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Algorithms in Excel

Detailed instruction (in PDF or without a reader), updated on January 26, 2021, are available for using the spreadsheets primal.xls, dual.xls, and primaldual.xls. The examples discussed in the thesis are contained in the compressed file examples.zip.

Brief Instructions for Solve the Von Neumann Model

The Von Neumann Model (VN)

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

Suppose we have three processes: the first process uses 280 quarters of wheat and 12 tons of iron as inputs to produce 575 quarters of wheat; the second process consumes 120 quarters of wheat and 8 tons of iron to produce 20 tons of iron; the third process, like the first, uses 280 quarters of wheat and 12 tons of iron but produces 400 quarters of wheat. The outputs are obtained at a time after the inputs are consumed. We will write these processes in tables or matrices of inputs A and outputs B, where each process occupies a row and each material a column:

Imagen con las matrices de insumos y productos

Solving VN consists of calculating how much each process must operate so that the system grows proportionally as much as possible. In proportional growth, the processes show the same rate of expansion, and for each material, the total consumption at a given time is equal to the total production from the previous time.

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 7 to 19 from the spreadsheet, and the number of materials to 2, deleting columns E to H.

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 =):

=-280+575/k =-12+0/k

=-120+0/k =-8+20/k

=-280+400/k =-12+0/k

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.

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

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 labeled X indicate the intensities; that is, the first process operates 0.4 times, the second process operates 0.6 times, and the third process operates 0 times or not at all.

Here, we have formulated VN assuming that free elimination is not available; if it were, which is how John von Neumann defined the model in the original article, we would add a free elimination process for each material. In our example, we would have to add the processes

=-1 =0
=0 =-1

To obtain prices, to pose problems in continuous time, or to illustrate some of the symmetries of the models, see the detailed instructions.

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