Optimal allocation

Optimal allocation theory

If information about the constraints and the objective to be achieved is available, it is possible (in theory) to calculate the optimal allocation.

Leonid V. Kantorovich , 1912-1986

Mathematical Methods of Organizing and Planning Production , https://www.jstor.org/stable/2627082 �The Best Use of Economic Resources , https://archive.org/details/bestuseofeconomi0000kant 

The optimal allocation of economic resources, https://www.dropbox.com/sh/pes5eth6td20cyo/AAD2dv7I1wtt8Vdh8JaNwVMWa/Kantorovich?dl=0&lst=&preview=La+asignaci%C3%B3n+%C3%B3ptima+de+los+recursos+econ%C3%B3micos.pdf&subfolder_nav_tracking=1

George Dantzig, 1914-2005

Linear Programming and Extensions

 https://archive.org/details/linearprogrammin0000dant

Economic calculation and constrained maximization

• In constrained maximization problems, the goal is to find variables that maximize an objective provided that certain constraints are met.
• Economic calculation seeks an allocation that maximizes a given objective, provided that such an allocation is possible. Calculating this optimal allocation is a problem of constrained maximization.
• objective is maximized (it is assigned for a specific purpose).
• restrictions are met (provided that the allocation is possible).
• (“Optima” here does not mean “good”, but simply that it maximizes something)

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Maximum conditions and Lagrange multipliers

• For logical reasons, in order to have a solution to a constrained maximization problem, certain maximum conditions must be met, involving numbers called Lagrange multipliers.
• For us to have a solution to a constrained maximization problem:
• Certain maximum conditions must be met
• These conditions indicate how the target increases with a small increase in the corresponding variable.
• There is a maximum condition for each variable.
• They play the role of accountants
• In which there are implicit auxiliary variables, the Lagrange multipliers.
• These numbers indicate how the target increases with a small increase in the corresponding constraint.
• There is one Lagrange multiplier for each constraint
• They play the role of values
• Therefore, in a constrained maximization problem we have:

Decision variables <-> Maximum conditions

Restrictions <-> Lagrange Multipliers

• Note that the maximum conditions and Lagrange multipliers depend on the objective we are maximizing .

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Economic calculation and �mathematical calculation

• The problem of calculating the optimal allocation is a problem of constrained maximization. In real-world economic calculations, concepts such as values and accounting play a role similar to that of Lagrange multipliers and maximum conditions in mathematical calculations.
• In mathematics texts dealing with constrained maximization, it is common to interpret Lagrange multipliers as the values of an economy (and the maximum conditions could also be interpreted as accounting). For this reason, mathematicians call Lagrange multipliers "shadow prices," "objectively determined valuations," or "dual values."
• When dealing with economics, we can interpret values and accounts as Lagrange multipliers and the verification of the maximum conditions of a constrained maximization problem.

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Economic calculation (real)

An optimal allocation implicitly involves accounting in which certain values are involved .

Mathematical (logical) calculation

constrained maximization problem, there are implicit maximum conditions involving Lagrange multipliers .

Economic calculation �in an economy or in a company

• (It is useful to imagine hypothetical socialisms, where people's needs are met, not "real socialisms" that pursue other goals, in order to view capitalisms with perspective. It is also useful to study capitalisms in their "pure" form, as allocations that maximize profit, without dwelling for the moment on the very complex "real capitalisms," where allocation follows complicated patterns.)
• Economic calculation can be applied to an entire economy or only to a single allocation unit. Optimal allocation theory is therefore applicable to the small scale of a business ( Kantorovich 's lumber mill ) or to an entire economy (socialist, Robinson's Island, or any other).
• In all cases, problems of the same type arise.
• The variables to be decided are those that determine the allocation, for example the level at which the possible production processes operate.
• The restrictions stipulate that, for the magnitude of those decision variables, the corresponding allocation is possible.
• For each material, the total amount consumed by the production processes cannot exceed the amount available (for example, because it has been produced).
• The objective is what is sought to be maximized, which may differ depending on the type of society.
• In a company, the company's profits.
• In a socialism (hypothetical, not a “real socialism”), attention to the needs of the people.
• On Robinson's Island, attention to Robinson's needs.
• In short:
• objective is maximized (it is assigned for a specific purpose).
• restrictions are met (provided that the allocation is possible).

