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Price, Value, and Exploitation using Input-Output Tables

Part 2: A Pure-Labor Economy

Now let’s begin with the simplest toy model: a pure labor economy.

The Set-Up

Suppose we have n economic, or industrial sectors, each producing one type of product indexed by j. Each economic sector has an associated worker-sector, which embodies all the N~j~ workers in that industry. We will not consider joint production, which is where one sector can produce more than one type of product.

(The set up with n industry-sectors and n worker-sectors)

If there are N~j~ workers in sector j, each working h hours per time-period, then the total labor-hours per time-period of sector j are N~j~ h, which we define as L~j~. We can think of each worker-sector as providing L~j~ hours to its industrial sector j per time-period. The units of h and L~j~ are flows: hours per unit of time. That may sound odd, but it’s familiar. We often talk about labor as “8 hours per day”, or “40 hours per week”. As long as the units are consistent across quantities, the time-period in question can be measured in any way. Typically it's determined by the time resolution of the data. In input-output analysis, I often see labor measured in hours per year.

In this framework, we distinguish two dual systems:

• The physical system, describing the flow of real products
• The price system, which we’ll cover next.

The Physical System

Let’s start with the physical system. In a pure labor economy, it’s simple: each j^th^ worker-sector contributes L~j~ labor hours to its sector, which produces a flow of q~j~ goods (of type j) per time-period. The units of q~j~ are in “units of product-j per time-period” (e.g. tonnes of iron per year, or pounds of apples per month, etc.).

If we denote the mean-productivity of sector j as ξ~j~ (units of product j per hour of labor) then:

q~j~ = ξ~j~ L~j~.

This assumes a linear relationship between labor input and output. More commonly, we use the direct labor-coefficient 𝓁~j~ which indicates how many labor-hours are needed to produce one unit of product j. For example, if it takes workers in a car industry 4 hours to build a single car, with the parts on hand, then 𝓁~car~ = 4 hours. The units of 𝓁~j~ are labor-hours per unit of product, and it’s the reciprocal of productivity:

𝓁~j~ = 1/ξ~j~.

We can gather the labor coefficients of all sectors in a 1 x n row-vector:

𝓁 = [𝓁~1~ 𝓁~2~ … 𝓁~n~]

Using our technical labor coefficient and rearranging our equation between gross output and labor time gives us the following important relationship

L~j~ = 𝓁~j~ q~j~

This tells us the labor in sector j given its output.

Note that 𝓁~j~ does not include the total embodied labor from other sectors that are needed to produce product j. For our car example, 𝓁~car~ excludes the labor needed to produce rubber, glass, steel, etc.. It only includes assembly occurring in the auto sector. Embodied labor is instead measured by value, and there can be different measures of value depending on what you mean by “embodied labor”. We’ll touch on value later, though. For a pure-labor economy, 𝓁 suffices as there are no intermediate inputs, an idealized abstraction for sure.

(Each sector j provides L~j~ hours of labor to produce a gross output of q~j~)

Often we don’t need to split the working class into n worker-sectors, and we can instead treat workers collectively as one big (happy) worker-sector. It depends on what level of resolution we're interested in for the question at hand.

Now if the labor in sector j is L~j~ = 𝓁~j~ q~j~, then we can sum the labor across all sectors to get the total labor in the economy:

L = 𝓁 q

This is a dot product of two vectors, 𝓁 and q. This is just a convenient way of expressing the sum of 𝓁~j~ q~j~ for all n sectors.

What is q? Well it's a n x 1 column-vector of produced goods for each sector:

q = [q~1~ q~2~ … q~n~]^T^

In this framework, we often treat the net product (here simplified as just final consumption) as given, and solve for the gross product q. In a pure labor economy, this is trivial—all products are directly consumed by workers:

q = c

Where c is an n x 1 column-vector of total consumption

c= [c~1~ c~2~ … c~n~]^T^

Each element c~j~ is the total amount of product-type j that is consumed by all workers regardless of what sector they may be working in.

(The system of production and consumption.)

Here because we are not interested in the exchange and distribution of products between workers, we can treat all workers as a single node and assume that this exchange succeeds under some ideal conditions.

If, or when, we are interested in this distribution between workers then we could subdivide consumption further. For instance, c~i,j~ could represent how much of product i is consumed by workers in sector j and we could use these to create a consumption network, but we’ll save that for another time.

The Price System

Now we can move into the price system for the pure-labor economy.

Suppose workers in sector j are paid a wage w~j~ per hour. For simplicity, we assume a uniform wage w across sectors. This could represent an average wage, or result from labor mobility equalizing pay across sectors (given suitable institutions). If we drop this assumption, we’d have to use a diagonal matrix W encoding wages w~j~, and supply data or theory for their variation.

