has tried to breath movement this framework.

Also, there is a range of natural prices for our system depending on how much surplus labor the capitalists are able to extract from us. And you can see that with the equation, the natural prices vary with profit rates even when the technical coefficients (𝓁, A) don't change.

We’ll explore this in more detail when we solve the price equation with an example. On to Part 5!

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