Von Neumann Model in Excel

The Von Neumann Model (VN)

Solving VN consists of calculating how much each process must operate so that the system grows proportionally as much as possible.

For an introduction to the model for non-mathematicians, see Chapter 1 of Computing Von Neumann.

The model studies production processes, each of which consumes a list of goods to produce another list of goods in the next moment. The model focuses on situations where all processes grow at the same rate of expansion, so that, for the economy as a whole and for each individual good, total consumption does not exceed total production from the previous moment. And of all the situations where this is possible, the model chooses the one that allows for the highest rate of expansion.

VN is a special case of optimal allocation, and the Leontief or Sraffa equations are special cases of VN. In optimal allocation the system changes over time, whereas in VN only a single point in time is studied because the system grows exponentially.

Example: calculating VN in our simulation

Let's assume we want to solve VN using data from our simulation in Capital; an experiment.

In the experiment, the first process uses 280 arrobas of wheat and 12 tons of iron as inputs to produce 575 arrobas of wheat; the second process consumes 120 arrobas of wheat and 8 tons of iron to produce 20 tons of iron; the third process consumes a certain amount of wheat and iron and produces half that amount. As explained in the presentation, this third process can be described in the matrices of inputs A and outputs B as two processes, so we have a total of four processes in the matrices. The outputs are obtained at a time after the inputs are consumed.

Solving VN in Excel

We will follow these steps:

1. We will check that our Excel has the Solver add-in loaded. Solver is available in all Excel programs on Windows or Mac, but it must be loaded to be able to use it.

2. We'll open the primal.xls file and see a spreadsheet like this (the appearance may vary slightly depending on the Excel version)

If a message similar to "Be careful: files from the Internet may contain viruses. If you don't need to edit it, it's best to stay in Protected View" appears, click the "Enable editing" button.

3. We'll adjust the number of rows to the number of processes and the number of columns to the number of materials. The sheet originally shows 16 processes or rows, and 6 materials or columns. In our example, we'll adjust the number of processes to 3, deleting rows 8 to 19 from the spreadsheet, and the number of materials to 2, deleting columns E to H.

To delete a column, right-click (on Windows; on Mac, hold down the CTRL key and click) in the cell with the column name and select "Delete" from the context menu. You can select multiple columns to delete them at once. To add a column, right-click on any column name, for example, column E, select "Copy" from the context menu, right-click again on the column name E, and select "Insert Copied Cells" from the context menu. The same procedure applies to rows.

4. We'll write the flow formulas for each process and each material in the yellow cells, updated according to the time at which they are performed, and adding a negative sign to consumption and a positive sign to production. Writing k as the expansion factor (1 plus the growth or expansion rate), the formulas are –A + B/k. Therefore, we will write them in the yellow cells (in Excel, formulas begin with = and in the US version the decimal separator is a period, while in the Spanish version it is a comma):

5. Click “Solver”. In modern versions of Excel, you will find it in the “Data” tab as an icon on the right, and in Excel 2004 and earlier versions, as an option in the “Tools” menu. A table will appear, whose parameters you do not need to define or modify, similar to

and click the “Solve” button.

If Solver has found a solution, a box similar to this will appear.

6. Click the "OK" button and the spreadsheet will be:

The blue cells inform us of the solution to VN. The cell labeled k tells us that the maximum expansion factor is 1.25, and that the system expands by 25% each time step. The cells under the label X indicate the intensities; that is, the first process operates 0.4 times, the second process operates 0.6 times, and the third process (indicated by the last two) operates 0 times or does not operate at all.

The green cells indicate that the constraints are met, because if they were not (for the 0.000001 margin of error), they would appear in red.

Advanced instructions

To formulate continuous-time problems, or to illustrate some of the symmetries of the models, please consult the detailed instructions (available in PDF or without a reader).

Accounting and values

If before clicking "OK" we click on "Sensitivity",

We can access this report in the tabs below:

In the table above, under the label "Final Value" you can find the magnitudes of the variables, and under the label "Reduced Gradient" the accounting balances of the processes: zero for operating processes and non-positive for non-operating processes.

In the table below, under the label "Final Value" you can find the magnitudes of the material balances, and under the label "Lagrange Multiplier" the values of the materials (with the sign reversed): 0.003810982 for wheat and 0.057164737 for iron. The value of iron is 15 times that of wheat.

VN is a special case of optimal allocation, as we can verify in Capital. VN como AO.xls. In VN the intensities increase, and the values decrease, with the expansion factor.

Duality

We can formulate the following equivalent problems, following the same instructions:

An assignment problem, which we call the primal: calculate the intensities that, while satisfying the material balances, achieve the highest magnitude of the expansion factor; this can be solved using primal.xls.

The blue cells represent the variables, the process intensities; the green cells represent the constraints, the material balances. The Lagrange multipliers are the material values, and the optimum conditions are the accounting balances.

A valuation problem, which we call dual: calculate the values that, while satisfying the accounting balances, yield the lowest expansion factor; you can use dual.xls.

The variables, the material values, are shown in blue; the constraints, the accounting balances, are shown in green. The Lagrange multipliers are the process intensities, and the minimum conditions are the material balances.

A joint allocation and valuation problem, which we call primaldual: calculate the intensities and values that satisfy the material and accounting balances, as well as the complementary slack; it is available in primaldual.xls.

The variables, the intensities X and the values Y, are shown in blue; the constraints, the material balances X F(k) and the accounting balances F(k) Y, are shown in green. Complementary slack conditions are also shown, which should be satisfied for the problem to have a solution, but which have not been included among the conditions that Solver must meet. If we write a potential solution in the blue cells of this sheet and no cell shows red color, then it is indeed a solution.

These sheets present the problem in its standard form, where material balances are written as equalities, X F(k) = 0, and prices can have any sign. The sheets can also be used for the canonical form, where material balances are written as inequalities, X F(k) ≥ 0, and prices must be non-negative, with primal forma canónica.xls, dual forma canónica.xls and primaldual forma canónica.xls.

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