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Optimal investment with vintage capital: equilibrium distributions

Author

Listed:
  • Silvia Faggian

    () (Department of Economics, University Of Venice Cà Foscari)

  • Fausto Gozzo

    () (Università LUISS Guido Carli)

  • Peter M. Kort

    () (CentER, Department of Econometrics & Operations Research, Tilburg University; Department of Economics, University of Antwerp)

Abstract

The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modelling optimal investment with vintage capital, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time τ but also by age s. Capital accumulation is hence described as a partial differential equation (briefly, PDE), and equilibrium points are in fact equilibrium distributions in the variable s of ages. Investments in frontier as well as non-frontier vintages are possible. Firstly a general method is developed to compute and study equilibrium points of a wide range of infinite dimensional, infinite horizon boundary control problems for linear PDEs with convex criterion, possibly applying to a wide variety of economic problems. Sufficient and necessary conditions for existence of equilibrium points are derived in this general context. In particular, for optimal investment with vintage capital, existence and uniqueness of a long run equilibrium distribution is proved for general concave revenues and convex investment costs, and analytic formulas are obtained for optimal controls and trajectories in the long run, definitely showing how effective the theoretical machinery of optimal control in infinite dimension is in computing explicitly equilibrium distributions, and suggesting that the same method can be applied in examples yielding the same abstract structure. To this extent, the results of this work constitutes a first crucial step towards a thorough understanding of the behaviour of optimal controls and trajectories in the long run.

Suggested Citation

  • Silvia Faggian & Fausto Gozzo & Peter M. Kort, 2019. "Optimal investment with vintage capital: equilibrium distributions," Working Papers 2019: 12, Department of Economics, University of Venice "Ca' Foscari".
  • Handle: RePEc:ven:wpaper:2019:12
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    References listed on IDEAS

    as
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    More about this item

    Keywords

    Equilibrium Points; Equilibrium Distributions; Vintage Capital Stock; Age-structured systems; Maximum Principle in Hilbert Spaces; Boundary control; Optimal Investment;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity

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