The Welfare Implications of Growth Regressions

The Welfare Implications of Growth Regressions Donal O’Neilly May 10, 2005 Abstract Regressions relating the growth rate in income to initial income have been the source of much recent debate in growth economics. Recent research has emphasised the importance of allowing for nonlinearities in these models when explaining the evolution of income over time. In this paper we argue these extended growth regressions are also useful in facilitating welfare comparisons across income distributions, in a way that is not possible using alternative measures of convergence. To do this we exploit the similaritites between the income convergence literature and work on tax progressivity in the public …nance literature. We illustrate our approach using both regional data across the United States, Japan and Europe and countrywide comparisons. Keywords: Growth Regressions, Welfare, Equality of Opportunity, Progressivity JEL Codes: O47 I would like to thank Olive Sweetman and Philippe Van Kerm for helpful comments on earlier versions of this paper. yEconomics Dept., NUI Maynooth, Co. Kildare, Ireland. e-mail:[email protected]. 1 1 Introduction The early literature on income convergence across countries was dominated by cross-section studies that regressed the growth rate of income on initial income to examine whether or not poor countries grew faster than richer countries. These regressions are sometimes called “Barro-regressions” (e.g Quah 1993a)1and faster growth among poor countries has become known as -convergence (e.g. Barro and Sala-i-Martin (1992) and Sala-i-Martin (1996a)). However, this approach has been the subject of much debate and has been criticised by many.2At a fundamental level a number of authors, including Friedman (1992) and Quah (1996), point out that, by itself, - convergence tells us little about the dynamic evolution of incomes. Friedman (1992), quoting Hotelling (1933), argues that "the real test of a tendency towards convergence would be in showing a constant decline in the variance...among individual enterprises." In the growth literature this type of convergence has been labelled as -convergence. As noted by Islam (2003) one of the main arguments for the rejection of Barro’s conclusions centered on the failure of traditional growth equations to accommodate non-linear speci…cations. As a result more recent developments in growth econometrics has emphasised nonlinearities in the growth process (Kalaitzidakis et al (2001), Fiaschi and Lavezzi (2003) and Maasoumi et al (2005)). These procedures provide a more detailed description of the evolution of the distribution of realised incomes over time than was possible using traditional linear Barro-regressions.3 Although there have been signi…cant econometric and theoretical devel1Some authors refer to these regressions as growth-initial level regressions and reserve the label "Barro-Regression" only for cases in which the growth regressions include other controls in addition to intitial income. This distinction is somewhat arbritary and unnecessary for our study. 2For recent summaries of this literature see Isalm (2003) and Durlauf et al (2005). 3Other critics of traditional growth regressions include Quah (1996), who argues that the speed of convergence estimated from growth regressions may simply re‡ect smallsample biases. However, he later acknowledges that the degree of precision reported in standard Barro-Regressions casts doubt on this explanation. Lee, Pesaran and Smith (1997) discuss the econometric problems that arise when using Barro-Regressions to estimate the structural parameters of a growth model. Although important, this issues is distinct from, and not relevant for the question we address in our paper. For detailed summaries of these issues and the alternative approaches to measuring convergence see de la Fuente (1997), Durlauf and Quah (1999) and Islam (2003). 2 opments in the analysis of growth models over time there has been very little empirical work devoted to characterising the welfare properties of the observed processes. When making welfare comparisons economists have traditionally focused on the distribution of observed outcomes. However in recent years a number of economists have argued that welfare comparisons should place greater emphasis on equality of opportunities rather than observed outcomes (e.g Fleurbay (1995), Roemer (1998)). The equal-opportunity framework stresses the link between the opportunities available to an agent and the initial conditions which are inherited or beyond the control of these agents. At the individual level these conditions may include characteristics such as race, gender or parental income. At a country-level analysis one may be interested in knowing to what extent the future opportunities of a country are determined by their initial income level. From