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Price, Value, and Exploitation using Input-Output Tables

Part 3: A Worker-Only Economy with Means of Production

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Let’s add in the input-output matrix.

Now we can start to include means of production into the system as circulating capital. We will not yet add profit and so no exploiting class. This system will allow us to introduce the input-output matrix.

The Input-Output Matrix

The input-output matrix is a tool which quantifies the material connections between different economic sectors. It tells us how many products each sector will require from all others when producing a good.

The set-up is almost identical to before, except that in order for each industrial sector j to produce its gross output q~j~ it must now use products produced by other sectors. The products that sector j requires as means of production can be expressed in the n x 1 column-vector z^(j)^, or sometimes written as z~★,j~. The star ★ in the row-index signifies that all sectors are included as n rows in the column-vector z~★,j~.

To clarify, sector 1 requires some quantity z~1,1~ of product-type 1 (produced by sector 1), some quantity z~2,1~ of product-type 2 (produced by sector 2), and so on. These are all encoded in sector 1’s necessary means of production vector z^(1)^.

Sector 2 similarly requires z~1,2~, z~2,2~, etc., forming z^(2)^.

More generally, sector j requires:

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

(Note the transpose).

We continue to assume linearity (constant returns to scale), so we divide each quantity z~i,j~ by sector j’s output q~j~, giving us the inter-sectoral input coefficients a~i,j~

a~i,j~ = z~i,j~/q~j~

or equivalently

z~i,j~ = a~i,j~ q~j~ .

(Helpful Note Pay attention to index order in two-index quantities. The order helps track the flow of inputs. For example z~i,j~ is the quantity of commodity i that flows into sector j. These are of product-type i, so they must be priced at p~i~. We'll see that the money that sector j must pay to sector i is then p~i~ a~i,j~ q~j~.)

The quantities a~i,j~ can be placed in an input-output matrix

which can also be written with the single generic term using index-notation

A = [a~i,j~].

A matrix can be thought of as a table of numbers where i indexes the row, and j the columns. For example, the third row and fourth column of A is denoted as a~3,4~. In our case there are at most n rows and n columns in our matrix. A matrix isn't just any table, though, has particular rules for how it multiples, adds, etc. with other tables. You can add two matrices if they're the same size (same number of rows and column). You can even take a matrix and multiply it by a vector as long as certain rules are met. We won't get into all the rules of linear algebra, there are plenty of introductions to that field. For the rest of the post I'll aim to give enough context to guide you through the math without explicitly explaining it all.

Economically, a~i,j~ represents the quantity of product i directly required in the creation of one unit of commodity j. This matrix must be non-negative (it can't take negative quantities to produce another), and for the economy to be productive A must also satisfy the Hawkins-Simon Condition.

We can also interpret A as a network of flows between sectors.

(n=3 industry sectors and the inter-industry coefficients between them. For example, in order for industry 3 to produce a single output it requires a~1,3~ from industry 1 and a~3,3~ of its own product to serve back as inputs.)

The Physical System

Let’s now describe the physical equations. As before, we treat the final consumption vector c as given and solve for the gross product q.

In equilibrium, sector j must produce enough to cover the desired final production c~j~, plus enough to serve as means of production for the entire economy. The total quantity it must supply to other sectors to serve as their inputs is z~j~.

So the gross product of sector j is

q~j~ = z~j~ + c~j~

We can relate z~j~ to the means of production requirements of the other sectors using the input-output matrix. Let’s express z~j~ in terms of other sectors’ needs: sector 1 requires z~j,1~ from sector j, sector 2 requires z~j,2~ from sector j, and etc. So:

z~j~ = z~j,1~ + z~j,2~ + … + z~j,n~

Substituting the input coefficients give us

z~j~ = a~j,1~ q~1~ + a~j,2~ q~2~ + … + a~j,j~ q~j~ + … + a~j,n~ q~n~

Now here’s the magic: note that the above is the j^th^ row of input-output matrix A times the column-vector q.

So the gross product for sector j can also be written as

q~j~ = a~j,★~ q + c~j~

where a~j,★~ is the j^th^ row of the input-output matrix A. Here the star ★ in the column-index means that all sectors are included as n columns in this row-vector.

