Abstract
In a seminar given in 1934 Abraham Wald was the first to use the weak axiom of revealed preference (1936a, b). However, in several ways his use of it looks odd to the post-Samuelson reader. First, since Wald’s main purpose was to establish a new condition for unique solution of the modified Walras–Cassel system of general equilibrium he had introduced earlier (1935), as originally stated it was a restriction on market rather than individual behaviour. Secondly, the axiom referred not to the (vector) market demand function z = f(p) but to its inverse p = f−1(z), whose existence is of course quite suspect. Finally, although later in the paper Wald did in fact invoke the individual version (wa) of the weak axiom as some ground – ‘a statistical probability’ – for belief in its market version (WA), he did not justify (wa) as did Samuelson (1938), as in its own right a sensible rule for consistent market behaviour. Instead, Wald derived if from an assumed additive Jevonian utility function for the individual, i.e. ui(zi) = Σ juij(zij) where in addition \( {\mathrm{d}}^2{u}_{ij}\left({z}_{ij}\right)/\mathrm{d}{z}_j^2<0 \) for each person i and good j; and so in Wald (wa) appeared as much more restrictive than it really is.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Baumol, W.J., and S.M. Goldfeld (eds.). 1968. Precursors in mathematical economics: An anthology, Series of reprints of scarce works in political economy, vol. 19. London: London School of Economics and Political Science.
Browder, F.E. 1976. Nonlinear operators and nonlinear equations of evolution in Banach Spaces. Proceedings of Symposia in Pure Mathematics, Vol. 18, Part 2. Providence: American Mathematical Society.
Deimling, K. 1985. Nonlinear functional analysis. Berlin: Springer.
Dolezal, V. 1979. Monotone operators and applications in control and network theory. Amsterdam: Elsevier Scientific.
Dorfman, R., P. Samuelson, and R. Solow. 1958. Linear programming and economic analysis. New York: McGraw-Hill.
Ghizzetti, A. (ed.). 1969. Theory and application of monotone operators. Gubbio: Edizioni Oderisi.
Hicks, J.R. 1946. Value and capital, 2nd ed. Oxford: Clarendon Press.
Houthakker, H.S. 1950. Revealed preference and the utility function. Economica NS 17: 159–174.
Hurwicz, L., and M.K. Richter. 1971. Revealed preference without demand continuity assumptions. Ch. 3. In Preferences, utility and demand, ed. J. Chipman, L. Hurwicz, M.K. Richter, and H. Sonnenschein. New York: Harcourt Brace, Jovanovich.
Jorgenson, D.W., and L.J. Lau. 1974. The duality of technology and economic behavior. Review of Economic Studies 41: 181–200.
Joshi, M.C., and R.K. Bose. 1985. Some topics in nonlinear functional analysis. New York: Wiley.
Kachurovskii, R.I. 1968. Non-linear monotone operators in Banach spaces. Russian Mathematical Surveys 23: 117–165.
Kusumoto, S.I. 1977. Global characterization of the weak Le Chatelier-Samuelson principle and its applications to economic behavior, preferences and utility-’embedding’ theorems. Econometrica 45: 1925–1956.
Minty, G.J. 1962. Monotone (nonlinear) operators in Hilbert space. Duke Mathematical Journal 29: 341–346.
Rockafellar, R.T. 1966. Characterization of subdifferentials of convex functions. Pacific Journal of Mathematics 17: 497–510.
Rockafellar, R.T. 1970a. Convex analysis. Princeton: Princeton University Press.
Rockafellar, R.T. 1970b. On the maximal monotonicity of subdifferential mappings. Pacific Journal of Mathematics 33: 209–216.
Samuelson, P.A. 1938. A note on the pure theory of consumer’s behaviour. Economica NS 5: 61–71.
Samuelson, P.A. 1946–47. Comparative statics and the logic of economic maximizing. Review of Economic Studies 14: 41–43.
Unger, K. 1974. Monotone operators and revealed preference. PhD dissertation, Johns Hopkins University.
Vainberg, M.M. 1973. Variational method of monotone operators in the theory of nonlinear equations. New York: Wiley.
Wald, A. 1935. Über die eindeutige positive Losbarkeit der neuen Produktionsgleichungen (I. Mitteilung). Ergebnisse eines Mathematischen Kolloquiums, 1933–4. 6: 12–18.
Wald, A. 1936a. Über die Produktionsgleichungen der ökonomischen Wertlehre (II. Mitteilung). Ergebnisse eines Mathematischen Kolloquiums, 1934–5 7: 1–6. Wald (1935, 1936) are translated as chapters 25 and 26, respectively, in Baumol and Goldfeld (1968).
Wald, A. 1936b. Über einige Gleichungssysteme der mathematischen Ökonomie. Zeitschrift für Nationalökonomie 7: 637–670. Trans. by O. Eckstein as Some systems of equations in mathematical economics. Econometrica 19, October 1951:368–403.
Wald, A. 1952. On a relation between changes in demand and price changes. Econometrica 20: 304–305.
Author information
Authors and Affiliations
Editor information
Copyright information
© 2018 Macmillan Publishers Ltd.
About this entry
Cite this entry
Newman, P. (2018). Monotone Mappings. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_779
Download citation
DOI: https://doi.org/10.1057/978-1-349-95189-5_779
Published:
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-349-95188-8
Online ISBN: 978-1-349-95189-5
eBook Packages: Economics and FinanceReference Module Humanities and Social SciencesReference Module Business, Economics and Social Sciences