An Answer To Ludwig Von Mises

The Start Of This Video Is An Anecdote About George Dantzig

1.0 Introduction

Ludwig Von Mises popularized the Socialist Calculation Problem and brought it to wider attention. (This problem is also known as the Economic Calculation Problem.) In Von Mises' 1920 paper, he argues rational economic planning is impossible without market prices for capital goods and unproduced resources. Thus, anybody advocating socialism is advocating a system that cannot obtain a high material standard of living. In this post, I want to offer a simple solution to Von Mises' statement of the problem.

2.0 Informal Statement of the Problem

Von Mises thinks that socialism must imply central planning. He assumes that markets for consumer goods exist. Households are given tokens with which they can use to purchase consumer goods, including from one another. He describes a process that will eventually result in equilibrium on markets for consumer goods. The planning authority can use these prices in their planning.

But markets do not exist for capital goods, for goods of higher order in the jargon of the Austrian school. Von Mises asserts that without prices on such markets, the planning authority cannot engage in rational economic accounting. It is an impossibility argument based on economic theory.

Many have provided models of socialism, including with full awareness of the SCP. Oskar Lange had the most well-known answer. Both Von Mises and Lange based themselves on marginalist equilibrium theory.

Price ... may have the generalized meaning of 'terms on which alternatives are offered.' ...It is only prices in the generalized sense which are indispensable to solving the problem of allocation of resources. The economic problem is a problem of choice among alternatives. To solve the problem three data are needed: (1) a preference scale which guides the act of choice; (2) knowledge of the 'terms on which alternatives are offered'; and (3) knowledge of the amount of resources available. These three data being given, the problem of choice is soluble.

Now it is obvious that a socialist economy may regard the data under 1 and 3 as given, in at least as great a degree as they are given in a capitalist economy. The data under 1 may be given by the demand schedules of the individuals or be judged by the authorities administering the economic system. The question remains whether the data under 2 are accessible to the administrators of a socialist economy. Professor Mises denies this. However, a careful study of price theory and of the theory of production convinces us that, the data under 1 and 3 being given, the 'terms on which alternatives are offered' are determined ultimately by the technical possibilities of transformation of one commodity into another, i.e., by the production functions. The administrators of a socialist economy will have exactly the same knowledge, or lack of knowledge, of the production functions as the capitalist entrepreneurs have. -- Oskar Lange

In this post, I, too, base myself on marginalist theory, but on a Linear Programming formulation. In demonstrating Von Mises mistaken, I put aside that he, like marginalists everywhere, does not have a price theory that applies to actually-existing capitalist economies.

Table 1: Parameters and Variables

Symbol	Variable or Parameter	Definition
k	Parameter	The number of consumer goods, also known as the number of goods for final demand.
m	Parameter	The number of activities or processes.
n	Parameter	The number of resources.
p	Parameter	A k-element row vector of prices of consumer goods.
x	Parameter	A n-element column vector of the available quantities of resources.
A	Parameter	n x m matrix. A column of A represents the resources used up by an activity, that is, a production process.
B	Parameter	A k x m matrix. A column of B represents the final goods produced by that activity.
q	Variable	A m-element column vector of levels of operations of the processes.
w	Variable	A n-element column vector of resource prices.

3.0 A Linear Programming Solution

I now turn to a formal statement and solution of the problem. I assume that the central planner knows the parameters listed in Table 1. The planner's task is to set the values of the elements of the vector q. It is not required to operate all processes. Some elements of q can be set to zero. Likewise, not all resources must be fully used. Some resources may be left in excess supply. But the planner cannot use more of a resource than exists. An important requirement of the problem is that the planner does not know w. Resources that are inputs to production do not have prices.

So the planner must choose q to maximize:

p B q

such that:

A q ≤ x

x ≥ 0

The above is a Linear Program. It has a dual program. This is where shadow prices come in. The dual problem is to choose w to minimize:

x^T w

such that:

A^T w ≥ B^T p^T

w ≥ 0

The value of the objective functions are equal for solutions of the primal and dual problems. Thus the solution of the primal, which does not require knowledge of resource prices, assigns prices to those resources. Von Mises did not understand duality theory.

Some theorems draw more connections between the solutions to the primal and dual problems. Suppose that a constraint in the primal problem is met with an inequality in the solution. Some resource is not fully used. Then its price in the solution to the dual is zero. And, contrawise, suppose a constraint in the dual is met with an inequality in its solution. The cost of running a process with the chosen prices exceeds the value of the goods produced by that process. Then that process will not be operated in the solution to the primal.

