Computing Von Neumann
Paper at the 8th Conference on Input-Output Analysis, https://io8.shaio.es/en/ , in September 2019, entitled as Tools for the Solution and Analysis of the Von Neumann Economic Model.
The model is described, highlighting its ability to reproduce many observable characteristics of capitalism based on an extremely simple mathematical structure. Its relationships with other models such as Leontief, Sraffa, game theory, and linear programs are studied, and generalizations are proposed to make it applicable to situations not covered by the original model. Several algorithms are developed to solve it, and its relationship with real-life economic allocation mechanisms is studied, showing how these algorithms can serve as abstract representations of real-life allocation mechanisms.
This text is provisional; a longer version will be published.
The paper can be viewed in PDF or without a reader.
Reading Guide for Non-Mathematicians
This text focuses on the relationship between real allocation mechanisms and algorithms for solving economic models. Very simple mathematics is used to achieve this, where necessary.
Even so, readers without a mathematical background may feel overwhelmed by some chapters. For this reader, I recommend the "Introduction," chapters 1 and 2, where there is no mathematics at all.
Optionally, you can look at section 3.1 for a cursory look at the mathematical approach to the model, but without having to delve into the equations. You can then skip to section 3.5, where an example of the equations and their solution is detailed, but again without worrying about the meaning of the symbols.
If you are interested in the connection with other models, you can take another look at chapter 4, but without dwelling on the equations either, which are very simple.
If the non-mathematician wishes to delve deeper, they can read the sections of the chapters dedicated to each algorithm, which contain the algorithm's description and economic interpretation (sections 1 and 2 of chapters 7, 10, 11, 12, 13, and 14, and also section 1 of chapters 8 and 9). From here, they can skip to section 1 of chapter 15, which briefly summarizes the economic interpretation of the algorithms. The rest of these chapters are more technical and are titled Appendix so they can be easily skipped.
To understand John von Neumann's role in contemporary science, some of the documentaries cited in the References section R.5 may be helpful.
Sections
Reading guide for non-mathematicians.
1: Why study the Von Neumann Model?
2: Why study algorithms?
3: Approach and conditions of the model.
4: Relationship with other models.
5: Generalizing the model.
Algorithms without internal variables.
6: Preliminary considerations on algorithms.
7: Eigenvalues.
8: Bisection.
9: Sequential linear programming.
10: Simplex.
Algorithms with internal variables.
11: Sampling.
12: Divide et impera.
13: Eigenvalues with internal variables.
14: Simplex with internal variables.
15: Final considerations on the algorithms.
References (with outdated links).
References
Links updated on 1/29/2023
R.1. Texts by John von Neumann
[1] Collected Works (edited by Abraham H. Taub), Pergamon Press, New York 1961-1963, https://vdoc.pub/author/von-neumann-j
R.1.1. The economic model
[2] “Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes”, Ergebnisse eines mathematischen kolloquiums (edited by Karl Menger), vol. 8, pages 73-83, Deuticke, Leipzig 1937, https://www.dropbox.com/sh/pes5eth6td20cyo/AABcGIewPGI07Lq0Fn2nl6Nfa/Neumann/%C3%9Cber%20ein%20%C3%B6konomisches%20Gleichungssystem%20und%20eine%20Verallgemeinerung%20des%20Brouwerschen%20Fixpunktsatzes.pdf?dl=0
[2b] “A Model of General Economic Equilibrium”, Review of Economic Studies nº 13, 1945, http://www.jstor.org/stable/2296111
[2c] “Un modelo de equilibrio económico general”, included in Barceló [1], https://www.dropbox.com/sh/pes5eth6td20cyo/AADBk1kROrF3UJ0jgpJSvrXua/Neumann?preview=El+modelo+de+Von+Neumann.pdf
R.1.2. Game theory
[3] With Oskar Morgenstern, Theory of Games and Economic Behavior, 3ª ed., Princeton University Press, 1953, https://www.pdfdrive.com/theory-of-games-and-economic-behavior-e185505671.html
[4] “Zur Theorie der Gesellschaftsspiele”, Mathematische Annalen, 100, 1928, https://gdz.sub.uni-goettingen.de/id/PPN235181684_0100?tify={%22pages%22:[299],%22view%22:%22info%22}
[4b] “On the Theory of Games of Strategy”, included in Albert W. Tucker y R. Duncan Luce (eds.), Contributions to the Theory of Games IV. Princeton University Press, 1959, https://archive.org/details/contributionstot0004unse/page/13
R.1.3. Theory of self-reproducing automata
[5] “The general and logical theory of automata”, included in Lloyd A. Jeffress (ed.), Cerebral mechanisms in behavior. The Hixon Symposium, John Wiley & Sons, New York 1951, https://archive.org/details/dli.scoerat.383cerebralmechanismsinbehavior/page/n13/mode/2up
[6] Theory of Self-Reproducing Automata, Arthur W. Burks (ed.), University of Illinois Press, Urbana y London 1966, https://archive.org/details/theoryofselfrepr00vonn_0
R.1.4. Other texts
[7] The Computer and the Brain, Yale University Press, 1958, https://archive.org/details/computerbrain00vonn
[7b] El ordenador y el cerebro, Bon Ton, Barcelona.