The total wage paid out to the workers of sector j is wL~j~, and the total wage paid to all workers is wL.

A Note on Institutions In this pure-labor economy, there's no exploiting class. We might imagine this economy as:

• A system of independent, or petty, producers paying themselves a "wage" from their revenue.
• A system where labor and property is completely socialized. All workers get paid a wage for their performed labor from the institutions (the state, co-ops, etc.)

The specific institutions are left vague, but if some institution is at play then it's treated as ideal, i.e. it is working smoothly and is self-correcting enough that we can ignore its inner-mechanisms. Real, messy, institutions must obviously exist to do the actual organizing of people and the economy, but a "pre-institutional" lens is our attempt at trying to abstract away form the details of the intuitions and to try to find something more general.

Another thing about institutions, a pre-institutional lens can't be fully abstracted away from specific institutions. Aside from the institutions that organize the productive units, we still posit some system of wages and monetary flows. Also we aren't discussing the market institutions, or even assuming that a market is what exists to give these products prices. But something is still acting as a means of accounting for costs, at least (here the true costs are labor). Perhaps there is an ideal market that is finding our natural prices, or perhaps there's a system of socialized distribution that keeps track of prices as purely an accounting mechanism. Again, we're trying to be as "pre-institutional" as we can (if that is possible). This framework doesn't really touch on the details of exchange, other than assuming that it proceeds, it isn't the focus.

Now before you walk away in disgust, though, I'd like to suggest that this approach leads to something like an immanent critique. If there's a market, then sure we'll take on the assumptions of a "perfect market" where supply can meet demand and the market price corresponds to the price of production/natural price. If there's a labor market, then sure we'll take on the assumptions that labor can smoothly move across sectors. We'll meet these assumptions and still show that the system is one of exploitation. We'll still show that profit is stolen labor even when the market is behaving as it should - even when the institutions are perfect.

Crony capitalism isn't the problem, it's just capitalism.

But not here in Part 2 just yet, let's stick with pure-labor before getting carried away.

Continuing, since there is no profit and also no capital inputs, each sector’s revenue, p~j~ q~j~, must just cover its wage bill:

p~j~ q~j~ = w L~j~

p~j~ q~j~ = w 𝓁~j~ q~j~

Dividing both sides by q~j~,

p~j~ = w 𝓁~j~

This holds for all j, so in vector form we have:

p = w 𝓁

where p is a 1 x n row-vector for each sector’s natural price.

Note that the physical system is given by column-vectors (q, c) and the price system is given by row-vectors (p). This convention, while not universal in the literature, helps express their duality and keeps the math clean.

Note that here in our pure labor economy the real cost of an item, its price, is the labor required for producing it, scaled by some wage-rate. Here, Adam Smith's quote rings out as obvious:

The real price of everything, what everything really costs to the man who wants to acquire it, is the toil and trouble of acquiring it.

We'll see though, that introducing profit muddles this a bit and results in something called the transformation problem. It doesn't mean that labor isn't involved in profit (it is and we'll show that), but it does mean that the direct link between labor and prices appears to break down. The real price of everything appears to not just be the toil and trouble of acquiring it, but instead that and a little bit extra.

When the transformation problem was pointed out our vulgar economists couldn't pass the chance to swoop in like vultures and interject that it was the toil and trouble of the poor bourgeoise that was that little bit extra. Fair and square.

But noticed I said appears. We can show, using this exact framework, how the cost of labor comes back into play even with profits. The connection between values and prices, i.e. costs, can be restored. Labor is the natural cost of something and origin of its value, even when capitalists make profit. But restoring that direct connection requires a little openness in thinking of how we define value for the question at hand. I won't get it into it, even in part 6. But I will most all of the pieces there for you. And I will show how you can still relate the profit rate directly to the amount of surplus labor extracted. That little extra on prices and values, perhaps in the future I'll come back to it.

The Circular Flow in a Pure Labor Economy

For now, Let’s summarize the flow of labor, products, and money in a pure labor economy.

• Workers supply labor L~j~ to sector j, summing to total labor L

• They receive total wages wL.

• They spend their entire wage on consumption c, which collectively costs them pc

Assuming no savings or debt, we have:

pc = wL

(The monetary flows involved. Wages are distributed to the working class for their labor and they spend it back on consumption items)

For purposes of visualization, we can further aggregate all n sectors together and represent this circular flow between labor, quantities, and money.

For the flow of labor and products we have

And for the flow of wages and payments we have

In Part 3 we’ll add circulating capital to this and introduce the input-output matrix.

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