this perspective one possible goal of policy makers could be to ensure that the opportunities available to agents are unrelated to initial endowments. This need not necessarily eliminate inequality in observed outcomes. Proponents of equality of opportunity accept inequality of outcomes that arise from genuine choice or shocks that are unrelated to initial conditions. The question that we address in this paper is whether growth regressions can contribute in a meaningful way to studies that focus, not on the the evolution of realised outcomes, but rather on the equality of opportunity across agents. We show that appropriate consideration of nonlinearites in the growth process is not only desirable when documenting the evolution of income over time but is also an essential component of a coherent equalopportunity based welfare framework. In particular, we extend the work of Benabou and Ok (2001) to show precisely how ‡exible form growth regressions can facilitate welfare comparisons in ways that are not possible using some of the alternative convergence concepts that have been proposed. 2 Progressivity, Growth and Welfare Transition probabilities, MT(xjy), specify the probability that an individual with income ytoday will earn at most xat time T. A number of authors have estimated associated transition matrices in the context of income convergence (e.g Quah (1993)). However, they are almost always presented as descriptive tools for understanding the evolution of observed incomes over time. However, in his survey of welfare theoretic approaches to the measurement 3 of mobility Maasoumi (1998) notes that "Mobility in any social hierarchy is an indication of opportunity." Benabou and Ok (2001) make a similar point when noting that many people care about mobility "not because income movements are intrinsically valuable, but primarily because of the hope that it helps attenuate the e¤ects of disparities in initial endowments on future income prospects (pg. 2)." In their paper they derive conditions under which a mobility process can be characterised as opportunity-equalising, as well as providing criteria to determine if one process is more equalising than another.4To do this they abstract from agents’aversion to risk and summarise future available opportunities or income prospects using the conditional expectation function determined by the underlying mobility process:5 eT(y) = Z1 0 xdMT(xjy)(1) Thus eT(y)summarises the opportunities available at time Tto an agent with current income y.6 7 While Benabou and Ok (2001) characterise future available opportunities using the underlying transition process, M, it is relatively straightforward to recast their results in terms of the underlying growth process. To do this we note that …nal realised income, yT, can always be written as the sum of initial income (y0), the expected change in income given the initial level (g(y0)) and a mean zero residual term (vT); that is: yT=y0+g(y0) + vT(2) In this case the opportunities available to agents at time T, with initial income y0, can be written as: eT(y0) = y0+g(y0)(3) 4They consider monotonic mobility processese such that for any y1; y2with y2> y1 then MT(xjy1)MT(xjy2)for all x. This implies eT(y2)> eT(y1). 5The use of conditional means to summarise opportunity sets is discussed in more detail in their paper. A major advantage of this approach is that it signi…cantly simpli…es the comparison of di¤erent opportunity sets. For a general discussion of some of the problems that arise when evaluating opportunity sets see Sen (1985). 6Benabou and Ok (2001) consider only monotonic mobility processese such that for any y1; y2with y2> y1then MT(xjy1)MT(xjy2)for all x. This implies eT(y2)> eT(y1). 7For extensions that consider discounted lifetime utilities see Benabou and Ok (2001) and Dardanoni (1993). 4 Writing the model in this way allows us to draw close parallels between the income convergence literature and the public economics literature on tax/bene…t progressivity (Lambert (1993)).8Following the tax literature we de…ne a growth process as progressive if dg(y0) y0 dy0<0, regressive if dg(y0) y0 dy0>0 and proportional if dg(y0) y0 dy0= 0. Intuitively a growth process is progressive if low income countries experience faster growth rates than higher income countries. This framework is su¢ cient to allow us to characterises the welfare properties of growth processes based on the progressivity or otherwise of the process. To see this let U(e)denote the utility accruing to an agent with future opportunities summarised by e:We assume U0(e)>0. Following an established tradition in public economics de…ne social welfare as the average utility across initial income levels. That is WF=ZU(eT(y)) f(y)dy (4) where f(y)is the distribution of initial incomes.9We can then establish the following theorem:10 Theorem 1 A monotone growth process increases (decreases) welfare more than an equal yield proportional growth process applied to the same pre-growth income distribution for all strictly concave Uand for all possible initial income distributions if and only if the growth process is progressive (regressive). Proof: See Appendix This theorem states that progression in the growth process, over the entire range of income, is a necessary and su¢ cient condition for the resulting 8In the tax/bene…t literature, y0would represent the tax/bene…t base, ewould represent …nal income and g(y0)would represent net bene…ts. See also Benabou and Ok (2001). 