Note that the equilibrium output of sector j depends on the produced outputs of all other sectors (given in q and related via the input-output matrix). So A helps to encode the inter-relationships between the sectors.

Since the above is true for any sector (again j is arbitrary) we can write our vector equation for the gross output of all sectors as

q = Aq + c

Since we take c as given data, we can solve for the gross output q

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

The matrix (I - A)^-1^ is commonly called the Leontief inverse, and it must exist for the matrix A to be considered productive.

The Price System

To find the price equation we can think about the costs that a sector j has when it produces its output q~j~.

Labor Costs: First it has the labor costs which we’ve previously discussed, w 𝓁~j~ q~j~.

Means of Production Costs: But now it must also be able to afford the means of production from other sectors. As discussed earlier, for sector j to make q~j~ it must productively consume a vector of products given by the n x 1 column-vector z^(j)^ = [z~1,j~ z~2,j~ … z~n,j~]^T^.

For this, sector j must pay an amount p~1~ z~1,j~ to sector 1, an amount p~2~ z~2,j~ to sector 2, …, etc.. Recall that z~i,j~ is a good of product-type i produced by sector i that flows to sector j. Since it is of product i it must be sold at sector i prices.

So the costs that sector j incurs to afford its means of production are

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

(An example of inputs and costs for some random sector 2)

Exploiting linearity, we can introduce the inter-sector coefficients again and write the above as

p~1~ a~1,j~ q~j~ + p~2~ a~2,j~ q~j~ + … + p~j~ a~j,j~ q~j~ + … + p~n~ a~n,j~ q~j~

or

(p~1~ a~1,j~ + p~2~ a~2,j~ + … + p~j~ a~j,j~ + … + p~n~ a~n,j~) q~j~

Note that the terms in the parentheses are a multiplication of the row vector of prices p times the j^th^ column of the input-output matrix A. So we can write the above as

p a~★,j~ q~j~

where a~★,j~ is the j^th^ column of input-output matrix A.

The price of product j when it sells q~j~ must cover these costs. So,

p~j~ q~j~ = p a~★,j~ q~j~ + w 𝓁~j~ q~j~

Divide both sides by q~j~ and we have our price equation for sector j

p~j~ = p a~★,j~ + w𝓁~j~

(An example of the unit-inputs and -costs of a three sector economy.)

Finally, note that the above applies for any sector j, and hence any column in A. So we can write the prices for all sectors as the following vector equation

p = pA + w 𝓁

And we can solve this for p to get

p = w 𝓁 (I - A)^-1^

Again, we rarely discussed the institutions involved in distribution and exchange. We can think that p represents natural prices that real market prices tend to gravitate toward for the system to have long-term viability. But perhaps there is no market mechanism and we have a type of planned economy that still keeps track of costs in this manner. A planned mechanism that keeps track of costs as labor would be nice as well, because then we could determine how much labor to reallocate across an economy to meet demand.

Value and Prices and the Fundamentals of Labor

Earlier we said that direct labor coefficients in 𝓁 only capture the immediate labor for each product. They do not account for the labor embodied in the inputs.

We can define value as the embodied labor of a product, i.e. a way of measuring the indirect labor required throughout the entire economy for the production of a commodity.

Pasinetti (1988) and Wright (2019) show there are multiple possible value definitions depending on how we define what is meant by “embodied”. For example,

• Do we want to count the labor required just to reproduce the good and its means of production?
• Do we want to count the above and and any extra labor required to grow the economy at a steady rate?
• Do we want to count both of the above and the labor required to produce the capitalists’ consumption goods?

The language of "embodied" has some ambiguity when we try to write it down as an equation. But let’s stick with the standard and common definition of value - the labor directly involved in production and the labor embedded in the means of production. To you give a brief taste, without deriving it, the standard measure of value is

v = 𝓁 (I - A)^-1^

where v is a row-vector of values for each product j.

Vector v contains the values of each unit product, i.e. v~j~ is the value of a one unit of product j. The total value of x~j~ units of product j is then v~j~ x~j~.

Notice that in our system of workers and means of production (with no capitalists), prices are proportional to these standard values.

p = w 𝓁 (I - A)^-1^

p = w v

The true cost, the real price … is still labor. But now we don’t measure labor using the direct labor coefficient, we must now move to using a measure of labor that takes into account the costs of the means of production, i.e. value as we’ve defined it.

