Skip to main content

The semi-martingale approach to the optimal resource allocation in the controlled labour-surplus economy

Part of the Lecture Notes in Mathematics book series (LNM,volume 1250)

Abstract

This paper deals with the characterization of an economic policy that over the longrun leads to the elimination of labour-surplus, i.e. one which directs the labour force from less productive or idleness to more highly productive employment.

Crucial is here the assumption that the rational response of individuals to economic development policies is uncertain and only partially known. Therefore, the dynamics of the labour participation in the more productive activities is stochastic and described by a stochastic differential equation of the diffusion type. We solve the associated stochastic optimization problem using semi-martingale techniques which we introduce in some detail with the purpose of making this paper self-contained. Our presentation is, however, intuitive and heuristic, since our emphasis lies in applications rather than in theories.

Keywords

  • Stochastic Differential Equation
  • Stochastic Control
  • Local Martingale
  • Labour Participation
  • Stochastic Control Problem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aoki, M. (1976). Optimal Control and System Theory in Dynamic Economic Analysis. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Arnold, L. (1974). Stochastic Differential Equations and its Applications. Wiley, New York.

    MATH  Google Scholar 

  • Arrow, K. and Kurz, M. (1970). Public Investment, the Rate of Return and Optimal Fiscal Policy. The John Hopkins University Press, Baltimore.

    Google Scholar 

  • Beněs, V.E. (1970). Existence of optimal strategies bases on specified information, for a class of stochastic decision problems, SIAM Journal on control, 9, pp. 354–371.

    Google Scholar 

  • Bensoussan, A. and Lions, J. (1978). Applications des Inéqualités Varionnelles en Contrôle Stochastique. Dunod, Paris.

    MATH  Google Scholar 

  • Bensoussan, A. (1982). Stochastic Control by Functional Analysis Methods. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Bourguignon, F. (1974). A particular class of continuous-time stochastic growth-models. Journal Economic Theory, 9, pp. 141–158.

    CrossRef  MathSciNet  Google Scholar 

  • Burmeister, E. and Dobell, R. (1970). Mathematical Theories of Economic Growth, Macmillan, London.

    MATH  Google Scholar 

  • Cass, D. and K. Shell (1976). The Hamiltonian Approach to Dynamic Economics, Academic Press, New York.

    MATH  Google Scholar 

  • Davis, M. (1977). Linear Estimation and Stochastic Control, Chapman and Hall, London.

    MATH  Google Scholar 

  • Davis, M. (1979). Martingale methods in stochastic control. Lect. Notes in Control and Information Sciences, Vol. 16, Springer Verlag, Berlin.

    MATH  Google Scholar 

  • Davis, M. (1980). Functionals of diffusion processes of stochastic integrals. Mathematical Proceedings of the Cambridge Philosophical Society, 87, pp. 157–166.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Davis, M. (1982). Stochastic control with tracking of exogeneous parameters. Lecture Notes in Control and Information Sciences, Vol. 43. Springer Verlag, Berlin.

    Google Scholar 

  • Davis, M. and P. Varaiya (1973). Dynamic Programming conditions for partially observable stochastic systems, SIAM Journal on Control and Optimization, 11, pp. 226–261.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Dellacherie, C. and P. Meyer (1980). Probabilité et Potentiel. Théorie des Martingales. Hermann, Paris.

    MATH  Google Scholar 

  • Duncan, T. and P. Varaiya (1971). On solutions of a stochastic control system. SIAM Journal on Control, 9, pp. 354–371.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Elliott, R. 1979). The Martingale Calculus and its Applications. Lecture Notes in Control, Vol. 16, Springer Verlag, Berlin.

    MATH  Google Scholar 

  • Elliott, R. (1982). Stochastic Calculus and Applications. Springer Verlag, Berlin.

    MATH  Google Scholar 

  • Fleming, W. and R. Rishel (1975). Deterministic and Stochastic Optimal Control. Springer Verlag, N.Y.

    CrossRef  MATH  Google Scholar 

  • Fujisaki, M.; G. Kallianpur and H. Kunita (1972). Stochastic differential equations for the nonlinear filtering problem. Osaka Journal of Mathematics, 9, pp. 19–40.

    MathSciNet  MATH  Google Scholar 

  • Gihman, I.I. and A.V. Skorohod (1972). Stochastic Differential Equations. Springer Verlag, Berlin.

    CrossRef  MATH  Google Scholar 

  • Gihman, I.I. and A.V. Skorohod (1979). The Theory of Stochastic Processes III. Springer Verlag, Berlin.

    CrossRef  MATH  Google Scholar 

  • Gómez, G. (1981). Controlling a dual economy. Part I. Div. of Appl. Math. Brown University.

