On the Generalized Consumption Externality with Heterogeneous Agents Chan, Ying Tung * November 8, 2016

Abstract This paper provides a novel explanation for the Easterlin paradox in the United States: Whereas the real GDP per capita has doubled in the past four decades, there has been almost no change in happiness. We propose an “inequality aversion” hypothesis to explain this puzzle. While the rise in average consumption increases agents’ utility, the increase in the variance of consumption lowers one’s utility due to the inequality aversion preference. We show that the inequality aversion hypothesis better explains the paradox than the standard relative income hypothesis. This paper presents a continuous time heterogeneous agent model with highly generalized consumption externality. Consumption externality is modeled as a dependence of agents’ utility on an arbitrary functional of consumption distribution which is endogenously determined. Our framework, first, generalises the “relative consumption” model in Dupor and Liu (2003) to a stochastic environment with heterogeneous agents. Second, it encompasses the inequality aversion preference by Fehr and Schmidt (1999) and other psychological behaviours such as symmetric desire, hostility to the rich, and sympathy for the poor. In contrast to the literature, the socially optimal consumption plan we derived reveals that negative externality does not always lead to over-consumption. Under-consumption or the co-existence of over- and under-consumption could happen, depending on the type of externality assumed. The model with an inequality averse preference is the most consistent with the observed US consumption patterns.

JEL Classification: E21, E62, H23 Keywords: Consumption externality, heterogeneous agents, inequality aversion, Easterlin paradox, progressive taxation *

Department of Economics, McGill University, Room 414, Leacock Building, 855 Sherbrooke Street West, Montreal, Quebec H3A 2T7 (email: [email protected]). I am grateful to Van Ngo Long, Francisco Alvarez-Cuadrado, Hassan Benchekroun, Fabian Lange, Chris Barrington-Leigh, Chi Man Yip, Bixi Jian and the seminar participants at McGill and CEA Annual Conference for their invaluable comments. All errors are my own.

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1

Introduction

Contrary to the permanent income hypothesis, the relative income hypothesis states that individual concerns not only her own consumption level, but also her consumption level relative to the average consumption level in an economy. The relative income hypothesis has widely been supported empirically (Luttmer, 2005) and experimentally (Solnick and Hemenway, 1998; Johansson-Stenman et. al., 2002; Alpizar et. al., 2005). Moreover, its applications are widely seen in macroeconomics, including the asset pricing theory and the equity premium puzzle (Gali, 1994; Campbell and Cochrane, 1999), the growth theory (Carroll et. al., 1997; Alvarez-Cuadrado et. al., 2004), the public policy models (Bowles and Park, 2005; Eckerstorfer and Wendner, 2013), etc. Nevertheless, the relative income hypothesis cannot stand alone in explaining the Easterlin paradox, which uncovers that the real GDP per capita is found increasing but the happiness remains steady over the past several decades.1 In the relative consumption literature, agent’s utility is assumed to increase with her consumption level and the relative consumption. According to Figure 1, the real GDP per capita nearly doubled in the past 40 years. Agents’ utility should increase even though the relative consumption remains unchanged. However, the fractions of individuals who report that they are “very happy” and “pretty happy” moves around 35 percent and 55 percent, showing no increasing trend.2 What explains the Easterlin paradox? This paper proposes a new explanation for this paradox, namely the inequality aversion hypothesis. Inequality aversion, also known as fairness concern, refers to an agent’s preference that depends on the dispersion of others’ rewards. Conditional on one’s reward level, the larger the difference between her reward and the others’, the lower is the utility owing to the fairness concern.3 According to Figure 2, the income inequality increases dramatically in the past four decades regardless of the measure of Gini coefficient and Theil index. While both the real GDP per capita and the inequality increase, the higher income disparity during the economic development probably balances out the increase in the happiness due to the income increase. Consequently, the happiness remains stable over the past four decades. While an extensive literature studies the optimal taxation and welfare analysis under 1

The explanation of Easterlin paradox using relative income hypothesis, proposed by Easterlin (1974) and Clark et. al. (2007). 2 Respondents answer the question “Taken all together, how would you say things are these days, would you say that you are very happy, pretty happy, or not too happy?” 3 The literature review for inequality aversion preference is in Section 2.

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# 104

Very Happy Pretty Happy Not too Happy

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(a) The real per capita GDP from 1972 to 2015 in US. This is chained 2009 dollars and is the quarterly data from Federal Reserve Bank of St. Louis.

(b) Happiness index in US from 1972 to 2015. Source: Smith et. al. (2015)

Figure 1: Real per capita GDP, Happiness level series in the U.S. from 1972 to 2015. the relative income hypothesis, there exists no theoretical paper in inequality aversion literature that studies its welfare implication. To fill the gap, this paper aims to answer the following questions. First, how is the decentralized equilibrium of an economy different from the first-best outcome when agents are inequality aversion? It is well documented that the relative income hypothesis leads to over-consumption (for example, Dupor and Liu, 2003). Under the “Keeping Up with the Joneses” assumption, the negative externality that is induced by the average consumption leads to a higher marginal utility of consumption. Agents thus consume more and the average consumption increases further. However, the utility depends on the consumption inequality, instead of the average consumption, under inequality aversion hypothesis. While the consumption of the rich creates the negative externality on others, the poor’s consumption increases others’ utility by narrowing down the consumption inequality. Therefore, the over- and the underconsumption coexist: Whereas the rich over-consume under a decentralized equilibrium, the poor under-consume. Second, this paper investigates possible policies that correct the externality. In other words, we examine the policies that potentially reduce the deadweight loss created by the relative consumption hypothesis and the inequality aversion preference. Third, this paper purposes to examine whether the relative consumption hypothesis and the inequality aversion hypothesis describe the US household preference better.

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Figure 2: Three income inequality measures of Equivalence-Adjusted Income Dispersion in US from 1972 to 2015. They are Gini Index, mean logarithm derivation of income and Theil Index. Source: U.S. Census Bureau, Current Population Survey.

To answer these questions, this paper proposes a heterogeneous agent model that incorporates both the relative consumption and the inequality aversion preference. First, our model features its generality of agents’ preference. In our model, agents’ utility is a function of the measure of consumption externality. Not only the preferences could adopt the relative consumption hypothesis in (Dupor and Liu, 2003) if the measure takes the value of the average consumption level, but they could also be inequality averse and status-seeking (Hopkin and Kornienko, 2004) if the measure takes the values of the standard derivation and the cumulative distribution function of one’s consumption, respectively.4 Not to mention, our model is the first macroeconomic model with the inequality aversion preference. Second, our model is the first heterogeneous agent model that endogenizes the relative consumption under a stochastic environment. The heterogeneity in wealth allows us to investigate the distributional impact of the relative consumption preference. Undoubtedly, the heterogeneity is required in any model to investigate how the relative consumption preference affects the equilibrium distribution in consumption. Furthermore, policy implications derived from the relative consumption hypothesis in principle reply heavily on the assumption of heterogeneity in income. Surprisingly, agents are homoge4

If the measure is the skewness of the consumption density, it captures the “symmetric desire” of the households. Interestingly, Hagerty (2000) finds that the skewness of the income distribution influences the level of “happiness”.

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neous in most the existing literature on the relative consumption hypothesis. Therefore, the welfare analysis on the consumption externality is largely limited. If one’s consumption level coincides with the average consumption level, taxes, regardless of progressive taxes or regressive taxes, derive no meaningful policy implication in correcting the negative externality. On the contrary, the heterogeneity in our model allows economists and policy makers to conduct meaningful welfare analysis on various kinds of policies. The generalization of the measure of the consumption externality provides economists a better understanding of how the wealth distribution influences the economic outcomes. First, we show that technology advance could lead to a fall in the welfare. While a positive technology shock increases both the labor and capital income, the wealthier households benefit more from the positive shock. This asymmetric benefit not only increases the average but also the variance of consumption. Our results show that the Gini coefficient has a greater percentage increase than the average consumption, thereby reducing the welfare for the inequality averse households. This explains why the real income per capita increased but the happiness remained steady in the United States in the past four decades, solving the Easterlin paradox. In this context, introducing a progressive tax narrows down the wealth gap and hence may increase the welfare even if the tax revenue is wasted. Those who are not at the upper end of the income distribution are happier even if their incomes are taken away, provided that more are taken away from the top income earners.5 Hence, our model identifies an additional benefit of progressive taxation: it satisfies the desire for equality of the whole population, even though the tax revenue does not transfer to the poor. Furthermore, our simulation analysis suggests that households with negative externalities of the first four moments have higher savings and consumption levels at a steady state than the models without externality. Under a positive productivity shock, a larger negative externality on the marginal utility of consumption in the new steady state is expected. To smooth out the larger externality, households cumulate their wealth faster. In other words, the presence of negative externalities could speed up the convergence rate, consistent with the literature (Alvarez-Cuadrado et. al., 2004). The paper proceeds as follows. Section 2 summarizes the related literature. In Section 3, we introduce the basic model and the generalized definition of consumption externality. In this section, both the time-varying and stationary equilibrium are defined. We 5

The argument that tax could potentially mitigate the impact of consumption externality and improve the welfare has already been suggested by Frank (2005). We justify his argument by our model.

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solve the closed form solution and compare our model with the existing models in section 3. In Section 4, we discuss the social planner’s problem and derive the conditions under which the centralized consumption path is greater, less than or coincides with the decentralized one. In Section 5, we perform the simulation exercises for both stationary equilibrium and transitional dynamics. We show how the Easterlin paradox could be explained by incorporating “inequality aversion”. Tax policy analysis will be conducted in this section. In Section 6, we introduce a procedure to select the types of externalities that best fits the US consumption data. Section 7 concludes the paper.

2

Literature Review

Our model is related to three strands of literature. First, our model is a generalization of the standard heterogeneous agent model (Aiyagari, 1994; Belwey, 1986; Huggett, 1993). The equilibrium in the Bewley-Huggett-Aiyagari-type model is determined solely by the interest rate, which is a function of the average wealth level in an economy. As long as the average wealth level is identical, the aggregate capital and interest rate are all the same in the equilibrium regardless of the wealth distribution. The model focuses on the interaction between firms and households and neglects the strategic interaction among the households. In contrast, our model provides a new mechanism for households to interact, namely, the externality concern. The strength and the way of how households interact can be manipulated by choosing different externality indices, such as the mean and the standard deviation of consumption. More importantly, we show that the dispersion of the distribution, measured by the standard deviation, describes the externality concern better than the mean to explain the Easterlin paradox. Technically, the paper follows the approach from Achdou et. al. (2015) who generalized the Bewley-Huggett-Aiyagari-type model into a continuous time framework by using mean field game technique from Lasry and Lions (2007). Second, our model is the first macroeconomic models with inequality aversion component. The existence of inequality aversion preference has been justified in many ways. In experimental economics, Fehr and Schmidt (1999) first proposes this preference and show in their experiment that players are willing to cooperate by imposing a costly punishment to free-riders. The result has been confirmed by Charness and Rabin (2002), Bolton and Ockenfels (2000) and Ho and Su (2009). All these frameworks are designed for experimental studies with a small number of agents and are intractable within the 6

macroeconomics models where the number of households is large. We provide a tractable macroeconomic model with a large number of households. Indeed, individuals’ preference towards income or consumption dispersion could lead to an aggregate impact. This is indirectly evidenced by the empirical findings that income dispersion is highly correlated to subjective well-being (hereafter, we use the terms subjective well-being and happiness interchangeably). Verme (2011) finds a significant negative impact of income inequality on life-satisfaction using cross-country data. Similar results are found by Schwarze and Härpfer (2007) using German data (see Clark and D’Ambrosio (2014) for a recent review).6 The importance of the preference is also evidenced by some surveys. For instance, Cowling and Harding (2007) document that in a large-scale survey with 22,000 respondents, 75 percent of them desire a fairer income distribution. Third, our model is one of the few heterogeneous agent models in the relative consumption literature. While most the existing works rely on a homogeneous agent model in this literature, notable exceptions include Eckerstorfer and Wendner (2013), GarciaPenalosa and Turnovsky (2008) and Barbar and Barinci (2009), both of them are in a deterministic framework. Also, recent works of Abel (2005) and Alvarez-Cuadrado and Long (2011, 2012) incorporate relative consumption preference into a deterministic overlapping generation model with heterogeneous agents. To investigate how the wealth distribution affects households’ consumption path via the externality concern, this paper, similar to Chan and Kogan (2002), relaxes the representative agent assumption. While agents are heterogeneous in preferences in Chan and Kogan (2002), agents are heterogeneous by having idiosyncratic shocks on their wealth in our paper. Our model is also different from theirs that the relative consumption is endogenously determined in the equilibrium, while it is exogenously assumed in Chan and Kogan (2002).

3

Model

We begin by first describing the households’ problem. 6

There are mixed empirical results of the relationships between income inequality and “happiness”. For instance, Knight et. al. (2009) obtain a positive correlation between income inequality and life satisfaction in rural China. Our model also captures the case of a positive relationship.

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3.1

Households’ Problem

Consider an economy with a continuum of agents indexed by i ∈ [0, 1]. In period 0, agent i has an initial endowment of wealth W0i ≥ 0 and an initial labor endowment z i0 ≥ 0. Assume that the interest rate r and the wage rate w are exogenous and constant over time.7 In period t, agent i with wealth level Wti and labor endowment level zti receives a labor income wzti and an interest payment rWti and has to decide how much to consume and save for the next period. We assume that the initial wealth W0i and an endowment z0i are drawn from some known density function g(W0 , z0 , 0) for every i. Denote the joint density of Wti and zti as g(Wt , zt , t) for any t ≥ 0. The household’s problem can be stated as follows: Given the initial wealth W0i , the initial labor endowment z0i , household i chooses the consumption levels cit for t ≥ 0 and the wealth levels Wti for t > 0 so as to maximize the lifetime utility function (1): ˆ

∞

e−ρt u(cit , Z[m])dt

E0

(1)

0

subject to the budget constraint (2) dWti = (rWti + wzti − cit )dt

(2)

and the borrowing constraint Wti ≥ Wmin

(3)

Similar to the Aiyagari model, the labor endowment process is assumed to follow the Ornstein–Uhlenbeck process: dzti = θ(¯ z − zti )dt + σdB

(4)

where u(cit , Z) : R+ × R → R is household i’s period utility, which is increasing and concave in cit . Denote σ > 0 as a diffusion coefficient, ρ > 0 as a discount rate, Wmin ≤ 0 as a borrowing limit and Bt as a standard Wiener process. Notice that zti ∼ N (¯ z , σ 2 /θ) asymptotically, so the parameters z¯ and θ respectively affect the mean and variance of the 7

Since the purpose of the paper is to illustrate the interaction of households through consumption externality, we ignore the firm side and only focus on the partial equilibrium in this paper. In general equilibrium, r and w are time varying and are jointly determined by the households and the firm side. The general equilibrium and its computational procedure are described in the Appendix. We leave the implementation of the procedure for further studies.

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endowment process. θ also controls the persistence of the endowment process. Assume the functional Z[.] : C 2 (R+ × R) → R.8 In period t, Z[m(c, t)] is a functional of the consumption density m(c, t) and is denoted as Z[m] hereafter for simplicity. As it will be discussed below, the optimal consumption choice is a function of wealth and endowment level so the consumption density depends on the wealth density g(W, z, t) ∈ C 2 (Ω1 × Ω2 × R), where Ω1 = [Wmin , Wmax ] and Ω2 = [zmin , zmax ]. Theoretically, Wmax = ∞, zmin = −∞ and zmax = ∞, but a finite range is set for simulation in Section 6. As in Clark and Oswald (1998), we consider two types of externality models in the analysis, namely, the ratio comparisons (RC) and the additive comparisons (AC) utilities, which have the functional forms given by u(c, Z[m]) = cφ Z[m]−γφ /φ and u(c, Z[m]) = (c − ηZ[m]γ )φ /φ respectively. Assume the risk aversion parameter φ < 1. We consider both of the utility specifications in our numerical study while in the theory part we only present the results for the RC utility. The corresponding results of the AC utility can be found in the Appendix. Notice that the functional Z[m] is the key component to our model, we will discussed it in details before proceeding to the model solution.

