A Note on Laws of Motion for Aggregate Distributions
- 1 Johannes Gutenberg University, Mainz, Germany
Abstract
I derive the law of motion for the aggregate distribution directly from the laws of motion for the individuals’ states. By relying on concepts from measure theory, the derivation is concise and intuitive. I address random shocks both at the micro level and at the macro level. Micro-level shocks completely cancel at the aggregate level provided that a law of large numbers applies. Therefore, the law of motion for the aggregate distribution is a deterministic process in the absence of macro-level uncertainty. If there are macro-level risks, the law of motion for the aggregate distribution exhibits a stochastic component additionally. I illustrate the formalism in a model of wealth accumulation with stochastic interest rates, deriving the law of motion for the aggregate wealth distribution.
- Achdou, Y., Han, J., Lasry, J. M., Lions, P. L., & Moll, B. (forthcoming). Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach. Review of Economic Studies.
- Bayer, C., & Walde, K. (2010). Matching and Saving in Continuous Time: Theory. CESifo Working Paper Series 3026, Munich: CESifo Group.
- Benhabib, J., & Bisin, A. (2018). Skewed Wealth Distributions: Theory and Empirics. Journal of Economic Literature, 56, 1261-1291. https://doi.org/10.1257/jel.20161390
- Cao, D., & Luo, W. (2017). Persistent Heterogeneous Returns and Top End Wealth Inequality. Review of Economic Dynamics, 26, 301-326. https://doi.org/10.1016/j.red.2017.10.001
- Chen, L., Peng, J., Liu, Z., & Zhao, R. (2017a). Pricing and Effort Decisions for a Supply Chain with Uncertain Information. International Journal of Production Research, 55, 264-284. https://doi.org/10.1080/00207543.2016.1204475
- Chen, L., Peng, J., & Zhang, B. (2017b). Uncertain Goal Programming Models for Bicriteria Solid Transportation Problem. Applied Soft Computing, 51, 49-59. https://doi.org/10.1016/j.asoc.2016.11.027
- Duffie, D., & Sun, Y. (2012). The Exact Law of Large Numbers for Independent Random Matching. Journal of Economic Theory, 147, 1105-1139. https://doi.org/10.1016/j.jet.2012.01.003
- Gabaix, X., Lasry, J. M., Lions, P. L., & Moll, B. (2016). The Dynamics of Inequality. Econometrica, 84, 2071-2111. https://doi.org/10.3982/ECTA13569
- He, W., Sun, X., & Sun, Y. (2017). Modeling Infinitely Many Agents. Theoretical Economics, 12, 771-815. https://doi.org/10.3982/TE1647
- Jacod, J., & Protter, P. (2004). Probability Essentials. 2nd Edition, Berlin: Springer. https://doi.org/10.1007/978-3-642-55682-1
- Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin: Springer.
- Merton, R. C. (1975). An Asymptotic Theory of Growth under Uncertainty. Review of Economic Studies, 42, 375-393. https://doi.org/10.2307/2296851
- Moscarini, G. (2005). Job Matching and the Wage Distribution. Econometrica, 73, 481-516. https://doi.org/10.1111/j.1468-0262.2005.00586.x
- Qiao, L., Sun, Y., & Zhang, Z. (2016). Conditional Exact Law of Large Numbers and Asymmetric Information Economies with Aggregate Uncertainty. Economic Theory, 62, 43-64. https://doi.org/10.1007/s00199-014-0855-6