Bounding the CRRA Utility Functions

Suen, Richard M. H. (2009): Bounding the CRRA Utility Functions.

Abstract

The constant-relative-risk-aversion (CRRA) utility function is now predominantly used in quantitative macroeconomic studies. This function, however, is not bounded and thus creates problems when applying the standard tools of dynamic programming. This paper devises a method for "bounding" the CRRA utility functions. The proposed method is based on a set of conditions that can establish boundedness among a broad class of utility functions. These results are then used to construct a bounded utility function that is identical to a CRRA utility function except when consumption is very small or very large. It is shown that the constructed utility function also satisfies the Inada condition and is consistent with balanced growth.

• Item Type: MPRA Paper
• Language: English
• Keywords: Utility Function; Elasticity of Marginal Utility; Boundedness
• Subjects: O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis

References

Aiyagari, S.R.: Uninsured Idiosyncratic Risk and Aggregate Saving. Q J Econ 109, 659-684 (1994)

Alvarez, F., Stokey, N.L.: Dynamic Programming with Homogeneous Functions. J Econ Theory 82, 167-189 (1998)

Barelli, P., Pessôa, S.: Inada Conditions Imply that Production Function must be Asymptotically Cobb-Douglas. Econ Letters 81, 361-363 (2003)

Becker, R.A., Boyd III, J.H.: Capital Theory, Equilibrium Analysis and Recursive Utility. Malden: Blackwell (1997)

Boyce, W.E., DiPrima, R.C.: Elementary Differential Equations and Boundary Value Problems, 6th edn. New York: John Wiley & Sons (1997)

Boyd III, J.H.: Recursive Utility and the Ramsey Problem. J Econ Theory 50, 326-345 (1990)

Durán, J.: On Dynamic Programming with Unbounded Returns. Econ Theory 15, 339-352 (2000)

Geweke, J.: A Note on Some Limitations of CRRA Utility. Econ Letters 71, 341-345 (2001)

Huggett, M.: The One-Sector Growth Model with Idiosyncratic Shocks: Steady States and Dynamics. J Monetary Econ 39, 385-403 (1997)

Huggett, M., Ospina, S.: Aggregate Precautionary Savings: When is the Third Derivative Irrelevant? J Monetary Econ 48, 373-396 (2001)

Le Van, C., Morhaim, L.: Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach. J Econ Theory 105, 158-187 (2002)

Miao, J.: Competitive Equilibria of Economies with a Continuum of Consumers and Aggregate Shocks. J Econ Theory 128, 274-298 (2006)

Ogaki, M., Reinhart, C.M.: Measuring Intertemporal Substitution: The Role of Durable Goods. J Political Econ 106, 1078-1098 (1998)

Palivos, T., Wang P., Zhang, J.: On the Existence of Balanced Growth Equilibrium. Int Econ Rev 38, 205-224 (1997)

Rincón-Zapatero, J.P., Rodríguez-Palmero, C.: Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case. Econometrica 71, 1519-1555 (2003)

Steger, T.M.: Economic Growth with Subsistence Consumption. J Development Econ 62, 343-361 (2000)

Stokey, N.L., Lucas, R.E., Prescott E.: Recursive Methods in Economic Dynamics, Cambridge: Harvard University Press (1989)

Vissing-Jorgensen, A.: Limited Asset Market Participation and the Elasticity of Intertemporal Substitution. J Political Econ 110, 825-853 (2002)

Widder, D.V.: Advanced Calculus, 2nd edn. New York: Dover Publications (1989)