Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T13:44:26.759Z Has data issue: false hasContentIssue false
  • English
  • Français

BEWLEY–HUGGETT–AIYAGARI MODELS: COMPUTATION, SIMULATION, AND UNIQUENESS OF GENERAL EQUILIBRIUM

Published online by Cambridge University Press:  27 March 2018

Robert Kirkby
Robert Kirkby*
Affiliation:
Victoria University of Wellington
*
Address correspondence to: Robert Kirkby, School of Economics and Finance, Victoria University of Wellington, Wellington, New Zealand; e-mail: robertdkirkby@gmail.com. Website: robertdkirkby.com.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides conditions under which an algorithm for the computation and simulation of Bewley–Huggett–Aiyagari models, based on state-space discretization, will converge to all true solutions. These conditions are shown to be satisfied in two standard examples: the Aiyagari model and its extension to endogenous labor. Bewley–Huggett– Aiyagari models are general equilibrium models with incomplete markets and idiosyncratic, but no aggregate, shocks. The algorithm itself is based on discretization, while the theory importantly allows for making simulations using the approximate computational solution of the value function problem rather than the true model solution. The numerical results of applying the algorithm to both models are provided and investigated in terms of replication, revealing that the Aiyagari model overestimates the degree of precautionary savings in the high-risk-and-high-risk-aversion case. The results also show that both models almost certainly have a unique general equilibrium. Theoretically, the existence of equilibria was known, but uniqueness remained an open question.

Keywords

Bewley–Huggett–Aiyagari ModelsNumerical MethodsConvergencePrecautionary Savings
Type
Articles
Information
Macroeconomic Dynamics , Volume 23 , Issue 6 , September 2019 , pp. 2469 - 2508
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

I appreciate the useful comments of the seminar participants at the Universidad Carlos III de Madrid, Australian National University, and the Australian Conference of Economists (2015). I am grateful to Manuel Santos and John Stachurski for their helpful discussion.

