Minimum consumption and transitional dynamics in wealth distribution

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Abstract

This paper investigates quantitatively how initial wealth holding differences across households are propagated through time in a one sector growth model economy. A key feature of the model is that household consumption cannot fall below a positive level each period. The existence of a minimum consumption requirement implies that the Intertemporal Elasticity of Substitution not only differs across households but also changes differently over time. This model is calibrated to match some key aggregate statistics of the U.S. economy. We find that, as in the data, the wealth distribution in our benchmark model economy exhibits a (brief) period of increasing inequality, a short period in which inequality diminishes and a steady level of inequality along the balanced growth path. However, our model illustrates that the evolution of inequality is very sensitive to the length of the transition path. Additionally, our model predicts an upsurge in wealth inequality following the productivity slowdown in the 1970s.

Introduction

There is an extensive literature that investigates the relationship between inequality and growth. The interest in this issue started with Kuznets (1955) who stated his famous conjecture on the non-monotonic relationship between income inequality and level of per capita income. Since then, a host of researchers have explored various channels through which growth affects inequality and vice versa.1 In this paper we contribute to the literature that examines the effects of growth on inequality. We study quantitatively the importance of differences in saving rates across households in driving changes in wealth inequality.

To analyze this channel we build a one sector growth model economy in which there is no uncertainty and capital markets are perfect. Households do not differ in labor earnings. The key feature we add to this otherwise standard neoclassical model is that household consumption cannot fall below a positive level in any period, i.e., we assume a minimum consumption requirement. The existence of this requirement implies that the intertemporal elasticity of substitution (IES) not only differs across households with different levels of wealth but also changes differently over time. This different behavior of the IES governs the evolution of saving rates and drives all changes in wealth inequality in our model economy. The existence of perfect capital markets and the quasi-homotheticity of preferences imply that the Engel curves are affine functions of the level of wealth. This property of the model ensures that the distribution of wealth does not affect the aggregate dynamics of the model—provided that no household's income falls below the consumption requirement—, whereas the wealth distribution does change along the transition path.

To assess the quantitative significance of this channel we focus our study on observed patterns of wealth inequality in the U.S. during the period 1870–1970. The model is calibrated to match some key aggregate statistics of the U.S. economy. Wealth distribution in the initial period is set to match historical data from the U.S. for that period. We simulate our model economy and compare the evolution of the wealth distribution of our model with that experienced in the U.S. We find that, as in the data, the wealth distribution in our benchmark model economy exhibits a (brief) period of increasing inequality, a short period in which inequality diminishes and a steady level of inequality once the economy reaches its balanced growth path. The intuition of these results is as follows: for low levels of aggregate income (therefore, of earnings) the existence of a minimum consumption level implies that poor households not only have smaller saving rates but also increase their savings more slowly than wealthy households decrease theirs. This implies a rise in wealth inequality. As aggregate income rises the behavior of the saving rates is reversed. Poor households increase their savings at a faster rate and inequality diminishes. Once the economy reaches its balanced growth path there are no changes in inequality.

Our experiments suggest that the evolution of wealth inequality is very sensitive to the length of the transition path. The reason for this is that the channel of differences in saving rates across households only operate along the transition path, not the balanced growth path. The farther from its steady state an economy is when it starts a process of sustained growth, the longer the period of increasing inequality and the higher its level. The length of the transition path depends on the contribution of total factor productivity growth (TPF, hereafter) and the initial level of aggregate capital. The higher the level of productivity growth, the shorter the period of increasing inequality, if there is any. The reason for this is that the larger the level of TFP growth the sooner the economy attains a level of income above which poor households start accumulating capital faster than wealthy households. Thus, our experiments suggest that differences in Total Factor Productivity growth may help us to understand the different wealth inequality experiences observed across countries. Finally, our model is able to predict an upsurge in wealth inequality following the productivity slowdown that started in the 1970s.