    Google Scholar 

  • Gómez, G. (1983). The intertemporal labour allocation inherent in the optimal stopping of the dual economy: the static case. European Meeting of the Econometric Society. Madrid, 3–7 September, 1984.

    Google Scholar 

  • Gómez, G. (1984a). Modelling the economic development by means of impulsive control techniques. Mathematical Modelling in Sciences and Technology, pp. 802–806, X.J. Avual and R.E. Kalman (eds.), Pergamon Press, N.Y.

    Google Scholar 

  • Gómez, G. (1984b). On the Markov Stopping Rule Associated with the Problem of Controlling a Dual Economy. Dynamic Modelling and Control of National Economies, 1983, pp. 197–204. T. Basar and L.F. Pau (eds.), Pergamon Press, N.Y.

    Google Scholar 

  • Gómez, G. (1985a). A mathematical dynamic model of the dual economy emphasizing unemployment, migration and structural change. Dept. of Mathematics, University of Erlangen-Nürnberg, Erlangen, FRG.

    Google Scholar 

  • Gómez, G. (1985b) Optimal stopping times in the economic development planning. Dept. of Mathematics, University of Erlangen-Nürnberg, Erlangen, FRG.

    Google Scholar 

  • Gómez, G. (1985c). Controlling a dual economy. Part II, III. Work in progress.

    Google Scholar 

  • Goméz, G. (1986). Attainability and Reversibility of a Golden Age for the Labour Surplus Economy: A Stochastic Variational Approach. Research Centre Bielefeld-Bochum-Stochastics, University of Bielefeld, Bielefeld, FRG.

    MATH  Google Scholar 

  • Hausmann, U. (1978). On the stochastic maximum principle. SIAM Journal on Control. 16, pp. 236–251.

    CrossRef  MathSciNet  Google Scholar 

  • Hausmann, U. (1979). On the integral representation of functionals of Itô process. Stochastics, Vol. 3, pp. 17–27.

    CrossRef  MathSciNet  Google Scholar 

  • Hausmann, U. (1980). Existence of partially observable optimal stochastic controls. Lecture Notes in Control, 36, Springer Verlag, Berlin.

    Google Scholar 

  • Hausmann, U. (1981). On the adjoint process for optimal control of diffusion processes. SIAM Journal of Control and Optimization, Vol. 19, pp. 221–243.

    CrossRef  MathSciNet  Google Scholar 

  • Hausmann, U. (1982). Extremal control for completely observable diffusions. Lecture Notes in Control, Vol. 42, Springer Verlag, Berlin.

    Google Scholar 

  • Ikeda, N. and S. Watanabe (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Jöreskog, K.G. and Wold, H. (1982). Systems Under Direct Observation, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Krylov, N.V. (1980). Controlled Diffusion Processes, Springer Verlag, N.Y.

    CrossRef  MATH  Google Scholar 

  • Kushner, H.J. (1967). Stochastic Stability and Control. Academic Press, N.Y.

    MATH  Google Scholar 

  • Kushner, H.J. (1977). Probability Methods for Approximations in Stochastic Control and for Elliptic Equations. Academic Press, N.Y.

    MATH  Google Scholar 

  • Kunita, H. and S. Watanabe (1967). On square integrable martingales, Nagoya Mathematical Journal, 30, pp. 209–245.

    MathSciNet  MATH  Google Scholar 

  • Liptser, R.S. and A.N. Shiryayev (1977). Stochastics of Random Processes I. Springer Verlag, Berlin.

    CrossRef  MATH  Google Scholar 

  • Malliaris, A. (1981). Stochastic Methods in Economics and Finance, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Marglin, S.A. (1976). Value and Price in the Labour-Surplus Economy. Oxford University Press, London.

    Google Scholar 

  • Merton, R. (1975). An asymptotic theory of growth under uncertainty, Review of Economic Studies, 42, pp. 375–393.

    CrossRef  Google Scholar 

  • Pitchford, D. (1974). Population in Economic Growth, North-Holland, Amsterdam.

    Google Scholar 

  • Rishel, R. (1970). Necessary and sufficient conditions for continuous-time stochastic optimal control, SIAM Journal on Control, Vol. 8, pp. 559–571.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations. Wiley, N.Y.

    MATH  Google Scholar 

  • Tintner, G. and Sengupta, J. (1972). Stochastic Economics. Academic Press, N.Y.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Davis, M.H.A., Gómez M., G.L. (1987). The semi-martingale approach to the optimal resource allocation in the controlled labour-surplus economy. In: Albeverio, S., Blanchard, P., Streit, L. (eds) Stochastic Processes — Mathematics and Physics II. Lecture Notes in Mathematics, vol 1250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077347

Download citation

  • DOI: https://doi.org/10.1007/BFb0077347

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17797-5

  • Online ISBN: 978-3-540-47835-5

  • eBook Packages: Springer Book Archive