THE MATHEMATICAL SOLUTION
As has been pointed out in the Introduction, discussion of these questions in the English literature began relatively late and at a comparatively high level. Yet it can hardly be said that the first attempts really met any of the main points. Two Americans, Professor F. M. Taylor and Mr. W. C. Roper, were first in the field. Their analyses, and to some extent also that of Mr. H. D. Dickinson in this country 1 were directed to show that on the assumption of a complete knowledge of all relevant data, the values and the quantities of the different commodities to be produced might be determined by the application of the apparatus by which theoretical economics explains the formation of prices and the direction of production in a competitive system. Now it must be admitted that this is not an impossibility in the sense that it is logically contradictory. But to argue that a determination of prices by such a procedure being logically conceivable in any way invalidates the contention that it is not a possible solution, only proves that the real nature of the problem has not been perceived. It is only necessary to attempt to visualize what the application of this method would imply in practice in order to rule it out as humanly impracticable and impossible. It is clear that any such solution would have to be based on the solution of some such system of equations as that developed in Barone’s article in the Appendix. But what is practically relevant here is not the formal structure of this system, but the nature and amount of concrete information required if a numerical solution is to be attempted and the magnitude of the task which this numerical solution must involve in any modern community. The problem here is, of course, not how detailed this information and how exact the calculation would have to be in order to make the solution perfectly exact, but only how far one would have to go to make the result at least comparable with that which the competitive system provides. Let us look into this a little further.
In the first place it is clear that if central direction is really to take the place of the initiative of the manager of the individual enterprise and is not simply to be a most irrational limitation of his discretion in some particular respect, it will not be sufficient that it takes the form of mere general direction, but it will have to include and be intimately responsible for details of the most minute description. It is impossible to decide rationally how much material or new machinery should be assigned to any one enterprise and at what price (in an accounting sense) it will be rational to do so, without also deciding at the same time whether and in which way the machinery and tools already in use should continue to be used or be disposed of. It is matters of this sort, details of technique, the saving of one material rather than the other or any other of the small economies which cumulatively decide the success or failure of a firm, and in any central plan which is not to be hopelessly wasteful, they must be taken account of. In order to be able to do so it will be necessary to treat every machine, tool, or building not just as one of a class of physically similar objects, but as an individual whose usefulness is determined by its particular state of wear and tear, its location, and so on. The same applies to every batch of commodities which is situated at a different spot or which differs in any other respect from other batches. This means that in order to achieve that degree of economy in this respect which is secured by the competitive system, the calculations of the central planning authority would have to treat the existing body of instrumental goods as being constituted of almost as many different types of goods as there are individual units. And so far as ordinary commodities, i.e. non-durable semi-finished or finished goods are concerned, it is clear that there would be many times more different types of such commodities to consider than we should imagine if they were classified only by their technical characteristics. Two technically similar goods in different places or in different packings or of a different age cannot possibly be treated as equal in usefulness for most purposes if even a minimum of efficient use is to be secured.
Now since in a centrally directed economy the manager of the individual plant would be deprived of the discretion of substituting at will one kind of commodity for another, all this immense mass of different units would necessarily have to enter separately into the calculations of the planning authority. It is obvious that the mere statistical task of enumeration exceeds anything of this sort hitherto undertaken. But that is not all. The information which the central planning authority would need, would also include a complete description of all the relevant technical properties of every one of these goods, including costs of movement to any other place where it might possibly be used with greater advantage, cost of eventual repair or changes, etc. etc.
But this leads to another problem of even greater importance. The usual theoretical abstractions used in the explanation of equilibrium in a competitive system include the assumption that a certain range of technical knowledge is “given”. This, of course, does not mean that all the best technical knowledge is concentrated anywhere in a single head, but that people with all kinds of knowledge will be available and that among those competing in a particular job, speaking broadly, those that make the most appropriate use of the technical knowledge will succeed. In a centrally planned society this selection of the most appropriate among the known technical methods will only be possible if all this knowledge can be used in the calculations of the central authority. This means in practice that this knowledge will have to be concentrated in the heads of one or at best a very few people who actually formulate the equations to be worked out. It is hardly necessary to emphasize that this is an absurd idea even in so far as that knowledge is concerned which can properly be said to “ exist ” at any moment of time. But much of the knowledge that is actually utilized is by no means “ in existence ” in this ready-made form. Most of it consists in a technique of thought which enables the individual engineer to find new solutions rapidly as soon as he is confronted with new constellations of circumstances. To assume the practicability of these mathematical solutions, we should have to assume that the concentration of knowledge at the central authority would also include a capacity to discover any improvement of detail of this sort.1
There is a third set of data which would have to be available before the actual operation of working out the appropriate method of production and quantities to be produced could be undertaken, data relative to importance of the different kinds and quantities of consumers’ goods. In a society where the consumer was free to spend his income as he liked, these data would have to take the form of complete lists of the different quantities of all commodities which would be bought at any possible combination of prices of the different commodities which might be available. These figures would inevitably be of the nature of estimates for a future period based upon past experience. But past experience cannot provide the range of knowledge necessary. And as tastes change from moment to moment, the lists would have to be in process of continuous revision.