3.2

Generalized Definition of Consumption Externality

According to Dupor and Liu (2003), preferences exhibit jealousy (admiration) if the marginal utility of the reference consumption ∂u/∂Z is less than (greater than) zero. Notice that for the RC and AC utilities, ∂u/∂Z = −γcφ Z −γφ−1 and ∂u/∂Z = −(c − ηZ γ )φ−1 γηZ γ−1 respectively. Both of them are less than zero if and only if η > 0 and γ > 0. Hence, the magnitude of η and γ determine+ the degree of (negative) consumption externality. Unless stated otherwise, assume η > 0 and γ > 0 throughout the analysis. There are two questions concerning the definition of Dupor and Liu (2003): First, the object that households are jealous about is not explicitly stated. Specifically, the term jealousy (admiration) in Dupor and Liu (2003) means the negative (positive) consumption externality on the average consumption in the economy. In our model terminology, this is to restrict ´∞ the functional Z[m] to be the consumption average. That is, Z[m] = 0 cm(c, t)dc = µt . A follow-up question is: How far could such a setting capture the behavior of consumption externality in reality? What if households have a lower utility level when the consumption distribution preserves the same mean but has a higher dispersion? The problem does not arise in the representative agent setting where the individual differences are assumed away. However, it is especially important in the heterogeneous agents 8

C 2 (R+ × R) is the set of twice-continuously differentiable functions defined on the domain R+ × R.

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setting under which households’ consumption levels are different. Adding different types of externalities could be an essential step in modeling the consumption and wealth distributions. In the spirit of this, we generalize the definition in Dupor and Liu (2003) as follows: Definition 1. Given any consumption density m(c), let the functional Z[m] be an index (or a functional) of m(c). The preference exhibits negative (positive) consumption exter< 0 (>0) . nality iff ∂u(c,Z) ∂Z The definition of consumption externality requires us to first define an index Z[m] on the consumption density m(c).9 Then, we insert the Z[m] into households’ utility functions so as to model the impact of a negative (positive) externality on an increase in Z[m]. For instance, the definition in Dupor and Liu (2003) is a special case of ours ´∞ by setting Z[m] = 0 cm(c)dc = µ to be the average consumption in the economy. In this case, there is a negative externality of the average consumption (or “jealousy”) if ∂u(c, Z)/∂Z < 0. Beyond the average consumption, Z[m] could be any functional or higher moment of m(c). If Z[m] is an index of dispersion of m(c) (for example, standard deviation ´∞ ´∞ ´c ( 0 (c − µ)2 m(c)dx)1/2 or Gini coefficient 0 M (c)(1 − M (c))dc/µ with M (c) = 0 m(x)dx), households’ decisions depend not only on the level, but also on the dispersion of the others consumption. The negative externality of a higher consumption dispersion reflects inequality aversion in consumption.10 Similarly, let Z[m] be the third or fourth moments of m(c) features the externality concerns on the skewness or kurtosis of the consumption density respectively. The negative externality of an increase in asymmetry and heavy tail of the consumption density respectively reflects households’ desire for a symmetric and thin-tailed consumption distribution. Besides, if Z[m] = Qc (q) for some quantile functions Qc (.) evaluated at q ∈ [0, 1], then households are affected positively or negatively 9

Below, in any period t it can be shown that consumption is solely a function of wealth and endowment c(W, z, t), so the consumption density m(c, t) can be expressed in term of the wealth density g(W, z, t). ˜ such that Hence, each type of consumption externality corresponds to a type of wealth externality Z[g] ˜ Z[g] = Z[m]. 10 Notice that this definition of inequality aversion is different from those in Fehr and Schmidt (1999). According to Deaton (2003), the preference of Fehr and Schmidt (1999) with infinitely many agents is ´∞ ´∞ U (c, Z[m, c]) = c − Z[m, c] = c − β1 0 1(x ≥ c)(x − c)m(x)dx + β2 0 1(x < c)(c − x)m(x)dx where 1(.) is an indicator function. This utility is much hard to model as the externality index Z[m, c] is a function of individual consumption c. Friedman (2008) generalizes the utility into U (c, Z[m, c]) = u ˆ(c) − Z[m, c] where u ˆ(c) is concave. Besides the utility specification, we are different from Friedman (2005) in two aspects: (i) Households in our model are solving the dynamic maximization problem; (ii) Our model is stochastics.

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by an increase in the q-th quantile of consumption in the economy. A negative externality of an increase in 90-th percentile (q = 0.9) and an admiration of an increase in 10-th percentile (q = 0.1) could respectively measure a hostility to the rich and a sympathy for the poor (or “compassion” in Hopkins (2008)).11 More generally, Z[m] could also be ´c a function of consumption. Say, denote Z[c, m] = M (c) = 0 m(x)dx where M (c) is the distribution function of c. This represents the scenario where households care about the ranking (social status) of their own consumption levels relative to the society. Hopkins and Kornienko (2004) and Frank (1985) have already examined this social status preference in a static and deterministic environment. The utility in our framework can serve as an extension of their works in a stochastic and dynamic environment. Furthermore, any linear and non-linear combination of these externalities can be captured in the present model as well.12 Hereafter, we will use the term jealousy and negative externality on Z[m] interchangeably.

3.3

Model Solution

Solving the problem (1) to (4) yields the following Hamiltonian-Jacobi-Bellman (HJB) equation. ρV (W, z, t) = maxc u(c, Z[m]) + ∂W V (W, z, t)(rW + wz − c∗ ) +θ(¯ z − z)∂z V (W, z, t) + 21 σ 2 ∂zz V (W, z, t) + ∂t V (W, z, t)

(5)

where V (W, z, t) is the value function at time t. ∂x V and ∂xx V are the first and second partial derivative of V with respect to x respectively. The superscript i and time subscript t are omitted hereafter for simplicity. Consider only the RC utility u(c, Z[m]) = 1 φ c Z[m]−γφ here. Since any particular individual’s decision has no influence on the denφ sity of consumption due to her infinitesimal size, the optimal consumption can be computed by treating m(c, t) as exogenous. The optimal consumption levels can be determined by the condition ∂c u(c, Z[m]) = ∂W V (W, z, t) 11

(6)

The definition of “compassion” from Hopkins (2008) is different from ours. Please refers to footnote 34 for the details. 12 We only focus on the cases where the functionals Z[m] are the first four moments and Gini coefficient in the numerical exercise below, while the theory part in the paper is applicable to the other functionals mentioned, except the one that Z[m, c] = M (c).

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This is simply the Euler equation where the marginal utility of consumption equals to the shadow value of the wealth. Under the RC utility, this reduces to c∗(φ−1) F [m] = ∂W V (W, z, t), where F [m] ≡ Z[m]−γφ . Substitute c∗ into (5) yields φ

1

ρV (W, z, t) = ∂W V (W, z, t) φ−1 F [m] 1−φ ( φ1 − 1) + ∂W V (W, z, t)[rW + wz]) +θ(¯ z − z)∂z V (W, z, t) + 21 σ 2 ∂zz V (W, z, t) + ∂t V (W, z, t)

(7)

Following Achdou et. al. (2015), the Kolmogorov Forward equation for g(W, z, t) can be written as:

1 ∂t g(W, z, t) = −∂W ((rW +wz−c∗ (W, z, t))g(W, z, t))−∂z (θ(¯ z −z)g(W, z, t))+ σ 2 ∂zz g(W, z, t) 2 (8) And the boundary conditions for the HJB equation are     

∂c u(wz + rWmin , Z[m]) ≥ ∂W V (Wmin , z, t) ∂z V (W, zmin ) = 0     ∂z V (W, zmax ) = 0

(9)

for any W and z.13 The initial density g(W, z, 0), the terminal function V ∗ (W, z) and ´ ´ the constraint Ω2 Ω1 g(W, z, t)dW dz = 1 for any t. Equations (7) and (8) characterize the equilibrium in which agents interact with each other through the distribution of state variables. Agents’ decisions are affected by the density g(W, z, t) and their decisions in turn constitute the density. With the terminal condition that V (W, z, t) = V ∗ (W, z) as t → ∞, equation (7) determines backwardly the individual’s optimal consumption and saving decisions and thus the evolution of the value function. While given the initial distribution of the state variable g(W, z, 0), equation (8) controls the evolution of the wealthendowment density when households respond optimally by choosing c∗ (W, z, t).14 With c∗ (W, z, t) obtained in (7) and g(W, z, t) obtained in (8), we are able to compute the consumption density m(c, t) by the transformation of variables technique.15 Contrary to the usual heterogeneous agent models where equilibrium is obtained by finding an equi13

The details of the boundary conditions can be referred to Achdou et. at. (2015). The backward and forward nature of (7) and (8) together formulate the so-called “mean field game", developed by Lasrey and Lions (2007). 15 According to the first order condition (6), the optimal consumption c∗ under the RC utility can be written as a function of W , z and t 14



∂W V (W, z, t) c (W, z, t) = F [m] ∗

12

1  φ−1

(10)

librium state variable distribution and a factor price, we close our model by finding an equilibrium consumption density m(c, t) and an externality index Z[m]. This is a key feature of our model and we will discuss its advantages in subsection 3.6.

3.4

Equilibrium

Below, we define the time varying equilibria and the stationary equilibrium respectively. Definition 2. The time-varying equilibria consist of a sequence of functions {V (W, z, t), g(W, z, t), m(c, t), c(W, z, t)} that satisfies the following conditions: 1. Given the initial wealth W0 from g(W, z, 0) and the density function of consumption m(c, t) for each t, households choose c(W, z, t) for t ≥ 0 to maximize (1) subject to (2). 2. Given the optimal choice c(W, z, t) and the initial density g(W, z, 0), the wealth density function g(W, z, t) solves (8) for t > 0 . 3. The optimal choice c(W, z, t) and the wealth density function g(W, z, t) together constitute the density function m(c, t). 4. The value function V (W, z, t) solves (7), subject to the boundary condition (9) and the terminal condition limt→∞ V (W, z, t) = V ∗ (W, z) for some V ∗ (W, z). The definition of stationary equilibrium is similar to the above except that the equilibrium consumption and state variables densities are invariant to t. We find functions V (W, z) and g(W, z) that satisfies the stationary HJB and KF equations

1 ρV (W, z) = max u(c, Z[m]) + ∂W V (W, z)[rW + wz − c] + θ(¯ z − z)∂z V (W, z) + σ 2 ∂zz V (W, z) c 2 (12) given the equilibrium density function m(·) and g(·) and under the assumption that F [m] 6= 0. ∂(c∗ (W, z, t), z) ∗ = g(W, z, t) m(c ˜ (W, z, t), z, t) (11) ∂(W, z) where m(c ˜ ∗ (W, z, t), W, t) is the joint density of (c∗ (W, z, t), W ) and ∂(c(W,z,t),z) is the determinant of ∂(W,z) ∂(c(W,z,t),z) ∂c(W,z,t) the Jacobian matrix. We have ∂(W,z) = ∂W . Thus, (11) is valid when ∂c(W,z,t) 6= 0. Indeed, ∂W differentiating both sides of (10) by W , we find that c(W, z, t) is strictly monotone in W iff V (W, z, t) is concave in W . The concavity of W is followed by the concavity of u(c, Z[m]) in c. This can be proved similar to the argument in Lemma 4.15 of Chang (2004).

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1 0 = −∂W ((rW + y − c∗ )g(W, z)) − ∂z (θ(¯ z − z)g(W, z)) + σ 2 ∂zz g(W, z) 2 respectively and the boundary condition is     

∂c u(wz + rWmin , Z[m(c, t)]) ≥ ∂W V (Wmin , z, t) ∂z V (W, zmin ) = 0     ∂z V (W, zmax ) = 0

(13)

(14)

for any W and z. The stationary consumption level c∗ (W, z) satisfies: "

∂W V (W, z) c (W, z) = φF [m] ∗

#

1 φ−1

(15)

Definition 3. The stationary equilibrium consists of a sequence of functions {V (W, z), g(W, z), m(c), c(W, z)} that satisfies the following conditions: 1. Given the initial wealth W0 from g(W, z) and the density function of consumption m(c), households choose c(W, z) to maximize (1) subject to (2). 2. Given the optimal choice c(W, z), the density function g(W, z) solves (13). 3. The optimal choice c(W, z) and the wealth density function g(W, z) together constitute the density function m(c). 4. The value function V (W, z) solves (12) subject to the boundary condition (14). The above definition of equilibrium resembles the concept of Nash equilibrium with finite players, but with an infinite number of players in the present setting. In the Nash equilibrium with finite players, each household i responds optimally given the decisions(s) of the other agent(s) −i. As the number of players tends to infinity, the density function m(c, t) would not be affected when the consumption of any particular household i is excluded. Therefore, in the game with infinite players, the equilibrium density m(c, t) should be a fixed point in the sense that given m(c, t), the value of response functional for each player constitutes a density function which is exactly m(c, t). The above definition of equilibrium is to find a path of density function m(c, t) that satisfies the households’ problem for every period t. Notice that in most of the cases, the model does not have a closed form solution, the algorithm for numerical solution is described in the Appendix.

14

3.5

Closed Form Solutions and Utility Specifications

Solving the model explicitly could provide a better understanding of its underlying mechanism. However, under our specified income process, there is no closed form solution for the system of partial differential equations (7) and (8). Next, we derive a closed form solution under a more restrictive income process. Theorem 4. Suppose the budget constraint in (2) follows the geometric Brownian motion dW = (rW + yt − c) + σW dB instead, where yt is an exogenous income process. When yt = 0, the closed form solution exists. The stationary equilibrium value function, consumption and saving are respectively: V (W, t) =

A0 F [m]W φ φ

(16)

1

c∗ = (A0 ) φ−1 W

(17)

1

s∗ = (r − (A0 ) φ−1 )W where A0 =

h

ρ−rφ 1−φ

+

σ2 φ 2

iφ−1

. Under the condition r < ρ +

(18) σ 2 φ(1−φ) , 2

the density of wealth

is α g ∗ (W ) = Wmin W −α−1

(19)

where Wmin is the lower bound of g ∗ (W ). The density of consumption is m(c) = cαmin c−α−1 where α = 1 +

2 ρ−r ( ) σ 2 1−φ

(20)

1

+ φ and cmin = (A0 ) φ−1 Wmin .

Proof. See Appendix. Remark. There is a closed form solution when yt = w0 W for some constant w0 . The solution is obtained by replacing r by r˜ = r + w0 in Theorem 4. In Theorem 4, the consumption to wealth ratio c/W = (A0 )1/(φ−1) is invariant to the wealth level. This leads to the result that both consumption and wealth are Pareto dis-

15

tributed with the same tail index α.16 By observing the formula of α directly, it is straightforward to identify the factors that affect the inequality of consumption and wealth: (1) σ 2 and r decrease with α; (2) φ increases with α for r < ρ and vice versa. The same result has already been obtained in Achdou et. al. (2015). Note that a drop in α implies a more heavy-tailed distribution. It is intuitive that a rise in the income shock volatility σ 2 would lead to a more fat-tailed wealth distribution as the dispersion of income realization increases. Besides, a higher interest rate benefits the wealthier households more by allowing them to earn more capital income rW , thus widening the wealth inequality. Since consumption is a linear function of the wealth level, this also implies a fatter-tailed consumption distribution. For r < ρ, households have no incentive to postpone their consumption to accumulate wealth, and therefore only the precautionary saving is operative. A drop in the risk aversion (i.e. an increase in φ) discourages this incentive and so decreases the saving level as well as the saving rate. For the same drop in saving rate, the wealthier households would face a larger drop in wealth, so the wealth distribution becomes less heavy-tailed. The main observation from Theorem 4 is that the optimal consumption in (17) does not depend on the value of Z[m]. It implies that the steady state consumption (and hence its distribution) is not affected by any type of externality Z[m] assumed. And indeed this is a special feature of RC utility function and the result is not driven by the specific income process assumed in Theorem 4. The following theorem shows that this feature continues to hold for any other income process. Theorem 5. Suppose u(c, Z[m]) = A(c)D(Z[m]) + B(c), for some differentiable function A(.), B(.), D(.): R → R and at least one of the A(.) and B(.) is a constant function. The steady state consumption c(W, z) is independent of the value of Z[m].17 Proof. See Appendix. If we assume B(c) = 0, A(c) = cφ /φ and D(Z[m]) = Z[m]γφ for any c, then u(c, Z[m]) reduces to the RC utility. Notice that Theorem 5 also applies to the additive separable utilityu(c, Z[m]) = ηZ[m]γ + B(c), by setting A(c) = η. At the steady state, m(c) does not vary over time, so Z[m] becomes merely a constant. Adding the externality term Z[m] to 16

This is not consistent with the data. As it is known that the consumption inequality is less than the wealth inequality in reality. The loss of reality is surely driven by y = 0 or y = wW . The assumption will be relaxed in the numerical exercise in Section 5. 17 Notice that only the steady state consumption are independent of Z[m]. The term of Z[m] has effects on the transitional dynamics.