9See Lambert (1993) section 4.2 for a rationalisation of this social welfare function. 10 See also Corollary 3 of Benabou and Ok (2001) . 5 distribution of opportunities to welfare dominate the distribution of opportunities derived from an equal yield proportional mobility process, irrespective of the initial income distribution. An immediate corollary of this theorem is that a distribution of future opportunities across agents generated by a progressive growth process will welfare dominate the initial distribution of opportunities provided average income does not decline. It is important to note the role of progressive growth in the above analysis. Since we are only considering monotone growth processes then progressivity must reduce the variance of future available opportunities across agents relative to those available in the initial distribution.11 However, in general it is possible for the process to be monotonic, for mean income to rise and for inequality (as de…ned by the variance or Gini coe¢ cient of opportunities) to fall and yet for Generalised Lorenz curves to cross so that unambiguous welfare rankings are not possible. A simple example which illustrates this possibility is given in Table 1. The …rst column shows the distribution of initial incomes (opportunities) and the second column shows a hypothetical distribution of future opportunities derived from this distribution. The last 4 rows summarise the respective distributions. The example is constructed so that on average opportunities have improved and dispersion in opportunities has fallen. This is true for each of the three standard measures of inequality reported. Furthermore the growth process is monotonic in that the rankings of countries in both distributions are preserved. Despite all of this it can be easily shown that the Generalised Lorenz Curves for these two distributions cross, which prevents unambiguous welfare rankings across the two distributions. The reason for this is that the growth process in this example is not progressive over the entire range. For example the growth rate for the second richest person is larger than the growth rate for the second poorest, which is a violation of progressivity. Lambert (1993) provides a more detailed discussion of the restrictions that must be imposed on preferences in order for the social welfare function to be completely summarised by mean income and a scalar index of inequality. He also discusses the limitations that these restrictions place on the type of inequality indices which could summarise social welfare. This latter discussion may have interesting implications for how one should measure - convergence in cross-country studies of income inequality. However, the key 11 As mentioned earlier, this need not imply a reduction in the dispersion of observed outcomes. 6 result that emerges from this analysis is that in order to make unanimous welfare comparisons across distributions of opportunities it matters how the reduction in inequality is generated. Simply comparing the variance of future available opportunities with current opportunities is not su¢ cient to establish welfare rankings. The above analysis shows how progressivity in the growth process can be used to facilitate welfare comparisons across alternative growth processes. We now establish the relationship between progressivity in the growth process and measures of convergence derived from a traditional growth regression. To determine the progressivity of the growth process we need to establish whether dg(y0) y0 dy00for all y0. Using the fact that dg(y0) y0 dy00for all y0 if and only if dg(y0) y0 dln(y0)0for all y0;we can use the following model of log income to characterise progressivity: ln yT= ln y0+m(ln y0) + "T(4) where "Tis a mean zero error term. Progressivity of the growth process requires dm(ln y0) dln y00everywhere. However, equation (4) is simply a ‡exible form Barro-regression and our progressivity condition is nothing more than a negativity condition on the slope of a non-parametric cross-sectional growth-initial level regression. Thus the progressivity requirements needed for welfare comparisons of alternative growth processes can be stated in terms of the convergence estimates obtained from a ‡exible speci…cation of a Barro-regression. This highlights a potentially important role for growth regressions that extends beyond their ability to distinguish between competing theories of growth or their capacity provide a useful summary of the evolution of realised outcomes. 