Oct 31, 2007 - There are huge differences between rich and poor people across the world. This is of primary concern ...
International Income Inequality: Measuring PPP bias by estimating Engel curves for food Ingvild Alm˚ as∗ October 31, 2007
Abstract Price-adjusted data on national incomes applied in cross-country comparisons are measured with bias. By studying micro data, this paper finds that the bias is systematic: the poorer a country is, the more its income tends to be overestimated. The price-adjusted data of the Penn World Table are applied in studies of poverty, inequality, growth and convergence. Hence, the bias alters the findings of these studies. This paper estimates both the bias and the subsequent consequences for estimates of inequality and convergence. The findings are that international income inequality tends to be underestimated whereas the convergence between poor and rich countries tends to be overestimated when analyses are based on the Penn World Table. The biases in the macro price variables are caused by factors analogous to those that create bias in consumer price indices. Exploiting this feature, the bias in the cross country comparable macro prices is measured by comparing estimated Engel curves for food, a method already established in measuring biases in consumer price indices.
1
Introduction
There are huge differences between rich and poor people across the world. This is of primary concern to economists, as well as to the public at large. Subsequently, ∗
Norwegian School of Economics and Business Administration, Helleveien 30, 5045 Bergen, Norway, e-mail:
[email protected]. Thanks to Gernot Doppelhofer, Bruce Hamilton, Steinar Holden, Timothy Kehoe, Jo Thori Lind, Branko Milanovic, Peter Neary, Xavier Sala-i-Martin and Bertil Tungodden for valuable comments and suggestions. The usual disclaimer applies.
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one of the main questions to discuss is whether these differences are getting larger or smaller, i.e., whether incomes diverge or converge across people and countries. In order to address this question, data from the Penn World Table (PWT) have been applied. This paper studies these data, and measures the bias in the country-specific per capita real incomes presented in the PWT. Moreover, the relationship between the bias and national per capita real income for a country is considered. This is done by first estimating the bias in the PWT macro prices. Based on the corrected prices, the unbiased real incomes are then calculated. By comparing the corrected incomes and the PWT real incomes, the questions of how the bias influences estimated inequality and estimated change in inequality, i.e., convergence or divergence, are answered. This paper reports four main findings: First, there are significant and substantial biases in the national incomes given in the PWT. Second, there is a systematic relationship between the PPP bias and the national income of a country: the poorer the country, the more its income tends to be overestimated relative to some base country.1 Third, the PPP bias causes a significant and robust underestimation of international inequality: the Gini index increases substantially when correcting for the bias and the uncorrected real incomes of the PWT Lorenz dominates that of the corrected PWT real incomes. Finally, a study of 22 countries reveals that the convergence between 1970 and 1995 is overestimated. When correcting for the PPP bias, the predicted convergence is relaxed. Although an enormous number of macroeconomic studies rely on the PWT data, very few studies focus on the measurement bias in this data set. There are some contributions, however, that focus on one part of the bias, the so-called substitution bias of the PWT variables, and they apply macro data to measure it (Dowrick and Akmal, 2005; Hill, 2000; Neary, 2004; Nuxoll, 1994). All of these studies find the same trend: international income differences tend to be underestimated in the PWT data. The finding of underestimation of inequality is in line with the findings in this study. The main methodological contributions of this paper are twofold. First, by applying micro data from household surveys, inaccuracies arising from aggregation techniques are avoided. Second, the specific method based on Engel curve estimation makes it possible to estimate the overall PPP bias, and to calculate the corrected real incomes by correcting for this bias. When we have estimates for the corrected incomes, we are able to discuss real inequality and convergence rates and compare 1
Throughout this paper, the phrase ‘PPP bias’ refers to the overall bias in the macro price variable for consumption given in the PWT and the subsequent bias in the measured national per capita real incomes.
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them to the uncorrected measures based on PWT data. The problems faced when constructing PPPs are in essence analogous to those faced when constructing consumer price indices (CPIs). One of the novelties of this paper is that it acknowledges and exploits this analogy by applying Hamilton’s method for estimating CPI bias to estimate PPP bias (Hamilton, 2001). This method utilizes the stable relation between the budget share for food and total expenditure to measure PPP bias. Engel curves for food are estimated by applying micro data from different countries and the macro price variables from the PWT. Household real incomes are made comparable by deflating household total expenditure by the macro price variable for consumption given in the PWT. As is standard in this method, the main assumption is that there is a stable relationship between the budget share for food and real incomes across countries, i.e., that there exists a unique Engel relationship for food in the world. Any systematic difference in the estimated Engel relationship between a country and a base country, reveals the PPP bias for the respective country relative to the base country. Several robustness checks are conducted and reported in Appendix C, one of which tests whether the functional form fits the data in the study. None of the robustness checks changes the main findings. We have no reason then to think that misspecification drives the results of this paper. The paper is organized as follows. Section 2 discusses the causes of the bias and why the Penn World Table is biased in a systematic way. Section 3 describes the empirical methodology in more detail. Section 4 describes both the micro data and the macro price variables from the PWT that are applied in the analysis. The analysis and main findings are presented in section 5. Section 6 discusses whether we have experience convergence or divergence in the last few decades, while section 7 compares the corrected measures to the exchange rate-based measures. Section 7 concludes.
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Explaining the Bias
The PPP bias stems from two problems well known in the price index literature: namely, bias caused by differences in quality, and substitution bias (Costa, 2001; Hamilton, 2001; Hill, 2000; Neary, 2004). Most PPP calculations, among them the Geary–Khamis calculations presented in the PWT, belong to the group of fixedbasket calculations. In a fixed-basket calculation, homogenous goods are needed for the purpose of comparison. The problem related to this is that the same goods are not consumed in all countries, and specifically, the quality of goods varies both over time and across countries. For example, it is not clear whether the observed 3
price difference in cars between Russia and the US reflects differences in the quality between the brands available in the two countries or some real price difference. Furthermore, there may be goods that exist in some but not all countries. For example, comparing the price of Pakistani gur, a sugar substitute, to that of Norwegian sugar substitutes is hard, because of the fact that sugar substitutes are not consumed in Norway. The same problem as when quality differs occurs in this situation: gur and sugar have to be put in the same broad goods category, and we are unlikely to pick up the quality difference between the two in a proper way. Furthermore, in a fixed-basket calculation, a set of cross-country comparable macro price variables is constructed and applied to evaluating the different countries’ realized consumption bundles. Hence, the fact that consumers would have substituted their consumption away from relatively more expensive goods towards relatively less expensive goods if faced with the constructed price level is not taken into account. Thus, if consumers do not have Leontief preferences, both PPP and CPI measures belonging to the group of fixed-basket calculations inherit a substitution bias. Both these problems, the problem of differences in quality and the problem related to substitution in consumption, deliver systematic biases. As poorer countries tend to have lower quality products than richer countries, we thus tend to overestimate poorer countries’ income because of the omittance of quality. Thus, we have reasons to expect that the problem of measuring quality correctly leads to a systematic overestimation of poorer countries’ income relative to richer countries’ income. Second, the further away from a country’s own price structure the reference price vector for comparison is, the greater its measured income tends to be (Nuxoll, 1994). The Geary–Khamis method, that underlies the Penn World Table applies a reference price vector that is calculated by a function weighting all prices and quantities. Thus, if the reference price vector is closer to the richer countries prices, the measured inequality will be lower than if the price vector gave equal weight to each country independent of its income level. The reference price for good i is given by: Πi =
N X qij ( PN j=1
pij ) P P P q j ij j=1
(1)
where qij is the quantity of good i consumed in country j, pij is the price of good i in country j and P P Pj is the overall price index of country j and N is the total number of countries in the system. This equation gives an indication that the Geary– Khamis method gives a higher weight to richer countries’ prices as the derivative 4
of the reference price of good i with respect to the local price of the same good in country j relative to the overall price level in this country, is given by: δΠi p δ P PijPj
qij = PN
j=1 qij
.
(2)
We see that this derivative is larger, the higher the quantity of this country is relative to other countries’ quantity of this good, i.e., the richer this country is in terms of consumption of this good. However, the overall price level of country j, P P Pj , is endogenous and given by2 : PM P P Pj = Pi=1 M
pij qij
i=1 Πi qij
.
