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Optional. Optimal allocation, maximum conditions, and Lagrange multipliers

Let us consider the problem of an optimal economy, an economy that maximizes an objective function (profit or any other).
Denoting Φ as the objective function to be maximized, X as the intensities of the processes, F as the matrix with the consumption and production flows, and D as the available quantities, the problem becomes

X represents the unknowns; the rest are given data. We denote Y as the Lagrange multipliers corresponding to the balances for each material, D + X F ≥ 0 , the maximum conditions are

We have that there are maximum conditions, where Lagrange multipliers are involved, one for each balance of each material, which must be satisfied for the allocation to maximize the objective.
The term refers to the rates at which the target increases directly with the variation in the intensities of the processes.
The Lagrange multipliers Y are the rates at which the objective increases with a variation in the constraints; they are the rates at which the objective would increase with the introduction of a small amount of the material; we will also call them the value of the material.
The term F Y is the rate at which the target would indirectly increase with the consumption and production of materials by the processes.
In order for the allocation to reach the greatest magnitude of the objective, while meeting the restrictions, it is necessary that what increases the objective directly plus indirectly is zero for the processes that operate, and not negative for those that do not operate.
This is true in every isolated optimal economy, without exchanges, also in Robinson Crusoe's economy, where for Robinson things have value, and the processes that operate must be those in which they either contribute directly to Robinson's goal, or contribute indirectly through their consumption and production, or contribute in both ways.

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Optimal economies and prices

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Optional. Values and exchange rates in an optimal economy ( updated Thünen )

If an economy that maximizes an objective function (profit or any other) can trade with its environment in such a way that the value of total trades is zero (or constant) and that trades are not an argument of the objective, the Lagrange multipliers must be proportional to the rates of trade.
Denoting Φ as the objective function to be maximized, X as the intensities of the processes, F as the matrix with the consumption and production flows, D as the available quantities, V as the net sales, and T as the exchange rates, the problem becomes

X and V are the unknowns; the rest are given. We denote Y as the Lagrange multipliers corresponding to the balances of each material, D + X F – V ≥ 0 , and as λ the Lagrange multiplier of the total exchanges, V T = 0 , the maximum conditions are

From the last expression, the Lagrange multipliers Y , or values, that correspond to each material (at each instant) are proportional to the exchange rates T.
Therefore, an allocator who maximizes an objective function and who can trade at given rates will trade until their values are proportional to the exchange rates.

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Values and exchange rates �in an optimal economy

If the allocator internally values a resource more than the cost of exchanging it, it is in their best interest to buy it.
If the assignor values it less, it's in his best interest to sell it.
Therefore, the allocator will tend to trade until the trade rates are proportional to the values, or Lagrange multipliers.
But note that values, Lagrange multipliers, depend on the objective being pursued. Things are valued internally based on the goal being sought.
We must "desacralize" value. Values depend on the objective and are proportional to exchange rates as a result of the actions of allocators exchanging in an attempt to maximize their objectives.
This is true in a capitalism that maximizes profit, but also for a castaway who could trade with other castaways and maximize their "utility", or for a socialist cooperative linked to others through exchanges.

If the allocator maximizes profit, they will trade until their Lagrange multipliers (the increase in profits from introducing small amounts of the materials, or marginal benefit) are proportional to the rates of exchange.
If the allocator maximizes his “utility”, he will trade until his Lagrange multipliers (which increase his “utility” with the introduction of small amounts of the materials, or marginal “utility”) are proportional to the rates of exchange.

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Optional. Values, exchange rates, and weightings in capitalist systems

Let's study a profit-maximizing firm, weighted physical production at certain prices, operating in a market. For simplicity, let's focus on a single time step. Denoting A as the input matrix, B as the output matrix, P as the weights used to determine profit, T as the prevailing market exchange rates, D as the available quantities, V as net sales, and X as the process intensities, the problem becomes

X and V are the unknowns; the rest are given. We denote as Y the Lagrange multipliers corresponding to the balances of each material, D – X A – V ≥ 0 , and as λ the Lagrange multiplier of the total exchanges, VT = 0 , the maximum conditions are

We have the three aspects of price: weights P , exchange rates T , and Lagrange multipliers or values Y .
From the last expression, the Lagrange multipliers Y , or values, corresponding to each commodity are proportional to the exchange rates T. The allocator's actions in trying to maximize profit tend to cause the values to be proportional to the exchange rates.
From the penultimate expression, with the processes that operate, marginal revenues B P are equal to marginal costs A Y .
The allocator has to set P as an estimate. And they will make that estimate based on what they observe in the market, based on exchange rates and values.
In capitalisms P , T , and Y tend to be proportional.