It is probably evident that the mere assembly of these data is a task beyond human capacity. Yet if the centrally run society were to work as efficiently as the competitive society, which as it were decentralizes the task of collecting them, they would have to be present. But let us assume for the moment that this difficulty, the “mere difficulty of statistical technique”, as it is contemptuously referred to by most planners, is actually overcome. This would be only the first step in the solution of the main task. Once the material is collected it would still be necessary to work out the concrete decisions which it implies. Now the magnitude of this essential mathematical operation will depend on the number of unknowns to be determined. The number of these unknowns will be equal to the number of commodities which are to be produced. As we have seen already, we have to take as different commodities all the final products to be completed at different moments, whose production has to be started or to be continued at present. At present we can hardly say what their number is, but it is hardly an exaggeration to assume that in a fairly advanced society, the order of magnitude would be at least in the hundreds of thousands. This means that, at each successive moment, every one of the decisions would have to be based on the solution of an equal number of simultaneous differential equations, a task which, with any of the means known at present, could not be carried out in a lifetime. And yet these decisions would not only have to be made continuously, but they would also have to be conveyed continuously to those who had to execute them.
It will probably be said that such a degree of exactitude would not be necessary, since the working of the present economic system itself does not come anywhere near it. But this is not quite true. It is clear that we never come near the state of equilibrium described by the solution of such a system of equations. But that is not the point. We should not expect equilibrium to exist unless all external change had ceased. The essential thing about the present economic system is that it does react to some extent to all those small changes and differences which would have to be deliberately disregarded under the system we are discussing if the calculations were to be manageable. In this way rational decision would be impossible in all these questions of detail, which in the aggregate decide the success of productive effort.
It is improbable that any one who has realized the magnitude of the task involved has seriously proposed a system of planning based on comprehensive systems of equations. What has actually been in the minds of those who have mooted this kind of analysis has been the belief that, starting from a given situation, which was presumably to be that of the pre-existing capitalistic society, the adaptation to the minor changes which occur from day to day could be gradually brought about by a method of trial and error. This suggestion suffers, however, from two fundamental mistakes. In the first instance, as has been pointed out many times, it is inadmissible to assume that the changes in relative values brought about by the transition from capitalism to socialism would be of a minor order, so permitting the prices of the pre-existing capitalistic system to be used as a starting-point, and making it possible to avoid a complete re-arrangement of the price-system. But even if we neglect this very serious objection, there is not the slightest reason to assume that the task could be solved in this way. We need only to remember the difficulties experienced with the fixing of prices, even when applied to a few commodities only, and to contemplate further that, in such a system, price-fixing would have to be applied not to a few but to all commodities, finished or unfinished, and that it would have to bring about as frequent and as varied price-changes as those which occur in a capitalistic society every day and every hour, in order to see that this is not a way in which the solution provided by competition can even be approximately achieved. Almost every change of any single price would make changes of hundreds of other prices necessary and most of these other changes would by no means be proportional but would be affected by the different degrees of elasticity of demand, by the possibilities of substitution and other changes in the method of production. To imagine that all this adjustment could be brought about by successive orders by the central authority when the necessity is noticed, and that then every price is fixed and changed until some degree of equilibrium is obtained is certainly an absurd idea. That prices may be fixed on the basis of a total view of the situation is at least conceivable, although utterly impracticable; but to base authoritative price-fixing on the observation of a small section of the economic system is a task which cannot be rationally executed under any circumstances. An attempt in this direction will either have to be made on the lines of the mathematical solution discussed before, or else entirely abandoned.
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