(3)
We therefore have to consider the overall effect of the price in country i in order to find the effect of being richer on the weight in the reference prices. It is more complicated to calculate the overall effect of country j’s prices on its own weight in the construction of the reference price level. Appendix B studies this question in the special case of two countries and two goods, and shows that the cross derivative of the reference price with respect to quantity and price is positive, i.e., the richer you are, the greater the weight given to your price level when constructing the reference price level. Thus, we can expect that the substitution effect causes a systematic bias: because the reference price vector is closer to the richer countries’ prices, the poorer countries’ incomes tend to be overestimated relative to the richer countries’ incomes. The substitution effect is similar to so called Gerschenkron effect in the growth literature. Gerschenkron (1947) made the observation that the earlier the base year, the higher the measured growth rate. The Gerschenkron effect arises with aggregation methods that use either a reference price structure or a reference volume structure to compare countries. For methods employing a reference price structure, e.g. the Geary–Khamis method underlying the PWT, a country’s share of total GDP (that is, the total for the group of countries being compared) will rise as the reference price structure becomes less characteristic of its own price structure. The Gerschenkron effect arises because of the negative correlation between prices and volumes. In other words, expenditure patterns change in response to changes in relative prices because consumers substitute their expenditure away from relatively expensive goods, towards relatively cheap goods (OECD, 2007; Gerschenkron, 1947; Hill, 2000; Nuxoll, 1994). The analogy to cross-country comparisons is evident: the further away a country’s price structure is from the reference price, the greater its measured income. 2
M being the total number of goods in the system.
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3
Empirical Methodology
This paper estimates national Engel curves for food, i.e., the relationship between the household budget share for food and household real income, based on household micro data from nine countries. The reason for estimating Engel curves for food and not other items is that food has two properties that are needed in order to identify the bias. First, food has an income elasticity different from unity. In order to identify the PPP bias, the coefficient for the logarithm of income is needed in addition to the country dummy coefficient. If the income elasticity was equal to unity, however, it would be impossible to estimate such a coefficient. Second, studies show that the Engel curve for food is stable, both over time and across societies (Banks et al., 1997; Beatty and Larsen, 2005; Blundell et al., 1998; Leser, 1963; Working, 1943; Yatchew, 2003). This stable relationship is exploited in order to measure the bias in the PPP-adjusted measure of national real incomes. In this study, the macro price variable for consumption, Pj , given in the PWT, is used to make household incomes comparable across countries. The bias in Pj is by definition country specific. Hence, the dummy coefficients can be utilized to measure the biases in the Pj ’s.
3.1
Empirical framework—Econometric specification
The standard almost ideal demand system (AIDS) specification is the following:
mh,r,j = a + b(ln yh,r,j − ln Pj ) + γ(ln Pf,r,j − ln Pn,r,j ) + θXh,r,j + ²h,r,j
(4)
where mh,r,j is the budget share for food of household h in region r, country j. yh,r,j is the nominal income of household h in region r, country j, and Pj is the composite price of consumption in country j. Pf,r,j is the price of food in region r, country j, and Pn,r,j is the price of nonfood items in region r, country j. Xh,r,j is a vector of demographic control variables for household h in region r, country j, which includes the age of the household head, the number of children and the number of adults in the household. There are no regional cross-country comparable price data available for the countries in the study and, therefore, the coefficient for relative prices, γ, cannot be estimated. Consequently, the main estimation excludes the relative prices between food and nonfood items, and thus, implicitly assumes that the budget share for food is unaffected by relative prices. However, we have observations on national relative
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prices for five countries and a robustness check for relative price effects is conducted in section 5. When excluding the relative price effect, (1) can be simplified to: mh,j = a + b(ln yh,j − ln Pj ) + θXh,j + ²h,j .
(5)
Denoting the biased macro price variable for consumption given in the PWT Pj0 and the PPP bias for country j, Ej , the unbiased corrected price variable Pj , can be expressed as follows. (6) Pj = Pj0 ∗ Ej . Equation (2) can, thus, be expressed as follows.
mh,j = a + b(ln yh,j − ln Pj0 − ln Ej ) + θXh,j + ²h,j = a + b(ln yh,j − ln Pj0 ) + θXh,j +
N X
dj Dj + ²h,j
j=1
(7) where Dj is the country dummy. The country dummy coefficient, dj , is a function of the PPP bias, Ej , and the coefficient for the logarithm of real income, b: dj = −b ln Ej .
(8)
The specification given in (4) is the preferred specification of this paper and the PPP bias is, thus, given by3 : dj
Ej = e− b .
(9)
Because the budget share for food is decreasing in income (i.e., b is negative), the estimated bias is larger than one as long as the estimated country dummy coefficient is positive. Whenever the bias is larger than one, the PWT consumption price is underestimated and the real income of the country is thus overestimated. The larger the estimated country dummy coefficient, the larger the estimated bias, and the larger the bias, the more the macro price level for consumption is underestimated. Subsequently, the larger the bias, the more national per capita real income is overestimated. 3
Alternative specifications are estimated in the robustness analysis of section 5.
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4
Data
The Engel curves are estimated from micro data in order to reveal biases in the macro price variable for consumption in the PWT. This section discusses the micro data and the macro price variables in turn. Section 4.1 discusses the micro data, section 4.2 discusses the macro price variables, while section 4.3 discusses the UN data used in the extension model of section 5.
4.1
Micro data from household surveys
The main estimation of the preferred specification includes 52,543 households from nine countries. Table 1 gives an overview of the different surveys. The household data for Azerbaijan, China, Nicaragua and Cˆote D’Ivoire are from the World Bank’s living standard measurement surveys (LSMS). The data for the USA are from the Consumer Expenditure Surveys (CES), US Bureau of Labor, and the Hungarian data are from the Hungarian Central Statistical Office, Section of Household Budget Survey. Luxembourg Income Studies (LIS) provide the data for France, United Kingdom and Italy.4 The nine countries have been picked from available nationally representative studies in order to maintain both a geographical spread and a combination of lower and higher income countries.5 It is demanding to harmonize data from different studies. Therefore, this analysis relies most heavily on surveys that are available from already harmonized sources, such as the LIS and the LSMS. There are no panel data available for the lower income countries, which limits the choice of estimation techniques. Moreover, scarcity of data for some of these countries also limits the inclusion of explanatory variables. In the main estimation of the preferred specification, all households are included, and the adult equivalence scaling of the OECD is applied in order to adjust for household composition and size. In the robustness analysis, other estimations are conducted—among them an estimation of the preferred specification using the 4,968 households with two adults and two children. This robustness check exploits one of the advantages of using micro data: it is possible to analyze households of the same composition and size in order to avoid inaccuracies because of heterogeneous household composition. [Table 1 about here.] 4
Detailed information on the different LSMS and LIS studies can be found on the World Bank and LIS websites, respectively (Luxembourg Income Studies, 2006; World Bank, 2005). 5 All data are constructed to be nationally representative, except those from China. For China, no nationally representative study is available. The Chinese data includes households from the provinces of Hebei and Liaoning, implying that only rural households are covered.
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Many of the households included in the analysis are farm households, and for these households, home-produced food amounts to a considerable part of total household consumption. In order to consider this, home-produced goods are included in the expenditure variables.
