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Front Page Titles (by Subject) 6: The Differential Equations of Mathematical Economics - Human Action: A Treatise on Economics, vol. 3 (LF ed.)

## 6: The Differential Equations of Mathematical Economics - Ludwig von Mises, Human Action: A Treatise on Economics, vol. 3 (LF ed.) [1996]

##### Edition used:

Human Action: A Treatise on Economics, in 4 vols., ed. Bettina Bien Greaves (Indianapolis: Liberty Fund, 2007). Vol. 3.

#### Part of: Human Action: A Treatise on Economics, in 4 vols. (LF ed.)

Liberty Fund, Inc. is a private, educational foundation established to encourage the study of the ideal of a society of free and responsible individuals.

### The Differential Equations of Mathematical Economics

In order to appraise adequately the idea that the differential equations of mathematical economics could be utilized for socialist economic calculation, we must remember what these equations really mean.

In devising the imaginary construction of an evenly rotating economy we assume that all the factors of production are employed in such a way that each of them renders the most highly valued services it can possibly render. No further change in the employment of any of these factors could improve the state of want-satisfaction under prevailing conditions. This situation, in which no further changes in the disposition of the factors of production are resorted to, is described by systems of differential equations. However, these equations do not provide any information about the human actions by means of which the hypothetical state of equilibrium has been reached. All they say is this: If, in this state of static equilibrium, m units of a are employed for the production of p, and n units of a for the production of q, no further change in the employment of the available units of a could result in an increment in want-satisfaction. (Even if we assume that a is perfectly divisible and take the unit of a as infinitesimal, it would be a serious blunder to assert that the marginal utility of a is the same in both employments.)

This state of equilibrium is a purely imaginary construction. In a changing world it can never be realized. It differs from today’s state as well as from any other realizable state of affairs.

In the market economy it is entrepreneurial action that again and again reshuffles exchange ratios and the allocation of the factors of production. An enterprising man discovers a discrepancy between the prices of the complementary factors of production and the future prices of the products as he anticipates them, and tries to take advantage of this discrepancy for his own profit. The future price which he has in mind is, to be sure, not the hypothetical equilibrium price. No actor has anything to do with equilibrium and equilibrium prices; these notions are foreign to real life and action; they are auxiliary tools of praxeological reasoning for which there is no mental means to conceive the ceaseless restlessness of action other than to contrast it with the notion of perfect quiet. For the theorists’ reasoning every change is a step forward on a road which, provided no further new data appear, finally leads to a state of equilibrium. Neither the theorists, nor the capitalists and entrepreneurs, nor the consumers, are in a position to form, on the ground of their familiarity with present conditions, an opinion about the height of such an equilibrium price. There is no need for such an opinion. What impels a man toward change and innovation is not the vision of equilibrium prices, but the anticipation of the height of the prices of a limited number of articles as they will prevail on the market on the date at which he plans to sell. What the entrepreneur, in embarking upon a definite project, has in mind is only the first steps of a transformation which, provided no changes in the data occur other than those induced by his project, would result in establishing the state of equilibrium.

But for a utilization of the equations describing the state of equilibrium, a knowledge of the gradation of the values of consumers’ goods in this state of equilibrium is required. This gradation is one of the elements of these equations assumed as known. Yet the director knows only his present valuations, not also his valuations under the hypothetical state of equilibrium. He believes that, with regard to his present valuations, the allocation of the factors of production is unsatisfactory and wants to change it. But he knows nothing about how he himself will value on the day the equilibrium will be reached. These valuations will reflect the conditions resulting from the successive changes in production he himself inaugurates.

We call the present day D1 and the day the equilibrium will be established Dn. Accordingly we name the following magnitudes corresponding to these two days: the scale of valuation of the goods of the first order V1 and Vn, the total supply9 of all original factors of production O1 and On, the total supply of all produced factors of production P1 and Pn, and summarize O1P1 as M1 and OnPn as Mn. Finally we call the state of technological knowledge, T1 and Tn. For the solution of the equations a knowledge of Vn, OnPnMn, and Tn is required. But what we know today is merely V1, O1P1M1, and T1.

It would be impermissible to assume that these magnitudes for D1 are equal to those for Dn, because the state of equilibrium cannot be attained if further changes in the data occur. The absence of further changes in the data which is the condition required for the establishment of equilibrium refers only to such changes as could derange the adjustment of conditions to the operation of those elements which are already operating today. The system cannot attain the state of equilibrium if new elements, penetrating from without, divert it from those movements which tend toward the establishment of equilibrium.10 But as long as the equilibrium is not yet attained, the system is in a continuous movement which changes the data. The tendency toward the establishment of equilibrium, not interrupted by the emergence of any changes in the data coming from without, is in itself a succession of changes in the data.