[8] First Draft of a Report on the EDVAC, University of Pennsylvania, 1945, https://archive.org/details/firstdraftofrepo00vonn
R.2. Algorithm codes
Afriat, Sydney N.
[1] “Von Neumann's Economic Model”, included in Dore, Chakravarty y Goodwin [1], https://archive.org/details/johnvonneumannmo0000unse/page/193 . Code in BASIC.
Bródy, András
[1] “The implicit dynamics of the von Neumann Growth Model”, Acta Oeconomica, Vol. 54 (1), 2004, https://www.jstor.org/stable/90002527 . Code in MATLAB.
Muiños, Manuel
[1] https://sites.google.com/site/manuelmuinhospan/ , 2011. Code in MATLAB.
Sargent, Thomas J. and John Stachurski
[1] Von Neumann Growth Model (and a Generalization), 2019, https://python.quantecon.org/von_neumann_model.html . Code in Python.
R.3. Other references on algorithms
Bose, Deb Kumar and Sanjit Bose
[1] “An Algorithm for Computing the von Neumann Balanced Growth Path”, Econometrica, Vol. 40, No. 4 (Jul., 1972), https://www.jstor.org/stable/1912972
Burley, S. Peter
[1] “Calculating Von Neumann Trajectories by Simulated Market Adjustments”, included in Bruckmann y Weber [1], https://archive.org/details/contributionstov0000unse/page/131
[2] “Dynamic Generalizations of the Von Neumann Model”, included in Łoś y Łoś [1], https://archive.org/details/mathematicalmode0000symp/page/27
Hamburger, Michael J., Gerald L. Thompson and Roman L. Weil Jr.
[1] “Computation of Expansion Rates for the Generalized von Neumann Model of an Expanding Economy”, Econometrica, Vol. 35, No. 3/4, jul.-oct. 1967, http://www.jstor.org/stable/1905656
Mantel, Rolf R.
[1] An efficient algorithm for the computation of a solution to von Neumann’s model. Instituto Torcuato Di Tella, DTE 68, Buenos Aires, junio 1969, https://repositorio.utdt.edu/handle/20.500.13098/12934
[2] “El modelo general de producción y crecimiento proporcional”, Económica, La Plata, Vol. XIV, Número especial, 1999, https://revistas.unlp.edu.ar/Economica/article/view/8850
Nikaidô, Hukukane
[1] "New aspects of Von Neumann's Model with special regard to computational problems", Annals of Institute of Statistical Mathematics, Vol. 6, No. 3, 1955, https://sci-hub.se/10.1007/bf02905916
Robinson, Stephen M.
[1] “Numerical solution of the irreducible von Neumann economic model”, Technical Summary Report No. 1142, Mathematics Research Center, University of Wisconsin, 1971, https://apps.dtic.mil/sti/citations/AD0732055
[2] “A linearization technique for solving the irreducible Von Neumann model”, included in Łoś y Łoś [1], https://archive.org/details/mathematicalmode0000symp/page/139
Schroeder, Roger G.
[1] “Linear Programming Solutions to Ratio Games”, Operations Research, Vol. 18, No. 2 (Mar. - Apr., 1970), https://www.jstor.org/stable/168686
Thompson, Gerald L. and Roman L. Weil
[1] “Von Neumann Model Solutions Are Generalized Eigensystems”, included in Bruckmann y Weber [1], https://archive.org/details/contributionstov0000unse/page/139
Thompson, Gerald L.
