###### Written by: Nasser Al Masri.

Across the left-wing political spectrum, the Soviet Union is often viewed as the prime example of a planned economy. However, despite the fascination with its perceived success, it is rare to find leftist political figures who possess a deeper understanding of how resources were actually allocated. The planned model is often dismissed as simply deciding the allocation of resources through "rational" means, without much consideration of how this rationality can be determined. A notable example of this is Hakim’s response to Economics Explained's video on the Soviet economy.1 Throughout the video, Hakim not only makes several factual mistakes (such as stating that only around 10,000 products were centrally planned2) but he also fails to provide any clear and concise explanation of how exactly a plan could be formulated. Instead, he only asserts that plans are formulated for "political reasons," which, if anything, would indicate the superiority of a market system with its clearer monetary incentive system driven by market signals. The goal, then, is to offer an informal introduction to the primary concepts of mathematical techniques—specifically Linear Programming—that emerged during the 1960s and 70s for formalizing plans and allocating resources.

# The Central Dilemma

Imagine yourself as a Soviet planner in the 1960s. Suddenly, you are handed a list of production quotas that must be met. The task is not only daunting, but you also have to figure out a way to create a plan that is both consistent and stable.3

Mathematically speaking, this means we need to find the plan that satisfies several logical conditions.

Imagine there exists a multi-commodity, multi-enterprise economy with discrete-time production. The problem of finding a consistent economic plan for any period involves finding the following matrix, which satisfies certain conditions:

where a_ij represents the amount of the ith good produced at the jth enterprise (noting that they can take negative values if they are required for production at other enterprises). For this plan to be consistent, it needs to satisfy the following two properties:

1

where a_j is a vector of all the good producing enterprises and A_j is the set of feasible plans for the jth enterprise. This constraint means is that every firm receives a feasible plan.4

2

where b_i is the current stock of the good and b_i prime is the desired stock at the following time period.

Failure to satisfy either of the two previous constraints will result in obvious problems with the functioning of the economy (such as delays in production, the need to alter plans, breakdowns in supplies, etc.).

Due to the sheer number of goods and enterprises that need to be managed, a consistent plan practically never existed. The reasons for this are numerous and not the main topic of this article, so instead, I direct the reader to the book "Planning Problems in the USSR: The Contribution of Mathematical Economics to their Solution 1960–1971" by Michael Ellman. In brief, a consistent plan could not exist due to the enormous number of calculations required for an economy comprising 61,000 enterprises and over 20,000,000 goods. Approximation, aggregation, and iterative methods were used; however, they always fell short, as noted by one GOSPLAN economist:

Because of the great labour intensity of the calculation of changes in the material balances and the insufficiency of time for the completion of such work in practice, sometimes only those balances which are linked by first order relationships are changed. As regards relationships of the second order, and especially of the third and fourth order, changes in the balance are made only in those cases where the changes are conspicuous.5

This often resulted in unfulfilled plans, firms adopting slack plans (where resources were wasted), and widespread material shortages. Not only were the calculations themselves lengthy, but more importantly, there was no consistent technique for determining resource allocations. Planners often relied on rules of thumb rather than any consistent method. This approach was largely influenced by a kind of "mathphobia" that emerged in the late 1930s and persisted well into the 1960s. As Kantorovich recalls, when discussing optimal planning with Soviet statistician Boris Yastremskii, he was told:

You are talking here about optimum. But do you know who is talking about optimum? The fascist Pareto is talking about optimum.6

As such, new mathematical techniques were needed to figure out ways to allocate resources efficiently and in a computationally tractable manner to achieve planned production.

# Kantorovich

In 1939, Soviet mathematician Leonid Kantorovich discovered a method which would be named *Linear Programming*7 for which he would later receive the Nobel prize in economics. Commenting on his discovery, Kantorovich stated that:

I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion. I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions8.

