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Level of development and income inequality in the United States: Kuznets hypothesis revisited once again
Accepted Manuscript Level of development and income inequality in the United States: Kuznets hypothesis revisited once again German Blanco, Rati Ram PII:
S0264-9993(18)30700-4
DOI:
https://doi.org/10.1016/j.econmod.2018.11.024
Reference:
ECMODE 4780
To appear in:
Economic Modelling
Received Date: 16 May 2018 Revised Date:
28 September 2018
Accepted Date: 27 November 2018
Please cite this article as: Blanco, G., Ram, R., Level of development and income inequality in the United States: Kuznets hypothesis revisited once again, Economic Modelling (2019), doi: https:// doi.org/10.1016/j.econmod.2018.11.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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German Blanco & Rati Ram∗
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Abstract
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Level of development and income inequality in the United States: Kuznets hypothesis revisited once again
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This study revisits the well-known Kuznets hypothesis that postulated an invertedU relation between the level of development of a country or region and its degree of income inequality. We address the widely-shared view that, in recent years, a regular-U, and not a Kuznets-type inverted-U, has characterized the relation between income and inequality in the United States and some other developed countries. The paper works with data for the US states and has two main distinguishing features. First, it permits cross-state spillovers, which seem conceptually relevant and statistically significant, but, to our knowledge, have not been included in the fairly extensive literature on the topic. Second, we use the most reliable recent annual panel covering the period 2006-2016, which has also not been included in any existing research. The pooled-OLS, as well as the two-way fixed-effects panel estimates, indicate the existence of a significant regular-U pattern which is consistent with the influential view in the literature. However, the estimates that adjust for cross-state spillovers indicate a weak regular-U pattern in which the (log) linear and quadratic terms lack statistical significance at the usual levels. We conclude that when cross-state spillovers are appropriately accounted for, statistically significant evidence is lacking to support the view that a regular-U pattern describes the relation between income and inequality in the United States in recent years.
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Keywords: Kuznets-hypothesis; US states; cross-state spillovers
∗ Email:
[email protected] (G. Blanco, corresponding author), and [email protected] (R. Ram).
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1. Introduction
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Kuznets (1955) famously hypothesized that, during the process of economic
growth, as the level of development (per capita income) of a country or region rises, income inequality first increases, reaches a peak, and declines afterwards. This proposition, commonly called the Kuznets inverted-U hypothesis, was suggested by him on the basis of the empirical observation of that pattern in a few
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highly developed countries. He thus articulated a “long swing” in inequality which increases in the early phases of economic growth, becomes stabilized for a while
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and then narrows in the later phases. In his characteristic humility, he notes (p. 26) that the “paper is perhaps 5 per cent empirical information and 95 per cent speculation.”
Kuznets-hypothesis has attracted an immense amount of scholarship and there are perhaps hundreds of studies that deal with some aspects of the hypothesis, notably its empirical status. Besides doing an empirical assessment of the Kuznetsian postulate in numerous different settings, this literature has considered its
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conceptual foundations, provided alternative explanations for the phenomenon, extended or modified the hypothesis, or explored different estimation procedures in the empirical applications.1
More recently, a widely-shared view suggests that the Kuznets-hypothesis has suffered a breakdown during the last few decades relative to some highly devel-
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oped countries like the United States and that such contexts are characterized by a regular-U, and not an inverted-U, relation between income and inequality. Some
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of the recent influential debates on increasing income inequality seem to derive from a thought of that kind. Although the literature on even this limited issue is considerable, we mention just a few selected studies that support the theme of a regular-U for highly-developed economies like that of the United States and provide a setting for our research. In some early works, Ram (1991, 1997) indicated the ex1 Kim,
Huang and Lin (2011, pp. 250-251) provide a compact summary of many studies
relating to the Kuznets hypothesis.
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istence of a regular-U structure for US and some other highly-developed countries.
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Similarly, List and Gallet (1999) suggested a positive correlation between income inequality and development in advanced economies, and attributed that partly
to a “shift away from a manufacturing base towards a service base”. Tribble (1999) proposed an “S-curve” model that includes agriculture-to-manufacturing and manufacturing-to-services structural transitions during the process of eco-
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nomic development. In their narrative on the evolution of income inequality in
the United States, Piketty and Saez (2003, pp. 1-2) have suggested a U-curve instead of the inverted-U proposed by Kuznets (1955). They note that “Today,
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the Kuznets curve is widely held to have doubled back on itself, especially in the United States, with the period of falling inequality observed during the first half of the twentieth century being succeeded by a very sharp reversal of the trend since the 1970s.” From data on US states, Kim, Huang and Lin (2011) concluded that the relation between inequality and development is “better characterized by a U shape rather than the inverted-U profile asserted by Kuznets (1955).” Piketty (2014) has documented a regular-U profile of income inequality in the US and some
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other countries. More recently, Costantini and Paradiso (2018) stated that their empirical results suggest the existence of a S-shaped relation between income and inequality in the US states.
The main purpose of the present study is to use a panel of US states to explore
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whether the recent relation between income and inequality in the United States is characterized by a regular-U pattern. A panel of US states offers some advantages besides the comparability of income and inequality data over time and across
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states. For instance, given the wide range of income across the states, it can mimic the evolution of inequality over a much longer period than that covered by
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the sample.2 Our work is marked by two distinguishing features. First, it accounts
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for cross-state spillovers which appear conceptually relevant and, as the empirical component of this paper shows, statistically significant, but, to our knowledge,
have not been considered in any study of Kuznets’ hypothesis despite their fairly extensive usage in other economic contexts. The importance of taking account of
such spillovers is primarily based on the thought that while Kuznets’ hypothesis
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relates the value of inequality with income (level of development) for each unit
(state), cross-state spillovers in inequality “contaminate” the own-state variable with the part that spills over from adjacent states, and it is appropriate to purge
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the inequality variable of such contamination. Our other contribution is the use of highly reliable state-level data from the American Community Survey (ACS) and Bureau of Economic Analysis (BEA) covering the most recent period 2006-2016 which has not been included in any existing study.
After a short recapitulation of some related studies in Section 2, we briefly outline the model specification, estimation method, and the data in Section 3. The main results are presented in Section 4, and the last section contains some
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concluding thoughts.
2. A Short Recapitulation of Some Related Studies The preceding section summarized several related works to motivate the present
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study. Those studies are mentioned in slightly greater detail in this section. The pioneering research by Kuznets (1955) indicating the existence of an inverted-U
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relation between income inequality and level of development over long periods in a few industrialized countries is, of course, well known and has generated enormous research on the topic. His position was summarized by saying (p. 18) that 2 For
example, real GDP per capita in 2016 varied from $32,334 in Mississippi to $65,168 (2009
dollars) in Connecticut, which is a much greater variance than in real GDP per capita in the United States from $29,503 in 1975 to $54,559 in 2016 (in 2012 dollars). The income variation in our sample is even greater from a minimum of $31,175 to a maximum of $70,876.
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there is a long swing in income-inequality which widens in the early phases of
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economic growth, becomes stabilized for a while, and then narrows in the later phases. His explanation (Kuznets, 1955, pp. 6-17) for the phenomenon was that in early stages of economic development, inequality tends to increase due to a shift from more equal and lower-income agriculture (rural) sector to less equal
and higher-income industrial (urban) sector, and concentration of savings in the
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hands of higher-income groups accentuates that increase in inequality. At later stages of development, several factors, including “legislative interference” and political decisions, mitigate the effect of savings being concentrated in the hands of
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higher-income groups. The inequality-increasing effect of the shift from more equal agriculture to less equal non-agriculture is offset by a rise in the income share of lower groups in the non-agriculture sector. He (1955, p. 20) also noted other trends related to that in income inequality, including those in population growth, urbanization, internal migration, savings and investment rates, international trade, and government activity.