[1] “Computing the Natural Factors of a Closed Expanding Economy Model”, Zeitschrift für Nationalökonomie / Journal of Economics, Bd. 34, H. 1/2 (1974), pp. 57-68, https://www.jstor.org/stable/41797656
Weil, Roman L.
[1] "Solutions to the Decomposable von Neumann Model", Econometrica 38, 1970, https://www.jstor.org/stable/1913009
[2] “An Algorithm for the von Neumann Economy”, Zeitschrift für Nationalökonomie; Jan 1, 1964, https://www.jstor.org/stable/41796805
Ye, Yinyu
[1] “On the von Neumann Economic Growth Problem”, Mathematics of Operations Research, Vol. 20, No. 3 (Aug., 1995), https://www.jstor.org/stable/3690174
R.4. General references
Acta Oeconomica
[1] Special issue about John von Neumann, Vol. 54 (1), 2004, https://www.jstor.org/stable/e90002523
Balinski, Michel L. and H. Peyton Young
[1] “Interpreting Von Neumann Model Prices as Marginal Values”, Journal of Economic Theory, 9, 1974, http://www.econ2.jhu.edu/people/young/scans/EG1.pdf
Barceló, Alfons
[1] (ed.), El modelo de Von Neumann, Departamento de Teoría Económica, Universidad de Valencia, 1975, https://www.dropbox.com/sh/pes5eth6td20cyo/AADBk1kROrF3UJ0jgpJSvrXua/Neumann?preview=El+modelo+de+Von+Neumann.pdf
Bródy, András
[2] Proportions, prices and planning: a mathematical restatement of labor theory of value, Akadémiai Kiadó, Budapest 1974, https://archive.org/details/proportionsprice0000brod
[3] “Growth or Development?”, Acta Oeconomica, Vol. 61 (2) pp. 131–142, 2011, https://www.jstor.org/stable/41318100
[3b] “¿Crecimiento o desarrollo?”, Revista de Economía Crítica, nº14, segundo semestre 2012, https://revistaeconomiacritica.org/index.php/rec/article/view/505
Bruckmann, Gerhart and Wilhelm Weber
[1] (ed.), Contributions to the Von Neumann Growth Model, Springer-Verlag, New York y Viena 1971, https://archive.org/details/contributionstov0000unse
Champernowne, David G.
[1] “A note on J. v. Neumann’s Article on ‘A Model of Economic Equilibrium’ ”, Review of Economic Studies, Vol. 13, 1945, https://www.jstor.org/stable/2296112
[1b] “Notas sobre el artículo de J. von Neumann ‘Un modelo de equilibrio económico’” included in Barceló [1], https://www.dropbox.com/sh/pes5eth6td20cyo/AADBk1kROrF3UJ0jgpJSvrXua/Neumann?preview=El+modelo+de+Von+Neumann.pdf
Dantzig, George B. and Mukund N. Thapa
[1] Linear Programming 1: Introduction, Springer, New York 1997, https://www.academia.edu/19935874/George_B_Dantzig_Mukund_N_Thapa_Linear_Programming_1_Introductio
[2] Linear Programming 2: Theory and Extensions, Springer, New York 2003, http://nozdr.ru/data/media/biblio/kolxoz/M/MOac/Dantzig%20G.,%20Thapa%20M.%20Linear%20Programming.%20Vol.2..%20Theory%20and%20extensions(Springer,2003)(475s)_MOc_.pdf
Dantzig, George B. and Philip Wolfe
[1] “The Decomposition Algorithm for Linear Programs”, Econometrica, Vol. 29, No. 4 (Oct., 1961), pp. 767-778, https://www.jstor.org/stable/1911818
Dore, Mohammed, Sukhamoy Chakravarty and Richard Goodwin
[1] (ed.) John von Neumann and modern economics, Clarendon Press, Oxford 1989, https://archive.org/details/johnvonneumannmo0000unse
Dorfman, Robert, Paul A. Samuelson and Robert M. Solow
[1] Linear programming and economic analysis, Courier Dover Publications, 1987, https://archive.org/details/linearprogrammin0000dorf
[1b] Programación lineal y análisis económico, 2ª ed., Aguilar, Madrid 1964.