4.2
Macro price variables
In the standard AIDS specification, three macro price variables are included. The first, Pj0 , is a composite price index for all consumption goods in country j, which is constructed by the Geary–Khamis method and presented in the PWT. The remaining macro price variables are the composite price index for food items, Pf,r,j , and nonfood items, Pn,r,j , respectively. The household surveys are conducted in different years, and thus, the macro price variable for consumption in the PWT has to be taken from different years. The consumption price reported in the PWT is given in current prices, and consequently the US exchange rate, as well as the US consumer price index, is applied in order to make the real income levels comparable across countries and time. The macro price variable for consumption and the exchange rate are taken from the Penn World Table 6.1 (Heston et al., 2002). The US CPI is taken from the World Bank’s World Development Indicators (WDI) online (World Bank, 2007). The preferred specification (equation (4)) does not include relative prices between Pf,r,j food and nonfood items (ln Pn,r,j ). The reason for this is simply the lack of data. Unfortunately, cross-country regional price data for food and nonfood items do not exist. Very few countries report regional price variation, and if they do, it is done relative to some base year. That is, the price in one region is compared to the price level of that same region in a different year. Thus, these cannot be used in crossregional comparisons for specific years. The same applies to national price indices, e.g., the food price index produced by the World Bank. These are also only defined relative to a base year, and thus, cannot be used to compare relative prices across countries. The International Comparison Project published cross-country comparable national prices for food and nonfood items for the year 1980 (phase IV, can be found at Neary, 2006). By combining these prices with the World Bank’s price indices, comparable national relative prices for Hungary, the USA, France, the United Kingdom, and Italy are calculated. It is, however, impossible to identify the coefficient of the relative price within the data set, because we do not have regional price data. To overcome this problem, Costa’s (2001) estimated coefficient for relative prices, γ, is applied in one of the robustness checks in section 5. Using Costa’s estimated coefficient, national relative price levels for the five countries are included and hence 9
the relative price effect is taken into account in this estimation.
4.3
UN data used in the extended model
Once the World Engel curve for food is estimated from the household studies included, other years and other countries can be included. Such an extension is done in this paper, and mean data from the UN Statistics Division, Common database, is applied in order to do so. 983 new mean consumption and budget shares are included, covering 46 different countries and observations from 1970 to 1995. Final household expenditure in national currency, at current prices is applied. The PWT price of consumption and exchange rate are used in order to make the final household consumption comparable between countries and years. In order to find the household mean of the logarithm of income from the mean household income, the estimated distributions of Sala-i-Martin (2006) are applied.6 The demographic controls are from the UN as well. Children and adults, and subsequently the OECD adult equivalence scaling, can be calculated directly. The age of the household head is approximated by using the relationship between life expectancy and age of household head estimated from the nine household surveys that serve as the background for the main analysis in this paper.
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Analysis and Findings
In this section, two models and two data sets are studied. First, the main model estimated based on household surveys from nine countries is studied, and the findings from this model are discussed in detail. Second, an extension of the model is presented, and aggregated UN household data are applied in order to study whether we can generalize from the findings of the nine countries.
5.1
Main model based on household surveys [Table 2 about here.]
The regression results are presented in Table 2. The preferred model is the model that applies the OECD adult equivalence scale in order to adjust for household composition and size.7 The estimated coefficients for the preferred specification are 6
It is necessary to utilize information on the distribution in addition to the mean of household expenditure, because we know that the mean of the logarithm of x is not equal to the logarithm of the mean of x (ln(mean(x)) 6= mean(ln(x))). 7 The OECD adult equivalence scale gives the value 1 to the first person in the household, 0.7 to each additional adult and 0.5 to each additional child (less than 16 years of age).The number
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given in column one. The estimated income elasticity of food is slightly smaller than in related studies (Costa, 2001; Hamilton, 2001). The US country dummy coefficient is by construction equal to zero, and the dummy coefficients for Azerbaijan, China, Nicaragua, Cˆote D’Ivoire, Hungary, France, the United Kingdom, and Italy are used to measure the PPP bias relative to the US bias. The first main finding in this paper is that the biases in the national incomes given in the PWT are substantial and significant: all country dummy coefficients are significantly different from zero. All countries except for the United Kingdom have a positive dummy coefficient; i.e., the macro price variables in the PWT underestimate the macro price levels compared to the US macro price level. Thus, all countries’ real incomes, except for the United Kingdom’s, are overestimated relative to the US real income in the PWT. The estimation shows that the group of non-OECD countries, China, Nicaragua, Azerbaijan and Cˆote D’Ivoire, have substantially higher dummy coefficients than the OECD countries. China has the highest dummy coefficient, whereas Nicaragua and Azerbaijan have slightly smaller dummy coefficients. The United Kingdom has a negative dummy coefficient. This implies that its real income is underestimated relative to US real income. [Figure 1 about here.] Figure 1 presents the second main finding of this paper: there is a decreasing relationship between the PPP bias and national real income levels. The measured bias is much higher for the poorest countries than for the richer. It is clear from Figure 1 that the overestimation of the poorest countries’ real income is substantial. For China, Azerbaijan and Nicaragua, the real income is overestimated by a factor of around four compared to the US. Hill (1994) finds that the PWT data overestimate some countries income by a factor of two. As Hill (1994) only measures the substitution bias, we see that the empirical findings support the expected effect of omitting quality: poorer countries’ income tends to be overestimated because of the failure to capture the fact that the products of poorer countries tend to have lower quality than those of richer countries. The observation that both omittance of quality and the substitution bias seems to make poorer countries’ incomes overestimated relative to the richer countries’ incomes, i.e., the overall bias is larger than the substitution bias for the poorer countries, supports the main intuition given in section 2. of households differs substantially among the countries. Despite this, the weight given to each household is the same. Two different weighting techniques have been conducted as a part of the robustness analysis, neither of which changes the main result: a weight equal to the population in the respective household’s country and a weight equal to the ratio of observations relative to the population of the country of residence.
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Table 3 reports different measures of international inequality all relying on the Gini index, where the first column reports the estimated inequality based on the PWT data and the second column reports the estimated inequality based on the corrected incomes. The third column reports the measured inequality based on the so-called exchange rate-based method that we will return to in section 7. We can see that the Gini increases substantially when correcting for the PPP bias, for both the unweighted and the population-weighted measures. While the unweighted Gini index increases from 0.45 to 0.58 when correcting for the bias, the population-weighted Gini increases from 0.58 to 0.73.8 [Table 3 about here.] [Figure 2 about here.] It is relevant to discuss whether this observed increase in inequality is robust; that is, will other inequality measures also find an increase in inequality, or is the choice of applying the Gini index essential for this finding? Figure 2 presents the Lorenz curves for uncorrected and corrected real incomes, respectively. The Lorenz curves show that the distribution of real incomes based on the biased macro price variables from the PWT Lorenz dominates that of the corrected real incomes. Hence, we have the robust conclusion that inequality is underestimated in the PWT according to any reasonable inequality measure.9
5.2
An extended model
We can apply the estimated Engel curve to estimate PPP bias for years and countries where we do not have detailed household surveys. If we have minimal information on household consumption from surveys without having the detailed surveys themselves, we can utilize this information to estimate the PPP bias for these countries and years. Instead of finding and utilizing the systematic deviation of households consumption pattern when deflated by the PWT prices in one country from the estimated Engel curve, we find and utilize the deviation of the mean of household consumption when deflated by the PWT price from the estimated Engel curve. We know that if there were no PPP bias and the estimated Engel curve represented the world Engel curve for food, the relationship between the budget share 8
Milanovic’s (2005), household survey application is an important contribution on the topic of inequality. The international inequality is equivalent to concept 1 inequality found in Milanovic (2005), whereas population-weighted inequality is equivalent to his concept 2 inequality. 9 All measures that satisfy the Pigou–Dalton criterion, which is uncontroversial, support this conclusion (Fields and Fei, 1978; Sen, 1997).
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for food and real income, when deflating by the PWT price level of consumption for a country, should be found on the world Engel curve. However, if the real income deviates from the estimated Engel curve, this can be utilized to find the bias in the PWT consumer price for this country. Moreover, if knowing the mean household budget share for food and the mean household nominal income in the currency of the estimated Engel curve, in addition to knowing the mean household demographic controls for a country, these data in addition to the estimated Engel coefficients can be applied to estimate the true price level of consumption for this country. From equation (2), we know that: mtk = a + bln ykt − b ln Pkt + θXkt
(10)
where mtk , ln ykt , and Xkt are the mean household budget share for food, the mean household logarithm of nominal expenditure and the mean household demographic characteristics, of z in country k at time t. If we normalize this, and set the true price index for country j equal to unity, Pj = 1, and apply the estimated coefficients of a, b and θ, the estimated corrected price of consumption for this same country can be expressed as follows: c c d b a + bbln ykt + θbXkt − mtk t d ln Pk = bb
(11)
c c where b a, bb and θb are the estimated coefficients for the Engel curve and Xkt and mtk are the mean household demographic controls and the mean household budget share d for food, respectively. ln ykt is the estimated mean household logarithm of nominal expenditure, by using the mean total household expenditure and some information on the distribution of income.
