Von Mises Wrong On Economic Calculation (Update)

1.0 Introduction

This post is an update, based on suggestions from a user on reddit. I have explained this before. Suppose one insists socialism requires central planning. In his 1920 paper, 'Economic calculation in the socialist commonwealth', Ludwig Von Mises claims that a central planner requires prices for capital goods and unproduced resources to successfully plan an economy. The claim that central planning is impossible without market prices is supposed to be a matter of scientific principle.

Von Mises was mistaken. His error can be demonstrated to follow from the theory of linear programming and duality theory. This application of linear programming reflects a characterization of economics as the study of the allocation of scarce means among alternative uses. This post demonstrates that Von Mises was mistaken without requiring, hopefully, anything more than high school mathematics to understand what is being claimed.

2.0 Technology, Endowments, and Prices of Consumer Goods as given

For the sake of argument, Von Mises assumes the central planner has available certain data. He wants to demonstrate his conclusion, while conceding as much as possible to his supposed opponent.

Accordingly, assume the central planner knows the technology with the coefficients of production in Table 1. Two goods, wheat and barley are to be produced and distributed to consumers. Each good is produced from inputs of labor, land, and tractors. The column for Process I shows the person-years of labor, acres of land, and number of tractors needed, per quarter wheat produced. The column for Process II shows the inputs, per bushel barley, for the first production process known for producing barley. The column for Process III shows the inputs, per bushel barley, for the second process known for producing barley. The remaining two processes are alternative processes for producing tractors from inputs of labor and land.

Table 1: The Technology

Input	Process I	Process II	Process III	Process IV	Process V
Labor	a1,1	a1,2	a1,3	a1,4	a1,5
Land	a2,1	a2,2	a2,3	a2,4	a2,5
Tractors	a3,1	a3,2	a3,3	0	0
Output	1 quarter wheat	1 bushel barley	1 bushel barley	1 tractor	1 tractor

A more advanced example would have at least two periods, with dated inputs and outputs. I also abstract from the requirement that only an integer number of tractors can be produced. A contrast between wheat and barley illustrates that the number of processes known to produce a commodity need not be the same for all commodities.

Von Mises assumes that the planner knows the price of consumer goods. In the context of the example, the planner knows:

• The price of a quarter wheat, p₁.
• The price of a bushel barley, p₂.

Finally, the planner is assumed to know the physical quantities of resources available. Here, the planner is assumed to know:

• The person-years, x₁, of labor available.
• The acres, x₂, of land available.

No tractors are available at the start of the planning period in this formulation.

3.0 The Central Planner's Problem

The planner must decide at what level to operate each process. That is, the planner must set the following:

• The quarters wheat, q₁, produced with the first process.
• The bushels barley, q₂, produced with the second process.
• The bushels barley, q₃, produced with the third process.
• The number of tractors, q₄, produced with the fourth process.
• The number of tractors, q₅, produced with the fifth process.

These quantities are known as 'decision variables'.

The planner has an 'objective function'. In this case, the planner wants to maximize the objective function:

Maximize p₁ q₁ + p₂ q₂ + p₂ q₃

The planner faces some constraints. The plan cannot call for more employment than labor is available:

a_1,1 q₁ + a_1,2 q₂ + a_1,3 q₃ + a_1,4 q₄ + a_1,5 q₅ ≤ x₁

More land than is available cannot be used:

a_2,1 q₁ + a_2,2 q₂ + a_2,3 q₃ + a_2,4 q₄ + a_2,5 q₅ ≤ x₂

The number of tractors used in producing wheat and barley cannot exceed the number produced:

a_3,1 q₁ + a_3,2 q₂ + a_3,3 q₃ ≤ q₄ + q₅

Finally, the decision variables must be non-negative:

q₁ ≥ 0, q₂ ≥ 0, q₃ ≥ 0, q₄ ≥ 0, q₅ ≥ 0

The maximization of the objective function, the constraints for each of the two resources, the constraint for the capital good, and the non-negativity constraints for each of the five decision variables constitute a linear program. In this context, it is the primal linear program.

The above linear program can be solved. Prices for the resources do not enter into the problem. So I have proven that Von Mises was mistaken.

4.0 The Dual Problem

But I will go on. Where do the prices of resources and of capital goods enter? A dual linear program exists. For the dual, the decision variables are the 'shadow prices' for the resources and for the capital good:

• The wage, w₁, to be charged for a person-year of labor.
• The rent, w₂, to be charged for an acre of land.
• The cost, w₃, to be charged for a tractor.

The objective function for the dual LP is minimized:

Minimize x₁ w₁ + x₂ w₂

Each process provides a constraint for the dual. The cost of operating Process I must not fall below the revenue obtained from it:

a_1,1 w₁ + a_2,1 w₂ + a_3,1 w₃ ≥ p₁

Likewise, the costs of operating processes II, and III must not fall below operating them:

a_1,2 w₁ + a_2,2 w₂ + a_3,2 w₃ ≥ p₂

a_1,3 w₁ + a_2,3 w₂ + a_3,3 w₃ ≥ p₂

The cost of producing a tractor, with either process for producing a tractor, must not fall below the shadow price of a tractor.

a_1,4 w₁ + a_2,4 w₂ ≥ w₃

a_1,5 w₁ + a_2,5 w₂ ≥ w₃

The decision variables for the dual must be non-negative also:

w₁ ≥ 0, w₂ ≥ 0, w₃ ≥ 0

In the solution to the primal and dual LPs, the values of their respective objective functions are equal to one another. The dual shows the distribution, in charges to the resources and the capital good, of the value of planned output. Along with solving the primal, one can find the prices of resources.

5.0 Conclusion

One could consider the case with many more resources, many more capital goods, many more produced consumer goods, and a technology with many more production processes. No issue of principle is raised. Von Mises was simply wrong.

One might also complicate the linear programs or consider other applications of linear programs. Above, I have mentioned introducing multiple time periods. How do people that do not work get fed? One might consider children, the disabled, retired people, and so on. Might one include taxes somehow? How is the value of output distributed; it need not be as defined by the shadow prices.

Or one might abandon the claim that socialist central planning is impossible, in principle. One could look at a host of practical questions. How is the data for planning gathered, and with what time lags? How often can the plan be updated? Should updates start from the previous solution? What size limits are imposed by the current state of computing? The investigation of practical difficulties is basically Hayek's program.