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Optional. Economic calculation in our game

• In our simulation of capitalism, each of you tries to maximize your expected profit. In our game we have:
• The variables to be decided.
• The production process to be used (and the quantities exchanged).
• objective is maximized (it is assigned for a specific purpose).
• You try to maximize profit.
• restrictions are met (provided that the allocation is possible).
• The amount of wheat that can be used in production is the amount available after the exchanges.
• The amount of iron that can be used in production is the amount available after the exchanges.
• Therefore, in the game you are trying to find the solution to a constrained maximization problem.
• Furthermore, if we consider the exchanges, the variables you have to decide are the amount of wheat you exchange and the amount of iron you exchange, based on the exchange rates offered to you, along with the production process you use.
• Furthermore, in our game, the wheat and iron production processes are described by what are technically known as fixed-coefficient production functions ( Leontief functions ). And in the third option, when these processes are not used, they are described by linear production functions.
• This is a fairly complex mathematical problem, a constrained maximization problem.

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You are amazing mathematicians �and you didn't even know it.

Worth

Important. Value (in any economy)

• In economic calculation problems, the values indicate how the objective being maximized would increase in response to a change in the constraints.
• The restrictions are that the amount consumed of each thing is the amount available of that thing.
• A variation in a constraint is that a slightly larger quantity of the thing is available.
• Therefore, the value of a thing is what would increase the goal we are maximizing if we had a slightly larger amount of that thing .

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Courage on Robinson's Island, in hypothetical capitalisms and socialisms

• The value category makes sense in any optimal allocation, in economies where an objective is maximized subject to certain constraints.
• Capitalism, socialism, and Robinsonian economics can all be understood as optimal allocations, and therefore the concept of value is relevant in all of them. In each case, it's about how the objective being maximized would increase if we had a slightly larger quantity of something.
• But values depend on the objective being maximized. As in capitalist, socialist, and Robinsonian economic systems, where objectives are different, values are also different.
• In a capitalist system, the value of something is what would increase profit if we had a slightly larger quantity of that thing.
• In a capitalist system, the value of a piece of land is the value of the successive harvests it yields. Land, like everything else, is valued for the profits it can produce.
• All things are worth the benefits they can produce, because in the company profit is maximized, and therefore for the company things are worth insofar as they contribute to that objective.
• In a socialist system, the value of something is what would increase attention to people's needs if we had a slightly larger quantity of that thing.
• In a socialist system, a piece of land can have value because of the crops that would be obtained from it, but alternatively also because it can form part of a natural park.
• Things can be valued not only for their potential production but also for other uses, like a nature park that doesn't generate profit. Things are valued insofar as they contribute to meeting people's needs. A piece of land unsuitable for cultivation can have (ecological) value for people.
• In Robinson's economy, the value of a thing is what would increase the attention to Robinson's needs if he had a slightly larger quantity of that thing.
• Robinson lives on his island in a one-man socialism, and therefore can value things according to criteria different from the production derived from them. We could say that the value of a thing is what increases Robinson's "subjective utility" when he possesses a slightly larger quantity of that thing.
• We must emphasize that the value category can make sense even where there are no exchanges, like Robinson's island, because it makes sense in all constrained maximization.
• An example of value in a non-market environment was already seen in Demography with reproductive values (see the slide on this). The constrained maximization problem here is to maximize the population growth rate under the constraints that the number of women in each age group grows (in the long run) at the same growth rate and in accordance with survival and reproduction rates.
• As we can see, the value category surpasses even Economics, and can make sense in contexts that are not specifically economic.

Optional. Example : calculating the value of wheat �in our game

• The value of something is the rate at which the objective we are maximizing (the profit in our game) increases with the addition of a small amount of that thing. We can use the calculator to set these values in our game.
• We will calculate the value corresponding to wheat if the first process is used.
• With 1000 of wheat and 1000 of iron, the expected income or profit of the first process is 2053,571429.
• With 1001 of wheat and 1000 of iron, the expected income from the first process is 2055,625.
• Then the Lagrange value or multiplier of wheat is �(2055,625 – 2053,571429) / (1001 – 1000) = 2,053571.
• We have increased the wheat by 1; in general we should have done the calculation with a smaller number.