The specific problem to which he applied this method was in the plywood industry: determining the most effective usage of machinery to maximize output. The details of the problem were as follows:

Suppose there is a final product that requires two inputs, A and B, which must be supplied in equal amounts. Additionally, there are three machines with the following productivities:

If all milling machines were used solely to produce A, they would produce 30 units per hour, or equivalently 60 units per hour of B. They can also produce 20 units of each if they spend 40 minutes producing A and 20 minutes producing B. Applying this same division to the other machines, they would produce a total of 77 units of each. However, this method does not maximize output, as it does not consider the comparative efficiency of each machine in producing A and B. For instance, the milling machines can produce A and B in a 1:2 ratio, the turret lathes in a 2:3 ratio, and the automatic lathes in a 3:8 ratio. Using these efficiencies to determine production levels yields the following results, which increase output compared to the previous method:

more technically, the problem we are attempting to solve is as follows:

We have **n** machines that produce **m** different parts which can produce **a_ik** per day (the kth part on the ith machine). Additionally, **h_ik** represents the fraction of the workday during which ith machine is used to produce the kth part, subject to the following conditions:

**h_ik** must be non-negative and machines must be used efficiently throughout the workday.

We further denote the number of the **k-th** part produced as

If one were to attempt to solve this using traditional mathematical methods, it would generally be unfeasible in any realistic setting. For example, attempting to solve the plywood example mentioned earlier would require solving for 32 unknowns and the complexity would exponentially increase, eventually demanding the solution of millions of equations—an impossible task to be done by hand. However, Kantorovich’s new method simplifies this to just 4 unknowns.

Returning to the plywood example, Kantorovich's method was used to solve the optimization problem as follows:

We can first define the following term:

This term measures the trade off of the ith machine producing the 1st or 2nd part. In the Plywood example, the ratios for each of the machines are 2 for milling machines, 1.5 for lathe, and 8/3 for automatics. This means that 1 unit of part 1 for the lathe machine is equivalent to 1.5 units of part 2 and so on for the other machines. If we arrange all the k_is in numerical order such that

It would become clear that it would be more efficient to produce more of part 1 on the first machine due to it having the lowest ratio of all the other machines. For those who are more economically inclined, this is a generalization of the classical economic view that shadow prices could be used for rational decision making. From this, we can set the following for the first machine (the one with the lowest ratio)

and set the following for the other machines:

It follows from the equality condition (that we must produce equal parts of both parts) that we select a number s such that the following holds:

Essentially, what these equations are saying is that assigning s-1 machines to part I will result in the equality condition not holding, but setting exactly s machines will result in either an equality or too much production. If we then set all machines below the number s to producing only part 1, and all those above to only produce part 2, we will have the solution needed for the problem by solving the following equations for machine number s:

To demonstrate, the solution for the plywood example mentioned earlier would be as follows:

If we take the number s=2, we get the following:

What these two equations tell us is that every machine below s will produce only part 1 while every machine above will produce only 2, this means the only unknown left to solve will be** h_s1** and** h_s2** since **h_1,1=1** and **h_1,2=0 **and the reverse is true, hence we solve for **h_s **from the following system:

Thus, an exact mathematical method was found for solving optimization problems without requiring an unfeasible amount of calculations.9

Beyond the Plywood industry, these mathematical methods were used across several industries and several different problems, from cost minimization to optimal transport problems and have proven very successful. To provide a brief example, the methods discussed were used in the optimization of the development and location of the cement industry, which, using the Ural 2 computer, was able to reduce the transport cost of meeting planning targets by around 30%10 and across all industries was able to achieve a cost lower by 10-15%. It is worth noting that the ideas of Kantorovich and other mathematicians were not implemented in full, as they were far more far reaching than was allowed in Soviet political discourse, such as the abolishment of rent and the slow introduction of markets11.

# The Economic Calculation Problem

Not only did Kantorovich’s technique (which he dubbed “resolving multipliers”) present a way to find technically efficient methods for calculating production, but he also unknowingly helped resolve a common critique of planned economies, specifically the economic calculation problem as articulated by Von Mises. Mises argued that "every step away from private ownership of the means of production and the use of money also distances us from rational economics.12" Without market prices, it would appear that any economic plan would lack rationality because there would be no monetary calculation to guide proper allocation.