5.3
Generalizing the results
In this section, based on the estimated Engel curve and UN data, we estimate the bias for 46 countries in different years, totaling altogether 983 observations. The PPP biases for these 983 observations are estimated. When pooling together these 983 observations, the main results hold: that is, the poorer countries tend to have a higher bias than richer countries, and the bias is systematic—the poorer the country, the higher the bias. Subsequently, the inequality among countries increases substantially when correcting for the PPP bias. Figure 3 shows the relationship between the bias and the corrected PWT real incomes. The line labeled lowess
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gives the smoothed line for the relationship between the bias and the real income of a country. 10 [Figure 3 about here.] Table 4 gives the Gini for the uncorrected and corrected measures when extreme values are excluded, as well as when they are included. We see that the Gini coefficient increases substantially, and more so when the extreme values are included. [Table 4 about here.] The findings in this section show that the bias is systematic and the inequality is also substantially underestimated for these 983 observations, and thus the findings of the main model are robust and not dependent on the nine countries in study.
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Convergence or Divergence?
Basing the analysis on means gives us data on real incomes over time. Thus, the highly debated question of convergence versus divergence can be properly analyzed. Based on the corrected data, we can investigate whether the per capita real incomes converged, i.e., inequality between countries decreased, or diverged, i.e., inequality increased, between 1970 and 1995. We have observations from 22 countries for all years from 1970 until 1995.11 Table 5 gives the Gini coefficients for the first and last years of this period. We can see that the PPP measure gives a very high convergence rate compared to the corrected measure. Again, the exchange rate-based method is discussed in the next section. [Table 5 about here.] 10
The ten percent with the highest bias and the ten percent with the lowest bias are removed. The analysis here will focus on these 80 percent of observations in the middle of the distribution. As shown, the results are strengthened even further if the extreme observations are included. 11 The countries that we have observations on are Belgium, Canada, Denmark, Ecuador, Finland, France, Hong Kong, India, Iran, Ireland, Israel, Italy, Japan, Republic of Korea, Mexico, Norway, Singapore, South Africa, Sweden, Switzerland, Thailand and the United States. There is here a possible self-selection problem here, as it may be more likely to have good data from richer countries and countries with higher growth rates. At some later stage, however, the analysis will be extended to cover more countries and an effort made to include countries from the poorest regions, such as Africa South of Sahara (i.e., countries other than fast-growing South Africa). This can be done because we have observations on many of the years for many of the omitted countries.
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Figure 4 displays the Lorenz curves for the uncorrected PWT measure, whereas figure 5 displays the Lorenz curves for the corrected measures. We can see that by applying the uncorrected measure, we obtain a more robust conclusion for convergence because the 1995 distribution Lorenz dominates that of 1970. When correcting for the PPP bias, however, the curves cross and we do not have the robust conclusion of convergence. [Figure 4 about here.] [Figure 5 about here.]
7
Exchange rate-based method
Throughout the paper, the PWT incomes have been compared to the corrected incomes. In this section, we will turn to a different method, namely the traditional exchange rate-based method, in order to investigate whether this method produces estimates closer to the corrected real incomes. The exchange rate-based method simply transforms each country’s income into a common currency, for example US Dollars. Applying the exchange rate-based method makes us implicitly assume that the classical assumption of purchasing power parity holds, as it does not correct for price differences not reflected in the exchange rate. We can see from both table 3 and table 4 that the measured inequality based on the corrected PWT incomes is closer to the measured inequality based on the exchange rate-based method, than measures based on the PWT incomes. Furthermore, we can see from table 5 that the measured rate of convergence of the exchange rate-based method is closer to that based on PWT data. This could indicate that the traditional assumption that purchasing power parity holds, however implausible, may be a better approximation than applying the PPP-adjusted incomes of the PWT. The PWT does not report exchange rate-based measures, and UN measures and the PWT measures may thus differ in other respects than the price adjustment. Hence, in order to avoid a comparison of ’apples and pears’ we focus on the UN measures in this section, and compare the UN PPP measures to the UN exchange rate-based measures. The UN PPP measures are found using the UN consumption estimates deflated by the consumption price from the PWT. The PPP bias of the UN real incomes have exactly the same trend as the PWT PPP bias; the bias decreases with real income. In order to compare the UN PPP measures, the UN exchange rate-based measures and the corrected UN measures, we have to normalize them. This is done by
15
applying the corrected measures as a base and then normalizing so that each country’s income in each year is measured as a share of the total income; total income being the sum of all country incomes over all years. Again, we pool all observations together, and thus the share has no specific interpretation; it is merely a tool to normalize measures so that they are comparable. Figure 6 presents a scatter plot of corrected UN real incomes and the ranking of the corrected PWT incomes. We can see that there is not an exact match between the corrected UN and the corrected PWT measures, because the scatter does not follow a monotonically increasing line. However, as shown, the smoothed line increases monotonically. [Figure 6 about here.] Figure 7 displays the relationship between the UN PPP measures and the ranking of the corrected PWT measures, whereas figure 8 displays the relationship between the UN exchange rate-based measures and the ranking of the PWT measures. As shown, none of the measures is exact. In order to compare the different measures to the corrected UN measures, we construct a lower and upper bound around the kernel of the corrected measures that corresponds to the 95 percent confidence interval around an estimated relationship. Figures 9 and 10 investigate the smoothed line for the PPP-based measures and the exchange rate-based measures, respectively, and compare these to the upper and lower bounds of the kernel for the corrected measures. We can see clearly from figure 9 that the UN PPP measures overestimate the incomes of poorer countries relative to richer countries. This illustrates the systematic relationship between the PPP bias and real income of a country: that is, the poorer a country is, the more its real income tends to be overestimated relative to the richest country. [Figure 7 about here.] [Figure 8 about here.] [Figure 9 about here.] In figure 10, the exchange rate-based method is compared to the lower and upper bounds, and we can see that the exchange rate-based method gives a smoothed line that is closer to the bounds than the smoothed line for the UN PPP measures. However, the exchange rate-based line also lies outside the bounds for most income levels, and thus there is considerable bias in the exchange rate-based measures. [Figure 10 about here.] 16
Why would we expect the exchange rate-based measures to be biased? First, the fact that we do not control for price differences delivers bias. As the poorer countries tend to have lower prices for many goods, we would expect the differences in prices to deliver an opposite bias to the ones we have studied before: that is, we would tend to underestimate poorer countries’ real incomes when assuming that purchasing power parity holds. However, this is not the only effect. Quality is not picked up properly in exchange rate comparisons either, and thus we have two opposing effects delivering bias. It is obvious that as long as the bias related to differences in prices is very large compared to the other bias generating factors, the fact that we have two opposing effects makes the exchange rate-based measures closer to the true real income levels than the PPP measures. As we do not decompose the bias, the magnitude of each factor is an empirical question and, consequently, whether the traditional framework of assuming that purchasing power parity holds gives better estimates than the PWT is an empirical question, too.