16

the above utility function is just a monotone transformation of the utility and so will not change the steady state consumption decision. Yet, it is not the case for the AC utility u(c, Z) = (c − ηZ[m]γ )φ /φ . Its marginal utility of consumption ∂u/∂c = (c − ηZ[m]γ )φ−1 which approaches infinity around the value ηZ[m]γ . Hence, changing the value of Z would affect the consumption decision.

3.6

Comparison of Our Model with the Existing Models

We end this section by pointing out some distinctive features compared to the existing models. Our model has a closed relationship between two classes of models in the literature, namely, the homogeneous agent model with consumption externality and the Bewley-Huggett-Aiyagari type heterogeneous agent model without externality. In this section, we compare our model with these models and illustrates that our model serves as a linkage between them. 3.6.1

Comparison of Our Model with the Representative Agent Consumption Externality Model

Notice that most of the current literature of consumption externality are in the representative agent framework,18 our extension of the models to a stochastic and heterogeneous framework has several advantages. In the homogeneous agent model with externality, it is assumed that there are many identical households choosing their consumption independently given a reference consumption c¯. At the equilibrium, households’ decision are exactly identical and matches the level of the reference consumption. The major problem of representative agent model is that at the equilibrium the consumption and wealth distributions are degenerate. The leads to at least two problems: First, it prevents us from analyzing other types of consumption externality like the inequality aversion. This is because the consumption variance is always equal to zero. Second, the distributional impact of policy or some parameter changes under the consumption externality cannot be analyzed. For instance, the policy impacts on the wealth or consumption inequalities in equilibrium could only be analyzed using the heterogeneous model. 18

A few exceptions include Garcia-Penalosa and Turnovsky (2008) who model agents with different initial capital shocks and different reference consumption levels in a deterministic framework. With only the difference in initial capital shocks, the equilibrium densities of consumption and wealth are degenerate.

17

3.6.2

Comparison of Our Model and the Standard Bewley-Huggett-Aiyagari model

Figure 3 shows the difference between the mechanisms of the standard Bewley-HuggettAiyagari model and our consumption externality model. In both models, households are making saving and consumption decisions in order to maximize their lifetime utility subject to the intertemporal budget constraints. In the Bewley-Huggett-Aiyagari model, there are firms that rent capital from the households. The aggregate capital supply is the aggregate saving from all the households and it is rented at a price (that is, the interest rate) equals to the marginal product of capital (MPK). A higher interest rate encourages households to save more and hence raises the capital supply while at the same time lowers the capital demand due to the higher capital cost. The general equilibrium is then determined by the interest rate which closes the capital market. More precisely, a stationary equilibrium can be characterized by a wealth density g ∗ (W, z) that is consistent with the households’ saving as well as the interest rate. In the transitional dynamics, the equilibrium is a path of g(W, z, t) that determines the paths of the interest rate that close the capital markets. While in our model, households’ saving and consumption decisions simultaneously determine the consumption and wealth distributions in the economy. The evolution of wealth density is determined by (8) while the consumption density can be deduced from the wealth density since the optimal consumption decisions are a function of wealth levels. The consumption distribution enters households’ utility through affecting the level of consumption externality Z[m]. More importantly, the change in Z[m] affects households’ marginal utility of consumption and so influences their saving and consumption decisions. The new consumption decisions would, in turn, constitute another consumption distribution. The equilibrium can then be obtained by finding an equilibrium path of consumption density m(c, t) or simply a path of the externality index Z[m] that satisfies households’ maximization problem microscopically and the evolution of wealth density by (8) macroscopically. And the stationary equilibrium is simply characterized by a consumption density m∗ (c) and so an externality value Z[m∗ ]. An advantage of our model is that the intensity of households’ interaction can be manipulated by picking different functionals Z[m]. Suppose that the equilibrium consumption distribution under the benchmark model without externality has a low average but a high variance. It is expected that models with the first and second moments externalities on the consumption distribution would yield substantially different equilibrium out-

18

comes.19 The model with average consumption externality is expected to have a similar equilibrium to the benchmark due to the low magnitude of the externality index. However, the equilibrium outcomes under the variance externality model would be very different. The large externality index under the benchmark equilibrium has a large impact on households’ marginal utility of consumption and hence their decisions. In contrast, the equilibrium in the standard Bewley-Huggett-Aiyagari model is only determined by the interest rate which only depends on the sum of the wealth of all the households. Any distribution of wealth that has the same mean yields the same aggregate capital and hence the same equilibrium interest rate. This restricts the flexibility of modeling, especially if we are interested in investigating the shape of wealth and consumption distributions. Another advantage of our model is that households can interact even when prices are exogenous and the production side of the economy is absent. To concentrate on the externality interaction, only the partial equilibrium is studied in the paper. Note that the firm side can be incorporated into our model, while this just complicates our setting, we describe the general equilibrium setting in the Appendix. Aggregate

Individuals

MPK

𝑲 = ∫ 𝑾𝒈(𝑾)𝒅𝑾

(a) The mechanism of Bewley-Huggett-Aiyagari model

Individuals

Externality 𝒁[𝒎]

Aggregate

(b) The mechanism of consumption externality model

Figure 3: Graphical representation of the difference between our model and the standard Bewley-Huggett-Aiyagari model 19

We prove below that equilibria under some types of utility functions are not affected by the presence of externality. The discussion here does not include those cases.

19

4

Social Planner Problem

To investigate the efficiency of the above decentralized problem, we follow the approach of Nuno and Moll (2015) to assume that there is a social planner who is able to determine the consumption paths of all the households in the economy. Since at any time t, households with the same wealth W and labor endowment level z should have the same level of consumption, the social planner problem is reduced to deciding a social optimal consumption function c∗sp (W, z, t) to maximize some social welfare functions. Notice that the social planner cannot violate the budget constraints for every household. That is, she still subjects to the budget constraint (2), (3) and (4) for each household. Macroscopically, this is equivalent to facing the evolution of the resource constraint specified by a Kolmogorov Forward equation similar to (8)

1 ∂t g(W, z, t) = −∂W ((rW +wz −c∗sp (W, t))g(W, z, t))−∂z (θ(¯ z −z)g(W, z, t))+ σ 2 ∂zz g(W, z, t) 2 (21) ∗ By choosing csp (W, z, t), the social planner tries to maximize the social welfare function subject to (21). Specifically, we define the welfare functional at time t as ˆ

ˆ

W el[c, g] =

u(c(W, z, t), Z[c, g])g(W, z, t)dW dz Ω2

(22)

Ω1

for any consumption function c(W, z, t) and wealth density g(W, z, t). The wealth density g(W, z, t) represents the frequency of population with the wealth W and endowment z at time t, so the welfare functional (22) represents the sum of the utility of all the households at time t. The social planner maximizes the discounted sum of the welfare flow (22) which can be written as ˆ J[c, g] =

ˆ

∞

e 0

ˆ

−ρt

u(c(W, z, t), Z[c, g])g(W, z, t)dW dzdt Ω2

(23)

Ω1

subject to the Kolmogorov Forward equation (8) which describes the evolution of the wealth density g(W, z, t) over time. Since the social planner is able to change the externality index by manipulating the consumption schedule, we denote the externality index by Z[c, g], instead of Z[m], to indicate that it is both a functional to c(W, z, t) and ´∞ g(W, z, t). Say, for the mean function Z[m] = 0 cm(c)dc, it is equivalent to write it as ´ ´ Z[c, g] = Ω2 Ω1 c(W, z, t)g(W, z, t)dW dz. Given the wealth density in each period t, the

20

social planner constructs a consumption schedule c(W, z, t) which determines how much the households with wealth W and endowment level z can consume. And with the consumption schedule, how does the wealth-endowment density g(W, z, t) evolve over time is determined by equation (8). Since the households ignore the externality impact of their consumptions to the others, it is expected that the optimal consumption plan of the social planner problem results is different from the decentralized one. This is justified by the following theorem. Theorem 6. Suppose the optimal consumption in the social planner problem c∗sp (W, z, t) exists and is monotone in W , then the marginal social value function j(W, z, t)20 solves

ρj(W, z, t) = u(c∗sp (W, z, t), Z[c∗sp , g]) + (rW + wz − c∗sp (W, z, t))∂W j(W, z, t) + ∂z j(W, z, t)(θ(¯ z − z)) ˆ ˆ ∗ δZ[csp , g] (W, z, t) ∂Z u(c∗sp (x, y, t), Z[c∗sp , g])g(x, y, t)dxdy + ∂t j(W, z, t) + 21 σ 2 ∂zz j(W, z, t) + δg Ω2 Ω1 |

{z

Change in Z by ↑ in g

}|

{z

Total impact of ↑ Zon the sum of utility 21

where δZ[c, g]/δg is the functional derivative of Z with respect to g. determined the following condition ˆ δZ[c∗sp , g] ∗ ∂c u(csp (W, z, t), Z[c, g])+ (W, z, t) | δc {z } Ω2 Change in Z by ↑ in c

And

}

(24) is

c∗sp (W, z, t)

ˆ

|

∂Z u(c∗sp (x, y, t), Z[c∗sp , g])g(x, y, t)dxdy = ∂W j(W, z, t) Ω1

{z

Total impact of ↑ Zon the sum of utility

}

(25) Proof. See the Appendix. By choosing the consumption schedule c(W, z, t), there are three channels that the social planner can change the utility level u(c, Z[c, g]) from (i) the directly impact of c(W, z, t); (ii) the indirect impact of c(W, z, t) on Z[c, g]; (iii) the indirect impact of c(W, z, t) on Z[c, g] through affecting g(W, z, t). Compared to the decentralized problem where only the direct effect is present, the social optimal planner takes the impact of consumption on the externality Z[c, g] into the accounts.The second term on the LHS of equation (25) is the aggregate externality impact of c directly on Z[c, g]. The term δZ[c,g] (W, z, t) δc is the change in the value of Z[c, g] in response to a small deviation of the function 20 21

Please check the interpretation of j(W, z, t) from Lucas and Moll (2014) and Nuno and Moll (2015). The definition of the functional derivative is in the Appendix.

21

c∗ (W, z, t). This is evaluated at the wealth-endowment level (W, z) and period t. And the ´ ´ term Ω2 Ω1 ∂Z u(c(x, y, t), Z[m, c])g(x, y, t)dxdy represents the total change in utility of all households due to the change in the value of Z[c, g]. In addition, the different consumption plan chosen by the social planner would results in a different saving schedule and hence a different endowment-wealth density which would affect the the value of Z[c, g]. The last term on the RHS of (24) captures this impact. It is the aggregate externality impact of Z[c, g] on u(c, Z[c, g]) through the change in g(W, z, t).22 The above social optimal condition encompasses two special cases. First, it includes the case without externality. Under this benchmark, the externality index Z[c, g] is absent and the social optimal condition can be obtained by setting both δZ[c∗sp , g](W, z, t)/δg = 0 and δZ[c∗sp , g](W, z, t)/δc = 0. Equations (24) and (25) then reduce to (7) and (6) respectively. The implies that in the absence of consumption externality, the decentralized equilibrium is socially optimal. Second, the social optimal condition can be applied to the wealth externality models (for example, the model in Long and Shimomura (2004)). In the standard wealth externality models, agents’ utility depends on the average wealth (instead of average consumption) of the other agents in the economy. A similar way to generalize it is to allow the reference wealth level to be any functional of the wealth density, denoted as Z[g]. Compared to consumption externality index Z[c, g], the wealth externality index Z[g] is independent of the consumption function c(W, z, t). Therefore, the social optimal condition for wealth externality can be obtained by simply setting δZ[c∗sp , g](W, z, t)/δc = 0 in (25), while keeping (24) the same. For simplicity, we ignore the third channel and assume the last term on the LHS in (24) equals to zero hereafter in this section.23 The following theorem shows under what conditions the negative externality preference would lead to over and under-consumption. > 0 (<0) at (W, z, t), houseTheorem 7. Assume that the last term in (24) equals zero. If δZ δc holds with wealth W and endowment z have social optimal consumption less (more) than the 22

Notice that the above optimal condition holds for any type of externality Z[m]. Not only that, as in Nuno and Moll (2015), it can be easily extended into a more general case in at least two ways: First, the utility function u(c(W, z, t)) appeared in (23) needed not be the same as the individual utility. it can be replaced by a more general function u ˜(c(W, z, t)). For instance, if u ˜(c(W, z, t)) = ω(W, z)u(c(W, z, t)) the social planner maximizes the weighted sum of households’ utility with weight ω(W, z) instead. Second, the labor endowment process need not follow the Ornstein–Uhlenbeck process. It can be more general as dz = h(z, θ)+σ(z, θ)dB for some drift function h(z, θ) and variance function σ(z, θ), given some parameters θ 23 In the third channel, consumption affect Z[c, g] by first affecting g(W, z, t) according to the KolmogorovForward equation in (8). Then, it influences the externality index Z[c, g] through the last term in (24). Since it is indirect and complicates the problem both computationally and analytically, we ignore it in this section.

22

decentralized one when the preference exhibits negative externality on Z[c, g]. Proof. See the Appendix. Remark. The case where preferences exhibit positive externality on Z[c, g] is just the opposite. Since the social planner takes the negative externality into consideration, a drop in c∗ is able to raise the utility by lowering Z[c, g]. It is optimal to consume at a lower level to avoid an excess “jealousy” created. In contrast to the literature that jealousy (i.e. Z[m] is the average consumption) always lead to over-consumption,24 Theorem 7 reveals that the result also depends on how the externality index Z[c, g] is specified. Under-consumption happens under a negative ex(W, z, t) < 0. A simple example is setting the externality ternality preference when δZ[c,g] δc index to be the negative of the Z[c, g], jealousy on it yields the same social planner solution as the admiration on Z[c, g]. Fortunately, many of the externality indices we used (W, z, t) > 0 for any (W, z, t), (For example, all the central moments that satisfies δZ[c,g] δc satisfy the condition). In this case, the social optimal would result in a lower externality index than the decentralized economy. This is consistent with the existing literature that jealousy leads to over-consumption, while we generalized the result into the heterogeneous framework. It is striking that over and under-consumption could co-exist under some negative externality preferences. For instance, inequality aversion preference with ´ ´ ´ ´ Z[c, g] = Ω2 Ω1 (c(x, y, t) − µ(t))2 g(x, y, t)dxdy where µ(t) = Ω2 Ω1 c(x, y, t)g(x, y, t)dxdy, then δcδ Z[c∗sp , g](W, z, t) = 2(c(W, z, t)−µ(t))g(W, z, t) >0 if and only if c(W, z, t) > µ(t). The social planner would assign more consumption to those households with c(W, z, t) < µ(t) and vice versa. In this way, the equilibrium consumption variance (inequality) in the social planner economy would be lower than those in the decentralized economy. By using (25), we could numerically compute the socially optimal consumption function (and hence the socially optimal consumption density). To focus on (25), we fix the value the density g(W, z, t) and the RHS of (25) equals to those under the decentralized economy, and plot the social optimal consumption by solving (25). Figure 4 plots the social optimal and decentralized consumption densities at the steady state under the relative consumption preference (Z[m] is average consumption) and inequality aversion preference (Z[m] is the standard derivation of consumption). The model parameters are shown in Table 1 and will be discussed in Section 5. The plots are consistent with the 24

Under the model with externality in average consumption and with production, Liu and Turnovsky (2005) (Proposition 1) show that there is no difference between the decentralized and social planner solution. In our model, there is no production.

23

result in Theorem 7. Under the relative consumption preference shown in Panel A, the social optimal consumption density lies on the left of the decentralized density, implying that the households are over-consumed in the decentralized economy. Under the inequality aversion preference in Panel B, the social planner density lies on a similar position as the decentralized density, but with thinner tails on both sides. This reveals that under inequality aversion preference, reducing consumption of the rich to subsidize the poor could improve the welfare.

