Mathematical Optimization and the Economic Calculation Problem

(Originally published by C4SS)

In 1939, Leonid Kantorovich, a Soviet mathematician, came up with a formulation for optimization problems called linear programming (LP) after being tasked with planning production in the plywood industry during World War 2. A linear program is a constrained optimization problem with a linear objective function that is maximized or minimized subject to linear equality and/or inequality constraints. There are various methods for solving linear programs including the famous Simplex method developed by George Dantzig in 1947. Linear programming is a useful lens through which to understand where exactly different versions of the economic calculation problem arise, including knowledge problems and problems of computational complexity, which are often conflated.

There are two frames of reference through which can consider the matter of economic calculation, centralization vs decentralization, and markets vs planning. In this article, I will focus on markets vs planning, which includes decentralized planning. I define economic planning as the allocation of resources without money, exchange, and prices, but with quantities of raw inputs.

A highly simplified example of an optimization problem that a producer might face is to determine the optimal combination of inputs to maximize crop yield. Each input is subject to scarcity and requirements of the production process, like a certain ratio of goods A and B. Inputs to this particular production process are water, sunlight, fertilizer, and land. The constraints stipulate minimum and maximum amounts of water, sunlight, and fertilizer for the crops to survive. Farmers also face a budget constraint for water, fertilizer, and land use (the opportunity cost of growing other crops). The LP framework is a way of expressing this setup, which can then be solved to determine optimal quantities of water, sunlight, fertilizer, and land in order to maximize crop yield.

Different formulations of a linear program are adopted by planners and market actors, based on the underlying economic system. A planned economy would maximize output subject to a scarce pool of resources, or minimize costs subject to a demand constraint. Conversely, the objective function that would be maximized by a market economy is the social utility function, where each actor maximizes their own utility (e.g. profit maximization). The key difference between the objective function of the planned economy and the market economy is the existence of prices, which function as the weights in the objective function and the budget constraint of the market economy. A price is the amount of compensation buyers and sellers agree on in exchange for a good or service.

The respective formulations reveal a difference in how planners and those who prefer markets define efficiency. If one defines efficiency as maximizing fixed ratios of output based on historical data, planning could be seen as perfectly efficient. However, this approach is one that defines itself into correctness, which is what many Marxian economists do when faced with the calculation problem. It is undoubtedly possible to plan an economy with predetermined goals in mind, but the question is whether or not it would lead to the misallocation of resources, expressed in terms of subjective measures like meeting consumer demand, compensating workers, and wasted time and resources. A better method is to look at which approach is better at most efficiently maximizing social utility, a subjective measure. Although this is the framework that market proponents usually adopt, it actually allows us to meaningfully compare the two systems.

Both markets and strictly planned economies use planning. A construction project in a free market, just like one in a planned economy will require the builders to determine the optimal amounts of scarce inputs like labor time, concrete and steel beams needed for the project. This could be formulated as a linear program in both economies. The key difference, as mentioned before, is that the budget constraint faced by the builders in the free market would have prices as weights for all inputs, whereas planners would use a proxy like labor time, surveys, and a cost index for raw materials that includes labor time, scarcity, transportation, and so on from down the supply chain.

The issue that comes up here is that of revealed vs stated preferences, where consumer preferences are best conveyed in terms of how much they are willing to spend on a good or service i.e. their actions, which differs from the data captured in surveys and other methods planners might rely on to capture the intensity of consumer preferences and labor costs. In a complex economy, a “cost index” used to determine labor costs that involves direct bargaining with workers is functionally an exchange, resulting in a price. The only way to access revealed preferences is through the price mechanism; for example, in cases of scarcity, it would mean that buyers would bid up prices as they compete with each other.

Any other methodology such as using labor hours as Marx suggested, inevitably misrepresents the preferences of workers and is abstracted from real human desires, following protocols set by the planner. Kantorovich proposed the use of objectively determined values, which ignore subjectivity by definition, and thereby imply authority, where planners use simplifying frameworks to capture viable data. This not only contradicts the principles of anarchy but also ignores the subjective disutility of workers, which socialism ought to value.

When constructing objectively determined values, the planner has to decide what variables to include and how much to weigh them. What factors determine how much a worker values their own labor? Effort, hours, safety, etc. the full list varies from worker to worker and is ultimately undiscoverable, even with prolonged struggle sessions between planners, workers, and consumers. How to quantify these factors? Are they simply scores from 1-10? If so, how to weigh these scores relative to labor time? This process flattens subjectivity, because the methodology for constructing objectively determined values is determined by the planners, and worker input is circumscribed by the process.

On a very small scale where supply chains are extremely short, informational volume is low, and all participants can directly communicate with each other on a fully peer to peer basis, planned, communistic economies without money and explicit exchange can be reciprocal and non-authoritarian because quantification and complex planning are unnecessary.

The issue of revealed vs stated preferences also relates to the dynamism of the respective systems. In a freed market, an increase in demand for a good or service would raise prices above the marginal cost of production, leading to economic profits in the short run, causing more producers to enter the market, leading to higher output and lower prices. Price signaling allows market actors to coordinate with each other without the need to directly communicate. In small communes, the ability to directly communicate and the lower volume of information also facilitate dynamism. However, at scales where communication is limited by time and space, information processing would be constrained by planning cycles, leading to longer periods in which resources get misallocated and consumer demand is not met.

Using the LP framework, it is clear that in decentralized models, the version of the economic calculation problem isn’t one of computational complexity on a decentralized scale, but of accessing information – the Hayekian formulation of the calculation problem. Prices capture large amounts of information, which cannot be captured to the same accuracy by systems that use planning. Therefore, linear programming doesn’t solve the issue, because both market actors and pure planners use similar planning methodologies, the difference is that weights on the budget constraint and objective for market actors contain data that more accurately reflects the preferences of both consumers and producers (workers).

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