Evstigneev, Igor V. and Klaus Reiner Schenk-Hoppé
[1] The von Neumann-Gale growth model and its Stochastic generalization, National Centre of Competence in Research, Financial Valuation and Risk Management, Working Paper No. 208, 2006, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=887409
Gale, David
[1] The Theory of Linear Economic Models, McGraw-Hill, New York 1960, https://archive.org/details/theoryoflinearec0000gale_v9s9
[2] “The closed linear model of production”, included in Kuhn y Tucker [1], https://archive.org/details/linearinequaliti00kuhn_0/page/n285
Giorgi, Gorgio
[1] “Eigenvalues and Eigenvectors in von Neumann and Related Growth Models: An Overview and Some Remarks”, Journal of Mathematics Research; Vol. 8, No. 1; 2016, https://ccsenet.org/journal/index.php/jmr/article/view/54604
Giorgi, Giorgio and Tinne Hoff Kjeldsen
[1] (editores) Traces and Emergence of Nonlinear Programming, Birkhauser, 2014, https://zlibrary.to/dl/traces-and-emergence-of-nonlinear-programming
Glimm, James, John Impagliazzo and Isadore Singer
[1] (ed.) The Legacy of John von Neumann, American Mathematical Society, 1990, https://vdoc.pub/documents/the-legacy-of-john-von-neumann-s5pjdgg4m400
Kantorovich, Leonid V.
[1] The best use of economic resources, Harvard University Press, 1965, https://archive.org/details/bestuseofeconomi0000kant
[1b] La asignación óptima de los recursos económicos, Ariel, Barcelona 1968.
Kemeny, John G., Oskar Morgenstern and Gerald L. Thompson
[1] “A Generalization of the von Neumann Model of an Expanding Economy”, Econometrica 24, 1956, http://www.jstor.org/stable/1905746
Koopmans, Tjalling C.
[1] (ed.) Activity Analysis of Production and Allocation, John Wiley and Sons, Inc., New York 1951, https://archive.org/details/activityanalysis0000cowl
[2] “Economic Growth at a Maximal Rate”, The Quarterly Journal of Economics, Vol. 78, No. 3, 1964, https://www.jstor.org/stable/1879473
Kuhn, Harold W. and Albert W. Tucker
[1] (ed.) Linear Inequalities and Related Systems, Princeton University Press, 1956, https://archive.org/details/linearinequaliti00kuhn_0
[2] "John von Neumann's work in the theory of games and mathematical economics", Bulletin of the American Mathematical Society, Volume 64 Number 3, Part 2 (1958), 100-122, https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10209-8/S0002-9904-1958-10209-8.pdf
Kurz, Heinz D. and Neri Salvadori
[1] Theory of production: a long-period analysis, Cambridge University Press, 1997, https://archive.org/details/theoryofproducti0000kurz_h1j6
Leontief, Wassily W.
[1] The Structure of American Economy, 1919-1939: an empirical application of equilibrium analysis, 2ª ed., Oxford University Press, New York 1951, https://archive.org/details/structureofameri0000leon
[1b] La estructura de la economía americana, 1919-1939: una aplicación empírica del análisis del equilibrio, Bosch, Barcelona 1958.
Łoś, Jerzy and Maria W. Łoś
[1] (ed.) Mathematical models in economics, North-Holland Pub. Co., 1974, https://archive.org/details/mathematicalmode0000symp
McKenzie, Lionel W.
[1] Classical General Equilibrium Theory, The MIT Press, Cambridge, Massachusetts, 2002, https://es.es1lib.org/book/1111188/d1f05b
[2] “Optimal economic growth, turnpike theorems and comparative dynamics" in Kenneth J. Arrow y M. D. Intriligator (ed.), Handbook of Mathematical Economics: Volume III, North-Holland, Amsterdam 1986, https://archive.org/details/handbookofmathem0003unse/page/1281
Medvegyev, Péter
[1] “A General Existence Theorem for von Neumann Economic Growth Models”, Econometrica, Vol. 52, No. 4, 1984, https://www.jstor.org/stable/1911193
Morgenstern, Oskar and Gerald L. Thompson
[1] Mathematical theory of expanding and contracting economies, Lexington Books, 1976, https://archive.org/details/mathematicaltheo0000morg
Morishima, Michio
[1] Theory of Economic Growth, Oxford University Press, 1969, https://archive.org/details/theoryofeconomic0000mori
[1b] Teoría del crecimiento económico, Tecnos, Madrid 1973.
