The Long-Run Behavior of Consumption and Wealth Dynamics in Complete Financial Market with Heterogeneous Investors

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 482314, 16 pages http://dx.doi.org/10.1155/2014/482314 Research Article The Long-Run Behavior of Consumption and Wealth Dynamics in Complete Financial Market with Heterogeneous Investors Darong Dai Department of Economics, Texas A&M University, College Station, TX 77843, USA Correspondence should be addressed to Darong Dai; Received 8 February 2014; Revised 19 June 2014; Accepted 24 June 2014; Published 14 July 2014 Academic Editor: Juan Manuel Peña Copyright © 2014 Darong Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A type of complete financial market with finite and countable heterogeneous investors, that is, investors equipped with heterogeneous elasticities of intertemporal substitution, heterogeneous time discount rates, and also heterogeneous beliefs, is constructed and two main results are established. First, long-run behaviors, specifically golden rules or modified golden rules, about consumption path and wealth accumulation are investigated under uncertainty and in the sense of uniform topology. Second, inefficacy of temporary taxation policies, which are chosen to be consumption tax and wealth tax, is confirmed in the current financial market. 1. Introduction Our goal in this paper is to explore the golden rule or modified golden rule properties of consumption and wealthaccumulation dynamics, as well as the effects of temporary taxation policies, which are chosen to be consumption tax and wealth tax, in a type of complete financial market with finite and countable heterogeneous investors (see, [1], e.g.), that is, investors with heterogeneous elasticities of intertemporal substitution (e.g., [2, 3]), heterogeneous time discount rates (e.g., [2–4]), and heterogeneous beliefs (see, [2, 3, 5–8], and among others), choosing optimal consumption and portfolio strategy in an economy of infinite horizon. Golden rules about the consumption path, the wealth dynamics, and the combination of both are proved under uncertainty and in the sense of uniform topology, which would be regarded as the first innovation of the current paper. Furthermore, inefficacy of temporary taxation policies has also been confirmed in the current complete financial market, leading us to the second inspiration of the current paper. In the past several decades, portfolio turnpikes (see [9–15], among others) in financial economics have been extensively studied and well-understood. Meanwhile, the concept of golden rule or modified golden rule (e.g., [16–21], among others) has been developed and plays a crucial role in studying optimal economic growth and optimal capital accumulation in macroeconomics. However, little attention up to the present has been paid to the golden rule or modified golden rule of consumption path and wealth accumulation in complete or incomplete financial market with heterogeneous investors. Noting that consumption strategy and wealth accumulation play the same, if not more, important role as that of portfolio choice in both capital asset pricing models (see [22–27], among others) and market selection theory (e.g., [2, 7, 28–35], among others), the current paper is encouraged to meet the gap and investigate the long-run behavior of consumption and wealth dynamics in a type of complete financial market with heterogeneous investors. Indeed, the current paper confirms the following strong conclusion: both optimal consumption path and optimal wealth dynamics are long-run golden rules in the sense of uniform topology and in the corresponding nonstationary environment, regardless of the fact that there are many heterogeneous investors in the economy. In other words, the uniform topology golden rules demonstrated in the present paper are robust to the types of investors in the market as long as they all exhibit the same type of CRRA preferences. Nonetheless, these golden rules are not turnpikes because they are sensitive to initial conditions of the corresponding dynamics [36–38]. And hence, naturally, an open question comes up: when these golden rules are also turnpikes? The exploration of this question will be left to future study. 2 The second goal of this paper is to study the effect of taxation policies, which are specifically chosen to be consumption tax and wealth tax, to optimal consumption strategy. As in the literatures of Yano [38, 39] and Kondo [40], the current paper proves the conclusion of inefficacy of temporary taxation policies in a type of complete financial market with heterogeneous investors in comparatively weak conditions, which are different from those of Yano [38, 39] and Kondo [40] due to the dynamic competitive-equilibrium framework they employed. In addition, although both this paper and Jin [13] investigate the long-run behavior of consumption process in a continuous-time finance model, it is worthwhile mentioning that our results are essentially different from those of Jin [13] in the following aspects. First, Jin mainly proves the portfolio turnpike theorems in a continuous-time model, which however is not our focus in this paper. In particular, Jin proves related turnpike properties, whereas our paper confirms the relevant golden rule properties, and we already know that golden run property is generally weaker than turnpike property. Second, Jin shows the related convergence between processes under different utility-function assumptions, that is, general utility functions, such that the inverse functions of the derivative of utility functions for consumption and investment belong to a special subclass of regularly varying functions and power utility functions, whereas we show the convergence between processes under arbitrary decisions and optimal decisions. Third, Jin just confirms the convergence of final wealth process while the present paper shows convergence for the entire path of wealth accumulation. And fourth, we show a much stronger convergence in the sense of uniform topology while this desired property generally cannot be satisfied in the paper of Jin. Finally, we would like to indicate the differences and also similarity between our investigation and the well-known papers of Sandroni [30] and Blume and Easley [7]. First, the fundamental issue investigated in these papers is distinct with our paper, that is, they all focus on the market-selection theory in complete or incomplete markets, while the current study purely characterizes the long-run behaviors of wealth and consumption processes of heterogeneous investors. Second, it is easy to see that the current model environment is different from the above papers; in particular, they use discretetime models and we use a continuous-time model driven by Brownian motions, and it is easy to see that our model intrinsically leads us (...truncated)