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Optional. Value theory

• In a socialist society (or on Robinson Crusoe's island) the value of a thing is what would increase attention to the needs of the people (or of Robinson Crusoe) if we had a slightly larger quantity of that thing.
• We could say that Robinson's economics is a one-person socialism.
• In these economies, what we might call a "subjectivist" view of value makes some sense, since things would be valued for how they contribute to satisfying needs (although in socialism we are not talking about personal or individual "subjective utilities" but rather collective or social ones).
• In these systems it can be maintained at least as an illustration that a material can be a "consumer good", because as such it contributes to directly increasing the satisfaction of people's needs (or Robinson's), or it can acquire value as a "means of production", because it contributes to producing "consumer goods".
• A "one-way street" view can also be maintained, in the sense that production is aimed at consumption to meet needs.
• But in a capitalist system, things are valued for how they contribute to increasing profits, not for how they contribute to satisfying people's needs. Therefore, a "subjectivist" view of value in capitalist systems is incorrect.
• In capitalist systems, all materials are "means of production," including those consumed by humans or even by humans themselves.
• There is no "one-way street," but rather circular production. The allocation is not aimed at consumption, but at production for its own sake.
• The “subjectivist” view errs in applying ideas that do make sense where the economy is geared toward satisfying needs, as in socialism or Robinsonian economics, to a system like capitalism where allocation does not serve that purpose. The “subjectivist” view of value in capitalism confuses an ideal with reality, confusing “what ought to be” with “what is.”

Accounting

Important. Accounting

• In economic calculation problems, accounting indicates how the objective being maximized would increase in response to a change in the decision variables.
• Decision variables are the levels of intensity with which the processes operate.
• A variation in an intensity level means that the variable increases slightly, that the process slightly increases its operation.
• Therefore, the accounting of a process is what would increase the objective we maximize if the levels of its operation were to increase slightly .
• Accounting uses values, calculating how processes contribute directly to increasing the objective sought, and also how they contribute indirectly through consumption and production.
• Regarding these indirect contributions, recall that the value of something is what would increase the objective we are maximizing if we had a slightly larger quantity of that thing. Therefore, consumption (which implies having less of what is consumed) will decrease the objective, and production (which implies having more of what is produced) will increase it, as will their respective values.
• Since the aim is to maximize the objective, the situation will be sought where increasing the levels of operation of the process does not allow increasing the objective (where accounting is canceled).

Important. Accounting �on Robinson Crusoe's Island, in capitalist and socialist systems.

• The accounting category makes sense in any optimal allocation, in economies where an objective is maximized subject to constraints.
• Capitalism, socialism, and Robinson's economics can all be understood as optimal allocations. All of them can be understood as optimal allocations, and therefore the category of accounting makes sense in all of them. In all of them, it is how the objective being maximized would increase if we slightly increased the operation of a process.
• But accounting methods depend on the objective being maximized. Since the objectives are different in capitalist and socialist systems, the accounting methods are also different.
• In a capitalist system, the accounting of a process is what would increase profit if we slightly increased the level of operation of the process.
• Accounting in a company measures the value of products minus the value of inputs. Since the value of things is what contributes to increasing profits, accounting measures how the operation contributes to increasing profits through the production and consumption of goods.
• In a socialist system, the accounting of a process is what would increase attention to people's needs if we slightly increased the level of operation of the process.
• Accounting in a socialist system doesn't just measure the value of outputs minus the value of inputs, but also how the process contributes to meeting people's needs. For example, the production processes for maintaining a nature park, which doesn't generate profit, represent a cost, but this cost can still be justified in a socialist system because the park is valued by the people.
• In Robinson's economics, the accounting of a process is what would increase attention to Robinson's needs if the level of operation of the process were to increase slightly.
• Note that accounting can be meaningful even where there are no exchanges or prices. Robinson can determine whether or not it is advantageous to increase a certain level of production, even though he doesn't trade with anyone.
• Capitalism has been described as the triumph of accounting. But in reality, it is the triumph of a certain kind of accounting, the kind that focuses on profit. Socialism could also use accounting, but it would be different from capitalist accounting because the objective being maximized is different.