5

Simulation

In the simulation study, the values of parameters are as follows: we set the discount rate ρ = 0.05. The interest rate r = 0.047 and wage w = 1.03. This is computed by using the marginal product of labor and the marginal product of capital of the production function ZK α L1−α with Z = 1 , α = 0.35 and capital per worker K/L = 3.8. Notice that the value of interest rate satisfies the assumption that r < ρ in the Aiyagari model. We set σ 2 = 0.12 and θ = 0.3. The range of endowment is set between 0.5 and 1.5. Hence, we assume z¯ = 0.6 to allow for the asymmetry of the endowment process (which is relevant for the analysis below). Table 1 summarizes the parameters values. We first analyze the stationary equilibrium with different functionals Z[m]. Then, we proceed to the transitional dynamics analysis and its implication in explaining Easterlin paradox. Finally, the taxation policy analysis will be examined. Additional results are presented in the Appendix. As shown in Theorem 5, the steady state consumption and wealth distribution are not affected by the presence of externality under the RC and additive separable utility function. Hence in the numerical exercise for stationary equilibrium, we assume the AC utility u(c, Z) = (c − ηZ[m]γ )φ /φ, where Z[m] can either be moments or Gini coefficient. Then, we switch back to the RC utility in the transitional dynamics. The advantage of using RC utility in transitional dynamics is that since the new and old steady states are the same for any type of externality, the difference of transition paths under different externalities can be compared more clearly.

24

5.1

Stationary Equilibrium

We set the risk aversion parameter φ = −1 and the degree of negative externality η = 0.5 and γ = 0.5 as in Carroll et. al. (1997).25 Notice that households with consumption that is closer to the value of ηZ[m] have utilities that are more affected by the externality. We first consider the case where Z[m] are the mean E(c), standard deviation [E(c − E(c))2 ]1/2 and skewness E(c − E(c))3 /[E(c − E(c))2 ]3/2 respectively. We also consider the benchmark Z[m] = 0. Figure 5 shows the consumption densities in these four cases. It is noting that all the other densities lie on the right of the benchmark one that without externality, which implies that negative externality results in a higher level of consumption on average. Under our specification, the marginal utility of consumption ∂u/∂c = 1/(c − ηZ[m]γ )2 . Notice that the marginal utility approaches infinity when c approaches F [m] = ηZ[m]γ > 0. Households with consumption levels close to F [m] under the benchmark would have an incentive to increase their consumptions. If Z[m] is the average consumption, this would further drive up Z[m] and induce even more consumption. This process continues until the equilibrium is reached. Therefore, consumption density under all the three types of externalities lie on the right of the benchmark density, and the one under average externality is at the rightmost end. 26

5.2

Transitional Dynamics

In the transitional dynamics, we turn back to the RC utility function u(c, Z) = (cZ[m]−γ )φ /φ. As before, we set φ = −1 and γ = 0.5. We divide the transition time from the terminal value function to the initial value function into 70 steps. To allow enough time for convergence, we set the number of periods to be 1000. Hence, the discretized HJB and Kolmogorov Forward equation involve 70 steps with the step size equals to 1000/70 ≈ 14.3. The initial wealth density g(W, z, 0) is the stationary wealth density from the above section. We assume there is a 10% increase in both wage and interest rate, this approximates the case of an 10% increase in total factor productivity when the interest rate and wage are the marginal product of capital and labor respectively. The terminal value function V ∗ (W, z) is simply the value function in the new stationary equilibrium. Since our model 25

Notice that ∂u/∂Z = −ηγ(c − ηZ γ )φ−1 Z γ−1 < 0, so the negative externality appears in all the households are not necessarily hold when φ 6= −1. 26 Technically, the steady state consumption with externality is higher the those without externality for all households when the two conditions are satisfied: (i) ∂cZ u(c, Z[m]) > 0 for any c, Z[m]; (ii) the functional derivative δZ[m]/δc > 0. The intuition of the story is similar to those in Clark and Oswald (1998).

25

is in partial equilibrium, we also assume exogenously the aggregate capital to increase from 4 to 4.3 to capture the downward pressure to interest rate due to the capital accumulation. As a result, the interest rate increases from 0.35 × 1 × 40.35−1 − 0.1 ≈ 0.042 to 0.35×1.1×4.30.35−1 −0.1 ≈ 0.049 and wage increases from (1−0.35)×1×40.35 −0.1 ≈ 0.96 to (1−0.35)×1.1×4.30.35 −0.1 ≈ 1.09. Another reason to increase K is to ensure that the new interest rate would still be below the discount rate (ρ = 0.05), as otherwise households would accumulate as many assets as possible. Besides setting Z[m] to be the moment functions, we also consider Z[m] to be the ´∞ Gini coefficient of consumption Gini = 0 M (c)(1 − M (c))dc/E(c) where M (c) is the distribution function of c. In the simulation exercise, we divide the consumption into 100 P grids [c1 , c2 , ..., c100 ] and use the sample counterpart Gini= 1 − 100 i=1 m(ci )(Si−1 + Si )/S100 , Pi where Si = j=1 m(ci )ci and S0 = 0. Here we only compare the case where Z[m] equals to mean and Gini coefficient to represent the jealousy and the inequality aversion preference respectively. And we compute the equilibrium where Z[m] = 1 as a benchmark where externality is absent. Panel A and B of Figure 6 plot the average level of consumption and wealth over time for different types of externalities. It is observed that both the mean and Gini externalities result in a faster growth in consumption and wealth. Households expecting a higher consumption level in the future would result in a higher level of negative externality, so they consume more in the earlier periods. Similar logic is applied in explaining the faster growth under the inequality aversion preference. The average consumption under the Gini externality grows faster at the earlier stage of the transition process and is caught up by the benchmark model later. It is because the RC utility is used here, the initial and terminal values must be the same according to Theorem 5. In summary, Panel A and B show the existence of negative externality could speed up the convergence rate during the economic growth if the externality index is greater in the new steady state. The result is consistent with the finding in Alvarez-Cuadrado et. al. (2004). As shown in the Panel C, the path of Gini coefficient is increasing. Since the increase in interest rate would benefit those with higher wealth levels more in absolute values and the higher wage rate would increase the variability of labor income under the same labor endowment process, Gini coefficient is greater in the new steady state. Under inequality aversion, the greater Gini coefficient in the new equilibrium could lower agents’ marginal utility of consumption due to the property of the RC utility that ∂cZ u(c, Z[m]) < 0. Agents expecting this would consume earlier and hence the convergence rate increase at the be-

26

ginning. Regarding a more than three times increase in Gini coefficient shown in Panel C compared to only less than one-time increase in average consumption in Panel A, it is reasonable to see that households with the Gini coefficient externality would react more dramatically to the shock compared to those with the mean externality. Indeed, the larger response of Gini coefficient than the average consumption is key to explain the so-called Easterlin paradox. It will be discussed in the following subsection.

5.3

Easterlin Paradox

Easterlin paradox (also known as the happiness-income paradox) refers to an empirical finding that happiness level does not associate with income level within a country in the long run.27 It is regarded as a paradox since if utility is increasing with consumption, higher income levels should result in higher consumption levels and hence higher utility values. Take the US as an example, Panel A of Figure 1 plots the survey result obtained from Smith et. al. (2015). It shows that from 1972 to 2015, the happiness levels in all the categories are more or less constant. The percentage of the population who declare “Very Happy”, "Pretty Happy and “Not very Happy” are stable at around 33%, 55% and 11% respectively. Panel B of Figure 1 shows the real per capita GDP has more than a double increase during this period (from $23,919 at the beginning of 1972 to $50,718 at the beginning of 2015). Easterlin (1974) points out that the Easterlin paradox could be explained by the relative income hypothesis. In addition to concerning the absolute value of consumption, agents care about their consumption levels relative to the others in the economy. This is equivalent to the relative consumption concern mentioned above. Although all the households may consume more during the economics development, the relative positions of all the households cannot increase together, this substantially limits 27

It is emphasised that the empirical finding only valid for the time series data within a country. At a given point of time, households with higher income are associated with a higher level of happiness. Also, for the cross-countries comparison, countries with a greater per capita real GDP would also have a higher level of happiness on average. Note that the paradox holds only in the long run, say more than 10 years. Happiness does vary with the business cycles in the short run. In addition, it does not mean that the happiness level is strictly constant over time. Precisely, it is the trend of happiness level not correlated with the trend of real income growth. The paradox is firstly documented by Easterlin (1974), and he confirms the finding in his subsequent papers (for instance, Easterlin (1995, 2005a, 2005b)). On the contrary, some papers find a positive relationship between happiness and income level for the time series data within a country. Examples include Stevenson and Wolfers (2008), Sacks et. al. (2013) and Veenhoven and Vergunst (2014). Easterlin (2016) recently replies to the criticism by pointing out that: (i) These papers inappropriately combines two different surveys into one dataset; (i) the time span of their dataset is long enough to conclude their results as a long-term relationship.

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the overall change in the utility levels. Clark et. al. (2008) formulate this idea by introducing the relative consumption utility as u˜(c, c˜) where c is the individual consumption and c˜ = c/¯ c is the ratio of c to the average consumption level in the economy c¯. They assume that u˜ is increasing and concave in both argument. That is, ∂c u˜(c, c˜)>0, ∂c˜u˜(c, c˜) > 0 and ∂cc u˜(c, c˜)<0, ∂c˜c˜u˜(c, c˜) < 0. Our RC utility belongs to the class of u˜(c, c˜). As explained by Clark et. al. (2008), introducing the component c˜ could slow down the increase in utility brings out by economic growth. To see this, suppose the economy grows from year t0 to year t1 . During these periods, suppose that the average consumption level increases from c¯t0 to c¯t1 . The agent consumes at the average level in both periods enjoys an increment in utility level equals to u˜(¯ ct1 , 1)− u˜(¯ ct0 , 1). Due to the diminishing marginal utility, the increment could be small when the level of c¯t0 and c¯t1 are already high. Clark et. al. (2008) then conclude that this illustrates why the happiness is not sensitive to the income growth in the developed countries. In addition, by comparing two agents in year t0 , if one of them consumes c¯t0 and the other consumes c¯t1 , their utility difference equals to u˜(¯ ct1 , c¯t1 /¯ ct0 ) − u˜(¯ ct0 , 1) which is greater than the utility difference u˜(¯ ct1 , 1) − u˜(¯ ct0 , 1) across periods. From this formulation, Clark et. al. (2008) verifies that during the process of economic development, the over-time utility improvement is less than the utility difference at any given point of time, which is consistent with the empirical finding. However, it seems that this specification is not consistent with the fact in the US that there is a double increase in real per capita GDP in the last 40 years, while without any improvement in happiness. Say, if the average consumption increases two times from c¯t0 to c¯t1 = 2¯ ct0 , we have the utility improvement equals to u˜(2¯ ct0 , 1) − u˜(¯ ct0 , 1), which is hardly close to zero. It is close to zero when the consumption level is so high that the utility is diminished to almost zero. Yet, if this is the case, it is hard to explain why people still give up leisure in exchange for consumption nowadays.28 Put differently, the relative income concern can only explain why happiness is growing slower than what the selfinterest utility predicted, but still cannot explain why the level of happiness is constant over time. Furthermore, some papers even find that in the US the relationship between happiness and economic growth is negative for the overall population (Firebaugh and Tach (2012) and Kenny (1999)) and for the female population (Stevenson and Wolfers (2009)). Utility only adding the relative consumption element cannot generate this neg28

In the extreme case, where the utility depends only on c˜ (that is, u ˜2 (˜ c) = u ˜(c, c˜) ), the utility level at c = c¯ always remains at u ˜(1). In this case, economic growth has no impact on the utility level. However, this assumption that agents only care about the status is quite restrictive and unrealistic.

28

ative relationship, we need another negative force on utility to balance the upward force during the economic growth.29 Instead, we suggest that the “inequality aversion” hypothesis can complement the relative income argument above. Figure 2 shows three time series of income dispersion measures, namely, Gini Index, Theil Index and the mean logarithm derivation of income from 1972 to 2015 in the US. The data is obtained from the US Census Bureau. Along with the economic growth during these periods, Gini Index, Theil Index and the mean logarithm derivation of income experiences a 28%, 70% and 106% growth respectively.30 The disutility due to the “inequality aversion” preference is thus increasing during this period. This can offset some of the rise in utility level brings out by the rise in the absolute value of consumption. Formally, consider the utility function u(c, Z[m]) where Z[m] is an externality index. By total differentiating the natural logarithm of the utility function, we get the formula: ∂ ln u d ln c ∂ ln u d ln Z[m] d ln u = + dt ∂ ln c dt ∂ ln Z[m] dt

(26)

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The formula reveals that the percentage change in utility d ln u/dt is the weighted sum of the percentage change in the individual consumption d ln c/dt and the externality index d ln Z[m]/dt. For the RC utility function, this becomes d ln c d ln Z[m] d ln u =φ − γφ (27) dt dt dt The externality index Z[m] could either be the average consumption or the measure of income dispersion. When it is the average consumption (i.e. Z[m] = c¯), we need the assumption that 0 < γ < 1 for the relative consumption hypothesis in Clark et. al. (2008) to hold. It is because (27) can be written as d ln u d ln c d ln c˜ = (φ − γφ) + γφ dt dt dt For the utility u to be increasing in both c and c˜, we need 0 < γ < 1. Under this restriction, when c increases and c˜ unchanges, the utility level would definitely increase, 29

As mentioned by Clark (2016), habit formation hypothesis is also another candidate in explaining the Easterlin paradox. This hypothesis subjects to the same problem as the relative income hypothesis. 30 We do not have data of consumption dispersion during this period. Recently, Aguia and Bils (2015) shows that consumption inequality tracks closely to the income inequality in the US. 31 One may argue that the externality index Z[m] is not unit invariant. For instance, the average consumption is clearly not measured in the same unit as the Gini coefficient. We get rid of the problem by expressing the change in utility in term of percentage rather than absolute value.

29

though it is in a smaller magnitude than the self-interested utility function where γ = 0. The result is not restrict to RC utility, any utility u˜(c, c˜) that is increasing in both c and c˜ would inevitably increase during the economic growth. However, under the “inequality aversion” preference, it only requires that γ > 0. Therefore, during the economic growth when both the individual consumption c and the consumption dispersion Z[m] increase together, the utility value can be kept as a constant if γ is large enough. In addition, the response of Z[m] also in favor of the “inequality aversion” argument. Back to our model, recall that in our transitional dynamic example, when there is a 1% increase in technology level, the average consumption increases by 56.5% (from 0.668 to 1.05), while the standard derivation and Gini coefficient increase by 235.5% and 117.9% respectively (the standard derivation increases from 0.073 to 0.244, and the Gini coefficient increases from 0.058 to 0.127). It shows that the dispersion measure, no matter it is the standard derivation of consumption or the Gini coefficient, is more sensitive to the technology shock. This is because the wealthier households benefits more from the rise in interest rate brings out by the positive technology shock. This encourages them to save more (and so saving function is increasing with wealth). Their higher wealth levels sustain greater consumptions in the new steady state. The unproportional benefits of the technological shock result in a more dramatic change in consumption inequality, compared to the increase in the average consumption. In this context, by (27), it can be shown that the percentage change in utility under the “inequality aversion” preference is less than those under the relative consumption preference, given the same values of parameters. This is because, first, by Theorem 5, the percentage change in individual c between the two steady states is independent of the externality index Z[m]. Second, the percentage change in the consumption variance is greater than the percentage change in the average consumption.32 To conclude, if we define the measure of happiness as the utility level at the average consumption level as in Clark et. al. (2008), clearly “inequality aversion” explains the paradox better. How about if we measure the happiness by the aggregate utility defined by (22)? We can come to the same conclusion. Figure 7 plots the aggregate utility defined by (22) against γ under the new and the old steady states with different types of externalities. By 32

The possibility that Easterlin paradox is due to the increase in consumption dispersion has also been claimed by Clark et. al. (2008). Consider the extreme case that the concave function u ˜2 (˜ c) = u ˜(c, c˜), an increase in the consumption variance would be lower than the aggregate utility defined in (22). This can be proved by the Jensen inequality. However, consider the general function u ˜(c, c˜), with both the average and variance of consumption rise as in the data. It is hard to claim that the aggregate utility remains constant.

