Fabbri, Giorgio and Gozzi, Fausto
(2006):
*Vintage Capital in the AK growth model: a Dynamic Programming approach. Extended version.*

Preview |
PDF
MPRA_paper_7334.pdf Download (484kB) | Preview |

## Abstract

This paper deals with an endogenous growth model with vintage capital and, more precisely, with the AK model proposed in [18]. In endogenous growth models the introduction of vintage capital allows to explain some growth facts but strongly increases the mathematical difficulties. So far, in this approach, the model is studied by the Maximum Principle; here we develop the Dynamic Programming approach to the same problem by obtaining sharper results and we provide more insight about the economic implications of the model. We explicitly ﬁnd the value function, the closed loop formula that relates capital and investment, the optimal consumption paths and the long run equilibrium. The short run ﬂuctuations of capital and investment and the relations with the standard AK model are analyzed. Finally the applicability to other models is also discussed.

Item Type: | MPRA Paper |
---|---|

Original Title: | Vintage Capital in the AK growth model: a Dynamic Programming approach. Extended version. |

Language: | English |

Keywords: | Endogenous growth; Vintage capital; AK model; Dynamic programming |

Subjects: | C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity |

Item ID: | 7334 |

Depositing User: | Giorgio Fabbri |

Date Deposited: | 26 Feb 2008 00:05 |

Last Modified: | 26 Sep 2019 13:54 |

References: | P. Aghion and P. Howitt. Growth and unemployment. Review of economic studies, 61:477–494, 1994. P. K. Asea and P. J. Zak. Time-to-build and cycles. Journal of Economic Dynamics and Control, 23(8):1155–1175, August 1999. P. Askenazy and C. Le Van. A model of optimal growth strategy. Journal of economic theory, 85:24–51, 1999. M. Bambi. Endogenous growth and time to build: the ak case. Journal of Economic Dynamics and Control, Forthcoming. V. Barbu and G. Da Prato. Hamilton-Jacobi equations in Hilbert spaces, volume 86 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1983. V. Barbu and G. Da Prato. Hamilton-Jacobi equations in Hilbert spaces: variational and semigroup approach. Ann. Mat. Pura Appl. (4), 142:303–349 (1986), 1985. V. Barbu, G. Da Prato, and C. Popa. Existence and uniqueness of the dynamic programming equation in Hilbert space. Nonlinear Anal., 7(3):283–299, 1983. M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control: Foundations and Applications. Boston, MA: Birkhauser. xvii, 570 p., 1997. R. J. Barro and X. Sala-i Martin. Economic growth. Mc Graw-Hill, 1995. E. Barucci and F. Gozzi. Optimal advertising with a continuum of goods. Ann. Oper. Res., 88:15–29, 1999. Optimal control and differential games (Vienna, 1997). E. Barucci and F. Gozzi. Technology adoption and accumulation in a vintage capital model. Journal of Economics, 1(74):1–30, 2001. A. Benhabib, J.; Rustichini. Vintage capital, investment, and growth. J. Econ. Theory, 1(55):323–339, 1991. A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter. Representation and control of Inﬁnite dimensional system. Birkh¨user Boston, 1992. a J.A. Benveniste, L.M.; Scheinkman. Duality theory for dynamic optimization models of economics: The continuous time case. J. Econ. Theory, 1(27):1–19, 1982. R. Boucekkine, D. de la Croix, and O. Licandro. Modelling vintage structures with ddes: principles and applications. Math. Popul. Stud., 11(3-4):151–179, 2004. R. Boucekkine, F. Del Rio, and B. Martinez. Technological progress, obsolescence and depreciation. Universite catholique de Louvain, D´partement des Sciences Economiques Working Paper 2006015, Universite catholique de Louvain, Departement des Sciences Economiques, March 2006. R. Boucekkine, M. Germain, and