How is this marginalism? This is certainly an example of the choice of the allocation of scarce resources among alternatives. Consider the impact on the solution of an incremental increase in the quantity of a specified resource. This increase introduces slack in the corresponding constraint in the primal. The value of the objective function is increased. The shadow price of that resource, if I recall correctly, is the value of the marginal product of the resource.

This math relates to an intervention in economics by Von Neumann and to Leontief's input-output analysis. The simplex algorithm is widely used to solve the above problems. Koopmans and Kantorovich shared the economics pseudo Nobel prize for Linear Programming. Leontief received a stand-alone pseudo Nobel.

4.0 Caveats

The above answers the problem, as stated by Von Mises (1920). Von Mises did not understand duality theory.

I can think of some difficulties to the above formulation of the problem. How would one accomodate economic growth? Can final demand include commodities not destined for consumption, and, if so, where would their prices come from? I suppose the planner could include such commodities, based on last mix of inputs last year in the industries that one wants to increase the output of. And their prices could then be last year's shadow prices. This approach leans toward input-output analysis, a better approach than the marginalist approach of Von Mises.

Another question is where do the consumers get their money-like tokens with which they purchase consumer goods? Are these labor vouchers? Some sort of income paid out of the total surplus regardless of contributions? Von Mises allows for a range of institutions for compensating various types of labor. Total equality need not be enforced.

Oskar Lange felt he had to go further than state the equations that must be solved. He had the planning authority simulating a tatonnement process, a trial and error solution. The managers of factories are instructed to treat the 'prices' issued by the central planner as given parameters. Lange, like others of his time, did not understand the difficulties in modeling capitalist markets by a static general equilibrium model.

Lange proposed a trial and error solution partly because he was responding to Robbins:

"On paper we can conceive the problem to be solved by a series of mathematical calculations... But in practice this solution is quite unworkable. It would necessitate the drawing up of millions of equations on the basis of millions of statistical data based on many more of millions of individual computations. By the time the equations were solved, the information on which they were based would have become obsolete and they would need to be calculated anew. The suggestion that a practical solution of the problem on planning is possible on the basis of the Paretian equations simply indicates that those who put it forward have not grasped what those equations mean." -- Lionel Robbins, as quoted by Lange (1938)

The analysis of the SCP nowdays should draw on the theory of computational complexity. One can count the computations that must be done to solve the problem. Presumably the matrices in the formulation of the problem are sparse. Cottrell and Cockshott are the authors I look to for these questions. I see the 2008 Gödel prize was awarded for work on the simplex method beyond me.

Von Mises does not consider the time or computational complexity for solving the equations. He says no calculations are possible. Likewise, Von Mises' objections are not about the difficulties of the central planner acquiring the data on prices for consumer goods, technological possibilities in physical units (in natura), or the difficulties in articulating dispersed tacit knowledge. These difficulties are later stated by Hayek when he and others changed the objection to an argument about the practicality, not possibility, of rational economic accounting under socialism. Along with this change in the statement of the problem, the Austrian school began to differentiate themselves from other marginalists. Von Mises attempted to pose the problem on a common ground shared by all marginalist economists.

References

• Allin Cottrell and W. Paul Cockshott. 1993. Calculation, Complexity and Planning: The Socialist Calculation Debate Once Again. Review of Political Economy 5: 73-112.
• Allin Cottrell and W. Paul Cockshott. 2007. Against Hayek. MPRA Working Paper No. 6062.
• Karras J. Lambert and Tate Fegley. 2023. Economic calculation in light of advances in big data and artificial intelligeence. Journal of Economic Behavior & Organization 206: 243-250.
• Oskar Lange and Fred M. Taylor. 1938. On the Economic Theory of Socialism. (ed. by Benjamin E. Lippincott). University of Minnesota Press.
• Tiago Camarinha Lopes. 2021. Technical or political? The socialist economic calculation debate. Cambridge Journal of Economics 45(4): 787-810.
• Ludwig Von Mises. 1920. Economic calculation in the socialist commonwealth. (Trans. and reprinted in Collectivist Economic Planning (ed. by F. A. Hayek). Routledge and Kegan Paul, 1935).
• Luigi L. Pasinetti. 1977. Lectures on the Theory of Production. Columbia University Press.
• John E. Roemer. 1994. A Future for Socialism. Harvard University Press.