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Time-to-build and cycles. (English) Zbl 1016.91068
Summary: We analyze the dynamics of a simple growth model in which production occurs with a delay while new capital is installed (time-to-build). The time-to-build technology is shown to yield a system of functional (delay) differential equations with a unique steady state. We demonstrate that the steady state, though typically a saddle, may exhibit Hopf cycles. Furthermore, the optimal path to the steady state is oscillatory. A counter-example to the claim that ‘models with a time-to-build technology are not intrinsically oscillatory’ is provided. We also provide a primer on the central technical apparatus – the mathematics of functional differential equations.

91B38 Production theory, theory of the firm
91B62 Economic growth models
34K60 Qualitative investigation and simulation of models involving functional-differential equations
90C90 Applications of mathematical programming
Full Text: DOI
[1] BĂ©liar, J.; Mackey, M., Consumer memory and price fluctuations in commodity marketsan integro-differential model, Journal of dynamics and differential equations, 1, 3, 299-325, (1989)
[2] Bellman, R., Cooke, K., 1963. Differential-delay Equations. Academic Press, New York. · Zbl 0105.06402
[3] Benhabib, J.; Nishimura, K., The Hopf bifurcation and the existence and stability of closed orbits in multi-sector models of optimal economic growth, Journal of economic theory, 21, 421-444, (1979) · Zbl 0427.90021
[4] Boldrin, M.; Woodford, M., Equilibrium displaying endogenous fluctuationsa survey, Journal of monetary economics, 25, 189-222, (1990)
[5] Chaffee, N., A bifurcation problem for a functional differential equation of finitely retarded type, Journal of mathematical analysis and applications, 35, 312-348, (1971) · Zbl 0214.09806
[6] Churchill, R., Brown, J., Verhey, R., 1976. Complex Variables and Applications. McGraw-Hill, New York. · Zbl 0299.30003
[7] Driver, R., 1974. Ordinary and Delay Differential Equations. Springer, Berlin. · Zbl 0374.34001
[8] Frisch, R.; Holme, H., The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics, Econometrica, 3, 225-239, (1935) · JFM 61.1326.05
[9] Gyori, I., Lada, P., Oscillation Theory of Delay Differential Equations: With Applications. Oxford Mathematical Monographs, Oxford.
[10] Hale, J., 1977. Theory of Functional Differential Equations. Springer, New York. · Zbl 0352.34001
[11] Howroyd, T.D.; Russell, A.M., Cournot oligopoly models with time lags, Journal of mathematical economics, 13, 97-103, (1984) · Zbl 0553.90019
[12] Ioannides, Y.M.; Taub, B., On dynamics with time-to-build investment technology and non-time-separable leisure, Journal of economic dynamics and control, 16, 225-241, (1992) · Zbl 0825.90158
[13] James, R.W.; Belz, M.H., The significance of characteristic solutions of mixed difference and differential equations, Econometrica, 6, 326-343, (1938) · JFM 64.0448.02
[14] Kalecki, M., A macroeconomic theory of business cycles, Econometrica, 3, 327-344, (1935) · JFM 61.1326.06
[15] Kydland, F.E.; Prescott, E.C., Time-to-build and aggregate fluctuations, Econometrica, 50, 1345-1370, (1982) · Zbl 0493.90017
[16] Mackey, M., Commodity price fluctuationsprice dependent delays and nonlinearities as explanatory factors, Journal of economic theory, 48, 497-509, (1989) · Zbl 0672.90022
[17] Mahaffy, J.; Zak, P.; Joiner, K., A geometric analysis of stability regions for a linear differential equation with two delays, International journal of bifurcation and chaos, 5, 3, 779-796, (1995) · Zbl 0887.34070
[18] Nishimura, K.; Sorger, G., Optimal cycles and chaosa survey, Studies in nonlinear dynamics and econometrics, 1, 1, 11-28, (1996) · Zbl 1078.91547
[19] Pontryagin, L.S. et al., 1962. The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York (Trirogoff, K.N., translator). · Zbl 0112.05502
[20] Reichlin, P., Endogenous cycles in competitive modelsan overview, Studies in nonlinear dynamics and econometrics, 1, 4, 175-185, (1997) · Zbl 1078.91548
[21] Rustichini, A., 1987. Hopf bifurcation for functional differential equations of mixed type. Unpublished Ph.D. Dissertation, Department of Mathematics, University of Minnesota. · Zbl 0684.34070
[22] Rustichini, A., Hopf bifurcation for functional differential equations of mixed type, Journal of dynamics and differential equations, 1, 20, 145-177, (1989) · Zbl 0684.34070
[23] Schmidt, D.S., 1976. Hopf bifurcation theorem and the center manifold of Lyapunov. In: Marsden, J.E., McCracken, M. (Eds.), The Hopf Bifurcation and its Applications, Applied Mathematical Sciences, vol. 19, Chapter 3C. Springer, Berlin.
[24] Walther, H-O., 1989. Hyperbolic Periodic Solutions, Heteroclinic Connections and Transversal Homoclinic Points in Autonomous Differential Delay Equations, Memoirs, American Mathematical Society, vol. 402. AMS, Providence, RI. · Zbl 0681.34062
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