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(27), the greater the γ, the less is the percentage change in the utility. Panel A shows the benchmark model where the utility function is not affected by the externality. Aggregate utility is independent of γ and rises from around -1.5 to -1 after the 1% technology shock. Panel B shows that under the inequality aversion preference, the aggregate utility under the old starts catching up the new one as γ increases. When γ is around 0.5, these two lines intersect and hence Easterlin paradox can be explained. Yet, for the “status seeking” externality in Panel C, the lines intersect only until γ is between 0.9 and 1. This is unrealistic since γ = 1 implies that the utility function entirely depends on the relative consumption. Under such a large γ, the absolute value of consumption has a minimal impact on utility, which is improbable. This provides evidences that the Easterlin paradox is more likely to hold under the “inequality aversion” preference at least in our model. We emphasize that both relative consumption and “inequality aversion” are not contradictory, it is believed that in reality both of them play role in explaining the movement of happiness.33 If inequality aversion has such a negative impact on the aggregate utility and prevent it from improving in the past decades, policies that reduce the income (or consumption) inequality is expected to be welfare-improving. This is discussed below.

5.4

Progressive Labor Income Tax

In this section, we introduce a progressive labor income tax to the households’ constraint (2). The new constraint is as follows: dW = (rW + (wz)1−τP − c)dt where τP ∈ [0, 1] is the progressive tax rate.34 To ensure that each household has a lower after-tax income, we set the range of z to be [1, 2] instead so that (wz)1−τP is de33 Hopkins (2008) also claims that inequality aversion could generate the Easterlin paradox. However, the concept of inequality aversion defined by Hopkins (2008) is different from this paper. The Hopkins (2008) define inequality aversion as a combination of “envy” and “compassion”, where the latter is the key element. “Envy” refers to the drop in one’s utility levels when someone with a higher consumption level then her consume more. “Compassion” refers to one’s utility level is higher when someone with lower consumption level than her consume more. Hopkins (2008) argue that with this preference, those with consumption growth less than the average growth rate could have no change in happiness level during the economic growth. However, this preference cannot explain why the happiness level in all categories remains constant as shown in Panel A of Figure 1. With this preference, the proportion of the “Very Happy” group should increase over time if everyone’s income increases with the rich increases more. Similar results are applied to the other example considered by Hopkins (2008). 34 By Ito’s Lemma, the stochasic differential equation of the labor income process IL ≡ (wz)1−τP is

31

creasing with τP for any τP > 0. Notice that the higher the progressive tax rate τP , the less is the income difference between households. For each τP , we compute the stationary equilibrium under the mean (status seeking), standard deviation (inequality aversion), skewness externalities and the benchmark model. Figure 8 shows the mean, standard deviation, skewness of consumption and the aggregate utility with different levels of τP from 0 to 0.5 under the stationary equilibrium. Since a higher τP lowers the average and the standard deviation of the labor income process wz simultaneously, both the consumption average and inequality decrease as shown in Panel A and B. This result is the same as the proportional capital and labor income tax shown in the Appendix (Table 4). Besides, progressive tax also lowers the skewness dramatically when τP ∈ [0, 0.4] and raises it slightly afterward. As τP approaches 0.5, the standard deviation and skewness of consumption shrink to around 0 and so the impacts of standard deviation and skewness externalities diminish. Models with both of the two externalities converge to the benchmark. The mean externality produces higher consumption average and inequality at the steady state compared to the other externalities, consistent with Figure 5, but the differences are diminishing as the tax rate increases. This reveals that introducing a progressive tax could narrow down the consumption gap between the one with and without negative consumption externalities. However, this does not imply that the welfare would converge to each other, since the utility functions are different. Panel D shows the aggregate utilities defined by (22) for each case. There are two forces that affect the change in u(c, Z[m]) when τP increases, namely: the change in the consumption level c, and the change in the externality index Z[m]. It is shown that the aggregate utility under the benchmark and standard deviation externality model are parallel to each other. On one hand, Panel A shows that the model with standard deviation externality has a steeper drop in average consumption than the benchmark model, lowering the utility with standard deviation externality more. One the other hand, Panel B shows that the standard deviation of consumption drops as τP rises, reducing the negative impact of the standard deviation externality on utility. These two opposite forces cancel out each other, resulting in approximately the same drop of standard deviation as the benchmark one according to (27). A more interesting case is that the aggregate utility increases under the skewness ex  σ 2 τP dIL = z −τP (1 − τP ) (θ(z − z¯) − )dt + σdB 2z

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ternality model when τP ∈ [0.08, 0.42]. This is because the impact from the decrease in the consumption skewness outweighs those from the reduction in the consumption level. Hence, rising progressive tax could increase the welfare level in the presence of externality, even if the tax revenue is not transferred and is wasted. Why does the welfare improvement only happen in the skewness externality? Table 2 reports the values and the percentage change in the mean, standard deviation and skewness of consumption when τP = 0 and 0.5 under different types of externalities. The percentage change in the skewness of consumption is the greatest, with a 82 % drop under the skewness externality, followed by a 55% drop of the standard deviation under the “inequality aversion” and only an around 11% drop of average consumption under the mean consumption externality. In other words, progressive tax leads to a greater percentage change in the higher moments of the stationary distribution of consumption than the mean, hence benefits the households with the higher moments externalities more. Under the AC utility specification u(c, Z) = (c − ηZ[m]γ )φ /φ, the individual and externality impact are respectively determined by the terms c and ηZ[m]γ . One may argue that the welfare is still improved under a progressive tax when η is large enough even if there is only a modest drop in Z[m] . Figure 9 plots the aggregate utility defined in (22) under mean, standard deviation and skewness externality against τP with different levels of η. It shows that under the standard deviation and skewness externality, the impact of welfare improvement increases along with the η. For η = 2.5, the aggregate utility has a 50% increment, from -2.65 to -2 when τP increases from 0.16 to 0.5. Similar results cannot be generated in the mean externality model. This is because the mean externality Z[m] is not sensitive enough to the change in τP .35 It is found that introducing a progressive tax could also speed up the rate of convergence under a positive technology shock, the result is presented in the Appendix.

6

Model Selection

To check which types of consumption externality best fits the data, we first compute the equilibrium consumption density from a model and treat it as a likelihood function. If the model is true, the density is the data generating process. The best externality model is the one that yields the highest likelihood. 35

The magnitude of ηZ[m]γ cannot exceed the consumption level c for any households as otherwise the utility is undefined. η = 0.8 in Panel C is almost the maximum feasible value.

33

We utilize the consumption data collected by Krueger and Perri (2006) and hence follow their definition of households’ total consumption expenditure.36 The data cd = [cd1 , cd2 , ..., cdT ] is from 1980 to 2003 with T = 458, 680 observations. For simplicity, we pool the data in all the years and estimate only one consumption density using the data. Given the parameters values θ, assume that the likelihood function is the stationary equilibrium consumption densities obtained from the models. It is clear that we do not have an analytical expression for the consumption densities, so to compute the likelihood function, we follow the approach in Fermanian and Salanié (2004): First, we generate a sample with the size S = 10, 000 from the stationary equilibrium consumption density of a model. Then, a consumption density m(c; θ) is estimated nonparametrically using the simulated data. And the actual data cd is plugged into m(c; θ) in order to compute the log-likelihood using the formula: L(θ) =

T 1X τ (m(cdt ; θ)) ln m(cdt ; θ) T t=1

(28)

where τ (.) is a regular function that screens out the extreme data. We follow Fermanian and Salanié (2004) to assume that τ (x) = 4(x − hd )/h3d − 3(x − hd )4 /h4d where x ∈ [hd , 2hd ] and h is the bandwidth of the nonparametric density estimation above. We set d = 2. When τ (x) < 0 for x < hd and τ (x) = 1 for x > 2hd . Rather than estimating the parameters, we only compare the likelihoods of the four functionals Z[m] (the mean, standard deviation, skewness and benchmark model) given the set of parameters specified in Table 1. The consumption data is normalized such that the minimum (maximum) of the data equals to the minimum (maximum) of the equilibrium consumption of the four Z[m]. Since the choice of the parameters values (η, σ 2 , θ) have not much literature to refer to, we try two set of them: (0.4, 0.072 , 0.4) and (0.5, 0.12 , 0.3). Table 3 shows the predicted mean, standard deviation, skewness, the sum of squared error of the three moments (SSE) and the simulated likelihood values for the four models under these two set of parameter values.37 The value of simulated likelihood suggests that the standard deviation externality best fits the data, with the likelihood value equals to 0.675 under the first set parameters. This is followed by the benchmark 36

The data can be download from the link: http://www.fperri.net/data/cexdataweb.zip. We use the variable “texp1” from the file. Please check Appendix A in Krueger and Perri (2006) for the details. 37 The sum of squared error is the sum of the square of the difference between the moments of the data and the model. That is, SSE = (µ1 − E(c))2 + (µ2 − [E(c − E(c))2 ]1/2 )2 + (µ3 − E(c − E(c))3 /[E(c − E(c))2 ]3/2 )2 where µ1 , µ2 and µ3 are respectively the mean, standard deviation and the skewness of the normalized consumption data.

34

model (0.6526) which also produces the minimum SSE among the models. The mean externality is able to generate a large standard deviation as in the data. Figure 10 plots the nonparametric estimates of the consumption densities using the simulated data under the four Z[m] and the data for (η, σ 2 , θ) = (0.4, 0.072 , 0.4). It is clear that only the standard deviation functional and the benchmark model out-perform the other two models. The data density is heavily right-tailed which cannot be matched by the four models, as we can see in Table 3 that the skewness generated from the models are much lower than those observed in the data. In summary, the best fit of the standard deviation externality suggests that agents are more likely to care about the dispersion of the consumption distribution than the level of it, raising questions to the suitability of the conventional status seeking preference (mean consumption externality) approach in the literature.

7

Conclusion

In the environment with heterogeneity, it is ultimately important to enrich agents’ interaction so as to provide a more flexible shape of distributions in equilibrium. Consumption externalities not only can shape the way that the households interact, but also can be interpreted as different psychological behaviors. Interestingly, sentiments from our understanding, like the desire for equality, the hostility to the rich and sympathy for the poor, are a rephrase of the consumption externality in economic terminology. This paper proposes a framework to model the consumption externality at a higher level of generality in a dynamic stochastic equilibrium framework. We also provide model solutions to the decentralized equilibrium and the social planner problem. With this general framework, we then narrow down our analysis on the “inequality aversion” preference. We the “inequality aversion” hypothesis complements the standard relative income hypothesis in providing a better explanation for the Easterlin paradox. There are two reasons behind. First, introducing the relative income preference could only slow down the utility growth, while “inequality aversion” preference is flexible to create a downward pressure on utility. Second, under the technology growth, the percentage change in the consumption variance is greater the percentage change in the consumption average. This creates a more significant externality impact on “inequality aversion” utility. In this context, our simulation shows that the aggregate welfare of the inequality aversion” households could drop even under a technology improvement. This inefficiency leads to an interesting policy implication: The aggregate utility levels 35

could increase with the progressive tax rate, even if the tax revenue is wasted and not transferred back to the households. This is true when households induce negative externalities on the skewness and standard deviation of consumption density which are lower under a higher progressive tax rate. Clearly, under “inequality aversion”, the social optimal consumption density is shown to have a smaller variance than those of the decentralized equilibrium. Put differently, poor are under-consumed and the rich are over-consumed under the “inequality aversion” preference. This contributes to the existing literature that negative consumption externality does not necessarily lead to over-consumption, whether households are overor under-consumed also depends on the type of externalities assumed. In addition, mean and Gini externality result in a faster rate of convergence under a permanent TFP shock and create a larger consumption inequality during the transition. Model comparison using the US data suggests that modeling utility with negative externality on the standard deviation of consumption fits the data the best, providing more evidence for the existence of the “inequality aversion” preference. In the relative consumption literature, there are two alternative ways of modeling that we have not considered here. First, the inward-looking households whose utility depends on their past consumption, as in the habit-formation model in Gali (1994) and the “deep habits” model in Ravn et. al. (2006). Our framework can easily be extended to these cases and even in a more general scenario where the past consumption inequality and other higher moments enter households’ utility. Second, households can choose their labor supply. The endogenous labor supply allows us to study the “keeping up with Joneses” and “running away from Joneses” defined in Dupor and Liu (2003). How different will be the households’ labor supply if they are “inequality averse”? We leave these extensions for further studies.

36

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[38] Friedman, D., and Ostrov, D. N. (2008): Conspicuous consumption dynamics. Games and Economic Behavior, 64, 1, pp. 121-145. [39] Gali, J. (1994): Keeping up with Joneses: Consumption externalities, portfolio choice, and asset prices. Journal of Money, Credit and Banking, 26, pp. 1-8. [40] Garcia-Penalosa, C. and Turnovsky, S. (2008): Consumption externalities: a representative consumer model when agents are heterogeneous. Economic Theory 37, 3, pp. 439-467. [41] Hagerty, M. R. (2000): Social comparison of income in one’s community: evidence from national surveys of income and happiness. Journal of Personality and Social Psychology, 78, 4, pp. 764-771. [42] Ho, T. H., and Su, X. (2009): Peer-induced fairness in games. The American Economic Review, 99, 5, pp. 2022-2049. [43] Hopkins, E. and Kornienko, T. (2004): Running to keep in the same place: consumer choice as a game of status. The American Economic Review, 94,4, pp. 1085-1107. [44] Hopkins, E. (2008): Inequality, happiness and relative concerns: What actually is their relationship?. The Journal of Economic Inequality, 6, 4, pp. 351-372. [45] Johansson-Stenman, O. and Carlsson, F. and Daruvala, D. (2002): Measuring future grandparents’ preferences for equality and relative standing. Economic Journal, 112, 479, pp. 362-383. [46] Kenny, C. (1999): Does growth cause happiness, or does happiness cause growth?. Kyklos, 52, 1, pp. 3-25. [47] Knight, J., Song, L. and Gunatilaka, R. (2009): Subjective well-being and its determinants in rural China. China Economic Review, 20, pp. 635–649 [48] Krueger, D. and Perri, F. (2006): Does income inequality lead to consumption inequality? evidence and theory. The Review of Economic Studies, 73, 1, pp. 163-193. [49] Lasrey, J. and Lions, P. (2007): Mean field games. Japanese Journal of Mathematics, 2, pp. 229–260.

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[50] Lucas Jr, R. E. and Moll, B. (2014): Knowledge growth and the allocation of time. Journal of Political Economy, 122, 1, pp. 1-51. [51] Liu, W. F. and Turnovsky, S. J. (2005): Consumption externalities, production externalities, and long-run macroeconomic efficiency. Journal of Public Economics, 89, 5-6 , pp.1097-1129. [52] Long, N. V. and Shimomura, K. (2004): Relative wealth, status-seeking, and catching-up. Journal of Economic Behavior & Organization, 53, 4, pp. 529-542. [53] Luttmer, E. F. (2005): Neighbors as negatives: relative earnings and well-being. The Quarterly Journal of Economics, 120, 3, pp. 963-1002. [54] Mehra, R. and Prescott, E. C. (1985): The equity premium: a puzzle. Journal of monetary Economics, 15, 2, pp. 145-161. [55] Moll, B. (2014): Productivity losses from financial frictions: can self-financing undo capital misallocation?. The American Economic Review, 104, 10, pp. 3186-3221. [56] Nuno, G. and Moll, B. (2015): Optimal control with heterogeneous agents in continuous Time. Working paper No. 1608 [57] Ravn, M., Schmitt-Grohé, S. and Uribe, M. (2006): Deep habits. Review of Economic Studies, 73, 1, pp.195-218. [58] Sacks, D. W., Stevenson, B. and Wolfers, J. (2013). Subjective well-being, income, economic development, and growth. In Annual World Bank Conference on Development Economics 2011: Development Challenges in a Post-crisis World. World Bank Publications. [59] Schwarze, J. and Härpfer, M. (2007): Are people inequality-averse, and do they prefer redistribution by the state? evidence from german longitudinal data on life satisfaction. Journal of Socio-Economics. 36, pp.233–249 [60] Smith, T. W., Son, J. and Schapiro. B. (2015): General social survey final report: trends in psychological well-being, 1972-2014. Chicago, Il, NORC at the University of Chicago. [61] Solnick, S. J. and Hemenway, D. (1998): Is more always better?: a survey on positional concerns. Journal of Economic Behavior & Organization, 37, 3, pp. 373-383. 41

[62] Stevenson, B. and Wolfers J. (2008): Economic growth and subjective well-being: reassessing the Easterlin Paradox. Brookings Papers on Economic Activity, 2008, 1, pp. 1-87. [63] Stevenson, B. and Wolfers, J. (2009): The paradox of declining female happines. American Economic Journal: Economic Policy, 1, 2, pp. 190-225. [64] Thurow, L. C. (1971): The income distribution as a pure public good. The Quarterly Journal of Economics, 85, 2, pp. 327-336. [65] Veblen, T. B. (1899): The Theory of the leisure class: an economic study of institutions. Modern Library, New York. [66] Veenhoven, R., and Vergunst, F. (2014): The Easterlin illusion: economic growth does go with greater happiness. International Journal of Happiness and Development, 1, 4,pp. 311-343. [67] Verme, P. (2011): Life satisfaction and income inequality. Review of Income and Wealth, 57, 1, pp. 111-127.