8
Concluding Remarks
This paper finds that there are significant and substantial biases in the national incomes given in the PWT. Furthermore, there is a systematic relationship between the PPP bias and the national income of a country: the poorer the country, the more its income tends to be overestimated relative to a base country. Consequently, the PPP bias causes the significant and robust underestimation of international inequality: the Gini index increases substantially when correcting for the bias and the uncorrected real incomes of the PWT Lorenz dominates that of the corrected PWT real incomes. The bias also influences the measured convergence rate. For 22 countries, the predicted convergence between 1970 and 1995 is relaxed when correcting for the PPP bias. The finding of convergence is robust for the uncorrected measures, as the Lorenz curve for 1970 Lorenz dominates that for 1995. However, when correcting for the PPP bias, the two Lorenz curves cross, and the finding of convergence is not robust. Several robustness checks reported in appendix C, shows that the main findings are not driven by specification of an incorrect functional form, differences in relative prices or household composition. However, this study, as well as other studies based on micro data (or macro data deduced from micro data), could have benefited if more studies were available and already harmonized. It will be interesting to consider future work that uses even more detailed data than what is readily available now. First, if panel data sets existed for poor countries, as it does for OECD countries, 17
it would be possible to use more sophisticated estimation techniques. Second, if one data set existed with harmonized data for both rich and poor countries, it would be less demanding to do cross-country comparisons based on micro data, and more than nine countries could be included in the estimation of the Engel curve. Section 2 and appendix A provide explanations of the bias, and show that we can expect the bias to be systematic in the way we find in the empirical analysis. In addition, Dowrick and Akmal (2005) and Nuxoll (1994) provide some theoretical support for the empirical finding that income differences are underestimated in the PWT. It is open for future research to generalize these insights for the Geary–Khamis method.
References [1] T. Beatty and E.R. Larsen, 2005. ”Using Engel Curves to Estimate Bias in the Canadian CPI as a Cost of Living Index”, Canadian Journal of Economics, 38(2): 482–99. [2] R. Blundell, A. Duncan and K. Pendakur, 1998. ”Semiparametric Estimation and Consumer Demand”, Journal of Applied Econometrics 13(5), Special Issue: Application of Semiparametric Methods for Micro-Data: 435–61. [3] D. Costa, 2001. ”Estimating Real Income in United States from 1888 to 1994: Correcting CPI Bias Using Engel Curves”, Journal of Political Economy, 109(6): 1288–1310. [4] A. Deaton and J. Muellbauer, 1980. ”An Almost Ideal Demand System”, American Economic Review, 70(3): 312–26. [5] S. Dowrick and M. Akmal, 2005. ”Contradictory Trends in Global Income Inequality: A Tale of Two Biases”, Review of Income and Wealth, 51(2): 201–29. [6] G. Fields and X. Fei, 1978. ”On Inequality Comparisons”, Econometrica, 46(2): 303–16. [7] A. Gerschenkron, 1947: ”The Soviet Indices of Industrial Production.” Review of Economics and Statistics 34, 217-26. [8] B. Hamilton, 2001. ”Using Engel’s Law to Estimate CPI Bias”, American Economic Review, 91(3): 619–30.
18
[9] A. Heston, R. Summers and B. Aten, 2002. Penn World Table Version 6.1, Center for International Comparisons at the University of Pennsylvania (CICUP), Philadelphia. [10] R.J. Hill, 2000. ”Measuring Substitution Bias in International Comparisons Based on Additive Purchasing Power Parity Methods”, European Economic Review, 44(1): 145–62. [11] C. Leser, 1963. ”Forms of Engel Functions”, Econometrica, 31(4): 694-703. [12] Luxembourg Income Studies, 2006. http://www.lisproject.org/. [13] B. Milanovic, 2005. Worlds Apart. Measuring International and Global Inequality., Princeton University Press. [14] B. Milanovic, 2002. ”True World Income Distribution, 1988 and 1993: First Calculation Based on Household Surveys alone, The Economic Journal, 112: 51–92. [15] P. Neary, 2004. ”Rationalizing Penn World Table: True Multilateral Indices for International Comparisons of Real Income”, American Economic Review, 94(5): 1411–28. [16] P. Neary, 2006. http://www.ucd.ie/economic/staff/pneary/gaia/gaia.htm [17] D.A. Nuxoll, 1994. ”Differences in Relative Prices and International Differences in Growth Rates”, American Economic Review, 84(5): 1423–36. [18] X. Sala-i-Martin, 2006. ”The World Distribution of Income: Falling Poverty and Convergence, Period, Quarterly Journal of Economics, 121,2: 351–97. [19] A. Sen, 1997. On Economic Inequality, Oxford University Press, Oxford. [20] World Bank, 2005. http://www.worldbank.org/lsms/. [21] World Bank, 2007. http://devdata.worldbank.org/dataonline/old-default.htm, 2007. [22] H. Working, 1943. ”Statistical Laws of Family Expenditure”, Journal of the American Statistical Association, 38(197): 43–56. [23] A. Yatchew, 2003. Semiparametric Regression for the Applied Econometrician, Cambridge University Press, Cambridge.
19
Appendix A
The Substitution Effect
... the direction and magnitude of bias in GK (Geary–Khamis) bilateral income ratios depends on whether the GK price vector corresponds most closely to the relative price structures of high income (high productivity) countries in which case most bilateral ratios will be underestimated or whether the GK price vector corresponds most closely to the relative price structures of low income (low productivity) countries in which case most bilateral ratios will be overestimated. The former situation is most likely to apply given that the GK method weights each country’s price vector by its share in total GDP, implying that more weight is given, ceteris paribus, to the price vectors of the richer countries. [Dowrick and Akmal, 2005, our emphasis.] This appendix shows it is not only more likely that the former situation will apply, but that this follows by construction (at least in the framework considered in this appendix).
A.1
Geary–Khamis Constructed World Prices
The Geary–Khamis method, underlying the PWT, calculates a vector of reference prices applied for comparison: Π = [Π1 , Π2 , ..., ΠM ]. The main question that we study here is whether the reference price vector resembles the prices of the richer countries more than that of the poorer countries. That is, we investigate whether the price vector of a rich country j, [pj1 , pj2 , ..., pjm ] is given a greater weight when constructing the reference price vector Π, than that of a poorer country, f , [pf 1 , pf 2 , ..., pf M ]. In the specific situation of two countries and two goods in the system, this boils down to finding whether the two prices in a rich country, 1, are given a greater weight in the construction of the two reference prices, than a poorer country, 2. The way this is investigated is to compare the influence of country 1’s prices on the reference vector Π in a situation where country 1 and 2 are equally rich, to a situation where country 1 is richer. Rephrasing this, we ask the following. From a situation where both countries are equally rich in the sense that they consume equal amounts of both goods, what happens if country 1 gets richer? Country 1 can get richer in three different ways. First, it can start consuming more of both goods. Second, it can start consuming more of one of the goods and not the other. Third, it can start consuming more of one good and less of the other good, but in such a way that the total expenditure level in the country is higher than that of the other 20
country, (π1 + ∆π1 )∆q11 + (π2 + ∆π2 )∆q21 > 0. The last situation can always be reformulated so that it can be analyzed in the same way as the second situation. The results for the first situation are shown below. The main results for the second situation are equal to that for the first situation and are not reported. The weight a specific price pij is given in the construction of a specific reference price, Πi , is given by the derivative of the reference price with respect to the specific price: δΠi wij = . (12) δpij Hence, in order to study the change in the weight of a country when it is getting richer than the other, the change in this weight of getting richer has to be considered. In the two-good, two-country situation under consideration here, this boils down to finding the change in the direct effect of p11 on the reference price of good 1 and the change in the indirect effect of the price of the other good p21 on the reference price of good 1, when country 1 is getting richer, studied by a situation where the consumption of both goods increase by an equal proportion, a: ∆q11 = ∆q21 = a > 0 Hence, the answer to the question of whether country 1’s price is providing a greater weight in the construction of the reference price vector, can be answered by finding the sign of the derivative: a(
δ 2 π1 δ 2 π1 + ). δq11 δp11 δq21 δp11
(13)
In the two-country, two-good situation, the Geary–Khamis system, given in equation (1) and (3), collapses to: Πi =
qi1 pi1 + qqi2 + pi2 (qi1 + qi2 )(P P P1 + P P P2 )
(14)
where qij is the quantity of good i consumed in country j, pij is the price of good i in country j and P P Pj is the overall price index of country j given by: P P Pj =
p1j q1j + p2j q2j . Π1j q1j + Π2j q2j
(15)
In this setup, the weight of country 1’s price of good 1 in the reference price of good 1 is given by:
w11 =
q22 q21 p21 q11 (q11 p12 q12 q21 + q11 p12 q12 q22 + q21 p22 q22 q11 + q21 p22 q22 q12 ) δπ1 = . δp11 (p11 q11 q12 q21 + p11 q11 q12 q22 + p21 q21 q11 q22 + p21 q21 q12 q22 )2 (16) 21
