Vintage Capital in the AK Growth Model: A Dynamic Programming Approach - Extended version

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.

[1]  Giuseppe Da Prato,et al.  Hamilton-Jacobi equations in Hilbert spaces , 1983 .

[2]  F. Smithies.  Linear Operators , 2019, Nature.

[3]  Michel C. Delfour,et al.  Stability and the Infinite-Time Quadratic Cost Problem for Linear Hereditary Differential Systems , 1975 .

[4]  Alexander J. Zaslavski.  Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems , 2000 .

[5]  Yi b'c,et al.  Can convergence regressions distinguish between exogenous and endogenous growth models ? , 2003 .

[6]  Alexander J. Zaslavski,et al.  Solutions for a class of optimal control problems with time delay, part 1 , 1996 .

[7]  R. Vinter,et al.  The Infinite Time Quadratic Control Problem for Linear Systems with State and Control Delays: An Evolution Equation Approach , 1981 .

[8]  M. C. Delfour,et al.  Status of the state space theory of linear hereditary differential systems with delays in state and control variables , 1980 .

[9]  V. Barbu,et al.  Hamilton-Jacobi equations in Hilbert spaces; variational and semigroup approach , 1985 .

[10]  Raouf Boucekkine,et al.  Vintage Capital , 2006 .

[11]  Peter M. Kort,et al.  Anticipation effects of technological progress on capital accumulation: a vintage capital approach , 2006, J. Econ. Theory.

[12]  Michel C. Delfour,et al.  CONTROLLABILITY AND OBSERVABILITY FOR INFINITE-DIMENSIONAL SYSTEMS* , 1972 .

[13]  Raouf Boucekkine,et al.  Replacement Echoes in the Vintage Capital Growth Model , 1997 .

[14]  T. C. Edens,et al.  Economic Growth , 1957, The Journal of Economic History.

[15]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[16]  Hitoshi Ishii,et al.  Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces , 1992 .

[17]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[18]  A. Manitius,et al.  Control systems with delays: Areas of applications and present status of the linear theory , 1977 .

[19]  Jose A. Scheinkman,et al.  Duality theory for dynamic optimization models of economics: The continuous time case☆ , 1982 .

[20]  Fausto Gozzi,et al.  Optimal strategies in linear multisector models: Value function and optimality conditions , 2008 .

[21]  M. Delfour.  The linear quadratic optimal control problem with delays in state and control variables: a state space approach , 1986 .

[22]  Silvia Faggian,et al.  Boundary-control problems with convex cost and dynamic programming in infinite dimension. I. The maximum principle , 2004, Differential and Integral Equations.

[23]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[24]  Mauro Bambi,et al.  Endogenous growth and time to build: the AK case , 2008 .

[25]  Silvia Faggian.  Boundary control problems with convex cost and dynamic programming in infinite dimension part II: Existence for HJB , 2004 .

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[27]  Piermarco Cannarsa,et al.  A boundary-value problem for Hamilton-Jacobi equations in hilbert spaces , 1991 .

[28]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[29]  Michel C. Delfour,et al.  Hereditary differential systems with constant delays. II. A class of affine systems and the adjoint problem , 1975 .

[30]  Fausto Gozzi,et al.  On Dynamic Programming in Economic Models Governed by DDEs , 2006, math/0606344.

[31]  Akira Ichikawa,et al.  Quadratic Control of Evolution Equations with Delays in Control , 1982 .

[32]  A. Haurie,et al.  Infinite horizon optimal control : deterministic and stochastic systems , 1991 .

[33]  Vladimir Borisovich Kolmanovskiĭ,et al.  Control of Systems with Aftereffect , 1996 .

[34]  Michel C. Delfour,et al.  Linear optimal control of systems with state and control variable delays , 1984, Autom..

[35]  Emilio Barucci,et al.  Optimal advertising with a continuum of goods , 1999, Ann. Oper. Res..

[36]  Giuseppe Da Prato,et al.  Second Order Partial Differential Equations in Hilbert Spaces: Bibliography , 2002 .

[37]  Niels Jacob,et al.  GIUSEPPE DA PRATO and JERZY ZABCZYK: 379 pp., £29.95 (LMS members' price £22.46), ISBN 0-521-77729-1 (Cambridge University Press, 2002) , 2004 .

[38]  G. Fabbri.  Viscosity solutions approach to economic models governed by DDEs , 2006 .

[39]  Dean A. Carlson,et al.  The existence of catching-up optimal solutions for a class of infinite horizon control problems with time delay , 1990 .

[40]  Viorel Barbu,et al.  Existence and uniqueness of the dynamic programming equation in Hilbert space , 1983 .

[41]  R. Steele,et al.  Optimization , 2005, Encyclopedia of Biometrics.

[42]  David de la Croix,et al.  MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS , 2004 .

[43]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[44]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV: Hamiltonians with unbounded linear terms , 1990 .

[45]  Emilio Barucci,et al.  Technology adoption and accumulation in a vintage-capital model , 2001 .

[46]  Daniel Tataru,et al.  Viscosity solutions for the dynamic programming equations , 1992 .

[47]  S. Rebelo,et al.  Long-Run Policy Analysis and Long-Run Growth , 1990, Journal of Political Economy.

[48]  Phillippe Michel,et al.  On the Transversality Condition in Infinite Horizon Optimal Problems , 1982 .

[49]  P. Zak,et al.  Time-to-Build and Cycles , 1997 .

[50]  Philippe Michel,et al.  Some Clarifications on the Transversality Condition , 1990 .

[51]  Dean A. Carlson.  Overtaking Optimal Solutions for Convex Lagrange Problems with Time Delay , 1997 .

[52]  Silvia Faggian,et al.  Regular Solutions of First-Order Hamilton–Jacobi Equations for Boundary Control Problems and Applications to Economics , 2005 .

[53]  Cuong Le Van,et al.  A model of optimal growth strategy , 1999 .

[54]  Suresh P. Sethi,et al.  Explicit Solution of a General Consumption/Investment Problem , 1986, Math. Oper. Res..

[55]  N. Salvadori,et al.  Existence of optimal strategies in linear multisector models , 2006 .

[56]  K. L. Cooke,et al.  Hereditary Differential Systems with Constant Delays , .

[57]  Giorgio Fabbri,et al.  Viscosity Solutions to Delay Differential Equations in Demo-Economy , 2008 .

[58]  Boyan Jovanovic,et al.  Solow vs. Solow: Machine Prices and Development , 1997 .

[59]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[60]  Raouf Boucekkine,et al.  Vintage capital and the dynamics of the AK model , 2005, J. Econ. Theory.

[61]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems, 2nd Edition , 2007, Systems and control.

[62]  James Tobin,et al.  Neoclassical Growth with Fixed Factor Proportions , 1966 .

[63]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[64]  Michael Gort,et al.  Measuring the Rate of Technological Progress in Structures , 1998 .

[65]  Stephen L. Parente.  Technology adoption, learning-by-doing, and economic growth , 1994 .

[66]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .

[67]  Narayana R. Kocherlakota,et al.  A Simple Time Series Test of Endogenous vs. Exogenous Growth Models: An Application to the United States. , 1996 .

[68]  Ewald Walterskirchen,et al.  Growth and Unemployment , 1994 .

[69]  Aldo Rustichini,et al.  Vintage capital, investment, and growth , 1991 .