O. Licandro. Replacement Echoes in the Vintage Capital Growth Model. Journal of Economic Theory, 74(2):333–348, 1997. R. Boucekkine, O. Licandro, L. A. Puch, and F. del Rio. Vintage capital and the dynamics of the AK model. J. Econom. Theory, 120(1):39–72, 2005. P. Cannarsa, F. Gozzi, and H.M. Soner. A boundary value problem for Hamilton-Jacobi equations in Hilbert spaces. Applied Mathematics and Optimization, 24(2):197–220, 1991. D.A. Carlson. The existence of catching-up optimal solutions for a class of inﬁnite horizon optimal control problems with time delay. SIAM Journal of Control Optimization, 28(2):402–422, 1990. D.A. Carlson. Overtaking optimal solutions for convex Lagrange problems with time delay. Journal of Mathematical Analysis and Applications, 208(1):31–48, 1997. D.A. Carlson, A. Haurie, and A. Leizarowitz. Inﬁnite horizon optimal control: deterministic and stochastic systems. Springer-Verlag, Berlin, 1991. M. Crandall and P.L. Lions. User’s guide to viscosity solution of hamilton-Jacobi equations. Bull. Amer. Math. Soc. (4), 142:303–349 (1986), 1985. M.G. Crandall and P.L. Lions. Viscosity solutions of Hamilton-Jacobi equations in inﬁnite dimensions. IV. Hamiltonians with unbounded linear terms. J. Funct. Anal., 90(2):237–283, 1990. G. Da Prato and J. Zabczyk. Second order partial differential equations in Hilbert spaces, volume 293 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2002. M. C. Delfour. Status of the state space theory of linear hereditary differential systems with delays in state and control variables. In Analysis and optimization of systems (Proc. Fourth Internat. Conf., Versailles, 1980), volume 28 of Lecture Notes in Control and Information Sci., pages 83–96. Springer, Berlin, 1980. M. C. Delfour. Linear optimal control of systems with state and control variable delays. Automatica J. IFAC, 20(1):69–77, 1984. M. C. Delfour. The linear quadratic optimal control problem with delays in the state and control variables: a state space approach. SIAM J. Contr. Optimiz., 24(1):835– 883, 1986. M. C. Delfour and A. Manitius. Control systems with delays: areas of applications and present status of the linear theory. In New trends in systems analysis (Proc. Internat. Sympos., Versailles, 1976), pages 420–437. Lecture Notes in Control and Informat. Sci., Vol. 2. Springer, Berlin, 1977. M. C. Delfour, C. McCalla, and S. K. Mitter. Stability and the inﬁnite-time quadratic cost problem for linear hereditary diﬀerential systems. SIAM J. Control and Optimization, 13:48–88, 1975. M. C. Delfour and S. K. Mitter. Controllability and observability for inﬁnite- dimensional systems. SIAM J. Control, 10:329–333, 1972. M. C. Delfour and S. K. Mitter. Hereditary diﬀerential systems with constant delays. II. A class of affine systems and the adjoint problem. Differential Equations, 18:18–28, 1975. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. Walther. Delay equations, volume 110 of Applied Mathematical Sciences. Springer-Verlag, New York, 1995. Functional, complex, and nonlinear analysis. N. Dunford and J.T. Schwartz. Linear operators, part I. Wiley-Interscience, New York, 1966. G. Fabbri. Viscosity solutions approach to economic models governed by ddes. sub- mitted, see http://150.146.2.4/amamef/Preprints/VGZ25.pdf. G. Fabbri, S. Faggian, and F. Gozzi. On the dynamic programming approach to economic models governed by dde’s. 2006. To appear in Math. Popul. Stud. S. Faggian. Boundary control problems with convex cost and dynamic programming in inﬁnite dimension. II. Existence for HJB. Discrete Contin. Dyn. Syst., 12(2):323– 346, 2005. S. Faggian. Regular solutions of ﬁrst-order Hamilton-Jacobi equations for boundary control problems and applications to economics. Appl. Math. Optim., 51(2):123–162, 2005. G. Feichtinger, R. Hartl, P. Kort, and