Consider the first situation, where country 1 is getting richer than country 2 by the following change: ∆q11 = ∆q21 = a. The weight of country 1’s price of good 1 in the construction of the reference price of good 1, is then given by:
∆
δπ1 δ 2 π1 δ 2 π1 |q11 =q12 ,q21 =q22 = a( + )|q =q ,q =q δp11 δq11 δp11 δq21 δp11 11 12 21 22
3 3 3 2 3 = a(q11 p12 q12 q22 p11 −q11 p12 q22 p21 q21 +q11 q21 p22 q22 q12 +q11 p12 q12 q21 p11 +2q21 p22 q22 q11 q12 2 3 2 3 −q11 p12 q12 q21 p21 q22 +q11 q21 p22 q21 p21 +2q11 p12 q12 q21 p21 q22 −q11 q21 p22 p11 q12 +2q11 p12 q12 2321 p21 3 2 p22 q22 p21 q12 ) p22 q22 q11 p11 q12 + q21 − q21 2 (q12 p21 q22 ) |q =q ,q =q (p11 q11 q12 q21 + p11 q11 q12 q21 + p11 q11 q12 q21 + p11 q11 q12 q21 )3 ) 11 12 21 22 2 2 3 2 2 3 = a(−p12 q12 q22 p21 +2q22 p22 q12 p11 +p12 q12 p11 −q22 p22 p11 q12 +2p12 q12 q22 p21 +q22 p22 p21 )p21 1 1 . (17) 4 q12 (p11 q12 + p21 q22 )3
a is positive by construction. Furthermore, we know that all prices and quantities are positive. Hence the fraction 41 q12 (p11 q121+p21 q22 )3 is positive. In order to show that getting richer gives you a greater weight in the constructed reference price vector, it is sufficient to show that 2 2 3 2 2 3 −p12 q12 q22 p21 +2q22 p22 q12 p11 +p12 q12 p11 −q22 p22 p11 q12 +2p12 q12 q22 p21 +q22 p22 p21 (18)
is positive. Applying maple to minimize the expression in (18), we find that this can never be negative, as the minimal value is zero. When forcing all quantities and prices to be strictly larger than zero, however, the minimal value is a small value, larger than zero. The maple output from the minimization procedure is given in table 6. We can see in the first column that the minimal value of the expression in (18) is zero when the price of both goods in country 2 is zero. In the situation where both prices are zero in country 2, country 2’s prices will have zero weight in the construction of the reference prices (this can be solved from equation (16)). Thus, the prices of country 1 are given maximal weight in this situation, and becoming richer will not influence the weight. However, the minimal value is never negative, and by forcing prices and quantities to be strictly larger than zero, the effect of getting richer is positive. [Table 6 about here.]
22
Appendix B
Robustness Analysis
This appendix conducts four different tests of robustness of the estimates from the main model based on nine countries. The main results are sustained in each of the tests. First, the preferred specification given in (4) is estimated on the subset of households with two children and two adults, in order to test whether differences in household composition influence the main results. Second, an alternative specification is considered by applying the OECD adult equivalence. Third, a semiparametric analysis is conducted in order to study whether the functional form fits the data in the study. Fourth, relative prices are included and the standard AIDS specification given in equation (1) is estimated on the subgroup of our sample where relative prices are available, i.e., on the households in the five countries in which cross-country comparable relative prices exist. Except for the semi-parametric analysis, the figures illustrating the results are presented in the appendix.
B.1
Household composition
The first robustness check is conducted in order to study whether inaccuracies caused by heterogeneous household composition affect the main findings. In order to avoid effects from household composition and size, we restrict our analysis to households consisting of two children and two adults. In the total sample, there are 4,968 such households, and equation (4) is estimated on this subsample. The regression results are reported in the second column of Table 2. The four non-OECD countries also have the highest bias in this estimation, and the main picture given in Figure 1, is sustained. Cˆote D’Ivoire has the highest measured bias in this estimation, and China follows as the country with the second largest bias. Nicaragua and Azerbaijan have slightly smaller measured biases than China. The dummy coefficient for France is no longer significantly different from zero; that is, we cannot statistically distinguish the bias of the macro price variable of France from that of the USA. The measured bias is greater the lower the real income as shown in figure 11. Table 7 reports the Gini indices for this robustness check. The Gini indices also increase substantially in this case, from 0.45 to 0.58, and we still obtain Lorenz dominance of the uncorrected measures (see figure 12). Hence, household composition does not seem to be crucial for our results. We also note that the estimated income elasticity is more in line with other related studies than the main estimation (Costa, 2001; Hamilton, 2001). [Figure 11 about here.] [Table 7 about here.] [Figure 12 about here.] 23
B.2
Functional form - does the semilog specification fit the data? [Figure 13 about here.]
A main concern with the method applied in this paper is to what extent the functional form specification is restrictive. In order to study the functional form, a semi-parametric analysis based on differencing is conducted. All variables, except the logarithm of real income, are included linearly in the regression. The robustness check, thus, investigates whether the log-linear relationship between the budget share for food and real income fits the data well. Figure 13 shows the kernel regression between the budget share for food and the logarithm of real income after the effect from the other variables is differenced away. The kernel regression function is linear where the curve is precisely defined, i.e., where the upper and lower bounds from the bootstrapping coincide with the kernel itself. The semi-parametric analysis, thus, confirms that the log-linear relationship between the budget share for food and real income assumed in equation (4) fits the data well. As we would expect, it fits better for the medium to high income levels where we have more observations, than for the fewer observations in the lower tail of the income distribution. For the lower incomes, the line does not seem to be completely linear and we have a group of observations to the southwest of the line of the specified functional form. If this were the true functional form, the fact that the curve was not completely linear would give a downward effect on the dummies for the poorest countries. The estimated dummies for the poorest countries are larger than unity, however, and thus our estimates would be conservative estimates if the functional form of figure 13 were the correct one and not the log-linear form. However, we consider the line given in figure 13 to be log-linear in the sense that we cannot reject the functional form assumed in this analysis.
B.3
Including relative price effects
Given that comparable national relative prices are available for five of the nine countries in the study, we examine whether including these relative prices changes the main results of this paper. This is done by estimating equation (1) on the subsample of households in the five countries that have these prices available, and applying the significant relative price effect estimated in Costa (2001). Given that the coefficient for the relative price is already estimated in Costa (2001), one new net variable is constructed. The left-hand side dependent variable is now the difference between the budget share for food and the effect of relative 24
prices given in equation (8). The coefficient of the logarithm of relative prices is assumed to be equal to 0.006, Costa’s estimated coefficient.12 The new left-hand side variable is defined as: mch,j = mh,j − 0.006 ∗ (ln(Pf,j ) − ln(Pn,j )).
(19)
Based on this variable, a new regression is run, and a new set of dummy coefficients, and subsequently a new set of PPP biases, are estimated. The estimation results are given in Table 7, column three. The PPP bias is also in this case a function of the coefficient of the logarithm of real income and the country dummy coefficient. In addition to this, it is now a function of the bias in the measured prices for food and nonfood items, and is given by: ln Ej =
γ dj (ln Ef,j − ln En,j ) − b b
(20)
where Ef,j and En,j are the biases in the measured prices for food and nonfood items, respectively. We have no method to identify the bias in all three prices at the same time, and it is assumed here that the bias in the price for food cancels out the bias in the nonfood price, i.e., that there is no bias in the relative price. Under this assumption, the bias is measured as expressed in equations (3) and (6). This assumption is quite strong, and because we cannot identify all the biases, we cannot test whether this assumption is valid. However, we know that the estimated coefficient of relative prices is far lower than the coefficient of real income and thus, the major effect picked up by the country dummy coefficient is from the PPP bias. Despite this, it should be kept in mind that if the bias for the food price is larger than the bias for the nonfood price, the bias is overestimated. The opposite will be true if the bias for the nonfood price is larger than the bias for the food price. For the five countries in the study, the measured PPP bias is higher the lower the real income (see figure 14). Also, in this case, a negative relationship between the PPP bias and real income can be displayed. Table 7 reports the Gini indices before and after the correction. The Gini index for these countries is 0.17 when using data from the PWT, whereas when correcting for the measured PPP biases, we get a substantial increase of the Gini index to around 0.30. As shown in figure 15, the uncorrected price measures for these countries also Lorenz dominate the corrected 12
The corresponding price elasticity is approximately 0.68. This price elasticity is calculated as −1 + [(γ − αb)/m], where α is the share of the food in the US total price index (Costa, 2001).