While Kuznets’ reasoning in support of the inverted-U hypothesis was largely
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verbal, Robinson (1976) provided a more rigorous and formal two-sector model of a developing economy that implied an inverted-U relation between income and inequality. He noted (1976, p. 439) that one implication of such a two-sector model is that, in the absence of explicit countervailing policies, a developing country will
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have increasing or unchanged inequality for a relatively long period. Among the more recent research that suggests the presence of a regular-U relation, instead of a Kuznets-type inverted-U, in highly developed countries during
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the last few decades, Ram (1991, 1997), List and Gallet (1999), Tribble (1999) and Kim, Huang and Lin (2011) have been mentioned in the preceding section, and the narrative by Piketty and Saez (2003) was noted. Piketty (2014) has also documented a regular-U pattern in the evolution of inequality in the United States and some other developed countries. While other scholars proposing a regular-U relation have used descriptive phrases like structural transformation from manufacturing to services, Piketty’s (2014) explanation is based on (a) rapid increases
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in the compensation of executives and (b) high returns to capital, which might
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be regarded as the contemporary version of concentration of savings in the highincome groups noted by Kuznets (1955). More recently, Costantini and Paradiso (2018), who worked with a panel of US states for the period 1960-2015, stated that “the empirical results seem to suggest the existence of a S-shaped relationship be-
tween inequality and per capita GDP.” However, their estimated rising segment
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of inequality (p. 117) seems to start when per capita GDP is at a low level of around $14,500. Although not directly related to the question about the existence
of an inverted-U in recent decades, the study by Doran and Jordan (2016) ana-
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lyzed changes in levels and composition of income inequality among US counties from 1969 to 2009. They found that income inequality has increased in the period studied, and between-state inequality has declined, but within-state inequality has increased.
While Kuznets’ postulate relates income inequality with the level of development (or per capita income) of a country or region, there has recently been considerable research on the relation between income inequality and economic growth.
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We mention a few selected studies in that strand of literature since some scholars seem to view that aspect as related to Kuznets hypothesis, and the substantial volume of research on that topic has led to two recent meta-analytic studies. The meta-analysis by de Dominicis, Florax and De Groot (2008) notes that although
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a considerable part of the literature found inequality detrimental to growth, more recent studies have claimed a positive effect of inequality on growth. They use meta-analysis to shed light on this puzzle and “systematically describe, identify
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and analyse the variation in outcomes of empirical studies.” They find “that estimation methods, data quality and sample coverage systematically affect the results.” The other meta-analysis by Neves, Afonso and Silva (2016) covers the studies published during 1994-2014 that estimate the effect on growth of inequality in income, land and human capital. After correcting for publication biases, they conclude that “the high degree of heterogeneity of the reported effect sizes is explained by study conditions, namely the structure of the data, the type of
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countries included in the sample, the inclusion of regional dummies, the concept
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of inequality and the definition of income.” Besides the meta-analytic research, we note just a few recent studies for the United States. Shin (2012) did a largely theo-
retical analysis of the relation between income inequality and economic growth and indicated that higher inequality may retard growth in early stages of economic de-
velopment, and can encourage growth near the steady state. Based on a dynamic
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spatial Durbin model, Atems (2013) estimated the effects on a county’s economic
growth of increased inequality in that county and in the neighboring counties. Rubin and Segal (2015) analyzed the relation between growth and income inequality
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in the US during 1953-2008 and concluded that “the income of the top income group is more sensitive to growth...compared to the income of the lower income groups.”
3. Model, Estimation Methodology, and Data Models and Estimation Methods
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We consider the following widely-used regression specification, which stacks observations of T successive cross-sections, each with N observations:3 IN EQ = α + β(lnY ) + γ(lnY )2 + ,
(1)
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where IN EQ is a N T × 1 vector of observations on states’ income inequality, α and lnY are conformable vectors containing ones and the natural log of income, respectively, and is assumed to be a N T × 1 vector of disturbances. In terms of
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the Kuznets hypothesis, we expect β to be positive and γ to be negative. After the pooled OLS estimation based on equation (1), our analysis starts with
the two-way fixed-effects model, which includes state-specific dummy variables to capture a limited amount of unobserved heterogeneity across the states and year3 In
our vector notation, we skip the usual i and t subscripts to refer to the cross-sectional
unit and the year of observation.
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dummies to account for the year-specific factors that may affect all states.4 The
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fixed-effects Kuznets quadratic may then be written as:
IN EQ = β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + ,
(2)
where 1T and 1N are vectors of ones of dimension T and N , respectively, δ is a N × 1 vector of “individual (state-specific) effects”, and η is a T × 1 vector of “time
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(year) effects”.
In addition to allowing for a limited amount of heterogeneity across the states and the years through the fixed-effects format of equation (2), we note that in-
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equality in the geographically neighboring states is likely to affect inequality in any given state through spatial “spillover effects”. Higher inequality in the adjacent states may be associated with higher inequality in a given state due to the “diffusion” of inequality across a state’s geographical boundaries which are necessarily porous. The effects of inequality-related policies or factors in the adjacent states are likely to spill over to a state, and a particular state’s measure of inequality
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might emulate that of its closest neighbors through imitation, social pressure, or other non-market externalities, and a measure of proximity plays a key role in determining the extent of influence exerted by other states. Inequality in a state may, therefore, be affected not merely by income in that state, as Kuznets hypothesis suggests, but also by inequality in the neighboring states. Such spillover effects
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have not been considered in the numerous empirical tests of Kuznets hypothesis, although these are widely recognized and have been considered important in several contexts in economics. A few examples of the study of such spillover effects
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include Atems (2013) who estimated the effect on a county’s economic growth of inequality in the county’s neighbors, Case (1991) who studied spatial relationships in Indonesian demand for rice, and Goel and Nelson (2007) who estimated the “contagious” effects of corruption and other crimes across US states. Baltagi 4 We
also consider the random-effects format, but the Hausman test described in Table 3 favors
the fixed-effects format.
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(2011, p. 436) noted the many areas in economics where spillover effects have been
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considered in panel data. To “purge” the inequality variable of such spillovers, and make the model closer to the Kuznets paradigm, we consider the following speci-
fication that conforms to the literature on spatial correlations and is expected to purge the spatial spillover effects that “contaminate” inequality in a state:
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IN EQ = ρ(IT ⊗ WN )IN EQ + β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + , (3)
where IT is an identity matrix of dimension T , W is a N ×N spatial weights matrix that formalizes the spatial structure of inequality of the N observations, and ρ is
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the spatial “autocorrelation” parameter, which is expected to be positive with the implication that higher inequality in a state has the spillover effect of tending to raise inequality in the neighboring states. The so-called spatial autoregressive model (Anselin, 1988; Anselin, Le Gallo and Jayet, 2008) specified in equation (3) models the spillovers from the proximate states through the first term on the right-hand side and makes the specification closer to the Kuznets paradigm. It is
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evident that (3) can be rewritten as:
IN EQ − ρ(IT ⊗ WN )IN EQ = β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + which can be interpreted as the Kuznets quadratic with state- and time-specific constants and a “net” inequality measure on the left-hand side that is free of the
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“contamination” caused by spillovers from the adjacent states. Given that the term (IT ⊗ WN )IN EQ on the right-hand side of (3) is endogenous, the estimation relies on straightforward extensions of the well-developed
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methods for single cross-section analysis, namely Maximum Likelihood (Ord, 1975; Anselin and Bera, 1998), Instrumental Variables, and Generalized Method of Moments (Kelejian and Prucha, 1998, 1999). Each procedure has its limitations and merit. For example, in contrast to Maximum Likelihood (ML), the InstrumentalVariable (IV) and the Generalized Method of Moments (GMM) estimators are consistent in the presence of heteroskedastic errors and do not employ the normality assumption (Kelejian and Prucha, 2010). 9
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Although our primary focus is on equation (3), we also consider a widely-used
such that
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alternative specification that allows the error term in (2) to be spatially correlated,
IN EQ =β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + υ, υ =λ(IT ⊗ MN )υ +
(4)
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where υ is the new N T ×1 vector of error terms and M is an N ×N spatial-weights
matrix, which reflects the spatial structure of the error term υ, and λ is the spatial autocorrelation parameter, which measures the strength of spatial dependence in
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the error and is expected to be positive. The rationale for spillover effects in the error term is similar to that for inequality spillovers, and is based on the thought that the unobserved factors that affect inequality in a state affect inequality in the neighboring states also, and that effect is independent of the direct spillover on inequality. The error term in (4) implies a non-spherical variance-covariance matrix, and, therefore, requires estimation via ML or GMM (Kelejian and Prucha, 1998, 1999, 2010).5
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It might be noted that the foregoing statements provide a conceptual rationale for the presence of spillover effects of inequality and unobserved factors from adjacent states. However, the significance and magnitude of such spillovers are testable through appropriate statistical procedures that are described in Section 4.
