Self-replacing subsystems

Previously we discovered that, at least in the special case of producing 1 unit of corn, vertical integration tells us exactly how many other commodities we also need to produce (e.g., iron) to ensure that the economy can, in principle, continue to produce corn indefinitely. The economy is then self-replacing because, in addition to producing 1 unit of corn as output, it also produces just the right quantity of other things to replace all the inputs that were used-up when producing that corn. We therefore made progress towards understanding precisely what activity levels solve the coordination problem.

Let’s briefly review the operation of vertical integration. We start with a technology, which is a network of direct input-output relationships between sectors of production:

Figure 1. A technology is a set of input-output relations between sectors of production.

We then imagine producing 1 unit of corn. So we start at the corn sector, and then ask: what inputs do we need to produce corn? This is the first step in vertical integration. And then we ask: what inputs do we need to produce the inputs needed to produce corn? This is the second step. And we continue, moving from direct to indirect and further indirect production, “unrolling” the technology graph as we go along:

Figure 2. Vertical integration “unrolls” the technology graph as we move from direct to more indirect production. Here we’ve unrolled three times.

Eventually the input coefficients vanish to zero. We then sum along all input paths, for each commodity, to calculate the self-replacing activity levels. In this example, to produce 1 unit of corn as output, and replace used-up stocks, we need to produce 1.25 corn, 5 iron and 3.875 units of labour.

Define a self-replacing subsystem as the set of vertically-integrated activity levels that produces 1 unit of a particular commodity as final output. The self-replacing corn subsystem is:

Figure 3. A self-replacing subsystem is a set of activity levels that produce 1 unit of a particular commodity as output (and replaces all used-up inputs).

Imagine a subsystem as a self-contained economic machine that continually produces 1 unit of corn as output.

Let’s also stipulate that the 3.875 workers consume 1 unit of corn (per unit of labour supplied). (Imagine drawing an arrow from the 1 unit of corn output to the labour input node in Figure 3). So whatever corn “comes out” of the subsystem is fed straight “back into” the worker households, who consume the corn. Then the subsystem should also reproduce the working capacity of the labourers.

To check this we’ll setup the economy to operate as a self-replacing corn subsystem:

Figure 4. This is the same economy shown in Figure 1 except setup to operate as a self-replacing subsystem that (i) produces 1 unit of corn as final output, (ii) replaces used-up corn and iron inputs, and (iii) where workers consume the final output, i.e. they consume corn only at a rate of 1/3.875 = 0.26 units of corn (per hour of labour supplied).

What happens when we run this economy for 1 unit of clock time? Activity occurs, but the net effect on stock levels is zero:

Figure 5. The corn subsystem continually uses-up and produces corn and iron, and the workers continually supply labour and consume corn. But the net effect of these productive flows on the stocks of corn, iron and labour is zero. The subsystem is perfectly self-replacing.

At last, we have flat lines indicating no change in stock levels. A self-replacing subsystem is like an economic perpetual motion machine (as per the picture of the self-filling cup that accompanies this blog post).

But what if we want to support a population of 10,000 workers?

Self-replacing subsystems have the elegant property of being scalable. They continue to be self-replacing when we scale all the activity levels by a constant. For example, let’s scale up this subsystem by a factor of 2580.65:

Figure 6. We’ve scaled all the activity levels by the same constant. Now we’re producing much greater quantities of corn and iron with a population of 10,000 workers.

And when we run the economy again, with these new activity levels, we get the same result: zero net change in stock levels:

Figure 7. Production on a much bigger scale is taking place (10,000 workers are supplying labour in parallel). But this remains a self-replacing subsystem, so all stocks are continually and perfectly replaced.

In summary, a self-replacing subsystem is a vertically integrated “slice” of an economy that produces 1 unit of a particular commodity as output, and replaces all the used-up inputs. Scaling a subsystem “up” or “down” doesn’t change its self-replacing properties. We can operate a subsystem at any scale of production. Think of them as coordinated, self-contained units of integrated production.

---

Next we’ll compose an economy from its self-replacing subsystems, and discover the overall activity levels that satisfy any (feasible) level of final demand, which yields the theoretical (not practical!) solution to the coordination problem.