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Optimal Control with State Constraint and Non-concave Dynamics: A Model Arising in Economic Growth

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Abstract

We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba and Askenazy–Le Van. The framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial capital \(k_{0}\ge 0\) and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton–Jacobi–Bellman equation.

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References

  1. Askenazy, P., Le Van, C.: A model of optimal growth strategy. J. Econ. Theory 85(1), 27–54 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barro, R.J., Sala-i-Martin, X.: Economic Growth. MIT Press, London (1999)

    MATH  Google Scholar 

  3. Carlson, D.A., Haurie, A., Leizarowitz, A.: Infinite Horizon Optimal Control. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  4. Cesari, L.: Optimization—Theory and Applications. Springer, New York (1983)

    Book  MATH  Google Scholar 

  5. Edwards, R.E.: Functional Analysis. Holt, Rinehart and Winston, New York (1995)

    Google Scholar 

  6. Fiaschi, D., Gozzi, F.: Endogenous Growth with Convexo-Concave Technology. Draft (2009)

  7. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, New York (1975)

    Book  MATH  Google Scholar 

  8. Freni, G., Gozzi, F., Pignotti, C.: Optimal strategies in linear multisector models: Value function and optimality conditions. J. Math. Econ. 44(1), 55–86 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lucas, R.E.: On the mechanics of economic development. J. Monet. Econ. 22, 3–42 (1988)

    Article  Google Scholar 

  10. Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38(152), 543–559 (1928)

    Article  Google Scholar 

  11. Romer, P.M.: Increasing return and long-run growth. J. Political Econ. 94(5), 1002–1035 (1986)

    Article  Google Scholar 

  12. Skiba, A.K.: Optimal growth with a convex-concave production function. Econometrica 46(3), 527–539 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yong, J., Zhou, X.: Stochastic Controls—Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    MATH  Google Scholar 

  14. Zabczyk, J.: Mathematical Control Theory—An Introduction. Birkhäuser, Boston (1995)

    MATH  Google Scholar 

  15. Zaslavski, A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Springer, New York (2006)

    MATH  Google Scholar 

  16. Zaslavski, A.J.: Turnpike Phenomenon and Infinite Horizon Optimal Control. Springer Optimization and Its Applications. Springer, New York (2014)

    MATH  Google Scholar 

Download references

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Correspondence to Paolo Acquistapace.

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Acquistapace, P., Bartaloni, F. Optimal Control with State Constraint and Non-concave Dynamics: A Model Arising in Economic Growth. Appl Math Optim 76, 323–373 (2017). https://doi.org/10.1007/s00245-016-9353-5

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  • DOI: https://doi.org/10.1007/s00245-016-9353-5

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