3 Empirical Analysis In this section we illustrate our approach using regional data sets taken from Barro and Sala-i-Martin (1995), as well as country level data taken from the Penn-World Tables Version 6.1. The regional data sets are those used by Salai-Martin (1996b) to study regional cohesion. Sala-i-Martin estimated linear Barro-regressions for the regions of the United States, Japan and Europe. In 7 order to apply Theorem 1 however we must consider ‡exible estimators of the growth process that allow for possible nonlinearities. To do this we extend Sala-i-Martin’s empirical analysis by estimating ‡exible nonparametric growth equations for each of these data sets. In particular we estimate the following ‡exible form growth equation: ln yi;T yi;0=N =m[ln(yi;0)] + i;T (5) In each case we use the Nadaraya-Watson kernel estimator to obtain a ‡exible estimate of m[ln(y0)].12 The dates for which the analysis is conducted depends on data availability and di¤ers across data sets. The data for the US refer to real annual personal income per capita for each of the 48 contiguous states from 1900 to 1990. The Japanese data measure real per capita income between 1955 and 1990 for the 47 prefectures, as collected by the Economic Planning Agency of Japan. Finally the European data measure GDP per capita in each of 90 regions of Europe covering Germany (11 regions), United Kingdom (11 regions), Italy (20 regions), France (21 regions), The Netherlands (4 regions), Belgium (3 regions), Denmark (3 regions) and Spain (17 regions).13 The nonparametric estimates, ^ m[ln(y0)], for the US states, the Japanese prefectures and the European regions are given in Figures 1-3 respectively. Our principal concern is the extent to which the growth process exhibits progression or regression over the income range; equivalently the extent to which the slope of ^ m[ln(y0)] is negative or positive at each value of y0. Recalling Theorem 1 we note that it is this feature of the growth process that facilitates welfare comparisons across the distribution of opportunities. Figures 1-3 show that all the regional growth processes exhibit progressivity over almost all of their respective income ranges. Indeed the only evidence of 12 For a more detailed discussion of kernel regresison techniques see Blundell and Duncan (1998). 13 Following Sala-i-Martin (1995) the European GDP …gures are expressed as deviations from country speci…c means. Thus the estimated growth process we present for the regions of Europe should be interpreted as a common, within country growth, process. More details on these data, including maps illustrating the regions under consideration, are available in Barro and Sala-i-Martin (1995). 8 regressive growth for these data occurs among high income Japanese prefectures. However, even then the con…dence intervals are such that we cannot rule out progressive growth over this income range. On the other hand we clearly reject the possibility that income growth is regressive over the entire income range for all the regional growth processes. In this case Theorem 1 implies that there exists at least one initial income distribution for which the observed growth process welfare dominates an equal yield proportional growth process. Furthermore, since the data strongly supports the hypothesis of progressivity over the entire range, our data are consistent with a scenario in which the distribution of future opportunities derived from the observed process unambiguously welfare dominates that obtained from a proportional growth process for all possible initial income distributions in all of the regions. Since average income has risen over this period in each of our regional data sets, and since we can always view the identity mapping as a proportional growth process, our data also support the hypothesis that the distribution of opportunities available today within each of these regions welfare dominates that available previously. We can also apply our approach to examine income growth across countries using the Penn World Table version 6.1. These data provide national incomes converted to international prices from 1950-2000. We use data for the period 1960-2000. We begin by looking at the unconditional growth process for the OECD countries and for a world sample of 83 countries for which there were no missing data.14 The non-parametric estimates of m[ln(y0)] for both these samples are given in Figures 4 and 5 respectively. The estimated growth process for the OECD countries exhibit a high degree of nonlinearity. Progressivity is most pronounced at low and high income levels. However, there