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Values, exchange rates and weightings in capitalisms

In capitalist systems, profit-maximizing companies tend to displace those pursuing any other objective. That is why profit maximization prevails.
Profits are calculated in prices, weighting the physical quantities, taking as a reference the exchange rates and values.

But it is clear that in systems other than capitalism, profit does not necessarily have to be maximized, and therefore there does not have to be any weighting factors.

This does not mean that the weights used to estimate profits are exactly proportional to current exchange rates and values. These weights are estimates that entrepreneurs have to make for the time when production is available, which is separate from the time when inputs are consumed by the production time. Therefore, they are not necessarily proportional.

In our game, you didn't think that the weightings, the price that would prevail in the next session, had to be exactly proportional to the prevailing exchange rates.

However, in the long term for most subjects, the weightings, if linked to exchange rates and values, do tend to be proportional, because entrepreneurs tend to think that what is expensive today will also be expensive when production can be carried out.
That is why in capitalist systems, in the long term, values, exchange rates and weightings do tend to be proportional.

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Optimal economies and accounting

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Optimal economy and maximum conditions

In constrained maximization problems:

For a solution to exist, in addition to the restrictions, certain maximum conditions must be met, involving auxiliary variables, the Lagrange multipliers.
Checking whether certain variables and multipliers form an optimal solution is simple by simply checking if the constraints and maximum conditions are met.
Many mathematical algorithms that attempt to solve constrained maximization problems use this fact systematically; they check if certain variables and multipliers satisfy the constraints and maximum conditions, and if not, they modify them, trying to make them violate them less.

In an optimal allocation, the maximum condition has two terms:

The direct marginal contribution indicates how much the magnitude of the objective increases with the operation of the process directly.
The indirect marginal contribution F Y increases due to the consumption and production of the processes; taking into consideration the Lagrange multipliers or values.

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Accounting and economic mechanisms

In real economic algorithms:

Values and accounts can be used in the same sense as Lagrange multipliers and maximum conditions in mathematical algorithms, because the real assignment problem logically corresponds to the constrained maximization problem.
Accounting indicates whether increasing the intensity of a process, or the level of an input, allows for increasing or decreasing the magnitude of the objective. If the accounting is positive, the intensity should be increased; if negative, it should be reduced. This is why accounting is essential in a real economic system.
But accounting must take into account both direct and indirect marginal contributions. And the direct contribution depends on the objective function.

In capitalist systems:

Values and accounting operate in the company in the same way as Lagrange multipliers and maximum conditions in mathematical algorithms for solving constrained maximization problems.
It has been rightly pointed out that capitalism is the triumph of accounting. But capitalism is the triumph of a particular kind of accounting: profit accounting.
In fact, capitalist accounting can be approximated as long-term maximization, and then, for processes prior to that long term, only the indirect marginal contribution, F Y , which are the marginal benefits, is considered.

In other systems:

Other systems can also be understood as optimal economies, and therefore there are also Lagrange multipliers and maximum conditions, specific values and accounting, although different from capitalist ones.
To build an economy distinct from capitalisms it will be necessary to use accounting and values ( in the atomized allocation units, which will be necessary to use ), but these have to take into consideration their specific objective function, different from profit.
In other systems, it will be necessary to consider not only the indirect marginal contribution but also the direct contribution, and how it influences the process's operation in achieving its objective. Accounting for other systems is not limited to marginal benefits.

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Mathematical algorithms and �economic mechanisms

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Algorithms and mechanisms

Mathematical algorithms for solving constrained maximization problems very often use:

The maximum conditions and Lagrange multipliers to find the solution to the problem.

For example, they start with some initial, more or less arbitrary, magnitudes of the variables and Lagrange multipliers, and try to obtain variables and multipliers that violate the restrictions and maximum conditions less.

Strategies like divide and conquer to try to tackle very big problems.

