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Homogeneity, Saddle Path Stability, and Logarithmic Preferences in Economic Models

  • Dirk Bethmann

    ()

    (Department of Economics, Korea University)

In a stylized Robinson Crusoe economy, we demonstrate the usefulness of homogeneity in initial conditions when solving and analyzing macroeconomic models. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimization problem. The model’s solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches and the structure of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuoustime Hamilton-Jacobi-Bellman equation. In this case the utility maximization problem can be transformed into an initial value problem for an ordinary differential equation.

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File URL: http://econ.korea.ac.kr/~ri/WorkingPapers/w0702.pdf
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Paper provided by Institute of Economic Research, Korea University in its series Discussion Paper Series with number 0702.

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Length: 20 pages
Date of creation: 2007
Date of revision:
Handle: RePEc:iek:wpaper:0702
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  1. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
  2. Caballe, Jordi & Santos, Manuel S, 1993. "On Endogenous Growth with Physical and Human Capital," Journal of Political Economy, University of Chicago Press, vol. 101(6), pages 1042-67, December.
  3. Danyang Xie, 2002. "Divergence in Economic Performance: Transitional Dynamics with Multiple Equilibria," GE, Growth, Math methods 0210002, EconWPA.
  4. Ortigueira, Salvador & Santos, Manuel S., 2001. "Equilibrium Dynamics in a Two-Sector Model with Taxes," Working Papers 01-17, Cornell University, Center for Analytic Economics.
  5. Abba P. Lerner, 1959. "Consumption-Loan Interest and Money," Journal of Political Economy, University of Chicago Press, vol. 67, pages 512.
  6. Mulligan, C.B. & Sala-i-Martin, X., 1992. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," Papers 651, Yale - Economic Growth Center.
  7. Eric W. Bond & Ping Wang & Chong K. Yip, 1993. "A general two-sector model of endogenous growth with human and physical capital: balanced growth and transitional dynamics," Research Paper 9324, Federal Reserve Bank of Dallas.
  8. Ladron-de-Guevara, Antonio & Ortigueira, Salvador & Santos, Manuel S, 1999. "A Two-Sector Model of Endogenous Growth with Leisure," Review of Economic Studies, Wiley Blackwell, vol. 66(3), pages 609-31, July.
  9. Bethmann, Dirk, 2008. "The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents," Journal of Macroeconomics, Elsevier, vol. 30(1), pages 396-414, March.
  10. Antonio Ladron de Guevara & Salvador Ortigueira & Manuel S. Santos, 1994. "Equilibrium Dynamics in Two-Sector Models of Endogenous Growth," Working Papers 9403, Centro de Investigacion Economica, ITAM.
  11. Abba P. Lerner, 1959. "Consumption-Loan Interest and Money: Rejoinder," Journal of Political Economy, University of Chicago Press, vol. 67, pages 523.
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