metadata: authors: 0: author: Van Der Weide,Roy
1: author: Lakner,Christoph
2: author: Ianchovichina,Elena
count: Egypt, Arab Republic of
docna: 0: docna: Is inequality underestimated in Egypt ?
            evidence from house prices
docty: Policy Research Working Paper
subtopic: Poverty Reduction Strategies,Urban Housing and Land Settlements,Urban Housing,Urban Governance and Management,Municipal Management and Reform,Labor & Employment Law,Inequality,Armed Conflict
teratopic: Law and Development,Communities and Human Settlements,Conflict and Development,Public Sector Development,Urban Development,Poverty Reduction
abstracts: cdata!: Household income surveys often fail to
            capture top incomes which leads to an underestimation of
            income inequality. A popular solution is to combine the
            household survey with data from income tax records, which
            has been found to result in significant upward corrections
            of inequality estimates. Unfortunately, tax records are
            unavailable in many countries, including most of the
            developing world. In the absence of data from tax records,
            this study explores the feasibility of using data on house
            prices to estimate the top tail of the income distribution.
            In an application to Egypt, where estimates of inequality
            based on household surveys alone are low by international
            standards, the study finds strong evidence that inequality
            is indeed being underestimated by a considerable margin. The
            Gini index for urban Egypt is found to increase from 36 to
            47 after correcting for the missing top tail.
geo_reg_key: 119229,517202,517191
repnme: repnme: Is inequality underestimated in Egypt ?
            evidence from house prices
display_title: Is inequality underestimated in
            Egypt evidence from house prices
listing_relative_url: /research/2016/06/26514568/inequality-underestimated-egypt-evidence-house-prices
topicv3: Poverty m1326687 695,Land and Housing m1327648 817,Revolutionary Violence m1327774 2465,Income Inequality m1503891 2668,Household Surveys m1504071 2761
publishtoextweb_dt: 2016-06-22T00:00:00Z
historic_topic: Law and Development,Communities and Human Settlements,Conflict and Development,Public Sector Development,Urban Development,Poverty Reduction
no_of_pages: 31
author_keys: 0: author_key: 677092
1: author_key: 1206398
2: author_key: 453205
model_extractions: WPS7727 Policy Research Working Paper 7727 Is Inequality Underestimated in Egypt? Evidence from House Prices _Roy van der Weide Christoph Lakner Elena Ianchovichina_ Development Research Group Poverty and Inequality Team & Middle East and North Africa Region Office of the Chief Economist June 2016
Policy Research Working Paper 7727

# **Abstract**

Household income surveys often fail to capture top incomes which leads to an underestimation of income inequality. A popular solution is to combine the household survey with data from income tax records, which has been found to result in significant upward corrections of inequality estimates. Unfortunately, tax records are unavailable in many countries, including most of the developing world. In the absence of data from tax records, this study explores the feasibility of using data on house prices to estimate the top tail of the income distribution. In an application to Egypt, where estimates of inequality based on household surveys alone are low by international standards, the study finds strong evidence that inequality is indeed being underestimated by a considerable margin. The Gini index for urban Egypt is found to increase from 36 to 47 after correcting for the missing top tail.

This paper is a product of the Poverty and Inequality Team, Development Research Group and the Office of the Chief Economist, Middle East and North Africa Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at rvanderweide@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Is Inequality Underestimated in Egypt? Evidence from House Prices

Roy van der Weide, Christoph Lakner and Elena Ianchovichina<sup>_∗_</sup> _JEL classification_ : D31.

_Keywords_ : inequality; top incomes; house prices; Egypt.

> _∗_ All authors are with the World Bank. Contact information: rvanderweide@worldbank.org, clakner@worldbank.org and eianchovichina@worldbank.org. This is a background paper for the report entitled “Inequality, Uprisings, and Conflict in the Arab World” led by the World Bank’s Chief Economist Office for the Middle East and North Africa region. The authors wish to thank Guoliang Feng and Youssouf Kiendrebeogo for excellent research assistance. We would like to thank Facundo Alvaredo, Francisco Ferreira, Nadine Ghobrial, Vladimir Hlasny, Aart Kraay, Peter Lanjouw, Branko Milanovic, Thomas Piketty, Martin Ravallion, Paolo Verme and participants of the World Bank workshop on the Arab Inequality Puzzle and the IARIW-CAPMAS Conference “Experiences and Challenges in Measuring Income, Wealth, Poverty and Inequality in the Middle East and North Africa” for useful comments. We are grateful to the UK Department for International Development for financial assistance through its Strategic Research Program.
# **1 Introduction**

Estimates of income inequality are conventionally derived from household income and expenditure surveys. Due to the sizeable cost of collecting accurate data on household standards of living the sample size of these surveys generally constitutes less than half a percent of the total population. Unfortunately, the rich are often missing or under-covered, either due to non-response or under-reporting of income or both; see the recent literature on top income shares (e.g. Atkinson et al., 2011). Surveys still permit accurate estimation of median income and measures of poverty, even when data on top incomes are poor or are missing altogether, since the rich make up a small percentage of the total population. For the estimation of income inequality however, having good data on top incomes is crucial.