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Data and a Descriptive Picture for the US To increase parametric homogeneity of the sample, we follow the usual practice
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of including the 48 US contiguous states, and the income inequality data are taken from the US Census Bureau’s American Community Survey (ACS) covering the most recent 11-year period 2006-2016 for which ACS provides comparable information on household Gini, which is employed as the measure of income inequality (IN EQ). The primary income variable (Y ) is real GDP per capita in 2009 dollars 5 Another
method for capturing spatial spillovers is the spatial Durbin model which considers
possible spillovers in the regressors also.
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reported by the Bureau of Economic Analysis (BEA). It is the state-level version
in empirical tests of Kuznets hypothesis.6
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of real GDP per capita for the US, and seems to be the most appropriate variable
We would like to note some nice features of our data set. First, we have taken real per capita GDP for each state directly from the BEA website. In most cases,
researchers have taken nominal per capita GDP and deflated it by a regional or
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US level deflator. Although state-level information on prices by industry is not
available, BEA follows a fairly elaborate procedure for estimating real GDP by state, and generates a measure that should be considerably superior to what one
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can obtain by applying the US GDP deflator to state-level nominal GDP. Second, we have taken state-level household Gini directly from Census Bureau’s annual ACS data, and not from a privately estimated source whose reliability is uncertain. Third, the dispersion in income and Gini measures in our data (see Table 1) is very large and mimics the dispersion in income and Gini for a much longer period than that covered by our sample. For example, the variation in real GDP per capita from $31,175 to $70,876 (in 2009 prices) in our sample is large as compared
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with the variation in real GDP per capita in the United States from $22,982 to $54,559 (in 2012 prices) between 1967 and 2016. The sample variation in Gini from 0.408 to 0.514 is also large as compared with the variation in household Gini for the United States from 0.397 in 1967 to 0.481 in 2016.7 Last, as already noted,
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the sample period is more recent than that used in probably any other study of Kuznets hypothesis from US data. A quick descriptive picture for the US may be of some interest at this point.
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Looking at the 2006-2016 period, it is noted that real GDP per capita shows a 6 In
addition to using state-level real GDP per capita from the BEA, which seems to be
the most appropriate variable, we also considered median household income from the ACS as an alternative measure of Y . The estimates based on that measure of income share some broad
similarities with those reported here, but are not the same. The unreported estimates are available from the authors. 7 Additional details related to these numbers are available from the authors.
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tendency to increase at a pace of approximately 0.3 percent per year and the
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t-statistic for the simple trend coefficient is 1.35. The Gini, however, shows a significant tendency to rise at the rate of 1.95 percent per year and the t-statistic for the trend coefficient is 5.73, which is consistent with the story about increasing inequality. However, the increased inequality is not accompanied by a significant
increase in income which would be expected at late stages of a regular U-shaped
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relation.8
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4. The Main Results
Table 1 provides descriptive statistics for the sample so as to indicate the broad order of magnitudes of the variables.
The OLS estimates in Table 2 from the pooled data indicate a fairly well-defined regular-U in which inequality first decreases with income and then increases with increased income. The implied “turning point” occurs when real GDP per capita is of the order of $45,940 in 2009 dollars, which occurs considerably later than the
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early 1970s when the start of increased inequality in the US is typically noted.9 However, the hazards of reliance on such pooled-OLS estimates are well known, and the postulate of identical parameters in the Kuznets quadratic for every state seems too stringent. It is, therefore, necessary to permit at least a limited amount of parametric variability across the states and years. As noted in the discussion of
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the models, a fixed-effects format which permits the constant term to vary across states and years provides a variability of that kind and that format is favored by
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8 It
is possible to worry that the presence of the recessionary period in our sample may distort
the picture. However, the broad scenario is very similar if year-dummies for the recessionary period (2008-2010) are included. In that case, the trend for real GDP per capita is 0.2 percent per year with a t-statistic of 0.90, and the trend for Gini is 1.89 percent per year with a t-statistic of 6.84. It is evident that the relation between the rates of increase of income and Gini is very loose. Additional details are available from the authors. 9 Our estimates imply the turning-point to be located at exp[3.82578/(2 ∗ 0.179364)].
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the Hausman test.10 Table 3 reports the fixed-effects estimates, which too support
stage when real GDP per capita is of the order of $44,558.
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a significant regular-U pattern with the turning point occurring at a fairly late
Before describing estimates that allow cross-state spillovers, we briefly discuss
ex ante tests for the presence of spatial dependence of the kind modelled in equa-
tions (3) and (4). Such testing requires a specification of the weighting matrix.
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We use a contiguity matrix W in which element wij equals 1 if states i and j are Pn contiguous and 0 otherwise.11 Further, W is row standardized, i.e., j=1 wij pj = 1 for all i, and we assume W = M for estimates based on (4), which is a common
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practice in many empirical applications.
Diagnostics for spatial spillovers are reported in Table 4. Based on Moran’s I statistic, which provides a general test for the presence of spillovers, one can clearly reject the null hypothesis of no-spatial-dependence and no-spillovers, indicating the importance of permitting cross-state spillovers. The robust Lagrange-multiplier test (Anselin et al., 1996; Anselin, 2001) strongly rejects the null of there being no spillovers in income inequality in terms of equation (3). The null of there being
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no cross-state spillovers in the error term in equation (4) is also rejected by the robust Lagrange-multiplier test, but at a lower significance level. The spatial panel specification results, which are perhaps the most important part of our work, are reported in Table 5. These show estimated parameters of
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the (log) linear and quadratic terms in Kuznets’ model of income inequality when adjustments for cross-state spillovers have been made. The estimation procedures are based on well-known criteria. The Jarque-Bera test rejects the null hypothesis
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of normality and the Hall-Pagan/Cook-Weisberg tests reject the null hypothesis 10 As
shown in Table 3, the Hausman chi-square test statistic is 6.22 with a p-value of 0.045.
The random-effects estimates are, however, similar to those from the fixed-effects format. 11 An alternative specification for the weighting matrix W has non-zero elements of the form wij = 1/(d)fij , where d is the euclidean distance between states i and j, and f is a “friction parameter” that is set to equal 2. We report results based on the contiguity matrix since these are similar to those based on the continuous distance matrix.
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of homoskedastic disturbances, which would favor IV estimation over the ML pro-
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cedure for the model of equation (3) and the GMM procedure for estimating the model of equation (4).12 To provide a complete picture, we report both IV and
ML estimates for equation (3) and ML and GMM estimates for equation (4). The estimates suggest the following points.
First, the estimates of the spillover parameters (spatial-rho and spatial-lambda)
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are fairly sizable and statistically significant in every case, and indicate the presence
of spillovers and the need to treat these appropriately. This scenario is consistent with the ex ante diagnostics reported in Table 4 which show that there are likely
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to be highly significant spillovers.
Second, more importantly, the IV as well as the ML estimates of equation (3), which postulates cross-state spillovers in inequality and is our core model, show lack of a significant association of the income terms with inequality. Each of the four income terms lacks statistical significance at the usual 5 percent level, and the estimated ML coefficients are hardly significant even at the 10 percent level. Third, the foregoing scenario provides a sharp contrast from the pooled OLS es-
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timates in Table 2 where both income terms are highly significant with t-statistics of -5.09 and 5.10. It is also a major and instructive contrast from the two-way fixed-effects format of Table 3 in which the income terms have high statistical significance and the t-statistics are -2.66 and 2.67. The fixed-effects format provides
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considerable parametric flexibility across states in the constant term and incorporates the “fixed” effects of years (time) on state-level inequality. However, it misses the role of cross-state spillovers.
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Fourth, the estimates of equation (4), which permits spillovers in the regression
errors, are broadly similar to those of equation (3). In particular, each of the ML estimates for the income terms lacks statistical significance even at the 10 percent level, and the GMM estimates are not significant at the usual 5 percent level. Fifth, therefore, it may be seen that consideration of spatial cross-state spillovers
12 Additional
details about these test statistics are available from the authors.
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is conceptually appropriate and statistically important, and an adjustment for the
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spillovers by purging the inequality variable (and the error term) of such spillovers makes an instructive difference to the estimates of the Kuznets-quadratic. While the fixed-effects (and pooled OLS) estimates show a significant regular-U for the
recent US data, as a highly influential view suggests, the panel estimates that adjust for the cross-state spatial spillovers, show lack of a significant relation between
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income and inequality in our recent sample that consists of high-quality data.
