We’re thinking about the problem of coordinating activity levels so that just the right amount of things get produced. And we saw that any particular coordination necessarily implies a breakdown of the the total working day into different activities.

So what are the right activity levels?

We’ll answer this question independently of how any particular set of economic institutions (e.g., capitalism or socialism) might discover and realise the right activity levels. That’s for later. For now, we just want to know what the right answer is.

First, let’s simplify the problem of discovering the right activity levels by commanding our small economy to only produce 1 unit of corn (per unit of clock time):

And, then, an experiment: we’ll run the economy for 1 unit of clock-time, and watch what happens to the stocks. The change in stocks will help us to begin to think about what activity levels are “right” for producing 1 unit of corn.

This is what happens:

As we expect the stock of corn increases (after all, we’re producing corn!) And, as expected, the iron and labour stocks reduce because they are used-up to produce the corn. Why no change in sugar? Simply because sugar is not needed to produce corn (there’s no arrow that links sugar to corn in the technology graph shown in Figure 1).

Clearly, production of corn at these activity levels is unsustainable. At some point in the future we’ll run out of iron and labour time, at which point production will stop. In contrast, if we produced with the right activity levels, then we should be able to produce a surplus of corn indefinitely.

The final stock levels, after producing 1 unit of corn, are:

Why have the stocks changed by precisely these amounts? That’s simply due to the technology in use. Here’s the relevant part of the technology graph from Figure 1:

So the stock of labouring capacity reduces by 0.7 hours (to 0.3), and iron by 0.2 units (to 0.8). Although we produce 1 unit of corn we need to subtract 0.2 units of seed corn giving a net addition of 0.8 units (and therefore the final stock is not 2 but 1.8 units of corn).

So we’ve used-up some stocks to produce corn. Why not replace these used-up stocks by simply producing more? If we did that, then we might be able to produce 1 unit of additional corn *and replace all the used-up inputs as well.*

Let’s do just that. We produce an additional 0.2 corn and 0.2 iron, which will exactly replace the used-up stock.

But … we have a problem.

Producing an additional 0.2 units of corn and iron *also* uses-up stocks of corn and iron (and labour). We cannot produce something from nothing.

We can visualise the difficulty by “attaching” the relevant parts of the technology (from Figure 1) on top of the graph (from Figure 4). The added parts are shown as dashed red lines. We immediately see that producing 0.2 units of corn uses-up 0.04 corn, 0.04 iron and 0.14 iron (by multiplying the respective technology coefficients). And producing 0.2 units of iron uses-up 0.19 units of iron and 0.12 units of labour. This is getting complicated!

So replacing used-up inputs requires additional production that … uses-up more inputs! Have we hit a hopeless infinite regress?

Let’s try one more time, and imagine further production to replace the stocks that were *indirectly* used-up to replace the stocks that were *directly* used-up to produce the 1 unit of corn. Again, we can “attach” the input requirements to the top of the previous graph (of Figure 5).

Notice something? As we recursively attempt to replace more and more indirect used-up stocks, the branches in this tree multiply exponentially. That seems bad.

However, the *quantities* of corn and iron that need replacing are clearly *decreasing*. As we recursively step backwards, replacing more and more indirectly used-up stocks, it appears that we might be approaching a limit where ultimately no stocks need replacing.

And, in fact, this is precisely what is happening. The reason the quantities get smaller and smaller is because this economy is capable of producing a surplus.

Every time we “attach” more indirect production to the graph we perform a theoretical operation known as *vertical integration*. “Vertical” because we are, in some sense, proceeding “upwards” to more and more indirect production. And “integration” because — as we shall see in a moment — we end up adding together all these numbers.

Just for fun this is what vertical integration looks like after 6 iterations:

Notice how those numbers at the top of Figure 7 are approaching zero.

We can’t visualise an infinite vertical integration. But we can certainly calculate it. That’s because the technology coefficients do reach zero in the infinite limit, and we know how to calculate convergent infinite sums.

So simply imagine Figure 7 extending out to infinity. And then imagine adding up all the quantities of corn, iron and labour on all the direct and indirect input paths. At the end of the summation we have:

**0.25**units of corn**5**units of iron**3.875**days of labour

So, according to the logic of vertical integration, to replace all the stocks used-up (both directly and indirectly) when producing 1 unit of corn, we need to produce an additional 0.25 corn, 5 iron and 3.875 days of labouring capacity.

Is this right? Let’s check by producing the 1 unit of corn again, but this time with the new “vertically integrated” activity levels:

Notice we’ve set the number of workers to zero (the activity level of labour is 0), and instead stipulated that 3.875 days of labour services are ready for use. Why? Temporarily, I want to ignore the complication of workers consuming a real wage. (We’ll return to this issue in a subsequent post). So, for now, assume that workers don’t consume.

What happens now when we run the economy?

In other words, vertical integration has calculated the right activity levels. And as will become increasingly clearer, vertical integration holds the key to the solution of the coordination problem.

In practical terms these activity levels mean that, if we were to replenish the workers capacity to supply labour, then we could produce a surplus of corn indefinitely, without over or under producing iron. The economy never runs out of stocks — because it always replaces them.

We’ve made good progress in answering the question: what are the right activity levels? We know now, at least in this restricted case of producing 1 unit of corn, that vertical integration tells us the right levels. And vertical integration has a straightforward economic meaning: we just add up (hypothetical) direct and indirect production as we recurse “backwards” or “upwards” through the technology graph. The final sums turn out to be exactly the activity levels that are self-replacing.

Next, we should probably turn to generalising this approach to find the right activity levels for an integrated economy where workers consume.

How would you mathematically notate the inputs (say, the iron input to corn production) as a series?

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To follow up, I understand what the corn input to itself would look like: ∞Σ[n=1] 0.2^n. (Pardon the formatting.) But the notation for iron, as an input to corn, isn’t as obvious to me, since it’s a direct input to corn as well as to itself.

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Hi Joe

The standard way to represent this situation mathematically is to use a n by n matrix, A, where each element of the matrix, a_{i,j}, represents the quantity of commodity i directly used-up to produce 1 unit of commodity j, and a separate vector of labour coefficients, where each element l_{i} represents the direct labour used-up to produce commodity i. Then the process of vertical integration to calculate the labour-value vector v, where each element v_{i} is the total direct and indirect labour used-up to produce 1 unit of commodity i, involves computing the Leontief inverse of matrix A and then pre-multiplying by vector l. You can get more details here: https://ianwrightsite.wordpress.com/2017/05/19/just-what-is-a-labour-value-anyway/

Hope this helps,

Ian.

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