Fabbri, Giorgio and Iacopetta, Maurizio (2007): Dynamic Programming, Maximum Principle and Vintage Capital.
Download (520kB) | Preview
We present an application of the Dynamic Programming (DP) and of the Maximum Principle (MP) to solve an optimization over time when the production function is linear in the stock of capital (Ak model). Two views of capital are considered. In one, which is embraced by the great majority of macroeconomic models, capital is homogeneous and depreciates at a constant exogenous rate. In the other view each piece of capital has its own ﬁnite productive life cycle (vintage capital). The interpretation of the time patterns of macroaggregates is quite different between the two cases. A technological shock generates an oscillatory movement in the time pattern of per capita output when capital has a vintage structure; conversely an instantaneous adjustment with no transitional dynamics occurs when capital is homogeneous. From a methodological point of view it emerges that the DP approach delivers sharper results than the MP approach (for instance it delivers a closed form solution for the optimal investment strategy) under slacker parameter restrictions. Cross-time and cross-country data on investments, income, and consumption drawn from the Penn World Table version 6.2 are used to evaluate the vintage and standard Ak model.
|Item Type:||MPRA Paper|
|Original Title:||Dynamic Programming, Maximum Principle and Vintage Capital|
|Keywords:||Vintage Capital; Penn World Table; Maximum Principle; Hilbert Space|
|Subjects:||E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E37 - Forecasting and Simulation: Models and Applications
C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61 - Optimization Techniques; Programming Models; Dynamic Analysis
E - Macroeconomics and Monetary Economics > E2 - Macroeconomics: Consumption, Saving, Production, Employment, and Investment > E22 - Capital; Investment; Capacity
O - Economic Development, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O47 - Measurement of Economic Growth; Aggregate Productivity; Cross-Country Output Convergence
O - Economic Development, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models
|Depositing User:||Giorgio Fabbri|
|Date Deposited:||03. Oct 2007|
|Last Modified:||12. Feb 2013 14:17|
Robert J. Barro and Xavier Sala-i-Martin (2004). Economic Growth, 2nd Edition. Cambridge: MIT Press.
Emilio Barucci and Fausto Gozzi (2001). Technology adoption and accumulation in a vintage capital model. Journal of economics. 74(1), 1-30.
Alain Bensoussan, Giuseppe Da Prato, Micheal C. Delfour and Sanjoy K. Mitter (1992). Representation and control of Inﬁnite dimensional system. Boston: Birkhäuser.
Alain Bensoussan, E. Gerald Hurst and Bertil Näslund (1974). Management applications of modern control theory. Amsterdam: North-Holland Publishing Co.
Raouf Boucekkine, Omar Licandro, and Alphonse Magnus (2001). Numerical solution by iterative methods of a class of vintage capital growth models, Journal of Economic Dynamics and Control 25, 255-269.
Raouf Boucekkine, Omar Licandro, and Christopher Paul (1997). Differential-difference equations in economics: On the numerical solution of vintage capital growth models, Journal of Economic Dynamics and Control 21, 347-362.
Raouf Boucekkine, Omar Licandro, Luis A. and Puch and Fernando del Rio (2005). Vintage capital and the dynamics of the AK model. Journal of Economic Theory, 120(1), 39-72.
Micheal C. Delfour (1986). The linear quadratic optimal control problem with delays in the state and control variables: a state space approach. SIAM journal of control and optimization, 24(1), 835-883.
Odo Diekmann, Stephan A. van Gils, Sjoerd M. Verduyn Lunel and Hans-Otto Walther (1995). Delay equations. Functional, complex, and nonlinear analysis. New York: Springer-Verlag.
Giorgio Fabbri and Fausto Gozzi (2006). Vintage Capital in the AK growth model: a Dynamic Programming approach. submitted.
Gustav Feichtinger, Richard F. Hartl, Peter M. Kort and Vladimir M. Veliov (2006). Anticipation Effects of Technological Progress on Capital Accumulation: a Vintage Capital Approach. Journal of Economic Theory, 126(1), 143-164.
Wendell H. Fleming and Raymond W. Rishel (1975). Deterministic and stochastic optimal control. Berlin: Springer-Verlag.
Charles I. Jones (1995). Time Series Tests of Endogenous Growth Models. The Quarterly Journal of Economics 110, 495-525.
Maurizio Iacopetta (2006). The Price of Capital in the AK Model. Available at SSRN: http://ssrn.com/abstract=909213
Maurizio Iacopetta (2007). Technology Adoption, Vintage Capital, and Inequality. Manuscript.
Ellen R. McGrattan (1998). A Defense of AK Growth Models. Federal Reserve Bank of Minneapolis Quarterly Review 22(4), 13-27.
Ammon Pazy (1983). Semigroups of linear operators and applications to partial differential equations. New York: Springer- Verlag.
Lev S. Pontryagin (1962). The Mathematical Theory of Optimal Processes. New York: Interscience Publishers.
Lauree F. Shampine and Skip Thompson (2000). Solving ddes in Matlab. Available at: “http://www.radford.edu/ thompson/webddes/manuscript.pdf”.
Jiongmin Yong and Xun Yu Zhou (1999). Stochastic control. Hamiltonian systems and HJB equations. New York: Springer-Verlag.
Kosaku Yosida (1995). Functional analysis. Berlin: Springer-Verlag.
Jerzy Zabczyk (1992). Mathematical control theory: an introduction. Boston: Birkhäuser.
William P.Ziemer (1990). Weakly diﬀerentiable functions. NewYork: Springer.