P1 is a set of magnitudes that do not correspond to today’s valuations. It is the outcome of actions which were guided by past valuations and faced a state of technological knowledge and of information about available resources of primary factors of production which was different from the present state. One of the reasons why the system is not in equilibrium is precisely the fact that P1 is not adjusted to present conditions. There are plants, tools, and supplies of other factors of production which would not exist under equilibrium, and other plants, tools, and supplies must be produced in order to establish equilibrium. Equilibrium will emerge only when these disturbing parts of P1, as far as they are still utilizable, will be worn out and replaced by items which correspond to the state of the other synchronous data, viz., V, O, and T. What acting man needs to know is not the state of affairs under equilibrium, but information about the most appropriate method of transforming, by successive steps, P1 into Pn. With regard to this task the equations are useless.

One cannot master these problems by eliminating P and relying only upon O. It is true that the mode of utilizing the original factors of production uniquely determines the quality and quantity of the produced factors of production, the intermediary products. But the information that could be won in this way refers only to the conditions of equilibrium. It does not tell us anything about the methods and procedures to be resorted to for the realization of equilibrium. Today we are confronted with a supply of P1 which differs from the state of equilibrium. We must take into account real conditions, i.e., P1, and not the hypothetical conditions of Pn.

This hypothetical future state of equilibrium will appear when all methods of production have been adjusted to the valuations of the actors and to the state of technological knowledge. Then one will work in the most appropriate locations with the most adequate technological methods. Today’s economy is different. It operates with other means which do not correspond to the equilibrium state and cannot be taken into account in a system of equations describing this state in mathematical symbols. The knowledge of conditions which will prevail under equilibrium is useless for the director whose task it is to act today under present conditions. What he must learn is how to proceed in the most economical way with the means available today which are the inheritance of an age with different valuations, a different technological knowledge, and different information about problems of location. He must know which step is the next he must make. In this dilemma the equations provide no help.

Let us assume that an isolated country whose economic conditions are those of Central Europe in the middle of the nineteenth century is ruled by a dictator who is perfectly familiar with the American technology of our day. This director knows by and large to what goal he should lead the economy of the country entrusted to his care. Yet even a full knowledge of today’s American conditions could not be of use to him in regard to the problem of transforming by successive steps, in the most appropriate and expedient way, the given economic system into the system aimed at.

Even if, for the sake of argument, we assume that a miraculous inspiration has enabled the director without economic calculation to solve all problems concerning the most advantageous arrangement of all production activities and that the precise image of the final goal he must aim at is present to his mind, there remain essential problems which cannot be dealt with without economic calculation. For the director’s task is not to begin from the very bottom of civilization and to start economic history from scratch. The elements with the aid of which he must operate are not only natural resources untouched by previous utilization. There are also the capital goods produced in the past and not convertible or not perfectly convertible for new projects. It is in precisely these artifacts, produced under a constellation in which valuations, technological knowledge and many other things were different from what they are today, that our wealth is embodied. Their structure, quality, quantity, and location is of primary importance in the choice of all further economic operations. Some of them may be absolutely useless for any further employment; they must remain “unused capacity.” But the greater part of them must be utilized if we do not want to start anew from the extreme poverty and destitution of primitive man and want to survive the period which separates us from the day on which the reconstruction of the apparatus of production according to the new plans will be accomplished. The director cannot merely erect a new construction without bothering about his wards’ fate in the waiting period. He must try to take advantage of every piece of the already available capital goods in the best possible way.

Not only the technocrats, but socialists of all shades of opinion, repeat again and again that what makes the achievement of their ambitious plans realizable is the enormous wealth hitherto accumulated. But in the same breath they disregard the fact that this wealth consists to a great extent in capital goods produced in the past and more or less antiquated from the point of view of our present valuations and technological knowledge. As they see it, the only aim of production is to transform the industrial apparatus in such a way as to make life more abundant for later generations. In their eyes contemporaries are simply a lost generation, people whose only purpose it must be to toil and trouble for the benefit of the unborn. However, real men are different. They want not only to create a better world for their grandsons to live in; they themselves also want to enjoy life. They want to utilize in the most efficient way those capital goods which are now available. They aim at a better future, but they want to attain this goal in the most economical way. For the realization of this desire too they cannot do without economic calculation.

It was a serious mistake to believe that the state of equilibrium could be computed, by means of mathematical operations, on the basis of the knowledge of conditions in a nonequilibrium state. It was no less erroneous to believe that such a knowledge of the conditions under a hypothetical state of equilibrium could be of any use for acting man in his search for the best possible solution of the problems with which he is faced in his daily choices and activities. There is therefore no need to stress the point that the fabulous number of equations which one would have to solve each day anew for a practical utilization of the method would make the whole idea absurd even if it were really a reasonable substitute for the market’s economic calculation.11

### The Government and the Market

[9. ]Supply means a total inventory in which the whole supply available is specified in classes and quantities. Each class comprehends only such items as have in any regard (for instance, also in regard to their location) precisely the same importance for want-satisfaction.

[10. ]Of course, we may assume that T1 is equal to Tn if we are prepared to imply that technological knowledge has reached its final stage.

[11. ]With regard to this algebraic problem, cf. Pareto, Manuel d’économie politique (2d ed. Paris, 1927), pp. 233 f.; and Hayek, Collectivist Economic Planning (London, 1935), pp. 207–14. Therefore the construction of electronic computers does not affect our problem.