References

REFERENCES

Aiyagari, S. Rao (1993) Uninsured Idiosyncratic Risk and Aggregate Saving. Working paper 502, Federal Reserve Bank of Minneapolis.Google Scholar
Aiyagari, S. Rao (1994) Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109 (3), 659–684.Google Scholar
Aldrich, Eric, Fernandez-Villaverde, Jesus, Gallant, Ronald, and Rubio-Ramirez, Juan (2011) Tapping the supercomputer under your desk: Solving dynamic equilibrium models with graphics processors. Journal of Economic Dynamics and Control 35 (3), 386–393.Google Scholar
Aruoba, S., Fernandez-Villaverde, Jesus, and Rubio-Ramirez, Juan (2006) Comparing solution methods for dynamic equilibrium economies. Journal of Economic Dynamics and Control 30 (12), 2477–2508.Google Scholar
Badel, Alejandro and Huggett, Mark (2014) Taxing Top Earners: A Human Capital Perspective. Working Papers of the Fed Reserve of St Louis, 2014-017.Google Scholar
Barillas, Fransisco and Fernandez-Villaverde, Jesus (2007) A generalization of the endogenous grid method. Journal of Economic Dynamics and Control, 31, 2698–2712.Google Scholar
Benhabib, Jess, Bisin, Alberto, and Zhu, Shenghao (2011) The distribution of wealth and fiscal policy in economies with finitely lived agents. Econometrica 79 (1), 123–157.Google Scholar
Benitez-Silva, Hugo, Hall, George, Hitsch, Gunter, Pauletto, Giorgio, and Rust, John (2005) A Comparison of Discrete and Parametric Approximation Methods for Solving Dynamic Programming Problems in Economics. Manuscript, Yale University.Google Scholar
Bewley, Truman (1983) A difficulty with the optimum quantity of money. Econometrica, 51, 1485–1504.Google Scholar
Bewley, Truman (1984) Notes on Stationary Equilibrium with a Continuum of Independently Fluctuating Consumers. Manuscript, Yale University.Google Scholar
Bhattacharya, Rabi and Lee, Oesook (1988) Asymmptotics of a class of markov processes which are not in general irreducible. The Annals of Probability 16 (3), 1333–1347. A correction to this article was published in the same journal in 1997, 25(3), 1541–1543.Google Scholar
Bhattacharya, Rabi and Majumdar, Mukal (2001) On a class of stable random dynamical systems: Theory and applications. Journal of Economic Theory 96, 208–229.Google Scholar
Cai, Yongyang and Judd, Kenneth L. (2014) Advances in numerical dynamic programming and new applications. In Schmedders, K. and Judd, K. L. (eds.), Handbook of Computational Economics, vol. 3, chapter 8. Amsterdam, Netherlands: Elsevier.Google Scholar
Caldara, Dario, Fernandez-Villaverde, Jesus, Rubio-Ramirez, Juan, and Yao, Wen (2012) Computing dsge models with recursive preferences and stochastic volatility. Review of Economic Dynamics 15 (2), 188–206.Google Scholar
Carroll, Christopher (2006) The method of endogenous gridpoints for solving dynamic stochastic optimization problems. Economics Letters 91, 312–320.Google Scholar
Castaneda, Ana, Díaz-Giménez, Javier, and Ríos-Rull, Jose Victor (2003) Accounting for the U.S. earnings and wealth inequality. Journal of Political Economy 111 (4), 818–857.Google Scholar
Clausen, Andrew and Strub, Carlo (2014) A General and Intuitive Envelope Theorem. Society of Economic Dynamic meeting papers, 235, 1–20.Google Scholar
Conesa, Juan Carlos and Krueger, Dirk (2006) On the optimal progressivity of the income tax code. Journal of Monetary Economics 53 (7), 1425–1450.Google Scholar
Conesa, Juan Carlos, Kitao, Sagiri, and Krueger, Dirk (2009) Taxing capital? Not a bad idea after all! American Economic Review 99 (1), 25–48.Google Scholar
Díaz-Giménez, Javier, Quadrini, Vicente, and Ríos-Rull, Jose Victor (1997) Dimensions of inequality: Facts on the U.S. distributions of earnings, income, and wealth. Quarterly Review of the Federal Reserve of Minneapolis 1997 (Spring), 3–21.Google Scholar
Díaz-Giménez, Javier, Glover, Andy, and Ríos-Rull, Jose Victor (2011) Facts on the distributions of earnings, income, and wealth in the United States: 2007 update. Quarterly Review of the Federal Reserve of Minneapolis 34 (1), 2–31.Google Scholar
Domeij, David and Heathcote, Jonathan (2004) On the distributional effects of decreasing capital taxes. International Economic Review 45 (2), 523–544.Google Scholar
Fella, Giulio (2014) A generalized endogenous grid method for non-smooth and non-concave problems. Review of Economic Dynamics 17 (2), 329–344.Google Scholar
Guner, Nezih, Lopez-Daneri, Martin, and Ventura, Gustavo (2016) Heterogeneity and government revenues: Higher taxes at the top? Journal of Monetary Economics 80, 69–85.Google Scholar
Hatchondo, Juan Carlos, Martinez, Leonardo, and Sapriza, Horacio (2010) Quantitative properties of sovereign default models: Solution method. Review of Economic Dynamics 13 (4), 919–933.Google Scholar