Two implications of our estimates might be mentioned. First, although there is an indication of a U-shaped relation between income and inequality for the United
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States in recent years, coefficients of income terms in the appropriate spatialspillover models lack statistical significance at the conventional levels. Therefore, increased income may be expected to be only loosely associated with increased inequality. Second, appropriate policy measures can be considered to attenuate the increase in inequality along the rising segment of the regular-U. Kuznets (1955, pp. 8-9) mentioned “legislative interference” and political decisions among the factors that attenuated the inequality-increasing effect of concentration of savings
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in upper-income groups. High returns to capital postulated by Piketty (2014) as a major factor in increasing inequality along the regular-U are somewhat analogous to concentration of savings in upper-income groups in Kuznets’ analysis, and Piketty (2014, p. 387) has suggested consideration of the possibility of a
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progressive tax on capital.
5. Concluding Reflections
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In the vast literature on the empirical status of the famous Kuznets’ (1955)
hypothesis, the primary contribution of our study lies in providing an illustration of the consideration of cross-state spatial spillovers in income inequality. We note the conceptual and the statistical basis for the presence of such spillovers, indicate a well-known procedure for purging the effects of such spillovers, and use good state-level panel data on income and inequality to compare estimates from two conventional models of the hypothesis with one that purges the inequality variable 15
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of the spillovers. While the two conventional models provide strong support to
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the influential view that a regular-U characterizes the recent relation between income and inequality in the United States, each of the four spatial-spillover models
indicates lack of a significant relation between income and inequality, and some of the estimates are not significant even at the 10 percent level.
We hope that future research on this important topic, which Kuznets (1955, p.
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26) indicated to be central to much of economic analysis and thinking, will give
Acknowledgement
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consideration to spatial spillovers and augment and refine our estimates.
Highly perceptive and useful comments by the Editor and two anonymous
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reviewers are gratefully acknowledged. The usual disclaimer applies.
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References
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Anselin, L. 1988. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers.
Anselin, L. 2001. Rao’s score test in spatial econometrics. Journal of Statistical Planning and Inference, 97: 113–139.
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Anselin, L., A. K. Bera, R. Florax, and M. J. Yoon. 1996. Simple diagnostic tests for spatial dependence. Regional Science and Urban Economics, 26: 77–104.
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Anselin, L., and A. K. Bera. 1998. Spatial dependence in linear regression models with an introduction to spatial econometrics. In Ullah, A. and Giles, D. (eds.), Handbook of Applied Economic Statistics. Marcel Dekker, 237–289. Anselin, L., J. Le Gallo, and H. Jayet. 2008. Spatial panel econometrics. In Maytas L. and Sevestre, P. (eds.), The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice. Springer, 625–660.
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Atems, B. 2013. The spatial dynamics of growth and inequality: Evidence using US county-level data. Economics Letters, 118: 19–22. Baltagi, B. H. 2011. Spatial panels. In Ullah, A. and Giles, D. (eds), Handbook of Empirical Economics and Finance. Chapman and Hall, 435–454.
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Case, A. C. 1991. Spatial patterns in household demand. Econometrica, 59: 953–
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Costantini, M., and A. Paradiso. 2018. What do panel data say on inequality and GDP? New evidence at US state-level. Economics Letters, 168: 115–117.
de Dominicis, L., R. J. Florax, and H. L. De Groot. 2008. A meta-analysis on the relationship between income inequality and economic growth. Scottish Journal of Political Economy, 55: 654–682.
Doran, J., and D. Jordan. 2016. Decomposing US regional income inequality from 1969 to 2009. Applied Economics Letters, 23: 781–784. 17
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Goel, R. K., and M. A. Nelson. 2007. Are corrupt acts contagious?: Evidence
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from the United States. Journal of Policy Modeling, 29: 839–850. Kelejian, H. H., and I. R. Prucha. 1998. A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with au-
toregressive disturbances. The Journal of Real Estate Finance and Economics,
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Kelejian, H. H., and I. R. Prucha. 1999. A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review,
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40: 509–533.
Kelejian, H. H., and I. R. Prucha. 2010. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics, 157: 53–67.
Kim, D., H. Huang, and S. Lin. 2011. Kuznets hypothesis in a panel of states. Contemporary Economic Policy, 29: 250–260.
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Kuznets, S. 1955. Economic growth and income inequality. The American Economic Review, 45: 1–28.
List, J. A., and C. A. Gallet. 1999. The Kuznets curve: What happens after
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the inverted-U? Review of Development Economics, 3: 200–206. ´ Afonso, and S. T. Silva. 2016. A meta-analytic reassessment Neves, P. C., O.
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of the effects of inequality on growth. World Development, 78: 386–400. Ord, K. 1975. Estimation methods for models of spatial interaction. Journal of the American Statistical Association, 70: 120–126.
Piketty, T. 2014. Capital in the Twenty-First Century. Harvard University Press, Cambridge, MA.
Piketty, T., and E. Saez. 2003. Income inequality in the United States, 1913– 1998. The Quarterly Journal of Economics, 118: 1–41. 18
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Ram, R. 1991. Kuznets’s inverted-U hypothesis: Evidence from a highly devel-
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oped country. Southern Economic Journal, 57: 1112–1123. Ram, R. 1997. Level of economic development and income inequality: Evidence from the postwar developed world. Southern Economic Journal, 64: 576–583.
Robinson, S. 1976. A note on the U hypothesis relating income inequality and
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economic development. The American Economic Review, 66: 437–440.
Rubin, A., and D. Segal. 2015. The effects of economic growth on income
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inequality in the US. Journal of Macroeconomics, 45: 258–273.
Shin, I. 2012. Income inequality and economic growth. Economic Modelling, 29: 2049–2057.
Tribble, R. 1999. A restatement of the S-Curve hypothesis. Review of Develop-
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ment Economics, 3: 207–214.
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Descriptive statistics for the major variables
Income inequality Gini coefficient (IN EQ):
Real GDP per capita in 2009 dollars (Y ):
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Table 1:
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Mean
Std. Dev.
Min.
Max.
N
0.458
0.019
0.408
0.514
528
46,706
8,452
31,175
70,876
528
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Note: Income inequality is measured by household Gini coefficient published by the US Census Bureau on the basis of American Community Survey (ACS). As is well known, it takes values from 0 to 1, and a larger number reflects greater inequality. Income (Y ) is represented by real GDP per capita in 2009 dollars, which is published by the US Bureau of Economic Analysis. The sample consists
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of 48 contiguous states and 11 years (2006 to 2016), leading to 528 observations.
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OLS estimates of equation (1): IN EQ = α + β(lnY ) + γ(lnY )2 + , from
the pooled sample Dependent Variable
IN EQ (Gini)
Constant
Coefficient of LnY
(LnY )2
13.283∗
-2.390∗
0.111∗
(5.27)
(-5.09)
(5.10)
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Table 2:
R2
N
0.049
528
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Note: Please see Table 1 for the variable definitions and the data sources. There are 48 observations for each of the 11 years (2006-2016) leading to a pooled
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sample of 528 observations. Relevant t-statistics based on robust standard errors are in parentheses. An asterisk denotes statistical significance of the estimate at
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least at the conventional 5% level.
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Table 3:
Two-way fixed-effects estimates from a model of the form defined in equa-
Dependent Variable
LnY
(LnY )2
-0.942∗
0.044∗
(-2.66)
(2.67)
R2
N
0.674
528
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IN EQ
Coefficient of
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tion (2) of the text that allows a separate constant term for each state and year
Note: Please see Table 1 for the variable definitions and the data sources. The estimates for the constant terms for different states can be retrieved, but are not
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useful and thus not reported. Year dummies are included in the
estimation, but their estimated coefficients are not reported for simplicity. The reported R-square is for the “within” component which is a more appropriate measure of the model fit. The numbers in parentheses are the t-statistics (based on robust standard errors) for the corresponding estimates. An asterisk indicates statistical significance at least at the 5% level. Hausman chi-square statistic for the null hypothesis that the random-effects
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model is appropriate is 6.22 with a p-value of 0.0447, thus favoring the fixed-effects format. However, the random-effects estimates are similar to
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those shown above and may be requested from the authors.