Other authors have explored the link between differences in saving rates and the dynamics of the wealth distribution. Stiglitz (1969) was among the first to discuss the distributional implications of non-linear consumption functions. Chatterjee (1994) investigates the wealth distribution implications of quasi-homothetic preferences. The main difference between his approach and ours is the measure of wealth used. He investigates the evolution of the distribution of life-time wealth, i.e., the present value of life-time earnings plus the present value of accumulated capital, whereas we define wealth as household net worth. Chatterjee shows analytically that when a minimum consumption requirement exists, the distribution of life-time wealth becomes more unequal along the transition path. We show that the distribution of household net worth can either become more egalitarian or unequal along the transition path, depending on the region of the parameters space considered. Caselli and Ventura (2000) study theoretically the evolution of wealth distribution in a framework similar to ours. They build a model in which households derive utility from a privately produced good and a public good. There are perfect capital markets. The public good in the utility function plays a role similar to a negative minimum consumption requirement. They find a qualitative result similar to ours but do not attempt to quantify its importance. Chatterjee and Ravikumar (1999) also study the link between capital accumulation and inequality along the transition path and introduce a minimum consumption requirement. They generate a Kuznets curve for wealth inequality. As opposed to us, they use a linear technology and calibrate their model to match some key statistics for the Indian economy. Finally, Obiols-Homs and Urrutia (2003) study the evolution of the distribution of wealth in a setup very close to ours. They also consider preferences where there is a minimum consumption but they restrict their analysis to the log-utility function, whereas we extend our analysis to the class of quasi-homothetic preferences used by Chatterjee (1994). Moreover, their study is analytical, whereas our focus is quantitative.

The key departure of our model from the standard neoclassical framework is that we introduce a minimum consumption requirement which implies that the household's IES increases with household's wealth. Using panel data on Indian villagers, Atkeson and Ogaki, 1996, Atkeson and Ogaki, 1997 find economically significant differences in the IES across rich and poor households. Rosenzweig and Wolpin (1993) also use Indian data and find a minimum consumption requirement to be statistically significant and to amount to a sizable fraction of consumption expenditures of the average household. Moreover, Rebelo (1992) and Ogaki et al. (1996) argue that low saving rates and low elasticity of savings with respect to the interest rate point to the existence of a minimum consumption requirement. Thus, we think that a minimum consumption requirement is important in understanding the process of capital accumulation and growth.

The rest of the paper is organized as follows: Section 2 describes the economic environment and discusses some theoretical properties of the class of economies considered. Section 3 describes the evidence we have about the evolution of wealth inequality in the U.S. economy for the period 1870–1970. Section 4 describes our calibration procedure. In Section 5 we present our main results. In Section 6 we study the sensitivity of the evolution of the wealth distribution with respect to the level of productivity growth. Section 7 concludes.

Section snippets

The model economy

We consider a infinite horizon production economy populated by a continuum of households of measure one that live forever. Sections 2.1 and 2.2 describe the technology, preferences and endowments, respectively. Section 2.3 presents the household problem and Section 2.4 provides a formal definition of equilibrium. In Section 2.5 we describe the class of economies to which we restrict our analysis. Section 2.6 shows the properties of the model that will allow us to analyze separately the

Historical evidence

The existing data on the evolution of the wealth distribution in the U.S. over the period 1870–1970 is highly fragmented but nonetheless offers some general guidance about the evolution of inequality over time. The studies that most thoroughly analyze the data available for that period are Lindert (2000) and Williamson (1991).

The evolution of wealth inequality in the late 19th and early 20th centuries appears to be controversial. Williamson (1991) reports that in 1870 the top 1 percent of all

Calibration

In this section we discuss the calibration of our model economy. We proceed in two steps. Section 4.1 describes the calibration of the technology and the preference parameters. We select parameter values so that the aggregate statistics of the balanced growth path of our model economy match some key features of the U.S. economy. In Section 4.2 we discuss our choice for the initial distribution of wealth across households.

Quantitative implications of the model

We have seen in the previous section that the aggregate behavior of the economy along the transition path is not affected by the level of wealth inequality. This allows us to study our three model economies in two steps: we analyze the evolution of prices and aggregate variables in their representative agent version in Section 5.1 and in Section 5.2 we study the evolution of their wealth distribution.

Wealth inequality and TFP growth

In this paper we have focused on wealth inequality. We have seen that once we introduce a minimum consumption requirement, the standard neoclassical growth model can account for the observed patterns of wealth inequality in the U.S. economy between 1870 and 1970. In this section we want to analyze which forces drive the evolution of the wealth distribution. Intuitively, we can see that the length of the transition path plays a central role. The length of the transition path depends on the level

Final comments

This paper has shown that differences in saving rates across households are important in accounting for the distributional data in the late 19th century, the increase in wealth inequality suggested by Williamson (1991) and Lindert (2000), and the stability of the wealth distribution after World War II. The only source of inequality in this model is the initial distribution of wealth. Households are identical in their levels of human capital and, thus, there is perfect equality in the earnings

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    Díaz thanks the Spanish Ministry of Education, DGI, project BEC2001-1653, for financial support. We are grateful to the editor and an anonymous referee for all their comments and suggestions.

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