But, as we will see, once exploitation is introduced prices do change as profits increase or decrease although values as defined above are not impacted. A transformation problem arises between these values and prices.

Some, like Samuelson, have argued that since values aren’t necessary to determine prices, they’re unnecessary altogether (”get that pesky Marx out-a-here!,” they say). Others, such as Wright (2019), building on Pasinetti (1988), have constructed definitions of value that are commensurate with prices. But for the most part this transformation problem has been a thorn of some sort, some people ignore it, some try to solve it, some say that it’s a waste of time. Some think it’s the Marxist version of debating pin dancing angels.

But I do think that talk of the transformation is important, and as this current demonstration continues it can given the impression (even to some “Marxian” economists) that value is redundant, unnecessary, and can be discarded.

The neo-Ricardians, post- and neo-Keynesians, and most "intelligent" economists at university certainly deem it as unnecessary baggage from that crazy ol’ Marx. Some Marxists even abandon hope and think labor values are irrelevant and give up on the labor theory of value. Some become a strange type of anti-empiricist who turn away from a scientific (and quantitative) approach to political economy. I think these are serious errors.

The following isn't necessarily part of the standard framework, but I want to rant and soapbox here and emphasize, without yet going into the math of it, that value is important as it is what grounds economic measures. It ties measures back to labor, the root of the economy. If there is no labor, and no labor allocation via the law of value, then there are no reproducible commodities. And hence there is no long-term economic system. We'd consume only the use-values that nature provides. Humans are not the type of organism who does this. You can see Shaikh (1982) for more on this.

Labor is foundational to economics, despite the attempts by economists to hide this. But this foundation is even deeper. Labor not only made the world, but as Engels argues labor made us human, i.e. even our biological evolution as a social species was done through labor. It is fundamental to our social-being, or our species-being.

If this isn't a glowing affirmation of the working class and its potential, then I don't know what is.

Any future existence of our species will hinge on how our social-being makes use of our laboring capacities. Climate change provides a clear example. Will we rise to the challenge and rearrange our society and production in order to prevent ecological collapse? Or will we make endless Funko pops till we die?

The Circular Flow

Before you panic, let’s recap the flow of labor, products, and money for this modeled economy.

• Workers provide total labor L, which is the sum of L~j~ from each sector j.

• Some of this labor goes into producing their consumption items c, some to producing means of production z = Aq.

• Regardless of the type of work, workers (in the aggregate) receive a wage wL for it.

• Workers use their wage to purchase their consumption items c, so pc = wL

• Industries use part of the product to produce inputs z = Aq, and these input costs sum to pAq. In the aggregate, these costs go out toward the industries and are received back by them.

Another Take on Natural Prices

Here is another structural way to think of natural prices. Note that if the natural prices were lower than what the system dictates, it may be possible for workers to purchase more consumption items, i.e. they could consume more than c. If this occurs then their consumption would be “eating into” the products that must serve as means of production. This would reduce the remaining product available as z, making the system unsustainable in the long-run unless workers can supply more labor to produce more means of production.

In other words, assuming all else was left constant (wL doesn’t change, and L and q doesn’t change) if p were lower, then c could increase but that would cut into q and cause a reduction in z which would be unsustainable for the production of c in the first place! (Remember, q = z + c).

So, the natural prices are such that allow for workers to eject c < q from the flow of commodities and ensures that z remains for future reproduction. The system can reproduce itself!

Another way of expressing it: the natural prices are those that take into account the labor costs required to produce the means of production, and so they result in prices which ensure that those supplies are set aside and not consumed with the workers’ wages.

These natural prices are not about markets per se, they represent reproduction-consistent values that close the loop on production and consumption.

Unfortunately, in the next post I won’t get into the solution for the transformation problem of how prices and values do relate (keep waiting). But I will clarify it. If interested, you can check out Wright (2019) for how this framework can resolve the transformation problem. At the very end at post 6 I'll give a taste of the solution.

Before that, I'll use this framework and it’s definition of value to show directly how profits are exploited surplus labor.

But first, let's talk introduce exploitation and discuss its impact on prices in Part 4

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