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Diagnostics for spatial spillovers based on models in equation (3) and (4)
of the text
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Table 4:
Degrees of Statistic
freedom
p-value
16.060
1
0.00
General test for spatial
Moran’s I Statistic
Specific tests:
Robust Lagrange multiplier
Spatial-error (error-spillover) Robust Lagrange multiplier
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Spatial-lag (inequality-spillover)
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dependence (spillovers):
22.213
1
0.00
3.668
1
0.06
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Note: Please refer to the text for information about the context of the tests.
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Table 5:
Estimates of the spatial-lag models that adjust for inequality-spillovers
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and error-spillovers across states, specified in equations (3) and (4) of the text A. Spatial-lag model: equation (3) of the text (inequality-spillovers) Coefficient of LnY
(LnY )2
(ρ)
-0.900
0.042
0.186∗
(-1.64)
(1.64)
(4.98)
-0.793
0.037
0.650∗
(-1.79)
(1.79)
(2.17)
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Maximum-likelihood estimates
spatial-rho
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Instrumental-variable estimates
B. Spatial-error model: equation (4) of the text (error-spillovers) Coefficient of
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Maximum-likelihood estimates
Generalized method of moments estimates
spatial-lambda 2
LnY
(LnY )
(λ)
-0.891
0.041
0.183∗
(-1.56)
(1.56)
(4.93)
-0.892
0.041
0.197∗
(-1.95)
(1.95)
(2.68)
Note: Please see Table 1 for variable definitions. State and Year dummies are
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included in the estimation. However, their estimated coefficients are not reported for simplicity. The numbers in parentheses are t-values for the corresponding
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estimates. An asterisk indicates the estimate is statistically significant at least at the 5% level.
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Level of development and income inequality in the United States: Kuznets hypothesis revisited once again
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Research Highlights: This is the first study to consider spatial spillovers in the estimation of Kuznets-type quadratic
•
We use high-quality panel data for the US states
•
Traditional OLS and fixed-effects models show a significant regular-U pattern
•
Spatial panel model estimates indicate the income terms to lack significance
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•
S0264-9993(18)30700-4
DOI:
https://doi.org/10.1016/j.econmod.2018.11.024
Reference:
ECMODE 4780
To appear in:
Economic Modelling
Received Date: 16 May 2018 Revised Date:
28 September 2018
Accepted Date: 27 November 2018
Please cite this article as: Blanco, G., Ram, R., Level of development and income inequality in the United States: Kuznets hypothesis revisited once again, Economic Modelling (2019), doi: https:// doi.org/10.1016/j.econmod.2018.11.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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German Blanco & Rati Ram∗
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Abstract
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Level of development and income inequality in the United States: Kuznets hypothesis revisited once again
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This study revisits the well-known Kuznets hypothesis that postulated an invertedU relation between the level of development of a country or region and its degree of income inequality. We address the widely-shared view that, in recent years, a regular-U, and not a Kuznets-type inverted-U, has characterized the relation between income and inequality in the United States and some other developed countries. The paper works with data for the US states and has two main distinguishing features. First, it permits cross-state spillovers, which seem conceptually relevant and statistically significant, but, to our knowledge, have not been included in the fairly extensive literature on the topic. Second, we use the most reliable recent annual panel covering the period 2006-2016, which has also not been included in any existing research. The pooled-OLS, as well as the two-way fixed-effects panel estimates, indicate the existence of a significant regular-U pattern which is consistent with the influential view in the literature. However, the estimates that adjust for cross-state spillovers indicate a weak regular-U pattern in which the (log) linear and quadratic terms lack statistical significance at the usual levels. We conclude that when cross-state spillovers are appropriately accounted for, statistically significant evidence is lacking to support the view that a regular-U pattern describes the relation between income and inequality in the United States in recent years.
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Keywords: Kuznets-hypothesis; US states; cross-state spillovers
∗ Email:
[email protected] (G. Blanco, corresponding author), and [email protected] (R. Ram).
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1. Introduction
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Kuznets (1955) famously hypothesized that, during the process of economic
growth, as the level of development (per capita income) of a country or region rises, income inequality first increases, reaches a peak, and declines afterwards. This proposition, commonly called the Kuznets inverted-U hypothesis, was suggested by him on the basis of the empirical observation of that pattern in a few
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highly developed countries. He thus articulated a “long swing” in inequality which increases in the early phases of economic growth, becomes stabilized for a while
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and then narrows in the later phases. In his characteristic humility, he notes (p. 26) that the “paper is perhaps 5 per cent empirical information and 95 per cent speculation.”
Kuznets-hypothesis has attracted an immense amount of scholarship and there are perhaps hundreds of studies that deal with some aspects of the hypothesis, notably its empirical status. Besides doing an empirical assessment of the Kuznetsian postulate in numerous different settings, this literature has considered its
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conceptual foundations, provided alternative explanations for the phenomenon, extended or modified the hypothesis, or explored different estimation procedures in the empirical applications.1
More recently, a widely-shared view suggests that the Kuznets-hypothesis has suffered a breakdown during the last few decades relative to some highly devel-
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oped countries like the United States and that such contexts are characterized by a regular-U, and not an inverted-U, relation between income and inequality. Some
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of the recent influential debates on increasing income inequality seem to derive from a thought of that kind. Although the literature on even this limited issue is considerable, we mention just a few selected studies that support the theme of a regular-U for highly-developed economies like that of the United States and provide a setting for our research. In some early works, Ram (1991, 1997) indicated the ex1 Kim,
Huang and Lin (2011, pp. 250-251) provide a compact summary of many studies
relating to the Kuznets hypothesis.
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istence of a regular-U structure for US and some other highly-developed countries.
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Similarly, List and Gallet (1999) suggested a positive correlation between income inequality and development in advanced economies, and attributed that partly
to a “shift away from a manufacturing base towards a service base”. Tribble (1999) proposed an “S-curve” model that includes agriculture-to-manufacturing and manufacturing-to-services structural transitions during the process of eco-
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nomic development. In their narrative on the evolution of income inequality in
the United States, Piketty and Saez (2003, pp. 1-2) have suggested a U-curve instead of the inverted-U proposed by Kuznets (1955). They note that “Today,
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the Kuznets curve is widely held to have doubled back on itself, especially in the United States, with the period of falling inequality observed during the first half of the twentieth century being succeeded by a very sharp reversal of the trend since the 1970s.” From data on US states, Kim, Huang and Lin (2011) concluded that the relation between inequality and development is “better characterized by a U shape rather than the inverted-U profile asserted by Kuznets (1955).” Piketty (2014) has documented a regular-U profile of income inequality in the US and some
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other countries. More recently, Costantini and Paradiso (2018) stated that their empirical results suggest the existence of a S-shaped relation between income and inequality in the US states.
The main purpose of the present study is to use a panel of US states to explore
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whether the recent relation between income and inequality in the United States is characterized by a regular-U pattern. A panel of US states offers some advantages besides the comparability of income and inequality data over time and across
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states. For instance, given the wide range of income across the states, it can mimic the evolution of inequality over a much longer period than that covered by
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the sample.2 Our work is marked by two distinguishing features. First, it accounts
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for cross-state spillovers which appear conceptually relevant and, as the empirical component of this paper shows, statistically significant, but, to our knowledge,
have not been considered in any study of Kuznets’ hypothesis despite their fairly extensive usage in other economic contexts. The importance of taking account of
such spillovers is primarily based on the thought that while Kuznets’ hypothesis
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relates the value of inequality with income (level of development) for each unit
(state), cross-state spillovers in inequality “contaminate” the own-state variable with the part that spills over from adjacent states, and it is appropriate to purge
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the inequality variable of such contamination. Our other contribution is the use of highly reliable state-level data from the American Community Survey (ACS) and Bureau of Economic Analysis (BEA) covering the most recent period 2006-2016 which has not been included in any existing study.
After a short recapitulation of some related studies in Section 2, we briefly outline the model specification, estimation method, and the data in Section 3. The main results are presented in Section 4, and the last section contains some
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concluding thoughts.
