Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-30T07:38:36.987Z Has data issue: false hasContentIssue false
  • English
  • Français

BRIDGING THE GAP BETWEEN GROWTH THEORY AND THE NEW ECONOMIC GEOGRAPHY: THE SPATIAL RAMSEY MODEL

Published online by Cambridge University Press:  01 February 2009

Raouf Boucekkine ,
Carmen Camacho  and
Benteng Zou
Raouf Boucekkine
Affiliation:
Catholic University of Louvain and University of Glasgow
Carmen Camacho*
Affiliation:
Catholic University of Louvain
Benteng Zou
Affiliation:
University of Luxembourg
*
Address Correspondence to Carmen Camacho, Department of Economics, Catholic University of Louvain, Place Montesquieu, 3, B-1348 Louvain-la-Neuve, Belgium. e-mail: carmen.camacho@uclouvain.be.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a Ramsey problem in infinite and continuous time and space. The problem is discounted both temporally and spatially. Capital flows to locations with higher marginal return. We show that the problem amounts to optimal control of parabolic partial differential equations (PDEs). We rely on the existing related mathematical literature to derive the Pontryagin conditions. Using explicit representations of the solutions to the PDEs, we first show that the resulting dynamic system gives rise to an ill-posed problem in the sense of Hadamard. We then turn to the spatial Ramsey problem with linear utility. The obtained properties are significantly different from those of the nonspatial linear Ramsey model due to the spatial dynamics induced by capital mobility.

Keywords

Ramsey ModelEconomic GeographyParabolic EquationsOptimal Control

Information

Type
Articles
Information
Macroeconomic Dynamics , Volume 13 , Issue 1 , February 2009 , pp. 20 - 45
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Bandle, C., Pozio, M.A., and Tesei, A. (1987) The asymptotic behavior of the solutions of degenerate parabolic equations. Transactions of the American Mathematical Society 303, 2, 487–501.CrossRefGoogle Scholar
Beckman, M. (1952) A continuous model of transportation. Econometrica 20, 643–660.CrossRefGoogle Scholar
Brito, P. (2004) The dynamics of distribution in a spatially heterogeneous world. Working Papers, Department of Economics, ISEG, WP13/2004/DE/UECE.Google Scholar
Fujita, M., Krugman, P., and Venables, A. (1999) The Spatial Economy. Cities, Regions and International Trade. Cambridge, MA, MIT Press.CrossRefGoogle Scholar
Fujita, M. and Thisse, J.F. (2002) Economics of Agglomeration. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Gaines, R. (1976) Existence of solutions to Hamiltonian dynamical systems of optimal growth. Journal of Economic Theory 12, 114–130.CrossRefGoogle Scholar
Gozzi, F. and Tessitore, M.E. (1998) Optimal conditions for Dirichlet boundary control problems of parabolic type. Journal of Mathematical Systems, Estimation and Control, January, Code 08113.Google Scholar
Hadamard, J. (1923) Lectures on the Cauchy Problem in Linear Partial Differential Equations. New Haven, CT: Yale University Press.Google Scholar
Isard, W. and Liossatos, P. (1979) Spatial Dynamics and Optimal Space-Time Development. Amsterdam: North-Holland.Google Scholar
Krugman, P. (1991) Increasing returns and economic geography. Journal of Political Economy 99, 483–499.CrossRefGoogle Scholar
Krugman, P. (1993) On the number and location of cities. European Economic Review 37, 293–298.CrossRefGoogle Scholar
Krugman, P. (1996) The Self-Organizing Economy, Appendix: A Central Place Model. Cambridge, MA: Blackwell.Google Scholar
Ladyzhenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type. Providence, RI: American Mathematical Society.Google Scholar
Lenhart, S.M. and Yong, J. (1992) Optimal Control for Degenerate Parabolic Equations with Logistic Growth. Institute for Mathematics and Its Application, University of Minnesota.Google Scholar
Mossay, P. (2003) Increasing returns and heterogeneity in a spatial economy. Regional Science and Urban Economics 33, 419–444.CrossRefGoogle Scholar
Pao, C.V. (1992) Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press.Google Scholar
Puu, T. (1982) Outline of a trade cycle model in continuous space and time. Geographical Analysis 14, 1–9.CrossRefGoogle Scholar
Raymond, J.P. and Zidani, H. (1998) Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls. SIAM Journal of Control and Optimization, 36 (6), 1853–1879.CrossRefGoogle Scholar
Raymond, J.P. and Zidani, H. (2000) Time optimal problems with boundary controls. Differential and Integral Equations, 13 (7–9), 1039–1072.CrossRefGoogle Scholar
Ten Raa, T. (1986) The initial value problem for the trade cycle in euclidean space. Regional Science and Urban Economics, 16 (4), 527–546.CrossRefGoogle Scholar
Wen, G.C. and Zou, B. (2000) Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients. Nonlinear Analysis TMA, 39, 937–953.CrossRefGoogle Scholar
Wen, G.C. and Zou, B. (2002) Initial-Boundary Value Problems for Nonlinear Parabolic Equations in Higher Dimensional Domains. Beijing/New York: Science Press.Google Scholar