Moreover, even if an objective function could be identified for maximization or minimization in monetary terms, the human mind would be too limited to handle its complexity. Von Mises' dilemma is algorithmic, depending on the human mind's inability to process all possible plans that could optimize the objective function. Kantorovich demonstrated that it is feasible to develop a mathematical technique independent of monetary terms, focusing instead on achieving planning targets and efficiency.

Kantorovich later developed a system of valuation known as objectively determined valuations (ODV), distinct from prices, used solely for economic calculations in planning, not for commercial purposes. However, practical implementation depends on the existence of algorithms capable of solving such equations at a sufficiently rapid pace. Clearly, solving such equations manually today would be impractical, potentially requiring the solution of millions or billions of equations. Yet, with modern computing power, computational constraints are no longer an issue.

For instance, a linear programming problem involving millions of variables can be solved in half an hour using a modest 4-CPU computer using interior point methods. In the context of a planned economy with billions of variables, large supercomputers possess adequate computational power to accomplish this task.

With the demise of the Soviet Union, it might seem to vindicate the Austrian attack against planned economies. However, with the recent advancements in computational power and algorithms, Kantorovich’s and CEMI’s vision of a planned economy is no longer a fantasy, but a viable alternative to the market system.

https://www.youtube.com/watch?v=nGm0u3UHDZM

This is patently false, I am not sure where he gets this claim, if I were to guess this is because Soviet planners did not plan for all 20,000,000 goods individually, but rather aggregated them to make the calculations more feasible, which if anything shows a serious flaw with the planning system of the time.

Meaning it does not change very often.

This should be read as “a belongs to the set A_j”.

Ellman, Michael. “The Consistency of Current Plans.” *Planning Problems in the USSR: The Contribution of Mathematical Economics to Their Solution, 1960-1971*, At the University Press, Cambridge, 1973, p. 28.

BOLDYREV, IVAN, and TILL DÜPPE. “Programming the USSR: Leonid v. Kantorovich in context.” *The British Journal for the History of Science*, vol. 53, no. 2, 8 Apr. 2020, pp. 255–278, https://doi.org/10.1017/s0007087420000059.

The naming is because its a function with a linear objective function and linear constraints.

Kantorovich, L. V. “Mathematical methods of organizing and planning production.” *Management Science*, vol. 6, no. 4, 1960, pp. 366–422, https://doi.org/10.1287/mnsc.6.4.366.

It is worth noting that there is a generalization of this for any **m** but it is far beyond the scope of this substack, an excellent exposition can be found in Kantorovich’s work referenced earlier.

Ellman, Michael. “Optimal planners and the methods of economic calculation.” *Planning Problems in the USSR: The Contribution of Mathematical Economics to Their Solution, 1960-1971*, At the University Press, Cambridge, 1973

More can be found about the ideas of Kantorovich and CEMI in Ellman’s book

Von Mises, Ludwig. *Economic Calculation in the Socialist Commonwealth*. Ludwig Von Mises Institute, Auburn University, 2008.

Which is where cybernetic control considerations come in. Both distributed and centralized planning need to coexist cooperatively to have a really effective economy, regardless whether it's socialist or capitalist. Modern information technologies make the socialist variation much more viable.

The Austrians were/are almost just the other side of the same coin, all of their recommendations (and I believe this is intentional as I Believe they were/are essentially just a PR firm for powerful special interests) led/lead to hyper centralized private sector central planning, not "market driven decision making". Also, in my personal opinion, a lack of computational power and telecommunications networking was the least of their problems and maybe wasn;t actually much of a problem at all. One example of this that a plan meeting logical conditions could be consistently applied in a many commodity, many enterprise economy with hundreds of thousands organizational agents and hundreds of millions of people ignores the hyper complexity of real world dynamics. The simplification into a matrix format doesnt account for qualitative diffs between goods, inter and intra regional dynamics, or the extremely large number of big, medium, and small unforeseen events.