A remedy that has gained considerable traction recently is to estimate the top tail of the income distribution using data from income tax records. This estimate of the top tail can then be combined with an estimate of the bottom part from the household survey to obtain an estimate of the complete income distribution (Atkinson, 2007; Alvaredo, 2011; Alvaredo and Londo ̃no V ́elez, 2013; Diaz-Bazan, 2014; Anand and Segal, 2015).<sup>1</sup> Income tax records denote the ideal source of data as far as top incomes are concerned. For lower incomes tax records may be less reliable, here the household income survey arguably denotes the ideal data. When household survey and tax data are combined in this way, the Gini index for (i) the United States in 2006 increases from 59 to 62 (Alvaredo, 2011), (ii) Colombia in 2010 from 55 to 59 (Alvaredo and Londo ̃no V ́elez, 2013), and (iii) Korea in 2010 from 31 to 37 (Kim and Kim, 2013).

For all the pros of income tax records, the availability of the data is unfortunately rather limited, particularly in developing and emerging economies. The World Top Incomes Database (Alvaredo et al., 2015) for example includes no countries from the Middle East and North Africa region. Furthermore, data derived from tax records are less useful in places where tax evasion is more pervasive, as is the case in many developing countries. It should also be noted that combining household survey data and tax records is not without complications because the two data sources use different income definitions (disposable versus taxable) and have different units of analysis (households versus tax units, which could be individuals).

In the absence of data from tax records, this study explores the feasibility of using data on house prices to estimate the top tail of the income distribution.

1Diaz-Bazan (2014) generalizes the method of Atkinson (2007) and Alvaredo (2011) by allowing for a more general choice of the cut-off level for joining up the distributions. Morelli et al. (2015) review the literature attempting to combine household surveys and tax data in rich countries.

2
Market house price data can often be obtained more easily and, most importantly, tend to be available in the public domain, in contrast to tax administration data which are subject to important confidentiality concerns and require cooperation from governments. Also, house sellers have no incentive to understate the value of their homes, in contrast to the income they report on their tax returns.

Using house prices as an alternative to income tax records demands two methodological innovations to the study of top incomes. Firstly, we will not be observing actual household income or expenditure (as is the case with tax record data), but rather a predictor of income. Secondly, a database with house price listings is generally not obtained using a particular sampling design. Therefore, the data are not guaranteed to provide a nationally representative sample, they will arguably be biased towards large urban centers. We will propose workable solutions to both these challenges that will hopefully contribute to a wider use of this approach. Note that the methodology is not restricted to the use of house prices, it can be applied to any database containing predictors of top incomes.

We illustrate our approach with an empirical application to Egypt which provides a good testing ground for our method. In addition to being a major Arab country, inequality in Egypt is of considerable interest not least because it has been cited as one of the factors behind the Egyptian revolution (Hlasny and Verme, 2013). Estimates of inequality based on household surveys suggest that inequality is low in Egypt and that it has declined in the last decade to a Gini of around 31 in 2009. Using house prices to capture top incomes we find that inequality may be significantly underestimated in Egypt. The Gini for urban Egypt in 2009 is estimated at 47 _._ 0 compared to a survey-only estimate of 36 _._ 4. Our results are in contrast with other studies using different methods of adjusting for top incomes in Egypt (Hlasny and Verme, 2013), which report a more modest effect.<sup>2</sup> Their correction however does not consult a second source of data. If the main problem is that high income earners are simply missing from the survey, then no adjustment that relies soley on the survey will resolve the downward bias in estimates of inequality. The only way to obtain a meaningful correction is to bring in a second source of data that carries the necessary information on top incomes and hence will permit for the consistent estimation of income inequality. This reasoning is shared by Alvaredo and Piketty (2014) who similarly argue that the household survey data by themselves are insufficient to estimate top incomes in Egypt. While they make an appeal for making data on income tax records available, we propose to work with house price data instead. It should be noted that relying on predictors of top incomes rather than actual incomes derived from

> 2The Gini coefficient of household expenditure per capita in 2009 increases from 30 _._ 5 to 31 _._ 8 which is found to be statistically significant, but not economically significant.

