The simplest economy would be production and consumption of two goods by one person. To imagine a typical neoclassical graph, we have the production possibility frontier (PPF) designating the trade-off between the goods and the optimum point on it that corresponds to the highest tangent indifference curve (IC).
In a real economy, there are many complications:
- There are multiple consumers, and so preferences cannot be aggregated into a single IC, nor interpersonal utility comparisons made.
- There are millions of projects producing consumer goods with millions of types of resources, so both the PPF and the IC are multi-dimensional. Technological recipes are complex.
- ICs shift due to changes in consumer preferences.
- PPFs shift due to new technologies and discoveries of new resources.
- Unpredictable acts of God occur all the time.
Let's abstract away from (1) by having only one person in our economy, Crusoe, who has somehow managed to build and operate an economy the size of our actual global economy. It is easy for him to rank bundles of consumer goods. (This will substitute nicely for our "heavenly" society, as well.) Let's suppose at some time Crusoe has an evenly rotating economy with a system of technological equations like this:
m*P1 = m*(5a1 + 7a2 + 10a15...)
n*P2 = n*(9a1 + 3a15 + 22a42...)
z*P1,000,000 = ...
P represents a product; a, some resource, whether original or produced. A utility is associated with each marginal P; Crusoe wants to arrange production by allocating resources in such a way as to maximize his happiness.
Resources like a1 are (1) scarce (Crusoe only has so many a1s), (2) heterogeneous (a1 cannot be fully substituted for a2), (3) partially specific (a1 can be used in production of many Ps). We do not grant Crusoe technological omniscience, so let's suppose he discovers a new way of producing P7. By taking 3 marginal units of a1 (say, 12a1) from 3*P1 for use in the new method, Crusoe unemploys 3 units of all the other resources: 21a2, 30a15, etc. Where shall they go in the whole scheme of things? Suppose we suggest that 8a2 go into P50, although there are many other possibilities. But that means that the factors complementary to a2 for producing P50, such as a104, a451, ..., also have to be increased. From what other projects shall they be taken away to be used in P50? And so on, in a branching fashion. The consequences of even a single change must result in the rearrangement of the entire production system. And there are numerous possibilities.
The problem is not to solve the system of equations; it’s to generate a new system that's superior to the old one upon gaining some new knowledge. Crusoe's problem is to deal with novelties, to improve his production as he learns things previous unknown and unsuspected. At every moment, Crusoe is not only ignorant of numerous things; he doesn't even know what sorts of things he does not know. He is constantly surprised by new data.
With the new discovery, there is a new PPF. There are two distinct problems here. One is to find any improvement on the new PPF.
The other is to find the optimal point or point of highest utility on the new PPF.
Suppose now that Crusoe has access to a powerful computer. Can he program it to solve either of these problems? What sort of problem is it to find a better / best allocation of resources, computationally? Is it tractable or not? I suspect that neither is a class P-problem. I think they are both exponential-time O(2n) problems. Moreover, while the first problem is an NP-problem, as in, can be easily verified given a solution (simply compare the total utilities of the solution and the original system); the second problem is not even that, because to verify that a solution is best, you'd have to sift through all production possibilities, i.e., verifying a solution is as hard as finding it.
As a result, a real-world economy cannot be run by a single man even with great computational resources. It seems that Crusoe requires more people to own and run his factories. There must be a division not just of labor of workers within factories but also of productive activities among profit-seeking entrepreneurs running the factories. This is relevant to the question "What constrains the size of a firm?" Firms in the market economy cannot get too big, lest they become unable to adjust to new market data.
I conclude that Crusoe cannot run the world even if he is the sole human on earth and has a computer the size of the moon to assist him.