V. Veliov. Anticipation effects of technological progress on capital accumulation: a vintage capital approach. Journal of Economic Theory, 20:645–668, 2006. W.H. Fleming and R.W. Rishel. Deterministic and stochastic optimal control. Springer-Verlag, Berlin, 1975. Applications of Mathematics, No. 1. G. Freni, F. Gozzi, and C. Pignotti. Optimal strategies in linear multisector models: Value function and optimality conditions, 2006. submitted. G. Freni, F. Gozzi, and N. Salvadori. Existence of optimal strategies in linear multi- sector models. Economic Theory, to appear, 2006. M. Gort, J. Greenwood, and P. Rupert. Measuring the rate of technological progress in structures. Rev. Econ. Dynam, 2:107–230, 1999. J.K. Hale and S.M. Verduyn Lunel. Introduction to functional differential equations - Applied Mathematical Sciences 99. Springer-Verlag, 1993. D. Henry. Geometric theory of semilinear parabolic equations, volume 840 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1981. A. Ichikawa. Quadratic control of evolution equation with delay in control. SIAM journal of control and optimization, 20:645–668, 1982. H. Ishii. Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces. J. Funct. Anal., 105(2):301–341, 1992. B. Jovanovich and R. Rob. Solow vs. solow: machine prices and development. NBER, 5871, 1997. I. Karatzas, J.P. Lehoczky, S.P Sethi, and S.E Shreve. Explicit solution of a general consumption/investment problem. Math. oper. res., 11:261–294, 1986. N.R. Kocherlakota and K.M. Yi. Can convergence regressions distinguish between exogenous and endogenous growth models? Economics Letters, 49(2):211–215, 1995. N.R. Kocherlakota and K.M. Yi. A Simple Time Series Test of Endogenous vs. Exogenous Growth Models: An Application to the United States. The Review of Economics and Statistics, 78(1):126–134, 1996. V. Kolmanovskii and A. Myshkis. Applied theory of functional-differential equations, volume 85 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1992. V. Kolmanovskii and A. Myshkis. Introduction to the theory and applications of functional-differential equations, volume 463 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1999. V. B. Kolmanovskii and L. E. Shaikhet. Control of systems with aftereffect, volume 157 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1996. X. Li and J. Yong. Optimal control theory for inﬁnite dimensional system. Birkhauser, 1995. P. Michel. On the transversality conditions in inﬁnite horizon optimal problems. Econometrica, 50 (4):975–985, 1982. P. Michel. Some clariﬁcations on the transversality condition. Econometrica, 58 (3):705–723, 1990. S. Parente. Technology adoption, learning by doing and economic growth. Journal of economic theory, 63:346–369, 1994. S. Rebelo. Long run policy analysis and lon run growth. Journal of Political Economy, 99:500–521, 1991. R.M. Solow, J. Tobin, C.C. von Weizsacker, and M. Yaari. Neoclassical growth with ﬁxed factor proportions. The Review of Economic Studies, 33:79–115, 1966. D. Tataru. Viscosity solutions for the dynamic programming equations. Appl. Math. Optim., 25(2):109–126, 1992. R. B. Vinter and R. H. Kwong. The inﬁnite time quadratic control problem for linear system with state control delays: An evolution equation approach. SIAM journal of control and optimization, 19:139–153, 1981. A.J. Zaslavski. Solutions for a class of optimal control problems with time delay. i. J. Optimization Theory Appl, 91(1):155–184, 1996. A.J. Zaslavski. Solutions for a class of optimal control problems with time delay. ii. J. Optimization Theory Appl, 91(2):455–490, 1996. A.J. Zaslavski. Existence and structure of optimal solutions of inﬁnite-dimensional control problems. Applied Mathematics and Optimization, 42(3):291–313, 2000. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/7334 |