42

8

Figures and Tables

Parameter ρ r w σ2 θ z¯ γ η φ

Value Description 0.05 Discount rate 0.047 Interest rate 1.03 Wage rate (0.1)2 Parameter of labor endowment process 0.3 Parameter of labor endowment process 0.6 Parameter of labor endowment process 0.5 Parameter of (negative) externality for both AC & RC utility 0.5 Parameter of externality for AC utility −1 Risk aversion index for both AC & RC utility Table 1: Parameters values for calibration

43

Without Tax

50 % Progressive Tax

Benchmark Mean Std. Dev. Skewness Benchmark Mean Std. Dev. Skewness

E(c)

%∆E(c)

p V ar(c)

1.194 1.249 1.202 1.218 1.054 1.072 1.055 1.056

-11.7% -14.2% -12.2% -13.3%

0.0877 0.111 0.091 0.0972 0.0432 0.0498 0.0436 0.0438

%∆

p V ar(c)

-50.8% -55.1% -52.0% -55.0%

Skewness

%∆Skewness

0.182 0.635 0.294 0.456 0.0402 0.4090 0.0677 0.0821

-78.0% -35.6% -77.0% -82.0%

Table 2: The change in mean, standard deviation and skewness of consumption under different progressive tax policies. The four models (column two) are compared under the progressive tax rate τP = 0 and 0.5. Column 3, 5 and 7 respectively show the expected, the standard deviation and the skewness of the stationary consumption levels. Column 4, 6 and 8 show their percentage changes compared to the case that without tax.

44

Mean Normalized Data 0.672 Benchmark 0.628 Mean 0.804 Standard Deviation 0.664 Skewness 0.746 Normalized Data 0.783 Benchmark 0.736 Mean 1.102 Standard Deviation 0.827 Skewness 0.920

Standard Deviation 0.148 0.092 0.140 0.109 0.132 0.230 0.142 0.193 0.172 0.187

Skewness 1.910 0.787 0.394 0.742 0.510 1.910 0.749 0.374 0.596 0.448

SSE Likelihood 1.267 0.6526 2.314 −0.09060 1.367 0.6750 1.966 0.3370 1.358 0.4847 2.462 -0.0610 1.733 0.2182 2.158 0.0481

Table 3: The predicted mean, standard deviation, skewness, sum of squared error and nonparametric likelihood value from four models. The first and the last five rows are respectively computed from the parameters (η, σ 2 , θ) = (0.4, 0.072 , 0.4) and (η, σ 2 , θ) = (0.4, 0.012 , 0.4). SSE = (µ1 − E(c))2 + (µ2 − [E(c − E(c))2 ]1/2 )2 + (µ3 − E(c − E(c))3 /[E(c − E(c))2 ]3/2 )2 where µ1 , µ2 and µ3 are respectively the mean, standard deviation and the skewness of the normalized consumption data.

45

4 decentralized consumption density social optimal consumption density 3.5

3

m(c)

2.5

2

1.5

1

0.5

0

0.5

1

1.5

c

(a) The consumption density under the relative consumption preference. 5 decentralized consumption density social optimal consumption density

4.5 4 3.5

m(c)

3 2.5 2 1.5 1 0.5 0

0.5

1

1.5

c

(b) The consumption density under the inequality aversion preference.

Figure 4: The decentralized and social optimal consumption density under the relative consumption (Z[m] is mean) and the inequality aversion preference (Z[m] is standard deviation).

46

4.5 Mean std. dev. skewness Benchmark

4 3.5

m(c)

3 2.5 2 1.5 1 0.5 0

0.5

1

1.5

c

Figure 5: Consumption densities in the stationary equilibrium. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

47

1.1

250

1.05 200

0.95

Average wealth

Average consumption

1

0.9 0.85 0.8

100

50

0.75

Mean Gini Benchmark

0.7 0.65

150

100

200

300

400

500 Time

600

700

800

900

Mean Gini Benchmark

0

−50

1000

100

200

300

400

500 Time

600

700

800

900

1000

(a) The transitional dynamics of the average

(b) The transitional dynamics of the average

consumption

wealth

0.13 0.12 0.11

Gini Coefficient

0.1 0.09 0.08 0.07 0.06 0.05

Mean Gini Benchmark

0.04 0.03

100

200

300

400

500 Time

600

700

800

900

1000

(c) The transitional dynamics of the Gini co-

efficient of consumption

Figure 6: The transitional dynamics of the average consumption, Gini coefficient of consumption and welfare level against time under a 10% increase in total factor productivity. The externality ´∞ indices Z[m] are mean Z[m] = E(c) and Gini coefficient Z[m] = 0 M (c)(1−M (c))dc/E(c) where M (c) is the cdf of c and benchmark Z[m] = 1 respectively.

48

-1

0 old utility new utility

-0.2 -1.1 -0.4

Aggregate Utility

Aggregate Utility

-1.2

-1.3

-0.6

-0.8

-1

-1.4 -1.2 -1.5 -1.4 old utility new utility

-1.6

-1.6 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

.

(a) Benchmark - The aggregate utility in the

(b) Inequality aversion - The aggregate

absence of externality: Z[m] = 1

utility when Z[m] is the Gini ´ ∞ coefficient of consumption: Z[m] = 0 M (c)(1 − M (c))dc/E(c) where M (c) is the cdf of c.

-1

-1.1

Aggregate Utility

-1.2

-1.3

-1.4

-1.5 old steady state new steady state

-1.6 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

.

(c) Relative consumption - The aggregate

utility when Z[m] is the average consumption: Z[m] = E(c).

Figure 7: The aggregate utility under the new and the old steady states with a 10% increase in technology shock.

49

1.25

0.12 Mean std. dev. skewness Benchmark

1.2

0.1

Consumption Inequality

Mean Consumption

Mean std. dev. skewness Benchmark

0.11

1.15

1.1

0.09

0.08

0.07

0.06

0.05

0.04

1.05 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Progressive Tax

Progressive Tax

(a) Average Consumption against the pro-

(b) Standard deviation of consumption

gressive tax rate with different externalities

against the progressive tax rate with different externalities -0.8

0.7 Mean std. dev. skewness Benchmark

-1

0.5 -1.2

Sum of Utility

Consumption Skewness

0.6

0.4

0.3

-1.4

-1.6 0.2 Mean std. dev. skewness Benchmark

-1.8

0.1

0

-2 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0

0.5

Progressive Tax

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Progressive Tax

(c) Skewness of consumption against the

(d) Aggregate utility against the progressive

progressive tax rate with different externalities

tax rate with different externalities

Figure 8: The impact of the progressive tax on the steady state variables. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

50

-0.8

-0.8

-1 -1 -1.2 -1.4

Sum of Utility

Sum of Utility

-1.2 -1.6 -1.8 -2

-1.4

-1.6 -2.2 -2.4

2 = 0.5 2 = 1.2 2 = 1.8 2 = 2.5

-2.6

2 = 0.2 2 = 0.4 2 = 0.6 2 = 0.8

-1.8

-2.8

-2 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

Progressive Tax

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Progressive Tax

(a) Aggregate utility against the progressive

(b) Aggregate utility against the progressive

tax rate with different values of η and Z[m] is the standard deviation consumption

tax rate with different values of η and Z[m] is the skewness standard deviation of consumption

-1

-1.5

Sum of Utility

-2

-2.5

-3

2 = 0.2 2 = 0.4 2 = 0.6 2 = 0.8

-3.5

-4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Progressive Tax

(c) Aggregate utility against the progressive

tax rate with different values of η and Z[m] is the mean of consumption

Figure 9: The impact of the progressive tax on the aggregate utility under different values of η. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

51

5 Data mean standard Derivation Skewness Benchmark

4.5 4 3.5

m(c)

3 2.5 2 1.5 1 0.5 0 0.4

0.6

0.8

1

1.2 c

1.4

1.6

1.8

2

Figure 10: Estimated consumption densities of the US data and the simulated data in the models. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

(η, σ 2 , θ) = (0.4, 0.072 , 0.4) and the other parameter values are as in Table 1.

52

9 9.1

Appendix Appendix A: Proof

Proof of Theorem 4 Following Chang (2004), given the consumption density m(c) and wealth density g(W ), we guess that the value function is of the form V (W ) =

A0 FWφ φ

for some constant A0 6= 0. We have ∂W V (W ) = A0 F W φ−1 and ∂W W V (W ) = (φ − 1)A0 W φ−2 . Substitute the value function and its derivatives into (12) yields φ ρ 1 1 A0 W φ F = (A0 ) φ−1 W φ F ( − 1) + rA0 W φ F + σ 2 A0 (φ − 1)W φ F φ φ 2

Divide both sides by W φ F and solves out A0 yields (16). Plug-in the value function (16) into (15) get (17). And saving s = (rW − c∗ ) when y = 0. Hence, (18) is obtained by using (17). (19) is obtained by first guess g(W ) = A1 W −α−1 for some constant A1 and α. Substitute this into (13): ∂W (A1 A2 W −α ) =

1 2 σ ∂W W (A1 W −α+1 ) 2

where we denote A2 such that s = (rW − c∗ ) = A2 W in (18). We get the two roots for α are 0 and 1 1 − 2A2 /σ 2 . The Pareto distribution requires A2 < 0 which is r < (A0 ) φ−1 . m(c(W ))c0 (W ) = g(W ) Using the formula m(c(W ))c0 (W ) = g(W ), equations (19) and (17) gives (20).

Proof of Theorem 5 Let B(.) = b for some constant b, the optimal consumption satisfies A0 (c∗ )F = ∂W V (W, z)

(29)

Suppose F = 1, the optimal consumption c˜∗ and the value function V˜ (W, z) satisfy the first order condition A0 (˜ c∗ ) = ∂W V˜ (W, z) and the HJB equation

1 ρV˜ (W, z) = A(˜ c∗ ) + b + ∂W V˜ (W, z)(rW + wz − c˜∗ ) + θ(¯ z − z)∂z V˜ (W, z) + σ 2 ∂zz V˜ (W, z) 2

(30)

For any value of F , the value function V (W, z) = F V˜ (W, z) − (F − 1)b/ρ and optimal consumption c = c˜∗ satisfies both (29) and the HJB equation ∗

53

1 ρV (W, z) = A(c∗ )F + b + ∂W V (W, z)(rW + wz − c∗ ) + θ(¯ z − z)∂z V (W, z) + σ 2 ∂zz V (W, z) 2

(31)

We have c˜∗ and hence c∗ is independent of F . Similarly, for u(c, F ) = aF + B(c) for some constant a, the first order condition B 0 (c∗ ) = ∂W V (W, z)

(32)

for any value of F . Let cˆ∗ and Vˆ (W, z) be the optimal consumption and value function when F = 0. It satisfies the first order condition B 0 (ˆ c∗ ) = ∂W Vˆ (W, z) and the HJB equation 1 ρVˆ(W, z) = B(ˆ c∗ ) + ∂W Vˆ(W, z)(rW + wz − cˆ∗ ) + θ(¯ z − z)∂z Vˆ (W, z) σ 2 ∂zz Vˆ(W, t) 2

(33)

The value function V (W, z) = Vˆ (W, z) + aF and c∗ = cˆ∗ satisfies the first order condition (32) and the HJB equation

1 ρV (W, z) = aF + B(c∗ ) + ∂W V (W, z)(rW + wz − c∗ ) + θ(¯ z − z)∂z V (W, z) + σ 2 ∂zz V (W, z) 2

(34)

Hence, c∗ = cˆ∗ is independent of F .

Proof of Theorem 6 The proof basically follows the approach in Bensoussan et. al. (2013) (Chapter 4) with some modifications.38 Below, we adopt their proof to our framework with the notations that are more familiar to the economists. We also present the Lagrangian approach in Moll and Nuno (2015) afterward. It will be shown that both approaches yield the same result. In the social planner problem, the social planner is able to control the consumption function c(W, z, t). The externality index is denoted as Z[c, g] to indicate that it is functionals of both c(W, z, t) and g(W, z, t). Therefore, the utility function u(c, Z[c, g]) is simultaneously a composte function and a functional of c(W, z, t) (through the functional Z[c, g]). Also, it is a functional of g(W, z, t). Re-state the discounted sum of the welfare flow (22) here: ˆ

∞

ˆ

ˆ e−ρt u(c(W, z, t), Z[c, g])g(W, z, t)dW dzdt

J[c, g] = 0

Denote plicitly by:

δZ[c,g] δg (ξ)

Ω1

(35)

Ω2

as the functional derivative of Z[c, g] with respect to the function g and defined im ˆ d Z[c, g + θˆ g ] = dθ Ω θ=0

38

δZ(c,g) (ξ, t)ˆ g (ξ, t)dξ δg

(36)

In Bensoussan et. al. (2013), the utility function is a composite function of the consumption function, but here it is both a composite function and a functional of the consumption function

54

for any test function gˆ(.) and ξ is a vector of variable in the domain Ω. For example, if Z[c, g] = ´´ d d Z[c, g + θˆ g ] θ=0 = dθ f (c(x, y, t))(g(x, y, t)+ f (c(x, y, t))g(x, t)dxdy, for some differentiable function f (.), then dθ ´´ δZ[c,g] θˆ g (x, y, t))dxdy = f (c(x, y, t))ˆ g (x, y, t)dxdy. By comparing it with the RHS in (36), we get δg (x, y, t) = ´´ 0 d c, g] = f (c(x, y, t))ˆ c(x, y, t)g(x, y, t)dxdy, so δZ[c,g] (x, y, t) = f 0 (c(x, y, t))g(x, y, t). f (c(x, y, t)). Similarly, Z[c + θˆ ´´

dθ

δc

θ=0

The approach of Bensoussan et. al. (2013) The social planner problem is equivalent to solving the first order condition: d =0 J[c + θˆ c, gc+θˆc ] dθ θ=0

(37)

where the subscript in gc is to denote that the density g(.) is implicitly related to the consumption function c by the KF equation in (8):

1 ∂t g(W, z, t) = −∂W ((rW + y − c∗ (W, z, t))g(W, z, t)) − ∂z (θ(¯ z − z)g(W, z, t)) + σ 2 ∂zz g(W, z, t) 2

(38)

Diverging c to c + θˆ c and differentiate both sides of the equation (38) with respect to θ yields

1 ∂t g˜(W, z, t) = −∂W ((rW +y−c(W, z, t))˜ g (W, z, t)−ˆ c(W, z, t)g(W, z, t))−∂z (θ(¯ z −z)˜ g (W, z, t))+ σ 2 ∂zz g˜(W, z, t) 2 (39) where we denote d g˜(W, z, t) = gc+θˆc (W, z, t) dθ θ=0 Expanding the LHS of (37) using (35) ˆ ∞ˆ ˆ d −ρt d e = dW dzdt J[c + θˆ c, gc+θˆc ] u(c(W, z, t) + θˆ c(W, z, t), Z[c + θˆ c, gc+θˆc ])gc+θˆc (W, z, t) dθ dθ 0 Ω2 Ω1 θ=0 θ=0 (40)

=

´∞´ 0

´ Ω2

Ω1

e−ρt

 ∂c u(c(W, z, t), Z[c, g])ˆ c(W, z, t) + ∂Z u(c(W, z, t), Z[c, g]) +u(c(W, z, t), Z[c, g])˜ g (W, z, t)} dW dzdt

d dθ Z[c

 + θˆ c, gc+θˆc ] θ=0 g(W, z, t) (41)

Using the formula ˆ ˆ d δZ[c, g] δZ[c, g] Z[c + θˆ c, gc+θˆc ] = (ξ)ˆ c(ξ)dξ + (ξ)˜ g (ξ)dξ dθ δc δg Ω Ω θ=0 (41) becomes

55

(42)