is a middle range of incomes for which the estimated growth process is approximately proportional. Nevertheless the con…dence intervals are such that the inferences that we can draw from the sample of OECD countries mirror those presented earlier for the regional data sets. We reject regressive income growth over the entire income range but cannot reject the hypothesis of progressivity over all initial income levels. Thus the data are consistent with welfare improving growth among the OECD countries. The situation for the entire world sample is di¤erent however. Figure 5 highlights important nonlinearities in the estimated growth process for the world sample. For this sample however, the overall tendency is for regressive 14 A complete list of these countries is given in Table 2. 9 Appendix: Proof of Theorem 1 De…ne the expected growth rate for person with initial income y0as (y0)g(y0) y0. =)Under proportional growth then the Lorenz curve of opportunities at time T, (eT(yi)), must equal the Lorenz curve for initial incomes or opportunities (e0y0). That is Leprop;T (p)=Le0(p) for all p2[0,1] By de…nition eT(y0) = y0+g(y0). Taking the average across agents we get that eT=e0(1+),where is the average expected growth rate across agents =Xg(y0) y0 N!and e0is average initial income or opportunities. Hence the Generalised Lorenz Curve for future opportunities derived from a growth process characterised by (y)can be expressed as : GLCeT(p)=e0(1+)LeT(p), where LeT(p) is the Lorenz curve of future opportunities. If our observed growth process is progressive, that is 0(y)<0for all y, then we can use the Jakobsson-Fellman theorem (Lambert (1993) page 150) and our assumption of monotonicity to conclude that : GLCeT (p)=e0(1+)LeT(p)e0(1+)Le0(p)= e0(1+)Leprop;T (p) all p2 [0,1]. The …rst inequality follows from our assumptions of monotonicity and progressivity and the last equality follows from step 1 of the proof. By de…nition this implies that: GLCeT (p)GLCeprop;T (p) all p2[0,1]. 16 Referring to Shorrocks’(1983) completes the proof in this direction. (Suppose GLCeT (p)GLCeprop;T (p) for all p and any pre-growth income distribution. Then following the logic above we can establish that LeT(p)Le0(p) for all p and all pre-growth income distributions From the Jakobsson-Fellman theorem we can then conclude that the mobility process is progressive for all y0. 17 Table 1: Ambiguous Welfare Rankings in the Presence of Declining Income Dispersion. yi;0e(yi;0) 10 17 20 20 30 21 40 45 50 48 Mean Income=30 Mean Income=30.2 Gini=.266 Gini=.23 Coe¢ cient of Var=.527 Coe¢ cient of Var.=.496 ln=.636 ln=.489 18 Table 2: Full Sample of 83 countries included in the analysis Argentina Costa Rica India Malawi Sweden Australia Denmark Ireland Malaysia Switzerland Austria Dominican Republic Iran Nigeria Syria Belgium Algeria Iceland Nicaragua Chad Benin Ecuador Israel Netherlands Togo Bangladesh Egypt Jamaica Norway Thailand Bolivia Jordan Nepal Trinidad and Tobago Brazil Finland Japan New Zealand Turkey Barbados France Kenya Pakistan Tanzania Canada Ghana Korea Panama United Kingdom Chile Gambia Sri Lanka Peru Uganda China GuineaBissau Lesotho Philippines Uruguay Cameroon Greece Mexico Portugal United States of America Congo, Republic of Guatemala Mali Paraguay Venezuela Colombia Hong Kong Mozambique Romania South Africa Spain Honduras Mauritius Rwanda Zambia El Salvador Indonesia Senegal Zimbabwe 19 Table 3: Summary Statistics Variable Average Minimum Maximum Hi4.6 years .5 (GNB) 10.8 (NZL) ni3.2% .3% (BEL) 11.68% (JOR) Si16.6% 2.06% (UGA) 31.80% (NOR) y1960 3699 381.5 (TZA) 14978.25 (CHE) y2000 9560 481.87 (TZA) 33292 (USA) 20 .005 .01 .015 .02 .025 .03 Growth Rate 1900-1990 0.5 11.5 Log GDP 1900 nonparametric estimates 95% lower CI 95% upper CI Figure 1: Nonparametric Estimates of the Growth Process across the US States. .04 .045 .05 .055 .06 Growth Rate 1955-1990 12.5 13 13.5 Log GDP 1955 nonparametric estimates 95% lower CI 95% upper CI Figure 2: Nonparametric Estimates of the Growth Process across the Japanese Perfectures. 21 -.02 -.015-.01 -.0050.005 Growth Rate 1950-1990 -.5 0.5 1 Log GDP 1950 nonparametric estimates 95% lower CI 95% upper CI Figure 3: Nonparametric Estimates of the Growth Process Across the European Regions. .02 .03 .04 .05 .06 Growth Rate 1960-2000 7.5 88.5 99.5 Log GDP 1960 nonparametric estimates 95% lower CI 95% upper CI Figure 4: Nonparametric Estimates of the Growth Process Across the OECD Countries. 22 .01 .015 .02 .025 .03 Growth Rate 1960-2000 6 7 8 9 10 Log GDP 1960 nonparametric estimates 95% lower CI 95% upper CI Figure 5: Nonparametric Estimates of the Unconditional Growth Process Across the World. -.04 -.03 -.02 -.01 0 Growth Rate 1960-2000 6 7 8 9 10 Log GDP 1960 nonparametric estimates 95% lower CI 95% upper CI Figure 6: Nonparametric Estimates of the Conditional Growth Process Across the World. 23