[2] “Economic Expansion and the Interest Rate in Generalized von Neumann Models”, Econometrica, Vol. 28, No. 2, 1960, https://www.jstor.org/stable/1907725
Muiños, Manuel
[2] Apuntes sobre la estructura y la evolución de las sociedades, doctoral thesis, Universidad Complutense de Madrid, 2011, https://eprints.ucm.es/12894/
Scarf, Herbert and Terje Hansen
[1] The Computation of Economic Equilibria, Yale University Press, 1973, https://archive.org/details/computationofeco0000scar
Schefold, Bertram
[1] “Von Neumann and Sraffa: Mathematical Equivalence and Conceptual Difference”, The Economic Journal, Vol. 90, No. 357, 1980, https://www.jstor.org/stable/pdf/2231661
Solow, Robert M. and Paul A. Samuelson
[1] “Balanced growth under constant returns to scale", Econometrica 21, 1953, https://www.jstor.org/stable/1905447
Sraffa, Piero
[1] Production of Commodities by Means of Commodities. Prelude to a critique of economic theory, Cambridge University Press, London 1960, http://pombo.free.fr/sraffa60.pdf
[1b] Producción de mercancías por medio de mercancías, Oikos-Tau, Barcelona 1966, https://es.scribd.com/document/375927835/Produccion-de-mercancias-por-medio-de-mercancias-Piero-Sraffa-pdf
Thompson, Gerald L.
[2] “On the solution of a Game-theoretic problem”, included in Kuhn y Tucker [1], https://archive.org/details/linearinequaliti00kuhn_0/page/275
Vegara, Josep María
[1] Economía política y modelos multisectoriales, Tecnos, Madrid 1979, https://es.scribd.com/document/401533622/Vegara-Economia-politica-y-modelos-multisectoriales-1979-EXCELENTE-MATRIZ-E-TEORIA-MARXISTA-pdf
R.5. Biographies and documentaries
ARTE France & BFC Productions
[1] John von Neumann, Prophetè du XXIe siècle, 2014, https://www.youtube.com/watch?v=X4cbFtbfm-8 (in French)
[1b] John von Neumann, Der Denker des Computer-Zeitalters, https://www.youtube.com/watch?v=2fB1RDIiLrE (in German)
[1c] John von Neumann, Prophet of the 21st Century, https://www.youtube.com/watch?v=97hfRcrYBtE (in German with English subtitles)
Blair, Clay, Jr.
[1] “Passing of a Great Mind”, Life Magazine, February 25, 1957, https://books.google.es/books?id=rEEEAAAAMBAJ&pg=PA89#
Heims, Steve J.
[1] John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death, MIT Press, Cambridge, Massachussets 1980, https://archive.org/details/johnvonneumannno00heim
[1b] John von Neumann y Norbert Wiener, Salvat, Barcelona 1989.
Israel, Giorgio and Ana Millán Gasca
[1] Il mondo come gioco matematico. La vita e le idee di John von Neumann, Bollati Boringhieri, Torino 2008, https://vdoc.pub/documents/il-mondo-come-gioco-matematico-la-vita-e-le-idee-di-john-von-neumann-6f42o4vg46i0
[1b] El mundo como juego matemático: John von Neumann, un científico del siglo XX, Nivola, Madrid 2001.
[1c] The World as a Mathematical Game: John von Neumann and Twentieth Century Science, Birkhäuser, 2009, https://archive.org/details/theworldasamathematicalgame
Kaldor, Nicholas
[1] “John von Neumann: A personal recollection”, in Dore, Chakravarty y Goodwin [1], https://archive.org/details/johnvonneumannmo0000unse/page/n9/
Macrae, Norman
[1] John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More, Pantheon Press 1992; republished by American Mathematical Society, 1999, https://archive.org/details/johnvonneumann0000macr
MAFILM
[1] Neumann János (John von Neumann), 1984, https://www.youtube.com/watch?v=NA0HYkh5I_I (in Hungarian with subtitles in English).
The Mathematical Association of America
[1] John von Neumann – A Documentary, MAA Video Classics 2, 1966, https://www.youtube.com/watch?v=q5SkVmZhnBw
Ulam, Stanisław M.
[1] “John von Neumann, 1903–1957”, Bulletin of the American Mathematical Society, 64, 1958, https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10189-5/S0002-9904-1958-10189-5.pdf