=

´∞´

´

nh ´ ´ δZ[c,g] −ρt e ∂ u(c(W, z, t), Z[c, g])ˆ c (W, z, t) + ∂ u(c(W, z, t), Z[c, g]) c(x, y, t)dxdy c Z δc o(x, y, t)ˆ 0 Ω1 Ω2 Ω2 Ω1 i ´ ´ δZ[c,g] + Ω2 Ω1 δg (x, y, t)˜ g (x, y, t)dxdy g(W, z, t) + u(c(W, t), Z[c, g])˜ g (W, z, t) dW dzdt (43) Introducing the value function j(W, z, t) that satisfies

z − z)∂z j(W, z, t) −∂t j(W, z, t) + ρj(W, z, t) − 21 σ 2 ∂zz j(W, z, t) − (rW + y − c(W, z, t))∂W j(W, z, t) − θ(¯ ´ ´ δZ[c,g] = u(c(W, z, t), Z[c, g]) + Ω2 Ω1 ∂Z u(c(x, y, t), Z[c, g]) δg (W, z, t)g(x, y, t)dxdy (44) Using the value function j(W, z, t) in (44), (43) becomes ´∞´

´

e−ρt [∂c u(c(W, z, t), Z[c, g])ˆ c(W,iz, t) ´Ω1 ´ δZ[c,g] +∂z u(c(W, z, t), Z[c, g]) Ω2 Ω1 δc (x, y, t)ˆ c(x, y, t)dxdy g(W, z, t)dW dzdt ´∞´ ´ −ρt + 0 Ω2 Ω1 e (−∂t j(W, z, t) + ρj(W, z, t) − (rW + y − c(W, z, t))∂W j(W, z, t)
−θ(¯ z − z)∂z j(W, z, t) − 21 σ 2 ∂zz j(W, z, t) g˜(W, z, t)dW dzdt =

0

Ω2

(45)

Before proceeding the derivation, we have the following Lemma

Lemma. The following equation holds ´∞´ 0

´ Ω2

e−ρt (−∂t j(W, z, t) + ρj(W, z, t) − (rW + y − c(W, z, t))∂W j(W, z, t)
−θ(¯ z − z)∂z j(W, z, t) − 21 σ 2 ∂zz j(W, z, t) g˜(W, z, t)dW dzdt ´∞´ ´ = 0 Ω2 Ω1 e−ρt j(W, z, t)∂W (ˆ c(W, z, t)g(W, z, t))dW dzdt

Ω1

Proof. Denote the operator 1 Aj = (rW + y − c(W, z, t))∂W j(W, z, t) + θ(¯ z − z)∂z j(W, z, t) + σ 2 ∂zz j(W, z, t) 2

(46)

And the (adjoint) operator 1 A∗ g = −∂W ((rW + y − c∗ (W, z, t))g(W, z, t)) − ∂z (θ(¯ z − z)g(W, z, t)) + σ 2 ∂zz g(W, z, t) 2

(47)

Applying integration by part, it is easy to show that ˆ

ˆ

ˆ

ˆ (A∗ g˜)j(W, z, t)dW dz

(Aj)˜ g (W, z, t)dW dz = Ω2

Ω1

Ω2

(48)

Ω1

Denote the last two terms of the RHS of equation (45) by Γ, we have ˆ

∞

ˆ

ˆ

ˆ

∞

ˆ

ˆ

e−ρt (∂t j(W, z, t) − ρj(W, z, t))˜ g (W, z, t)dW dzdt −

Γ=− 0

Ω2

Ω1

e−ρt (Aj)˜ g (W, z, t)dW dzdt 0

Using (48),

56

Ω2

Ω1

ˆ

∞

ˆ

ˆ

ˆ

∞

ˆ

ˆ

∂t (e−ρt j(W, z, t))˜ g (W, z, t)dW dzdt −

Γ=− 0

Ω2

Ω1

e−ρt (A∗ g˜)j(W, z, t)dW dzdt 0

Ω2

Ω1

Applying integration by part to the first term above, we have Γ=

´ Ω2

´ Ω1

´∞´ ´ j(W, z, 0)˜ g (W, z, 0)dW dz + 0 Ω2 Ω1 e−ρt j(W, z, t)∂t g˜(W, z, t)dW dzdt ´∞´ ´ − 0 Ω2 Ω1 e−ρt (A∗ g˜)j(W, z, t)dW dzdt

(49)

The first term above can be ignored (See Nuno and Moll (2015)). Rewrite the equation (39) as ∂t g˜(W, z, t) = A∗ g˜ + ∂W (ˆ c(W, z, t)g(W, z, t))

(50)

Combining equation (49) and (50), the result follows.

Using the Lemma above, (45) can be written as ´∞´

´

e−ρt [∂c u(c(W, z, t), Z[c, g])ˆ c(W,iz, t) ´Ω1 ´ δZ[c,g] +∂z u(c(W, z, t), Z[c, g]) Ω2 Ω1 δc (x, y, t)ˆ c(x, y, t)dxdy g(W, z, t)dW dzdt ´∞´ ´ + 0 Ω2 Ω1 e−ρt j(x, y, t)∂x (ˆ c(x, y, t)g(x, y, t))dxdydt =

0

Ω2

´∞´

´

e−ρt [∂c u(c(W, z, t), Z[c, g])ˆ c(W,iz, t) ´Ω1 ´ δZ[c,g] +∂z u(c(W, z, t), Z[c, g]) Ω2 Ω1 δc (x, y, t)ˆ c(x, y, t)dxdy g(W, z, t)dW dzdt ´∞´ ´ −ρt c(x, y, t)g(x, y, t)dxdydt − 0 Ω2 Ω1 e ∂x j(x, y, t)ˆ =

0

(51)

Ω2

(52)

(52) equals to zero for any cˆ(W, z, t), we can set cˆ(W, z, t) = δ(W,z) where δ(W,z) is a Dirac delta function equals to zero except the point (W, z). We have ˆ

ˆ

∂z u(c(x, y, t), Z[c, g])

∂c u(c(W, z, t), Z[c, g]) + Ω2

Ω1

δZ[c, g] (W, z, t)g(x, y, t)dxdy − ∂W j(W, z, t) = 0 (53) δc

Therefore, the optimal consumption plans for the social planner is determined by (53) and its value function j(W, z, t) can be solved by the (44). This completes the proof.

The approach of Nuno and Moll (2015) Here we use the Lagrangian approach in Nuno and Moll (2015) to show that the result is consistent to those in the approach of Bensoussan et. al. (2013). First, we set up the Lagrangian function

+

L[c, g] = ´∞´ ´ 0

Ω2

´∞´ 0

Ω1

e

´

e−ρt u(c(W, z, t), Z[c, g])g(W, z, t)dW dzdt j(W, z, t) (−∂t g(W, z, t) + A∗ g(W, z, t)) dW dzdt

Ω2 −ρt

Ω1

(54)

where j(W, z, t) is a Lagrangian multiplier which is also the value function of the social planner as shown by Nuno and Moll (2015). The operator A∗ is defined in (47). (54) can be re-written as (check Nuno

57

and Moll (2015) for the derivation): ´∞´ ´ L[c, g] = 0 Ω2 Ω1 e−ρt u(c(W, z, t), Z[c, g])g(W, z, t)dW dzdt ´∞´ ´ + 0 Ω2 Ω1 e−ρt (∂t j(W, z, t) − ρj(W, z, t) + Aj(W, z, t)) g(W, z, t)dW dzdt

(55)

where the operator A is defined in (46). Diverging the function g(.) to g(.) + θˆ g (.) and differentiate (55) with respect to θ yields ´∞´ ´ e−ρt u(c(W, z, t), Z[c, g])ˆ g (W, z, t)dW dzdt Ω2 Ω1 ´ ∞ ´ ´ 0 −ρt d = + 0 Ω2 Ω1 e L[c, g + θˆ g ]) (∂t j(W, z, t) − ρj(W, z, t) + Aj(W, z, t)) gˆ(W, z, t)dW dzdt dθ ´∞´ ´ θ=0 d −ρt + 0 Ω2 Ω1 e ∂Z u(c(W, z, t)Z[c, g]) dθ Z[c, g + θˆ g ]) θ=0 g(W, z, t)dW dzdt

(56)

At optimum, (56) should equal to zero for any gˆ(.), we have

u(c(W, z, t), Z[c, g]) + ∂t j(W, z, t) − ρj(W, z, t) + 21 σ 2 ∂zz j(W, z, t) + (rW + y − c(W, z, t))∂W j(W, z, t) ´ ´ δ +θ(¯ z − z)∂z j(W, z, t) + Ω2 Ω1 ∂Z u(c(x, y, t)Z[c, g]) δg Z[c, g](W, z, t)g(x, y, t)dxdy = 0 (57) The next step is to maximize the Lagrangian function with respect to c. Similarly, we diverge the function c(.) to c(.) + θˆ c(.): ´∞´ ´ L[c + θˆ c, g] = e−ρt u(c(W, z, t) + θˆ c(W, z, t), Z[c + θˆ c, g])g(W, z, t)dW dzdt ´ ∞ ´ ´ 0 −ρtΩ2 Ω1 + 0 Ω2 Ω1 e (∂t j(W, z, t) − ρj(W, z, t) + Ac+θˆc j(W, z, t)) g(W, z, t)dW dzdt

(58)

and differentiate it with respect to θ: ´∞´

d = L[c + θˆ c, g] dθ θ=0

´

e−ρt ∂c u(c(W, z, t), Z[c, g])ˆ c(W, z, t)g(W, z, t)dW dzdt Ω1 ´ ∞´ −ρt − e cˆ(W, z, t)∂W j(W, z, t)g(W, z, t)dW dzdt ´ ∞ ´ ´ 0 −ρtΩ2 Ω1 d + 0 Ω2 Ω1 e ∂z u(c(W, z, t), Z[c, g]) dθ Z[c + θˆ c, g] θ=0 g(W, z, t)dW dzdt 0

Ω´2

(59)

Since it equals to zero for any cˆ(W, z, t), we have ˆ

ˆ

∂c u(c(W, z, t), Z[c, g]) − ∂W j(W, z, t) +

∂z u(c(x, y, t), Z[c, g]) Ω2

Ω1

δZ[c, g] (W, z, t)dxdy = 0 δc

(60)

Equations (57) and (60) respectively corresponds to equations (44) and (53).

Theorem 7 δ Notice that if Z[c∗ , g] is linear in g, the functional derivative δg Z[c∗ , g](W, z, t) is independent of g and only   ∂ δ ∗ depends on c∗ . By envelope theroem, we have ∂W δg Z[c , g](W, z, t) = 0.

58

Since the last term on the LHS of (24) equals to zero, if we substitute the social optimal consumption plan c∗sp (W, z, t) into the equation (5), the social optimal and the decentralized value function coincides: V (W, z, t) = j(W, z, t). When ∂Z u(c(x, y, t), Z[m, c]) < 0 for any (x, y), we have ´ ´ ∂ u(c(x, y, t), Z[m, c])g(x, y, t)dxdy < 0. In addition, if δZ[c,g] δc (W, z, t) > 0, Ω2 Ω1 Z ´ ´ δZ[c,g] (W, z, t) ∂ u(c(x, y, t), Z[m, c])g(x, y, t)dxdy < 0. If we substitute c∗sp (W, z, t) into (6), we δc Ω2 Ω1 Z get ∂c u(c∗sp (W, z, t)) > ∂W V (W, z, t) Hence the households with (W, z) at time t have incentive to rise their consumption and the decentralized consumption level will be greater than the social optimal consumption level, that is, c(W, z, t) > ´ ´ c∗sp (W, z, t). On the contrary, if δZ[c,g] δc (W, z, t) Ω2 Ω1 ∂Z u(c(x, y, t), Z[m, c])g(x, y, t)dxdy > 0, c(W, z, t) < c∗sp (W, z, t)

59

9.2 9.2.1

Appendix B: Algorithm and Other Model Specifications Algorithm

Given the initial wealth density g(W, z, 0) and terminal value function V ∗ (W, z), the timevarying equilibria of an economy can be computed by finding {V (W, z, t), g(W, z, t)} using ´ ´ (7), (8) with the constraint Ω2 Ω1 g(W, z, t)dW dz = 1 for all t. The optimal consumption and its density m(c, t) can be computed by (6) and the transformation of variables. The terminal value function V ∗ (W, z) is the value function evaluated at the stationary equilibrium. We first compute the stationary distributions {g ∗ , m∗ } and value functions V ∗ by the following procedure: 1. Guess a number Z0 , use it to substitute out Z[m] in (12). Then, compute the HJB equation by the Finite Difference Method.39 2. Use the solution Vˆ obtained in Step 1 to compute the optimal consumption c∗ by (15). Then, use c∗ to compute the Kolmogorov Forward equation in (13) by the Finite Difference Method. 3. Use the solution c∗ in Step 2 and gˆ in Step 2 to compute the estimated m(c). ˆ ˆ m]. 4. For each m(c) ˆ in Step 3, compute the estimated number Z[ ˆ



5. Check if Zˆ − Z0 < e for some norm and tolerance level e. If not, replace Z0 by Zˆ and go to Step 1.

The transitional dynamics can then be computed similarly as follows: 1. Compute the new and old stationary equilibria using the above procedure. Decide a terminal period T where T is large. Denote the value function in the new equilibrium as V ∗ (W, z) and the density at the old equilibrium as g(W, z, 0). Decide scales Z00 and Z0T to be the equilibrium F in old and new steady states. −1 2. Guess a sequence of number {Z0t }Tt=1 , use {Z0t }Tt=0 to compute HJB equation in (7) by the Finite Difference Method to obtain Vˆ with the terminal value function V ∗ (W, z). Use Vˆ to compute c∗ (W, z, t) by (6).

3. Use the solution c∗ (W, z, t) found in Step 2 with the initial density g(W, z, 0) to solve the Kolmogorov Forward equation in (8) by the Finite Difference Method. 39

The exact implementation of the Finite Difference Method can be found in Achdou et. at. (2015).

60

4. Use the solution c∗ (W, z, t) in Step 2 and gˆ(W, z, t) in Step 3 to compute the estimated sequence of m(c, ˆ t). ˆ m]. 5. For each m(c, ˆ t), compute the estimated number Z[ ˆ



6. Check if maxt { Zˆt − Z0t } < e for some norm k·k and tolerance level e > 0. If not, replace the sequence {Z0t }Tt=0 by {Zˆt }Tt=0 and go to Step 2.

9.2.2

Additiive Comparison Model Specification

We consider an alternative specification of utility u(c, Z) = (c − ηZ[m]γ )φ /φ with φ > 1. This incorporates the commonly used case where u = ln(c−η¯ c) by setting Z[m] = cm(c)dc , γ = 1 and φ → 1. The HJB equations is 1 ρV (W, z, t) = max u(c, Z[m])+∂W V (W, z, t)(rW +wz−c)+∂z V (W, z, t)θ(¯ z −z)+ σ 2 ∂zz V (W, z, t)+∂t V (W, z, t) c 2

Therefore, the HJB can be written as ρV (W, z, t) =

1−φ φ

φ

(∂W V (W, z, t)) φ−1 + ∂W V (W, z, t)(rW + wz − ηZ[m]γ )

+∂z V (W, z, t)θ(¯ z − z) + 21 σ 2 ∂zz V (W, z, t) + ∂t V (W, z, t)

(61)

The optimal consumption satisfies 1

c∗ (W, z, t) = (∂W V (W, z, t)) φ−1 + ηZ[m]γ

(62)

Since the second term in (62) is the same across agents, this implies that the individual consumption differences in each period is caused by the first term in (62) only. The Kolmogorov Forward equation is the same as (8) and re-stated as follows:

1 ∂t g(W, z, t) = −∂W ((rW + wz − c∗ )g(W, z, t)) − ∂z (θ(¯ z − z)g(W, z, t)) + σ 2 ∂zz (g(W, z, t)) 2 (63) The density function of consumption can be found by using (62) and with the method of transformation of variables. And the equilibrium can be computed by using (61) - (63). Similarly, the stationary equilibrium is when ∂t V (W, z, t) = 0 and ∂t g(W, z, t) = 0. This implies the HJB and Kolmogorov Forward equation are 61

ρV (W, z) =

φ 1−φ 1 (∂W V (W, z)) φ−1 ∂W V (W, z)(rW +wz−ηZ[m]γ )+∂z V (W, z)θ(¯ z −z)+ σ 2 ∂zz V (W, z) φ 2 (64)

1 0 = −∂W ((rW + wz − c∗ )g(W, z)) − ∂z (θ(¯ z − z)g(W, z)) + σ 2 ∂zz g(W, z) (65) 2 With g(W, z) and V (W, z) solved in (64) and (65), the stationary optimal rule for consumption c∗ (W, z) and the resulting density m(c∗ ) are determined by (62) and the transformation of variables respectively. 9.2.3

General Equilibrium Framework

The general equilibrium setting and the computation method basically follow Achdou et. al. (2015). Households maximize the discounted lifetime utility (1) subject to (2), (3) and (4). The only difference is that the wage and interest rate are endogenous. Denote K(t) as the aggregate capital at time t and F (.) is the production function. Assume that both labor and capital market are competitive, we have w(t) = ∂L F (K(t), 1) and r(t) = ∂K F (K(t), 1) − δ where δ is a depreciation rate. The HJB and Kolmogorov Forward equa´ ´ tion are the same as (7) and (8). The aggregated capital is K(t) = Ω2 Ω1 W g(W, z, t)dW dz for every time t. The optimal consumption c∗ (W, z, t) is still determined by (6) and the consumption density m(c, t) is obtained from the transformation of variables. The computation procedure is similar to those in the main text except that for each time t, we first compute w(t) and r(t) using the initial K0 (t). Then, solve out {Vˆ (W, z, t), gˆ(W, z, t), m(c, ˆ t)} given ´´ K0 (t). Finally, guess the new Knew (t) = θK0 (t) + (1 − θ) W g(W, z, t)dW dz. Repeat the procedure until Knew (t) is close to K0 (t).