25
measures. Thus, all four main findings are also sustained when relative prices are included in the analysis. [Figure 14 about here.] [Figure 15 about here.]
26
4 3 Relative bias, adeq 2 1 0 0
5000
10000 Real income
15000
20000
0
.2
.4
.6
.8
1
Figure 1: PPP bias and real income. The figure presents a declining relationship between PPP bias and corrected real income per capita in international dollars for the nine countries. The bias for country j is measured relative to the US bias, −dj relativebiasj = Ej − 1 = e b − 1.
0
.2
.4 .6 Cum. Pop. Prop. Cum. Pop. Prop. PWT
.8
1
adeq
Figure 2: Lorenz dominance for the main model, applying the OECD adult equivalence scale. The distribution of the biased real incomes (PWT) Lorenz dominates the Lorenz curve for the distribution of the corrected real incomes (adeq).
27
20 15 10 5 0 0
5000
10000 Real income
Relative bias, adeq
15000
20000
lowess rel_bias corrcons
0
.2
.4
.6
.8
1
Figure 3: The relationship between the PPP bias and real income— extended model. The figure shows exactly the same as figure 1. The relationship between the bias relative to the bias of the United States and the real per capita income of a country is displayed. We can also see that the extended model shows a declining relationship between bias and the per capita real income of a country.
0
.2
.4 .6 Cum. Pop. Prop.
.8
1
Cum. Pop. Prop. Cum. Dist. of totalconsumption_1970/_N Cum. Dist. of totalconsumption_1995/_N
Figure 4: The Lorenz curves for 1970 and 1995 based on the uncorrected UN measures of real income.
28
1 .8 .6 .4 .2 0 0
.2
.4 .6 Cum. Pop. Prop.
Cum. Pop. Prop. Cum. Dist. of corrcons_1995/_N
.8
1
Cum. Dist. of corrcons_1970/_N
0
.002
.004
.006
Figure 5: The Lorenz curves for 1970 and 1995 based on the corrected UN measures of real income.
0
200
400 600 Ranking corrected PWT incomes
UN_corrected
800
lowess UN_corrected obs
Figure 6: The corrected UN measures. The figure displays a scatter plot of the corrected UN incomes as a function of the ranking of the corrected PWT incomes. The line labeled lowess gives the smoothed line of the observations.
29
.004 .003 .002 .001 0 0
200
400 600 Ranking corrected PWT incomes
UN_PPP
800
lowess UN_PPP obs
0
.002
.004
.006
Figure 7: The UN PPP measures. The figure displays a scatter plot of the UN PPP incomes as a function of the ranking of the corrected PWT incomes. The line labeled lowess gives the smoothed line of the observations.
0
200
400 600 Ranking corrected PWT incomes
UN_exchange
800
lowess UN_exchange obs
Figure 8: The UN exchange rate-based measures. The figure displays a scatter plot of the UN exchange rate-based incomes as a function of the ranking of the corrected PWT incomes. The line labeled lowess gives the smoothed line of the observations.
30
.004 .003 .002 .001 0 0
200
400 obs lowess UN_PPP obs lower
600
800 upper
Figure 9: The UN PPP and the corrected UN measures. The figure displays the smoothed lines for the UN PPP measures as well as the upper and lower bounds for the corrected UN measures. The upper and lower bound defines the confidence interval at a five percent significance level. Bootstrapping around the kernel for the corrected UN measures is applied in order to display the bounds. The kernel relation is found using the Epanechnikov kernel smoother. The bandwidth used is found from the formula bandwidth = 0.15 ∗ (maxcorrectedU N − −mincorrectedU N ) where maxcorrectedU N and mincorrectedU N are the maximum and minimum values of the corrected UN measures of real income, respectively. The bounds correspond to 95% confidence intervals.
31
.004 .003 .002 .001 0 0
200
400 obs
600
lowess UN_exchange obs lower
800 upper
0
Relative bias, same composition 2 4
6
Figure 10: The UN exchange rate-based and the corrected UN measures. The smoothed line for the exchange rate-based measures is displayed together with the upper and lower bounds of the corrected UN measures. Details for the upper and lower bounds are given in the description of figure 6.
0
5000
10000 Real income
15000
20000
Figure 11: Relative bias and per capita real income: The curve illustrates the relationship between measured bias and per capita real income when only households with two children and two adults are applied in the estimation.
32
1 .8 .6 .4 .2 0 0
.2
.4 .6 Cum. Pop. Prop. PWT Cum. Pop. Prop.
.8
1
samecomp
0
.2
budget share for food .4 .6 .8
1
Figure 12: Lorenz dominance when including households with two children and two adults. The distribution of the biased real incomes (PWT) Lorenz dominates the Lorenz curve for the distribution of the corrected real incomes (samecomp).
0
2
4 logarithm of real income
budget_share_for_food upper_bound
6
8
lower_bound
Figure 13: Kernel regression. The Kernel relation using the Epanechnikov kernel smoother: the relationship between the budget share for food and the logarithm of real income when the effects from the other explanatory variables are differenced away. A tenth order differencing is conducted based on the optimal differencing weights proposed in Yatchew (2003). The bandwidth is equal to 1.36149. The bandwidth used is found from the formula bandwidth = 0.15 ∗ (maxlogrealcons − −minlogrealcons ) where maxlogrealcons and minlogrealcons are the maximum and minimum values of the real logarithm of expenditure, respectively. The bounds correspond to 95% confidence intervals. We see that for values of the logarithm of real income that are below 2.7 the curve is not precisely defined because of few observations and it is not clear whether the curve is linear in this part or not. However, to the right of this point, the curve is precisely defined and it is linear.
33
1 .8 Relative bias, Costa .4 .6 .2 0 0
5000
10000 Real income
15000
20000
0
.2
.4
.6
.8
1
Figure 14: Relative bias and per capita real income: The curve presents the same relationship for the five OECD countries when national relative prices are included.
0
.2
.4 .6 Cum. Pop. Prop. Cum. Pop. Prop. PWT
.8
1
Costa
Figure 15: Lorenz dominance when including relative price effects. The distribution of the biased real incomes (PWT) Lorenz dominates the Lorenz curve for the distribution of the corrected real incomes (Costa).
34
Azerbaijan China Cˆote D’Ivoire France Hungary Italy Nicaragua United Kingdom United States
Survey year Institution No. of hh 1995 SORGU / World Bank 1966 1994 Min. of Agg./World Bank 786 1986 Inst. Nat. Stat. / World Bank 1573 ´ ´ 1995 Inst. Nat. Stat. Etud. Ec. / LIS 9627 1996 Hungarian Cent. Stat. Off. 7528 1995 Bank of Italy / LIS 8116 1993 INEC / World Bank 3185 1995 UK Data Archive / LIS 6789 1995 CES, US Bureau of Labor 12973
Nat. Repr. Yes No Yes Yes Yes Yes Yes Yes Yes
Table 1: The different surveys. This table gives an overview of the nine different surveys included in the study and the institutions that conducted the different studies.
35
log of real income (adeq adj.)