2. A Short Recapitulation of Some Related Studies The preceding section summarized several related works to motivate the present
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study. Those studies are mentioned in slightly greater detail in this section. The pioneering research by Kuznets (1955) indicating the existence of an inverted-U
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relation between income inequality and level of development over long periods in a few industrialized countries is, of course, well known and has generated enormous research on the topic. His position was summarized by saying (p. 18) that 2 For
example, real GDP per capita in 2016 varied from $32,334 in Mississippi to $65,168 (2009
dollars) in Connecticut, which is a much greater variance than in real GDP per capita in the United States from $29,503 in 1975 to $54,559 in 2016 (in 2012 dollars). The income variation in our sample is even greater from a minimum of $31,175 to a maximum of $70,876.
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there is a long swing in income-inequality which widens in the early phases of
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economic growth, becomes stabilized for a while, and then narrows in the later phases. His explanation (Kuznets, 1955, pp. 6-17) for the phenomenon was that in early stages of economic development, inequality tends to increase due to a shift from more equal and lower-income agriculture (rural) sector to less equal
and higher-income industrial (urban) sector, and concentration of savings in the
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hands of higher-income groups accentuates that increase in inequality. At later stages of development, several factors, including “legislative interference” and political decisions, mitigate the effect of savings being concentrated in the hands of
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higher-income groups. The inequality-increasing effect of the shift from more equal agriculture to less equal non-agriculture is offset by a rise in the income share of lower groups in the non-agriculture sector. He (1955, p. 20) also noted other trends related to that in income inequality, including those in population growth, urbanization, internal migration, savings and investment rates, international trade, and government activity.
While Kuznets’ reasoning in support of the inverted-U hypothesis was largely
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verbal, Robinson (1976) provided a more rigorous and formal two-sector model of a developing economy that implied an inverted-U relation between income and inequality. He noted (1976, p. 439) that one implication of such a two-sector model is that, in the absence of explicit countervailing policies, a developing country will
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have increasing or unchanged inequality for a relatively long period. Among the more recent research that suggests the presence of a regular-U relation, instead of a Kuznets-type inverted-U, in highly developed countries during
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the last few decades, Ram (1991, 1997), List and Gallet (1999), Tribble (1999) and Kim, Huang and Lin (2011) have been mentioned in the preceding section, and the narrative by Piketty and Saez (2003) was noted. Piketty (2014) has also documented a regular-U pattern in the evolution of inequality in the United States and some other developed countries. While other scholars proposing a regular-U relation have used descriptive phrases like structural transformation from manufacturing to services, Piketty’s (2014) explanation is based on (a) rapid increases
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in the compensation of executives and (b) high returns to capital, which might
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be regarded as the contemporary version of concentration of savings in the highincome groups noted by Kuznets (1955). More recently, Costantini and Paradiso (2018), who worked with a panel of US states for the period 1960-2015, stated that “the empirical results seem to suggest the existence of a S-shaped relationship be-
tween inequality and per capita GDP.” However, their estimated rising segment
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of inequality (p. 117) seems to start when per capita GDP is at a low level of around $14,500. Although not directly related to the question about the existence
of an inverted-U in recent decades, the study by Doran and Jordan (2016) ana-
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lyzed changes in levels and composition of income inequality among US counties from 1969 to 2009. They found that income inequality has increased in the period studied, and between-state inequality has declined, but within-state inequality has increased.
While Kuznets’ postulate relates income inequality with the level of development (or per capita income) of a country or region, there has recently been considerable research on the relation between income inequality and economic growth.
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We mention a few selected studies in that strand of literature since some scholars seem to view that aspect as related to Kuznets hypothesis, and the substantial volume of research on that topic has led to two recent meta-analytic studies. The meta-analysis by de Dominicis, Florax and De Groot (2008) notes that although
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a considerable part of the literature found inequality detrimental to growth, more recent studies have claimed a positive effect of inequality on growth. They use meta-analysis to shed light on this puzzle and “systematically describe, identify
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and analyse the variation in outcomes of empirical studies.” They find “that estimation methods, data quality and sample coverage systematically affect the results.” The other meta-analysis by Neves, Afonso and Silva (2016) covers the studies published during 1994-2014 that estimate the effect on growth of inequality in income, land and human capital. After correcting for publication biases, they conclude that “the high degree of heterogeneity of the reported effect sizes is explained by study conditions, namely the structure of the data, the type of
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countries included in the sample, the inclusion of regional dummies, the concept
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of inequality and the definition of income.” Besides the meta-analytic research, we note just a few recent studies for the United States. Shin (2012) did a largely theo-
retical analysis of the relation between income inequality and economic growth and indicated that higher inequality may retard growth in early stages of economic de-
velopment, and can encourage growth near the steady state. Based on a dynamic
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spatial Durbin model, Atems (2013) estimated the effects on a county’s economic
growth of increased inequality in that county and in the neighboring counties. Rubin and Segal (2015) analyzed the relation between growth and income inequality
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in the US during 1953-2008 and concluded that “the income of the top income group is more sensitive to growth...compared to the income of the lower income groups.”
3. Model, Estimation Methodology, and Data Models and Estimation Methods
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We consider the following widely-used regression specification, which stacks observations of T successive cross-sections, each with N observations:3 IN EQ = α + β(lnY ) + γ(lnY )2 + ,
(1)
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where IN EQ is a N T × 1 vector of observations on states’ income inequality, α and lnY are conformable vectors containing ones and the natural log of income, respectively, and is assumed to be a N T × 1 vector of disturbances. In terms of
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the Kuznets hypothesis, we expect β to be positive and γ to be negative. After the pooled OLS estimation based on equation (1), our analysis starts with
the two-way fixed-effects model, which includes state-specific dummy variables to capture a limited amount of unobserved heterogeneity across the states and year3 In
our vector notation, we skip the usual i and t subscripts to refer to the cross-sectional
unit and the year of observation.
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dummies to account for the year-specific factors that may affect all states.4 The
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fixed-effects Kuznets quadratic may then be written as:
IN EQ = β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + ,
(2)
where 1T and 1N are vectors of ones of dimension T and N , respectively, δ is a N × 1 vector of “individual (state-specific) effects”, and η is a T × 1 vector of “time
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(year) effects”.
In addition to allowing for a limited amount of heterogeneity across the states and the years through the fixed-effects format of equation (2), we note that in-
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equality in the geographically neighboring states is likely to affect inequality in any given state through spatial “spillover effects”. Higher inequality in the adjacent states may be associated with higher inequality in a given state due to the “diffusion” of inequality across a state’s geographical boundaries which are necessarily porous. The effects of inequality-related policies or factors in the adjacent states are likely to spill over to a state, and a particular state’s measure of inequality
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might emulate that of its closest neighbors through imitation, social pressure, or other non-market externalities, and a measure of proximity plays a key role in determining the extent of influence exerted by other states. Inequality in a state may, therefore, be affected not merely by income in that state, as Kuznets hypothesis suggests, but also by inequality in the neighboring states. Such spillover effects
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have not been considered in the numerous empirical tests of Kuznets hypothesis, although these are widely recognized and have been considered important in several contexts in economics. A few examples of the study of such spillover effects
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include Atems (2013) who estimated the effect on a county’s economic growth of inequality in the county’s neighbors, Case (1991) who studied spatial relationships in Indonesian demand for rice, and Goel and Nelson (2007) who estimated the “contagious” effects of corruption and other crimes across US states. Baltagi 4 We
also consider the random-effects format, but the Hausman test described in Table 3 favors
the fixed-effects format.
