A theoretical solution to the coordination problem

To solve the coordination problem we need to operate the economy at the correct activity levels such that final demand is satisfied without over or under production. A fully coordinated economy produces the population’s consumption bundle without wasteful accumulation, or unsustainable depletion, of stocks.

Of course, final demand continually changes, and so do the techniques of production. So a complete solution of the coordination problem requires discovering, and re-discovering, the correct activity levels, and ensuring that the economy conforms to those levels.

We won’t tackle the full problem quite yet. First, we need to determine the correct activity levels for a given scale and composition of final demand.

And we’ll solve this problem by composing together a set of self-replacing subsystems (or simply subsystems) where each subsystem is dedicated to producing one component of final demand.

This will be a theoretical solution. We as yet say nothing about possible economic institutions that may in fact reach (or continually grope toward) the coordinated state.

So let’s begin. Consider the following small economy that produces corn, iron and sugar using the following technologies:

Figure 1. A small economy.

Worker households consume 0.2272 corn and 0.10013 of sugar per hour of labour they supply. (You may wonder why these particular numbers were chosen. For now, just take them as given, and we’ll make a note to discuss feasible consumption another time).

Let’s say there are 100 workers in the population. We therefore need to find the right activity levels for corn, iron and sugar production to produce the consumption requirements of all 100 workers (i.e., produce as output the right amounts of corn, in the form of bread, and sugar, in the form of cake).

(Of course, workers consume different consumption bundles. And we may include this finer-grained modelling granularity later. But, for now, think of the consumption coefficients as defining aggregate consumption across all households).

What are the right activity levels then? We first construct the subsystems by vertical integration.

Recall that a subsystem is the set of activity levels that produces 1 unit of output of a specific commodity, and perfectly replaces all the stock used-up. Here’s the corn subsystem (post vertical integration and summing):

Figure 2. The self-replacing corn subsystem. (If we operate at these activity levels we produce 1 unit of corn as output, and replace all used-up stocks).

And here’s the sugar subsystem:

Figure 3. The self-replacing sugar subsystem.

OK, we’ve now constructed the two subsystems that correspond to the two commodities that workers demand.

Workers demand 0.2272 corn per hour of labour they supply. And 100 workers supply labour in parallel, and therefore supply 100 hours of labour per unit of clock time. Recall that we can scale subsystems “up” or “down” and they remain self-replacing. So if we re-scale the corn subsystem by 0.2272 x 100 = 22.72 then it will produce sufficient corn to satisfy all 100 workers.

Similarly, workers demand 0.10013 sugar per hour of labour. So if we re-scale the sugar subsystem by 0.10013 x 100 = 10.013 then it will produce sufficient sugar for everyone.

Scaling the coefficients in each subsystem gives:

Figure 4. The input quantities of the re-scaled corn and sugar subsystems.

Subsystems have another remarkable property. Not only can we re-scale them, we can compose them.

The corn subsystems produces the right amount of corn. And the sugar subsystem produces the right amount of sugar. What if we add the subsystems together to form a composite subsystem that produces the right amount of sugar and corn, and replaces all the used-up stock when producing both these outputs?

Adding the subsystems in Figure 4 gives:

Figure 5. The composed and re-scaled corn and sugar subsystems.

The activity levels in Figure 5 should be exactly the answer we’re looking for. We expect — by the logic of vertical integration, and the fact we can re-scale and compose subsystems — that if we run our economy at these levels we’ll neither over or under produce any commodity, and all 100 workers will consume exactly what they need.

Let’s check by running the economy at these levels:

Figure 6. The stocks of a perfectly coordinated economy are constant over time. But underneath this smooth surface a huge amount of activity continually occurs: each hour 100 workers supply labour to produce ~31 units of corn, ~125 units of iron and ~11 units of sugar. And each hour the 100 workers consume ~23 units of corn and ~10 units of sugar. The initial (quite small) stock levels are maintained throughout (including some spare capacity to perform work without consumption).

As we can see from Figure 6 the economy is perfectly coordinated.

Although this is a simple example, with only three commodities, we shouldn’t underestimate what we’ve achieved. We can now calculate the coordinated activity levels for any economy of any size (such as, for example, economies consisting of many millions of commodities and workers).

In summary, to solve the coordination problem we re-scale and compose the self-replacing subsystems that correspond to each commodity present in final demand.

Next, we’ll linger a little longer on the idea of a subsystem. Subsystems have something very important to tell us about Marx’s concept of labour values. And they can also tell us something new about the total working day. After lingering here, we should probably turn our attention to how a market economy solves the very same coordination problem.

(The operation of constructing subsystems, by vertical integration, and then re-scaling and composing them, is mathematically equivalent to solving certain matrix inverse problems. For example, the method relates to the kind of problems posed and solved by the Soviet mathematician Kantorovich who, in the 1930s, pioneered the technique of linear programming in order to solve problems of economic planning. He was awarded both a Nobel and Stalin prize for these advances. The method also directly relates to the work of Leontief who, around the same time, pioneered the input-output approach to economics, for which he was also awarded a Nobel prize.

An exposition in terms of vertically-integrated subsystems is hopefully simpler and clearer compared an exposition in terms of linear algebra. The concept of a subsystem originates with the Cambridge economist and Marxist, Piero Sraffa, and the concept was further elaborated and developed by his pupil, Luigi Pasinetti. In addition, we can visualise vertical integration as recursing “backwards” through a technology graph, which I hope readers will find intuitive. And, as we shall see later, the concepts of vertical integration and self-replacing subsystem lead to deeper insights into the structure of an economy.)


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