Models with a large number of ex-ante identical agents with standard preferences subject to uninsurable, idiosyncratic shocks to earnings are a key tool used to study, among else, two questions that are relevant for macroeconomics, that is, what is the size of precautionary savings and what accounts for the large observed differences in wealth holdings among households. In this paper we use the general framework mentioned above to assess the role played by non-homothetic preferences that make a distinction between basic and luxury consumption, in determining the volume of precautionary savings and in shaping the distribution of wealth, and hence, their contribution to answering those two important macroeconomic questions.

Non homothetic preferences have been recently used to improve the performance of portfolio choice models by Wachter and Yogo (2010) and to help understanding the equity premium puzzle by Ait-Sahalia, Parker, and Yogo (2004). Other authors like Ikeda (2006) studied models where luxury consumption induces a preference for wealth and explored the consequences of taxation of luxury consumption for capital accumulation. The distinction between luxury and basic consumption has even been used by Dalgin, Trindade, and Mitra (2008) to improve the understanding of international trade patterns. Despite all this work with non-homothetic preferences, their implications for the determination of precautionary savings and for the shape of the wealth distribution have been neglected. Filling up this gap is precisely the purpose of the present paper.

The economies studied in this research are in most respect standard. They feature a continuum of infinitely lived ex-ante identical agents. Agents receive an exogenous stochastic stream of earnings that cannot be insured due to incomplete markets. They have access to a single risk-free asset that they can use to smooth consumption in the face of variable earnings, subject to a borrowing constraint. As a result of different histories of realizations of the earnings shock they will be ex-post heterogeneous. The model is closed by a standard neoclassical production function. It does not feature any aggregate uncertainty, hence we can focus on studying stationary equilibria. The key innovation of the model is the assumption that there are two consumption goods: a basic good and a luxury good. The basic good has an expenditure elasticity less than unity while the luxury good has an expenditure elasticity greater than unity so that its share is rising in total expenditures. In order to obtain this result we use an additive logarithmic intra-temporal utility function defined over the two goods. Following Wachter and Yogo (2010) we calibrate its parameters so as to match the median share of basic consumption in total expenditures and the gradient of the share with respect to quartiles of the expenditure distribution. We then simulate several economies characterized by a different amount of earnings risk under the standard assumption of homothetic preferences over a single good and under the assumption of non-homothetic preferences over two goods. These economies are otherwise equally parameterized. We find that precautionary savings is reduced under all earnings processes. The reduction is always significant and may be very large when earnings risk is large. With respect to the wealth distribution, the sign of the effect of introducing this class of preferences is not univocally determined: wealth concentration decreases for moderate levels of earnings risk but increases when earnings risk is large. In both cases though, quantitatively the change in wealth concentration is not big. Overall then we can say that these preferences are important to understand aggregate savings but do not seem to represent a step forward towards the explanation of the long standing issue of wealth concentration, especially at the top of the distribution.

The intuition for the reduction in the size of precautionary savings is that if the model with homothetic utility and the one with non-homothetic utility are equally parameterized, agents in the latter model will be on average less risk-averse because risk aversion declines from a common value as wealth, hence the share of luxuries in total consumption, increases. Wealthy agents, who carry out most of the saving are less risk-averse in the non-homothetic model, hence they reduce their precautionary savings, thus depressing savings in the aggregate. As for the wealth distribution, the two goods with non-homothetic utility introduce two forces operating in opposite directions. On the one hand poor agents who consume a smaller proportion of luxuries are more risk averse than richer agents hence have stronger precautionary motives. This works towards a more equal distribution of wealth. On the other hand the elasticity of inter temporal substitution is higher for wealthier agents who can thus more easily cut consumption in the face of negative earnings shocks. This factor works towards a more concentrated wealth distribution. Which of the two forces prevails is a quantitative issue that turns out not to be univocally determined.

The current paper is related to several strands of literature. First it is related to the literature that has tried to quantify the impact of earnings risk on wealth accumulation in the context of models with a large number of ex-ante identical agents subject to uninsurable idiosyncratic risk. This literature includes early work in partial equilibrium like Hubbard, Skinner, and Zeldes (1994) and Carroll and Samwick (1997) and in general equilibrium like Aiyagari (1994) and Huggett (1996). It also includes more recent work that uses a structural estimation approach in life-cycle partial equilibrium economies like Cagetti (2000) and Gourinchas and Parker (2002). Second it is related to the large body of literature that, in the same framework, has tried to understand the wealth distribution. This was started by the already mentioned works by Aiyagari (1994) and Huggett (1996) who first showed how models with uninsurable idiosyncratic shocks to earnings in general equilibrium could generate the observed fact that wealth is more concentrated than earnings. The failure to match the observed wealth concentration spurred a number of papers that, building on the original framework, added several possible explanatory mechanisms. These included entrepreneurship, like in Quadrini (2000) and Cagetti and De Nardi (2006), bequest motives like in De Nardi (2004), heterogeneous preferences like in Krusell and Smith (1998) or the assumption of earnings risk large enough to match the observed earnings variability, like in Castañeda, Díaz-Giménez, and Ríos-Rull (2003). Among the body of research mentioned above, the current paper is particularly related to works of Díaz, Pijoan-Mas, and Ríos-Rull (2003), Carroll (2000), and Francis (2009). The relationship with Díaz, Pijoan-Mas, and Ríos-Rull (2003) stems from the fact that both papers analyze the effect of an alternative preference specification on both precautionary savings and the concentration of wealth. The main difference with that paper is that they explore the role of habit formation while here we focus on the introduction of multiple consumption goods with non-homothetic preferences defined over the goods.^{1} On the other hand Carroll (2000) and Francis (2009) both propose non-homothetic preferences as a possible explanation of wealth inequality. In their work though, these are introduced in the form of having wealth in the utility function, which gives another saving motive on top of precautionary saving. Also, Carroll (2000) does not explore the quantitative implications of his idea, while Francis (2009) calibrates the model by picking the parameters of the utility function over wealth to match the Gini index of the wealth distribution itself. The present paper, besides focusing only on the precautionary saving motive, also chooses a rigorous calibration procedure where the parameters of the utility function are based on independent empirical evidence on the share of the two goods in total consumption.

The third strand of literature that is related to the current research consists of macroeconomic models that have put to different uses the assumption of multiple, non-durable consumption goods. Among them Ait-Sahalia, Parker, and Yogo (2004) using a more restrictive definition of luxury goods – high end luxuries – show that the empirical correlation between the equity return and consumption of these goods may reconcile the observed equity premium with a limited degree of risk aversion. Wachter and Yogo (2010) use non-homothetic preferences to explain why the share of risky assets is increasing in wealth. Finally Aguiar and Bils (2015) and Aguiar and Hurst (2013) use the multiple goods framework to address issues of consumption inequality. Aguiar and Bils (2015) find empirically that once this distinction is made consumption inequality tracks very closely the increase in income inequality observed in the recent decades. Aguiar and Hurst (2013) focus on goods with different elasticity of substitution of time and expenditures in their production to explain the evolution of consumption inequality over the life-cycle.

The rest of the paper is organized in the following way. 2 presents the model, 3 is devoted to explaining the calibration strategy, 4 presents the results. In Section 5 a brief conclusion is outlined. Finally, an appendix describes the numerical methods used to solve the model.

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