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(2011, p. 436) noted the many areas in economics where spillover effects have been
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considered in panel data. To “purge” the inequality variable of such spillovers, and make the model closer to the Kuznets paradigm, we consider the following speci-
fication that conforms to the literature on spatial correlations and is expected to purge the spatial spillover effects that “contaminate” inequality in a state:
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IN EQ = ρ(IT ⊗ WN )IN EQ + β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + , (3)
where IT is an identity matrix of dimension T , W is a N ×N spatial weights matrix that formalizes the spatial structure of inequality of the N observations, and ρ is
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the spatial “autocorrelation” parameter, which is expected to be positive with the implication that higher inequality in a state has the spillover effect of tending to raise inequality in the neighboring states. The so-called spatial autoregressive model (Anselin, 1988; Anselin, Le Gallo and Jayet, 2008) specified in equation (3) models the spillovers from the proximate states through the first term on the right-hand side and makes the specification closer to the Kuznets paradigm. It is
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evident that (3) can be rewritten as:
IN EQ − ρ(IT ⊗ WN )IN EQ = β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + which can be interpreted as the Kuznets quadratic with state- and time-specific constants and a “net” inequality measure on the left-hand side that is free of the
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“contamination” caused by spillovers from the adjacent states. Given that the term (IT ⊗ WN )IN EQ on the right-hand side of (3) is endogenous, the estimation relies on straightforward extensions of the well-developed
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methods for single cross-section analysis, namely Maximum Likelihood (Ord, 1975; Anselin and Bera, 1998), Instrumental Variables, and Generalized Method of Moments (Kelejian and Prucha, 1998, 1999). Each procedure has its limitations and merit. For example, in contrast to Maximum Likelihood (ML), the InstrumentalVariable (IV) and the Generalized Method of Moments (GMM) estimators are consistent in the presence of heteroskedastic errors and do not employ the normality assumption (Kelejian and Prucha, 2010). 9
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Although our primary focus is on equation (3), we also consider a widely-used
such that
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alternative specification that allows the error term in (2) to be spatially correlated,
IN EQ =β(lnY ) + γ(lnY )2 + (1T ⊗ δ) + (1N ⊗ η) + υ, υ =λ(IT ⊗ MN )υ +
(4)
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where υ is the new N T ×1 vector of error terms and M is an N ×N spatial-weights
matrix, which reflects the spatial structure of the error term υ, and λ is the spatial autocorrelation parameter, which measures the strength of spatial dependence in
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the error and is expected to be positive. The rationale for spillover effects in the error term is similar to that for inequality spillovers, and is based on the thought that the unobserved factors that affect inequality in a state affect inequality in the neighboring states also, and that effect is independent of the direct spillover on inequality. The error term in (4) implies a non-spherical variance-covariance matrix, and, therefore, requires estimation via ML or GMM (Kelejian and Prucha, 1998, 1999, 2010).5
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It might be noted that the foregoing statements provide a conceptual rationale for the presence of spillover effects of inequality and unobserved factors from adjacent states. However, the significance and magnitude of such spillovers are testable through appropriate statistical procedures that are described in Section 4.
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Data and a Descriptive Picture for the US To increase parametric homogeneity of the sample, we follow the usual practice
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of including the 48 US contiguous states, and the income inequality data are taken from the US Census Bureau’s American Community Survey (ACS) covering the most recent 11-year period 2006-2016 for which ACS provides comparable information on household Gini, which is employed as the measure of income inequality (IN EQ). The primary income variable (Y ) is real GDP per capita in 2009 dollars 5 Another
method for capturing spatial spillovers is the spatial Durbin model which considers
possible spillovers in the regressors also.
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reported by the Bureau of Economic Analysis (BEA). It is the state-level version
in empirical tests of Kuznets hypothesis.6
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of real GDP per capita for the US, and seems to be the most appropriate variable
We would like to note some nice features of our data set. First, we have taken real per capita GDP for each state directly from the BEA website. In most cases,
researchers have taken nominal per capita GDP and deflated it by a regional or
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US level deflator. Although state-level information on prices by industry is not
available, BEA follows a fairly elaborate procedure for estimating real GDP by state, and generates a measure that should be considerably superior to what one
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can obtain by applying the US GDP deflator to state-level nominal GDP. Second, we have taken state-level household Gini directly from Census Bureau’s annual ACS data, and not from a privately estimated source whose reliability is uncertain. Third, the dispersion in income and Gini measures in our data (see Table 1) is very large and mimics the dispersion in income and Gini for a much longer period than that covered by our sample. For example, the variation in real GDP per capita from $31,175 to $70,876 (in 2009 prices) in our sample is large as compared
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with the variation in real GDP per capita in the United States from $22,982 to $54,559 (in 2012 prices) between 1967 and 2016. The sample variation in Gini from 0.408 to 0.514 is also large as compared with the variation in household Gini for the United States from 0.397 in 1967 to 0.481 in 2016.7 Last, as already noted,
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the sample period is more recent than that used in probably any other study of Kuznets hypothesis from US data. A quick descriptive picture for the US may be of some interest at this point.
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Looking at the 2006-2016 period, it is noted that real GDP per capita shows a 6 In
addition to using state-level real GDP per capita from the BEA, which seems to be
the most appropriate variable, we also considered median household income from the ACS as an alternative measure of Y . The estimates based on that measure of income share some broad
similarities with those reported here, but are not the same. The unreported estimates are available from the authors. 7 Additional details related to these numbers are available from the authors.
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tendency to increase at a pace of approximately 0.3 percent per year and the
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t-statistic for the simple trend coefficient is 1.35. The Gini, however, shows a significant tendency to rise at the rate of 1.95 percent per year and the t-statistic for the trend coefficient is 5.73, which is consistent with the story about increasing inequality. However, the increased inequality is not accompanied by a significant
increase in income which would be expected at late stages of a regular U-shaped
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relation.8
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4. The Main Results
Table 1 provides descriptive statistics for the sample so as to indicate the broad order of magnitudes of the variables.
The OLS estimates in Table 2 from the pooled data indicate a fairly well-defined regular-U in which inequality first decreases with income and then increases with increased income. The implied “turning point” occurs when real GDP per capita is of the order of $45,940 in 2009 dollars, which occurs considerably later than the
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early 1970s when the start of increased inequality in the US is typically noted.9 However, the hazards of reliance on such pooled-OLS estimates are well known, and the postulate of identical parameters in the Kuznets quadratic for every state seems too stringent. It is, therefore, necessary to permit at least a limited amount of parametric variability across the states and years. As noted in the discussion of
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the models, a fixed-effects format which permits the constant term to vary across states and years provides a variability of that kind and that format is favored by
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8 It
is possible to worry that the presence of the recessionary period in our sample may distort
the picture. However, the broad scenario is very similar if year-dummies for the recessionary period (2008-2010) are included. In that case, the trend for real GDP per capita is 0.2 percent per year with a t-statistic of 0.90, and the trend for Gini is 1.89 percent per year with a t-statistic of 6.84. It is evident that the relation between the rates of increase of income and Gini is very loose. Additional details are available from the authors. 9 Our estimates imply the turning-point to be located at exp[3.82578/(2 ∗ 0.179364)].
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the Hausman test.10 Table 3 reports the fixed-effects estimates, which too support
stage when real GDP per capita is of the order of $44,558.
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a significant regular-U pattern with the turning point occurring at a fairly late
Before describing estimates that allow cross-state spillovers, we briefly discuss
ex ante tests for the presence of spatial dependence of the kind modelled in equa-
tions (3) and (4). Such testing requires a specification of the weighting matrix.
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We use a contiguity matrix W in which element wij equals 1 if states i and j are Pn contiguous and 0 otherwise.11 Further, W is row standardized, i.e., j=1 wij pj = 1 for all i, and we assume W = M for estimates based on (4), which is a common
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practice in many empirical applications.
Diagnostics for spatial spillovers are reported in Table 4. Based on Moran’s I statistic, which provides a general test for the presence of spillovers, one can clearly reject the null hypothesis of no-spatial-dependence and no-spillovers, indicating the importance of permitting cross-state spillovers. The robust Lagrange-multiplier test (Anselin et al., 1996; Anselin, 2001) strongly rejects the null of there being no spillovers in income inequality in terms of equation (3). The null of there being
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no cross-state spillovers in the error term in equation (4) is also rejected by the robust Lagrange-multiplier test, but at a lower significance level. The spatial panel specification results, which are perhaps the most important part of our work, are reported in Table 5. These show estimated parameters of
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the (log) linear and quadratic terms in Kuznets’ model of income inequality when adjustments for cross-state spillovers have been made. The estimation procedures are based on well-known criteria. The Jarque-Bera test rejects the null hypothesis
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of normality and the Hall-Pagan/Cook-Weisberg tests reject the null hypothesis 10 As
shown in Table 3, the Hausman chi-square test statistic is 6.22 with a p-value of 0.045.
The random-effects estimates are, however, similar to those from the fixed-effects format. 11 An alternative specification for the weighting matrix W has non-zero elements of the form wij = 1/(d)fij , where d is the euclidean distance between states i and j, and f is a “friction parameter” that is set to equal 2. We report results based on the contiguity matrix since these are similar to those based on the continuous distance matrix.
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of homoskedastic disturbances, which would favor IV estimation over the ML pro-
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cedure for the model of equation (3) and the GMM procedure for estimating the model of equation (4).12 To provide a complete picture, we report both IV and
ML estimates for equation (3) and ML and GMM estimates for equation (4). The estimates suggest the following points.
