Dynamic Programming, Maximum Principle and Vintage Capital

Fabbri, Giorgio and Iacopetta, Maurizio (2007): Dynamic Programming, Maximum Principle and Vintage Capital.

Abstract

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
• Language: English
• 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 - Consumption, Saving, Production, Investment, Labor Markets, and Informal Economy > E22 - Investment ; Capital ; Intangible Capital ; Capacity; O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O47 - Empirical Studies of Economic Growth ; Aggregate Productivity ; Cross-Country Output Convergence; O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models

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