62

9.3

Appendix C: More Simulation Results

This Appendix, as a supplement to Section 5, reports several interesting results found in the numerical exercise. It is found that households with negative externality save more and thus results in a more dispersed wealth density in the equilibrium. The difference between the benchmark and the externality equilibrium is greatly affected by the persistence of the endowment shock. In the policy analysis, we find that both (proportional) capital and labor income tax reduce the average consumption level and consumption dispersion in the stationary equilibrium. And the extents of reduction are greatly influenced by the types of externalities assumed. Furthermore, both the proportional capital income tax and progressive labor income tax speed up the convergence rate of consumption growth when there is a positive permanent TFP shock. 9.3.1

Stationary Equilibrium

Figure 11 shows the density of wealth conditional on different labor endowments: g(W |z) = g(W, z)/f (z) where g(W, z) and f (z) are joint density of (W, z) and the marginal density of z respectively. We show the cases where z = 0.5, 1, 1.5. For the households experiencing low labor endowment shocks, a higher portion of them is accumulated around the borrowing constraint and the average wealth is thus lower. The order of the average wealth for different externalities is the same as those of the average consumption. It is noticed that the wealth dispersion varies among different types of externalities. We find that negative externality could increase the wealth inequality and the jealousy preference results in the largest effect. Under the jealousy preference, almost no household is willing to stay around the borrowing constraint even if their endowment shocks realization z = 0.5. Figure 12 shows the saving against wealth under different labor endowments. Under all of the cases, households’ saving is decreasing with the wealth level. It is because under our assumption that r < ρ, there is no incentive to defer the consumption. The only saving motive is to prepare for the low endowment shocks and would obviously decrease with the level of wealth. Households dissave when z = 0.5 and increase their savings along the level of endowment shock. This reveals their expectations on the future income and their intension to save less when the endowment shock is below its expected level. The saving functions are also affected by different types of externalities. Generally speaking, negative externality could increase saving at any level of wealth and endowment shock. The order of saving functions is consistent with the average level of consumption according to the consumption densities shown in Figure 2: a higher saving rate results 63

in a higher average consumption level in the steady state. It is due to two facts: (i) a higher saving rate increases the wealth accumulated and (ii) the optimal consumption is an increasing function of wealth. The gap between the saving functions is closed when endowment level is high, revealing that the effects of externalities are more prominent for those who are having a low income realization. Since the autocorrelation function of the Ornstein–Uhlenbeck process is Corr[zt , zt+s ] = exp(−θs) (see Moll (2014) and Dixit and Pindyck (1994)), the parameter θ is inversely related to the persistence of the endowment process. It is believed that high persistence of endowment shocks widen the wealth and consumption inequality which would, in turn, affect households’ utility in the presence of inequality externality. Figure 13 shows the consumption densities of mean and Gini coefficient externality with different endowment persistencies. As expected, model with “inequality averse” households results in a more concentrated consumption density, this is similar to the case when Z[m] equals to the standard deviation. As the persistence of endowment increases from exp(−0.9s) to exp(−0.1s), both wealth and consumption densities become more dispersed. This is intuitive as high-income households have a higher probability of maintaining their high income in the future. But indeed the consumption dispersion under inequality externality increases more than those of the mean externality, this is illustrated by Figure 14 which plots the Gini coefficient of consumption densities under mean and inequality externalities against different endowment persistencies. It is clear that the negative inequality externality would reduce the Gini coefficient in the steady state. However, as the persistence increases, its Gini coefficients face a larger increment and start catching up the case of mean externality, revealing that externalities are less effective on reducing the consumption dispersion when the consumption dispersion is high. 9.3.2

Transitional Dynamics

Figure 15-17 show how consumption and wealth densities under different Z[m] evolve over time. They basically follow the discussion in section 4.2 that the consumption densities converge faster to the new steady state under the model with externalities. As we can see in Figure 15, a higher percentile of consumption spreads greatly to the right, implying that a group of households is now able to sustain a higher amount of consumption. Comparing Figure 16 and 17, wealth densities conditional on z = 0.5 faces a more dramatic change. Their initial densities are more concentrated around the borrowing constraint, but now a large group of households is out of the borrowing constraint in the 64

presence of permanent TFP shock. 9.3.3

Policy Implication

Proportional Capital Income and Proportional Labor Income Tax To understand how different taxation policies affect the households’ behaviours, we introduce the proportional capital income tax τC and the proportional labor income tax τI into the households’ constraint (2), it then becomes dW = ((1 − τC )rW + wz − c)dt with the capital income tax, and dW = (rW + (1 − τI )wz − c)dt with the labor income tax. It is obvious that increasing a capital income tax rate is equivalent to lowering an interest rate in our setting. Under the assumption in Theorem 4 that w = 0, we can utilize the closed form solution in Theorem 4. Theorem 5 implies that a higher capital income tax rate would reduce the consumption and wealth inequality. This is consistent with the result in Achdou et. al. (2015). It is more interesting to see how the two policies could have different impacts without this assumption. Here, we simulate the model with taxes in the steady states and transitional dynamics. We set τC = 0.01 to indicate the 1% of households’ capital income is taxed in each period. And we set τI = 0.05 to represent there is a 5% labor income tax in each period. Compared to the stationary consumption densities in Figure 5, Panel A and B in Figure 18 show that the capital and labor income taxes have similar impacts on consumption densities, namely: they both lower the average consumption and consumption inequality. Charging a capital tax have the same impact as reducing the interest rate, this would discourage households to save. The drop in saving rate would then reduce the portion of the wealthy households. Similarly, a higher income tax is equivalent to a reduction of wage rate, which would reduce the difference of the productivity realization z between households and hence narrow down the dispersion of wealth and consumption distributions. Table 4 shows the expected value and standard deviation of consumption under the two tax policies and the model without tax. Comparing the drops in mean and standard 65

deviation of consumption, it is clear that the reduction in the standard deviation is much greater under the capital income tax than those under the labor income tax. This is because the households’ capital income dispersion is much greater than the labor income dispersion. Hence, capital income tax is more effective in reducing consumption inequality than labor income tax. Table 4 also reveals that the impact of tax policies could be very different under different types of externalities. Compared to the benchmark case, introducing a capital income tax would narrow down the consumption inequality more under the case with negative externality. However, the average consumption reduces more at the same time. In addition, it is not surprising that mean externality results in the greatest percentage change in average consumption, while the standard deviation externality lead to the greatest reduction in consumption inequality. The impact of labor income tax is opposite, mean externality leads to the least percentage change in average consumption. This reveals the importance of considering externality in the policy analysis since the effectiveness of tax policies on the wealth and consumption distribution is greatly influenced by the types of externalities assumed. Transitional Dynamics with Capital Income and Labor Income Tax We revisit the transitional dynamics exercise using RC utility function with the presence of tax. All parameters follow those in Section 5.2. We only compare the model with mean externality and the one without externality with 1% capital income tax and 5% labor income tax. Figure 19 shows the transitional dynamics when there is a 10 % increase in total factor productivity. Panel A and B in Figure 19 reveals that the average consumption and wealth are consistently higher in the presence of negative externality, no matters whether there is a tax or not. The intuition is explained in Section 5.2. Notice that introducing either of the taxes would reduce the consumption and wealth levels. Yet, as shown in Panel C, the Gini coefficient for consumption inequality under the mean externality with labor income tax is now higher than those without tax. This is different from the steady state result in Table 4 that labor income tax narrows down the consumption inequality. Hence with mean externality, although the drop in average wealth and consumption under the labor income tax is much less than those of the capital income tax, the rise in consumption inequality creates an extra adverse impact on introducing a labor income tax. Another advantage of introducing capital income tax is that it could speed up the transitional dynamics of technology growth. As shown in Theorem 5, the steady state is

66

not affected by the presence of externality in the RC utility function. It is noticing that in all the four Panels, both of the lines with capital tax converge much closer at time = 1000 and much flatter than the other two cases, implying a faster rate of convergent to the new steady state. When there is a technology improvement, it takes time for the high-income households to accumulate wealth in order to smooth their consumptions. Higher capital income tax discourages this wealth accumulation and so shorten the time of convergence. Panel D shows the aggregate utility defined by (22) for each period. Overall, the welfare impacts of 1% capital income tax and 5% labor income tax are similar, with the utility in labor income tax slightly higher both in the cases of benchmark and mean externality. The welfare difference between the benchmark and mean externality are enlarged in the presence of both taxes. In summary, both taxes enlarge the welfare difference between externalities, while capital income tax speeds up the transitional dynamics of TFP growth. How does the progressive labor income tax affect the rate of convergence? It has been shown in the section 5.4 that introducing progressive labor income tax could reduce both the average consumption and consumption inequality. It is more interesting to know whether (and how) the progressive tax could affect the transitional dynamics when there is a productivity growth. Same as the above, we assume there is a 10% total factor productivity growth from z = 1 to z = 1.1. And we compute transition paths of different variables under the presence of progressive tax at various rate τP from 0 to 0.5. To see how the speed of convergence is affected by the progressive tax rate, we calculate the halflife tHL for each transition paths. For any variable xt that transits from the old steady state x0 to the new steady state x∞ , the half-life, denoted as tHL , is a time index that satisfies 1 xtHL − x0 = x∞ − x0 2

(66)

We pick the variables xt to be the average consumption, the standard deviation of consumption and the aggregate utility defined by (22). The results are presented in Figure 20. The half-life of average consumption is way shorter than those of the standard deviation of consumption, revealing the process of economic growth of a permanent increase in technology: the consumption level growth is on the earlier state while the inequality rise is in the later state of the process. It is noting that the half-lives for both the average and the standard deviation of consumption are shorter under the mean externality than the benchmark. As we mentioned, this is because households expect a higher 67

mean consumption in the new steady state, which would create a lower utility level and the marginal utility of consumption. Hence, they will put forward their increase in consumption. It is clear that the higher the progressive tax, the faster the rate of convergence. Technology growth disproportionally benefits those with high income realization due to the persistence of the income process. High-income households save more and hence lead to a more wealth dispersion in the new steady state. A higher progressive tax discourages this saving incentive of the wealthy households and hence shorten the time of convergence. As we can see in Panel A that the impact is huge, a 20% of progressive tax could shorten the half-life from 46 periods when τP = 0 to 36 periods when τP = 0.2 under the benchmark case. Also, the gap between the half-lives of the two cases converges as τP increases. Hence, not only can the progressive tax speed up the rate of convergence, it minimizes the impact of externality on the half-life.

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Without Tax

1 % Capital Income Tax

5% Labor Income Tax

Benchmark Mean Std. Dev. Skewness Benchmark Mean Std. Dev. Skewness Benchmark Mean Std. Dev. Skewness

E(c)

% change in E(c)

0.71 0.89 0.74 0.83 0.70 0.86 0.72 0.80 0.67 0.86 0.70 0.79

-1.6% -4.1% -2.2% -3.0% -5.1% -2.9% -4.8% -3.7%

p

V ar(c)

0.094 0.14 0.11 0.13 0.087 0.13 0.097 0.12 0.090 0.136 0.103 0.130

% change in

p

V ar(c)

-6.5% -7.7% -8.5% -8.4% -3.6% -3.5% -2.9% -2.5%

Table 4: The change in mean and standard deviation of consumption under (proportional) capital and labor income tax policies. The four models are compared under a 1% capital income tax, 5% labor income tax and without tax. Column 3 and 5 respectively show the expected and the standard deviation of the stationary consumption level. Column 4 and 6 show their percentage changes compared to the case that without tax.

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0.45

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Figure 11: Wealth densities in the stationary equilibrium conditional on different labor endowments. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

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Figure 12: Saving functions against wealth level W given different labor endowments. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 , skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

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7 Mean Gini 6

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(b) Corr(zt , zt+s ) = exp(−0.1s)

Figure 13: Consumption densities in the stationary equilibrium. The labor endowment z is assumed to follows Ornstein–Uhlenbeck process dz = θ(¯ z −z)dt+σdB with different persistencies θ = 0.1, 0.9. The externality indices Z[m] are mean Z[m] = E(c) and Gini Coefficient Z[m] = ´∞ 0 M (c)(1 − M (c))dc/E(c) where M (c) is the cdf of c respectively.

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0.09 0.085 0.08 0.075

Gini

0.07 0.065 0.06 0.055 0.05 mean Gini

0.045 0.04

0.45

0.5

0.55

0.6

0.65

0.7

Corr Figure 14: Gini Coefficient against the labor endowment shock persistence θ in the process dz = θ(¯ z − z)dt + σdB

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Figure 15: The transitional dynamics of consumption densities over time time under an 10% increase in total factor productivity. The externality indices Z[m] are mean Z[m] = E(c) and Gini ´∞ coefficient Z[m] = 0 M (c)(1−M (c))dc/E(c) where M (c) is the cdf of c and benchmark Z[m] = 1 respectively.

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0.3 0.25 0.2 0.15 0.1 0.05 0 30 1000

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800 600

10

400

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200 0

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(c) Z[m]: Gini coefficient

Figure 16: The transitional dynamics of wealth densities conditional on z = 0.5 over time under an 10% increase in total factor productivity. The externality indices Z[m] are mean Z[m] = E(c) ´∞ and Gini coefficient Z[m] = 0 M (c)(1 − M (c))dc/E(c) where M (c) is the cdf of c and benchmark Z[m] = 1 respectively.

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0.1 0.08 0.06 0.04 0.02 0 30 1000

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800 600

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200 0

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(c) Z[m]: Gini coefficient

Figure 17: The transitional dynamics of wealth densities conditional on z = 1.5 over time under an 10% increase in total factor productivity. The externality indices Z[m] are mean Z[m] = E(c) ´∞ and Gini coefficient Z[m] = 0 M (c)(1 − M (c))dc/E(c) where M (c) is the cdf of c and benchmark Z[m] = 1 respectively.

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(b) Stationary consumption densities with

1% capital income tax

5% labor income tax

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(c) Stationary wealth densities conditional

(d) Stationary wealth densities conditional

on z = 1 with 1% capital income tax

on z = 1 with 5% labor income tax

Figure 18: Stationary consumption and wealth distributions with income and capital taxes. The externality indices Z[m] are mean Z[m] = E(c), standard deviation Z[m] = [E(c − E(c))2 ]1/2 and skewness Z[m] = E(c − E(c))3 /[E(c − E(c))2 ]3/2 and benchmark Z[m] = 1 respectively.

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Average consumption

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(b) Average wealth paths under the models

models with labor income and capital income taxes and without tax.

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Welfare

Gini Coefficient

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(c) Stationary wealth densities with 1% capi-

(d) Aggregate Utility paths under the models

tal income tax (z = 1)

with labor income and capital income taxes and without tax.

Figure 19: Transitional dynamics of a 10% TFP growth with 5% labor income and 1% capital income taxes. The externality indices Z[m] are mean Z[m] = E(c) and benchmark Z[m] = 1 respectively.

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50 Benchmark Mean

Half Life of Average consumption

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progressive tax rate τP

Half Life of standard deviation consumption

340 Benchmark Mean

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(b) Half-life of the path of standard deviation of Consumption

against the progressive tax rate τP

Figure 20: Half-lives of the transition paths of the mean and standard deviation of consumption when there is a 10% TFP growth against the progressive tax rate

79