Dividing them into smaller problems, seeking the solution to these smaller problems, and attempting to solve the overall problem by coordinating the solutions.

Economic mechanisms can approximate an optimal allocation:

Using accounting and values,
Using division and coordination: firms and pricing mechanisms.

Economic mechanisms must be real algorithms in order to tackle the extremely complex economic problem.

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Example: gradient algorithm and capitalist mechanism

One of the classic algorithms for solving a constrained maximization problem is the gradient algorithm. Let's proceed:

We assign initial magnitudes to the variables and the Lagrange multipliers,
We increase the variables according to their maximum conditions, increasing those with positive maximum conditions and reducing those with negative ones.
We decrease the Lagrange multipliers according to the restrictions, increasing those with negative restrictions, and reducing those with positive restrictions.

In an optimal economy, the variables are the intensities, the maximum conditions are the accounts, and the Lagrange multipliers are the values. If we apply the gradient algorithm to the problem of an optimal economy, we will proceed as follows:

We are increasing the intensities whose accounts are positive and reducing those with negative accounts.
By increasing the values of the subjects with a deficit and decreasing the values of the subjects with a surplus.

In a capitalist system:

We would have to increase profitable processes and decrease loss-making processes.
Increase the values of things with a deficit and reduce the values of things with a surplus.

It hardly seems necessary to emphasize the similarity between the way the capitalist mechanism operates and the gradient algorithm. But we could construct another, non-capitalist mechanism, taking into account the accounting appropriate to its objective.

Note that we have to modify how the processes change, because we have a different accounting system, but not the way the values change.

(At https://sites.google.com/site/manuelmuinhospan/computando-a-von-neumann , algorithms for solving the Von Neumann model and its link to economic mechanisms are described ; see especially the simplex algorithm and divide et impera.

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Optional. Example: simplex algorithm and process selection

Dantzig's simplex algorithm for linear programs (and analogous algorithms for other problems, such as the Von Neumann model) can be understood as parallel to the process of process selection in capitalism.
In capitalist systems:

The processes that operate determine prices and interest rates;
If a process is not profitable, it is discontinued;
If a process becomes profitable, it is put into operation;
With the new processes in operation, we return to point 1.
Here, the role of prices and accounting is essential.

In the simplex algorithm (for the Von Neumann model):

The basic processes (the processes that operate) determine the Lagrange multipliers (the prices) and the factor (1 plus the interest rate);
Exit criterion: basic processes that comfortably meet the maximum conditions (those that are not profitable for prices and interest rates) are withdrawn from the basic processes;
Entry criterion: non-basic processes that do not meet the maximum conditions (which are profitable for prices and interest rates) are included in the basic processes.
With the new basic processes, we return to point 1.
Here the role of Lagrange multipliers and maximum conditions is essential.

In a non-capitalist system, one could use a procedure similar to the simplex method , but would not use capitalist accounting (profitability or earnings) but rather the corresponding maximum condition taking into account the objective function.

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Optional. Example: divide et impera algorithm; division and coordination

Divide and rule algorithms can be understood as parallel to the process of division and coordination of allocation in capitalisms (and potentially in other systems).
In capitalist systems:

By dividing the problem into companies, the cost of obtaining the information, performing the calculation, and putting the calculation into practice becomes possible in each company.
Each company solves its own economic problem, without needing to solve it for the economy as a whole. Companies have control over prices and interest rates to maximize their profits.
Given the production of the companies, the global coordination of local allocations is developed through the price mechanism, obtaining new prices and interest rates.
The procedure is repeated.

In the divide et impera algorithm (with the Von Neumann model):

Dividing the general problem into subproblems makes it possible to solve each subproblem.
Each subproblem is solved in such a way as to maximize the benefit, for a factor of interest and prices, obtaining the magnitudes of the internal decision variables and flows for each subproblem;
Knowing the flows of each subproblem, the general problem is solved, and the interest factor and general prices are obtained.
The procedure is repeated.

A non-capitalist system could also use divide and rule ; it could also divide the allocation into units and coordinate them in some way (with the market or otherwise).

But then: either the local units would not maximize profit but the corresponding objective function, or the accounting would have to be modified to take the objective into account (even if a market is used and the units compete with each other).