Heathcote, Jonathan, Storesletten, Kjetil, and Violante, Giovanni (2009) Quantitative macroeconomics with heterogeneous households. Annual Review of Economics 1 (5), 319–354.Google Scholar
Hopenhayn, Hugo and Prescott, Edward C. (1992) Stochastic monotonicity and stationary distributions for dynamic economies. Econometrica 60 (6), 1387–1406.Google Scholar
Huggett, Mark (1993) The risk-free rate in heterogenous-agent incomplete-insurance economies. Journal of Economic Dynamics and Control 17, 953–969.Google Scholar
Huggett, Mark (2003) When are comparative dynamics monotone? Review of Economic Dynamics 6 (1), 1–11.Google Scholar
Kamihigashi, Takashi and Stachurski, John (2014) Stochastic stability in monotone economies. Theoretical Economics 9 (2), 383–407.Google Scholar
Kamihigashi, Takashi and Stachurski, John (2016) Seeking ergodicity in dynamic economies. Journal of Economic Theory 163, 900–924.Google Scholar
Kinderman, Fabian and Krueger, Dirk (2014) High Marginal Tax Rates on the Top 1 Percent? CFS working paper series 473.Google Scholar
Kirkby, Robert (2017a) Convergence of discretized value function iteration. Computational Economics 49 (1), 117–153.Google Scholar
Kirkby, Robert (2017b) A toolkit for value function iteration. Computational Economics 49 (1), 1–15.Google Scholar
Le Van, C. and Stachurski, John (2007) Parametric continuity of stationary distributions. Economic Theory 33, 333–348.Google Scholar
Li, Huiyu (2015) Numerical policy error bounds for eta-concave stochastic dynamic programming with non-interior solutions. Computational Economics 46 (2), 171–187.Google Scholar
Maliar, Lilia and Maliar, Serguei (2013) Envelope condition method versus endogenous grid method for solving dynamic programming problems. Economics Letters 120, 262–266.Google Scholar
Marcet, Albert, Obiols-Homs, Francesc, and Weil, Philippe (2007) Incomplete markets, labor supply and capital accumulation. Journal of Monetary Economics 54 (8), 2621–2635.Google Scholar
Miao, Jianjun (2006) Competitive equilibria of economies with a continuum of consumers and aggregate shocks. Journal of Economic Theory 128 (1), 274–298.Google Scholar
Pál, Jeno and Stachurski, John (2013) Fitted value function iteration with probability one contractions. Journal of Economic Dynamics and Control 37, 251–264.Google Scholar
Peralta-Alva, Adrian and Santos, Manuel (2014) Analysis of numerical errors. In Schmedders, K. and Judd, K. L. (eds.), Handbook of Computational Economics, volume 3, chapter 9. Amsterdam, Netherlands: Elsevier.Google Scholar
Pijoan-Mas, Josep (2006) Precautionary savings or working longer hours? Review of Economic Dynamics 9 (2), 326–352.Google Scholar
Quadrini, Vicenzo (2000) Entrepreneurship, saving and social mobility. Review of Economic Dynamics 3 (1), 1–40.Google Scholar
Rincón-Zapatero, Juan Pablo, and Santos, Manuel (2009) Differentiability of the value function without interiority assumptions. Journal of Economic Theory 144 (5), 1948–1964.Google Scholar
Ríos-Rull, José Víctor (1995) Models with heterogenous agents. In Cooley, T. (ed.), Frontiers in Business Cycle Research, chapter 4. Princeton, NJ: Princeton University Press.Google Scholar
Ríos-Rull, José Víctor (2001) Computation of equilibria in heterogenous agent models. In Márimon, R. and Scott, A. (eds.), Computational Methods for the Study of Dynamic Economies, chapter 11. Cambridge, UK: Oxford University Press.Google Scholar
Santos, Manuel (2000) Accuracy of numerical solutions using the euler equation residuals. Econometrica 68 (6), 1377–1402.Google Scholar
Santos, Manuel and Peralta-Alva, Adrian (2005) Accuracy of simulations for stochastic dynamic models. Econometrica, 66, 409–426.Google Scholar
Santos, Manuel and Vigo-Aguiar, Jesus (1998) Analysis of a numerical dynamic programming algorithm applied to economic models. Econometrica, 66, 409–426.Google Scholar
Stachurski, John (2008) Continuous state dynamic programming via nonexpansive approximations. Computational Economics 31 (2), 141–160.Google Scholar
Stenflo, O. (2001) Ergodic theorems for markov chains represented by iterated function systems. Bulletin of the Polish Academy of Sciences: Mathematics 49, 27–43.Google Scholar
Stokey, Nancy, Lucas, Robert E., and Prescott, Edward C. (1989) Recursive Methods in Economic Dynamics. Boston, MA: Harvard University Press.Google Scholar
Tauchen, George (1986) Finite state Markov-chain approximations to univariate and vector autoregressions. Economics Letters 20, 177–181.Google Scholar
Tsyrennikov, Viktor, Maliar, Lilia, Maliar, Sergei, and Arellano, Christina (2016) Envelope condition method with an application to default risk models. Journal of Economic Dynamics and Control 69, 436–459.Google Scholar