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Optional. Accounting in a “hard” planning

Accounting is important not only in capitalism, but also in alternative societies. However, some, while recognizing the importance of accounting, held very naive ideas about how it could be implemented. ��Lenin, The State and Revolution , 1918 https://archive.org/details/obras-completas-lenin-tomo33/page/102/mode/2up?view=theater

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Vladimir Ilyich Ulyanov, Lenin, 1870-1924

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Optional . Economic theory, �capitalism and socialism

• The price equations of Leontief , Sraffa , or Von Neumann are the mathematical translation of the accounting and values of capitalist systems. Therefore, they describe capitalist systems, but not socialist ones.
• Sraffa 's equations describe capitalist accounting. They state that, for every process,

Cost of the process (1 + profit rate) = revenue from the process

• Process costs are taken as physical inputs multiplied by their prices; process revenues as physical outputs multiplied by their prices. Inputs and outputs are assumed to be known, and prices and the rate of profit are the unknowns.
• Leontief price equations are a special case of the Sraffa equations, and these are equivalent to the maximum conditions of the Von Neumann growth model for operating processes.
• But we have already pointed out that socialist accounting and values are different from capitalist ones.
• Therefore, a theory of socialism must consist of equations distinct from those of a theory of capitalism. In a socialist system, these equations would not hold true because the objective would prioritize addressing people's needs, not just profit, and therefore accounting, values, and allocation would also be different.
• Therefore, in a socialist system, these equations should not be used without further consideration to establish values, because if these values are used, capitalist accounting and values would be reproduced, and with them, capitalist allocation.
• Even less should equations be used that correspond to economies more primitive than capitalism, such as Smith's "crude and primitive state" or "simple commodity production," where prices are labor values (the quantities of labor directly and indirectly necessary to produce goods). Using labor values would simply reproduce the allocation of resources in those societies.
• See the debate between Proudhon (who advocated using labor values) and Marx (who advocated not using them). https://archive.org/details/marx-karl.-miseria-de-la-filosofia-ocr-1987

Food for thought. �The productivist Robinson Crusoe

• To illustrate the difference between the allocation of capitalism and that of socialism, let us imagine a hypothetical Robinson Crusoe on his island who maximized profit, the final production weighted at given prices, not the attention to his needs.
• This productivist Robinson Crusoe would work as much as possible and consume as little as possible, because this is how he would obtain the greatest benefits, the highest final output. The accounting and values he would use, and the allocation he would obtain with them, would be those of the capitalists.
• However, the Robinson Crusoe of Defoe's novel does not behave this way; he saves on his labor and maximizes attention to his needs. Robinson Crusoe lives a one-person socialism, not a one-person capitalism; he does not behave like a productivist Robinson Crusoe.
• If we observed a castaway on his island behaving in a productive manner, we would tend to think that there is some force compelling him to do so.
• However, humanity on this planet behaves largely like a productivist Robinson Crusoe, not like Robinson Crusoe himself. Capitalism maximizes profit, not the fulfillment of human needs.
• Why does humanity live in a capitalist system, in an economy that fundamentally functions to maximize profit and not to meet its needs? What force compels it to do so?

Optional . Glossary

• To avoid confusion, we detail the terms we use:
• Economic problem : This arises from the need to allocate resources in a complex economy with advanced technology.
• Economic calculation : these are the procedures used (real or hypothetical) to establish an allocation, to try to solve the economic problem, such as accounting and value.
• Optimal allocation : is that allocation (real or hypothetical) that meets the conditions of maximizing an objective whenever the allocation is possible.
• Economic calculation problem or optimal allocation theory : it is the theory of calculation of the previous concept.
• Constrained maximization problem : This is a problem in which variables are sought that maximize an objective provided that certain constraints are met.
• Lagrange multipliers : are auxiliary numbers in the calculation that are defined in all constrained maximization problems.
• Maximum conditions : These are conditions that must be met for us to have a solution to a constrained maximization problem.

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