First, the estimates of the spillover parameters (spatial-rho and spatial-lambda)
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are fairly sizable and statistically significant in every case, and indicate the presence
of spillovers and the need to treat these appropriately. This scenario is consistent with the ex ante diagnostics reported in Table 4 which show that there are likely
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to be highly significant spillovers.
Second, more importantly, the IV as well as the ML estimates of equation (3), which postulates cross-state spillovers in inequality and is our core model, show lack of a significant association of the income terms with inequality. Each of the four income terms lacks statistical significance at the usual 5 percent level, and the estimated ML coefficients are hardly significant even at the 10 percent level. Third, the foregoing scenario provides a sharp contrast from the pooled OLS es-
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timates in Table 2 where both income terms are highly significant with t-statistics of -5.09 and 5.10. It is also a major and instructive contrast from the two-way fixed-effects format of Table 3 in which the income terms have high statistical significance and the t-statistics are -2.66 and 2.67. The fixed-effects format provides
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considerable parametric flexibility across states in the constant term and incorporates the “fixed” effects of years (time) on state-level inequality. However, it misses the role of cross-state spillovers.
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Fourth, the estimates of equation (4), which permits spillovers in the regression
errors, are broadly similar to those of equation (3). In particular, each of the ML estimates for the income terms lacks statistical significance even at the 10 percent level, and the GMM estimates are not significant at the usual 5 percent level. Fifth, therefore, it may be seen that consideration of spatial cross-state spillovers
12 Additional
details about these test statistics are available from the authors.
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is conceptually appropriate and statistically important, and an adjustment for the
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spillovers by purging the inequality variable (and the error term) of such spillovers makes an instructive difference to the estimates of the Kuznets-quadratic. While the fixed-effects (and pooled OLS) estimates show a significant regular-U for the
recent US data, as a highly influential view suggests, the panel estimates that adjust for the cross-state spatial spillovers, show lack of a significant relation between
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income and inequality in our recent sample that consists of high-quality data.
Two implications of our estimates might be mentioned. First, although there is an indication of a U-shaped relation between income and inequality for the United
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States in recent years, coefficients of income terms in the appropriate spatialspillover models lack statistical significance at the conventional levels. Therefore, increased income may be expected to be only loosely associated with increased inequality. Second, appropriate policy measures can be considered to attenuate the increase in inequality along the rising segment of the regular-U. Kuznets (1955, pp. 8-9) mentioned “legislative interference” and political decisions among the factors that attenuated the inequality-increasing effect of concentration of savings
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in upper-income groups. High returns to capital postulated by Piketty (2014) as a major factor in increasing inequality along the regular-U are somewhat analogous to concentration of savings in upper-income groups in Kuznets’ analysis, and Piketty (2014, p. 387) has suggested consideration of the possibility of a
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progressive tax on capital.
5. Concluding Reflections
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In the vast literature on the empirical status of the famous Kuznets’ (1955)
hypothesis, the primary contribution of our study lies in providing an illustration of the consideration of cross-state spatial spillovers in income inequality. We note the conceptual and the statistical basis for the presence of such spillovers, indicate a well-known procedure for purging the effects of such spillovers, and use good state-level panel data on income and inequality to compare estimates from two conventional models of the hypothesis with one that purges the inequality variable 15
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of the spillovers. While the two conventional models provide strong support to
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the influential view that a regular-U characterizes the recent relation between income and inequality in the United States, each of the four spatial-spillover models
indicates lack of a significant relation between income and inequality, and some of the estimates are not significant even at the 10 percent level.
We hope that future research on this important topic, which Kuznets (1955, p.
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26) indicated to be central to much of economic analysis and thinking, will give
Acknowledgement
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consideration to spatial spillovers and augment and refine our estimates.
Highly perceptive and useful comments by the Editor and two anonymous
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reviewers are gratefully acknowledged. The usual disclaimer applies.
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Descriptive statistics for the major variables
Income inequality Gini coefficient (IN EQ):
Real GDP per capita in 2009 dollars (Y ):
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Table 1:
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Mean
Std. Dev.
Min.
Max.
N
0.458
0.019
0.408
0.514
528
46,706
8,452
31,175
70,876
528
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Note: Income inequality is measured by household Gini coefficient published by the US Census Bureau on the basis of American Community Survey (ACS). As is well known, it takes values from 0 to 1, and a larger number reflects greater inequality. Income (Y ) is represented by real GDP per capita in 2009 dollars, which is published by the US Bureau of Economic Analysis. The sample consists
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of 48 contiguous states and 11 years (2006 to 2016), leading to 528 observations.
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OLS estimates of equation (1): IN EQ = α + β(lnY ) + γ(lnY )2 + , from
the pooled sample Dependent Variable
IN EQ (Gini)
Constant
Coefficient of LnY
(LnY )2
13.283∗
-2.390∗
0.111∗
(5.27)
(-5.09)
(5.10)
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Table 2:
R2
N
0.049
528
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Note: Please see Table 1 for the variable definitions and the data sources. There are 48 observations for each of the 11 years (2006-2016) leading to a pooled
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sample of 528 observations. Relevant t-statistics based on robust standard errors are in parentheses. An asterisk denotes statistical significance of the estimate at
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least at the conventional 5% level.
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Table 3:
Two-way fixed-effects estimates from a model of the form defined in equa-
Dependent Variable
LnY
(LnY )2
-0.942∗
0.044∗
(-2.66)
(2.67)
R2
N
0.674
528
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IN EQ
Coefficient of
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tion (2) of the text that allows a separate constant term for each state and year
Note: Please see Table 1 for the variable definitions and the data sources. The estimates for the constant terms for different states can be retrieved, but are not
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useful and thus not reported. Year dummies are included in the
estimation, but their estimated coefficients are not reported for simplicity. The reported R-square is for the “within” component which is a more appropriate measure of the model fit. The numbers in parentheses are the t-statistics (based on robust standard errors) for the corresponding estimates. An asterisk indicates statistical significance at least at the 5% level. Hausman chi-square statistic for the null hypothesis that the random-effects
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model is appropriate is 6.22 with a p-value of 0.0447, thus favoring the fixed-effects format. However, the random-effects estimates are similar to
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those shown above and may be requested from the authors.
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Diagnostics for spatial spillovers based on models in equation (3) and (4)
of the text
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Table 4:
Degrees of Statistic
freedom
p-value
16.060
1
0.00
General test for spatial
Moran’s I Statistic
Specific tests:
Robust Lagrange multiplier
Spatial-error (error-spillover) Robust Lagrange multiplier
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Spatial-lag (inequality-spillover)
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dependence (spillovers):
22.213
1
0.00
3.668
1
0.06
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Note: Please refer to the text for information about the context of the tests.
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Table 5:
Estimates of the spatial-lag models that adjust for inequality-spillovers
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and error-spillovers across states, specified in equations (3) and (4) of the text A. Spatial-lag model: equation (3) of the text (inequality-spillovers) Coefficient of LnY
(LnY )2
(ρ)
-0.900
0.042
0.186∗
(-1.64)
(1.64)
(4.98)
-0.793
0.037
0.650∗
(-1.79)
(1.79)
(2.17)
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Maximum-likelihood estimates
spatial-rho
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Instrumental-variable estimates
B. Spatial-error model: equation (4) of the text (error-spillovers) Coefficient of
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Maximum-likelihood estimates
Generalized method of moments estimates
spatial-lambda 2
LnY
(LnY )
(λ)
-0.891
0.041
0.183∗
(-1.56)
(1.56)
(4.93)
-0.892
0.041
0.197∗
(-1.95)
(1.95)
(2.68)
Note: Please see Table 1 for variable definitions. State and Year dummies are
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included in the estimation. However, their estimated coefficients are not reported for simplicity. The numbers in parentheses are t-values for the corresponding
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estimates. An asterisk indicates the estimate is statistically significant at least at the 5% level.
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Level of development and income inequality in the United States: Kuznets hypothesis revisited once again
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Research Highlights: This is the first study to consider spatial spillovers in the estimation of Kuznets-type quadratic
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We use high-quality panel data for the US states
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Traditional OLS and fixed-effects models show a significant regular-U pattern
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Spatial panel model estimates indicate the income terms to lack significance
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•