Adeq -.094∗∗∗
Same-Composition
Costa
-.097∗∗∗
-.089∗∗∗
(.003)
(.0009)
(.001)
log of real income Azerbaijan
∗∗∗
.144
(.005)
China Nicaragua Cˆote D’Ivoire
∗∗∗
.153
Italy children adults
.153∗∗∗
.147∗∗∗
.147∗∗∗
(.004)
(.011)
.142∗∗∗ ∗∗∗
.051
∗∗∗
.006
(.001)
United Kingdom
(.018)
(.012)
(.002)
France
.151
(.005)
(.005)
Hungary
∗∗∗
-.031
∗∗∗
.186∗∗∗ (.019)
.046∗∗∗
.055∗∗∗
(.007)
(.002)
-.0008
.018∗∗∗
(.004)
(.001)
-.019
∗∗∗
(.001)
(.004)
(.001)
.075∗∗∗
.086∗∗∗
.064∗∗∗
(.002)
(.005)
(.002)
-.0002
.021∗∗∗
(.0005)
(.0006)
-.005
∗∗∗
.029∗∗∗
(.0005)
age cons Number of observations Adjusted R-squared
-.006∗∗∗
.0006
∗∗∗
(.0006) ∗∗∗
.001
.0006∗∗∗
(.00003)
(.0002)
(.00003)
.663∗∗∗
.727∗∗∗
.621∗∗∗
(.006)
(.020)
(.006)
52543 .564
4968 .619
45033 .383
Table 2: Regression results. The table reports three sets of estimates. The estimates for the main model given in equation (2) and the preferred specification are reported in the first column. The second column gives the estimates for the main model on the sub-sample of households with two children and two adults. Finally, the third column reports the estimates when relative prices are included. The model specification including relative prices is deduced from equation (1) by applying Costa’s (2001) estimated relative price coefficient as specified in (8).
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Main model Population weighted
PWT 0.45 0.58
Corr Gini 0.58 0.73
Gini UN exchangerate 0.54 0.72
Table 3: Gini Indices. The table shows the Gini index measured by the PWT data, the corrected real incomes and the UN exchange rate measures. Row 1 gives the unweighted Gini indexes, i.e., the indexes that will give equal weight to each country irrespective of country size. The measures in row 2 are weighted measures that give each country a weight proportional to the population size.
Gini Pop weighted Gini Gini (whole sample) Pop weighted Gini (whole sample)
PWT 0.37 0.56 0.49 0.59
Corr incomes 0.46 0.70 0.88 0.85
UN Exch rate based 0.48 0.64 0.59 0.68
Table 4: The Gini coefficient for the uncorrected and corrected real incomes.
Measure Exchr based PWT PPP PWT corrected
1975 0.39 0.32 0.42
1995 0.36 0.25 0.38
Convergence / Divergence C C C
Table 5: The Gini coefficient for the different measures of 1970 and 1995, respectively.
Restrictions Minimal value p11 p12 q11 q22 p21 p22 q21 q22
q’s ≥ 0 and p’s≥ 0 0 2.596 0 1.770 1.770 0.847 0 1.029 1.029
q’s >0 and p’s> 0 0.114e − 19 3.904 0.100e − 8 2.400 2.400 0.100e − 8 0.100e − 8 1.015 1.015
q’s≥ 1 and p’s≥ 1 0.107e − 9 1742.8 1.000 837.4 837.4 1.00 1.00 37.15 37.15
Table 6: The maple output of the minimization. The table shows the minimal value of () and the values for prices and quantities at the minimal value for three different restriction on prices and quantities.
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Unweighted Pop weighted
Gini PWT Gini main 0.45 0.59 0.58 0.73
Gini same comp 0.58 0.72
Gini OECD 0.17 0.06
Gini Costa 0.30 0.14
Table 7: The Gini indexes. The table shows the Gini indexes for the uncorrected PWT measures, the corrected measures based on the main model, the corrected measure based on the model estimating on the subsample of household with two children and two adults. In addition the two last columns give the uncorrected Gini index for the sample of OECD countries and the corrected Gini index when including relative prices for these countries.
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Curriculum Vitae Ingvild Alm˚ as E-mail:
[email protected] Telephone: +47 559 59349 Fax: + 47 559 59543 Title: PhD student/fellow Nationality: Norwegian Date of birth: 03.10.1978
Education • Cand. Oecon, Department of Economics, University of Oslo • Development studies, one year (grunnfag), Oslo University College • One year as a visiting student/scholar at Cornell University, USA
Field of interests • International inequalities: Focus on PPP measures. • Fairness and Redistribution: Measurement of inequality and fairness. And Experimental approaches. • Demographics, Pensions and International relations.
Doctoral Project Income Inequality and Fairness measures. Focus on PPP measures.
Working papers • International Income Inequalities: Measuring PPP bias by estimating Engel curves for food. NHH department of Economics working paper No 17/07. • Investment to serve future consumption needs - Trade theory applied to demographic challenges. SNF Working Paper No 72/05. 1
Work in progress • International Income Inequality: Measuring PPP bias by estimating Engel curves for food. • Equalizing income versus equalizing opportunity - A comparison of the United States and Europe. • Measuring Unfair Inequality - evidence from Norwegian data. With Aleaxander W. Cappelen, Jo Thori Lind, Erik Ø. Sørensen and Bertil Tungodden. • Measuring Wealth Inequality and Poverty: A Life-Cycle Perspective. With Magne Mogstad. • Development of fairness preferences in children. With Aleaxander W. Cappelen, Erik Ø. Sørensen and Bertil Tungodden.
Presentations and Conference participation • University of Minnesota, forthcoming, November, 2007: International Income Inequality: Measuring PPP bias by estimating Engel curves for food. • Federal Reserve Bank of Minneapolis, forthcoming, November, 2007: International Income Inequality: Measuring PPP bias by estimating Engel curves for food. • Measuring and comparing unfair inequality across countries. ECINEQ conference, Berlin, 2007. • International Income Inequalities: Measuring PPP bias by estimating Engel curves for food. Spring Meeting of Young Economists, Hamburg, 2007. • Measuring Unfair Inequality - evidence from Norwegian data. Forskermøtet, Tromsø, 2007. • International Income Inequality: Measuring PPP bias using Engel curves for food. University College Dublin, 2006. • Student presentation at Cornell University: Measuring PPP bias by estimating Engel curves for food. • Measuring PPP bias using Engel curves for food. Social Choice and Welfare, Istanbul, 2006.
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• Measuring PPP bias using Engel curves for food. Nordic Conference for Development Economics, Oslo, 2006. • Investment to serve future consumption need - Trade theory applied to demographic challenges. Nordic Conference for Development Economics, Helsinki, 2005. • Participation at 1st Lindau Meeting of the Winners of the Bank of Sweden Prize in Economics Sciences in Memory of Alfred Nobel, 2004.
Publications • Internasjonal handel - Gevinster til alle? RØST: Økonomisk teori og politisk praksis, 2005, ISBN-82-996845-I-X (Norwegian only). • Investment to serve future consumption needs. Master Thesis University of Oslo, 2003. • Den internasjonale debatten om pensjonssystemer. RØST: Pensjoner, 2003, ISBN-82-996845-0-I (Norwegian only).
References • Bertil Tungodden, Professor, Supervisor and Head of Department of Economics, Norwegian School of Economics and Business Administration. e-mail:
[email protected]. Further contact details: http://www.nhh.no/sam/cv/tungodden-bertil.html • Gernot Doppelhofer, Associate professor, Department of Economics, Norwegian School of Economics and Business Administration. PhD Columbia University 2000. e-mail:
[email protected]. Further contact details: http://www.nhh.no/sam/cv/doppelhofer-gernot.html
Past experience • Research assistant, Frisch Centre, project on pensions. • Research assistant, Department of Economics, University of Oslo, project on Ragnar Frisch Biography. • One-year-scholarship, Norwegian Institute of International Affairs. • Seminars in mathematics and economics for undergraduate students, Department of Economics, University of Oslo. 3
Other experience Referee for Scandinavian Journal of History and Økonomisk Forum.
Feature articles (Norwegian only) • Dagens Næringsliv, 16.04.07, Oljepenger og sykepleiere fra Afrika. • Klassekampen economics feature series: – 31.03.05: Internasjonal inntektsulikhet - et spørsm˚ al om selvrespekt? – 24.02.05: Spillets regler - Er der noen forskjell p˚ a ikke-diskriminering og likestilling? – 27.01.05: Hønen eller egget - Er der noen sammenheng mellom ˚ apenhet og økonomisk framgang? – 30.12.04: Søppel p˚ a Upsete. – 25.11.04: Ting tar tid.
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