3
tax records is not without caveats. For example, we need to make assumptions about the functional form of the relationship between the house price and household income, and about the functional form of the upper tail of the house price distribution. In addition it is assumed that one house constitutes one household and that all houses are domestically owned. Therefore, in cases where tax record data are available these should undoubtedly be considered first. However, we certainly believe that our approach provides more reliable estimates of inequality than estimates obtained using survey data alone. The perfect should not be the enemy of the good.

This paper is related to a number of other studies which have tried to correct household surveys for the problem of missing or underreported top incomes.<sup>3</sup> Korinek et al. (2006) exploit geographic variation in response rates to correct for selective non-response in the United States. Lakner and Milanovic (2015) exploit the gap between household surveys and national accounts to adjust the top end of the income distribution.<sup>4</sup> This paper is structured as follows. The methodology is presented in Section 2. In Section 3 we introduce the data used in the empirical application to Egypt. The empirical application itself is presented in Section 4. Finally, Section 5 concludes.

# **2 Methodology**

## **2.1 Combining income survey with top income data**

The objective is to estimate the level of income inequality for a given population. We will refer to database 1 (DB-1) as the primary data source for the estimation of inequality. It is assumed that top incomes are mostly missing from this database. Database 2 (DB-2), which we will refer to as the secondary data source, primarily contains data on top incomes but generally not on lower incomes. Estimates of income inequality will be biased if computed using any single one of these databases. It takes a combination of the two to obtain consistent estimates of inequality. DB-1 commonly represents a household income or expenditure survey. For DB-2 researchers often look at tax record data, as is discussed in the introduction.

> 3Recently, the EU-SILC survey in some countries began using register-based information (including tax records) for some questions (J ̈antti et al, 2013). This is of course preferable to any ex-post combination of these different data sources, as we use in this paper. In the year after the introduction of the register data, the Gini index for France increased from 39 to 44, which is consistent with the previously used household data underestimating top incomes (Burricand, 2013).

> 4See also the study on global interpersonal inequality by Anand and Segal (2015) who append for every country the estimated top 1% share to the household survey distribution. The latter is assumed to represent the bottom 99%. For the majority of countries, the top 1% share is predicted from a cross-country regression using the top 10% share in the household survey.

4
Let us denote household income by _y_ and its cumulative distribution function by _F_ ( _y_ ). Let _τ_ denote the income threshold above which we will refer to incomes as “top incomes”, and let _λ_ measure the share of the population enjoying a top income, i.e. _λ_ = _Pr_ [ _Y > τ_ ] = 1 _− F_ ( _τ_ ). It is assumed that DB-1 permits a consistent estimator for _F_ 1( _y_ ) = _Pr_ [ _Y ≤ y|Y ≤ τ_ ], and that DB-2 permits a consistent estimator for _F_ 2( _y_ ) = _Pr_ [ _Y ≤ y|Y > τ_ ]. By the same token it is assumed that DB-1 does not permit a consistent estimator for _F_ 2( _y_ ), while DB-2 does not permit a consistent estimator for _F_ 1( _y_ ). Suppose also that an estimate of _λ_ is available.<sup>5</sup> Given estimates of _F_ 1( _y_ ), _F_ 2( _y_ ) and _λ_ , an estimator for the complete income distribution function _F_ ( _y_ ) can be obtained as follows:

Given _F_ ( _y_ ), any measure of income inequality can readily be computed. Alternatively, one may appeal to the sub-group decomposition of one’s inequality measure of choice, which would by-pass the need for evaluating the income distribution for the population ( _F_ ( _y_ )). We have two sub-groups, those with income below _τ_ (sub-group 1) and those with income above _τ_ (sub-group 2). Let _Pk_ denote the population share of sub-group _k_ , and let _Sk_ denote their corresponding income shares, i.e. _Sk_ = _Pkμk/μ_ , where _μk_ and _μ_ measure average income in sub-group _k_ and the total population, respectively. Note that _P_ 1 = 1 _− λ_ and _P_ 2 = _λ_ . Let us also define _S_ 1 = 1 _− s_ and by extension _S_ 2 = _s_ . It can be verified that income inequality as measured by the Gini coefficient satisfies the following decomposition (see e.g. Alvaredo, 2011): where _Ginik_ measures the Gini coefficient for population sub-group _k_ . A similar decomposition can be obtained for the mean-log-deviation _MLD_ (see e.g. Shorrocks, 1980):

> 5It is generally assumed that DB-2 contains the total number of units (i.e. households or tax units) whose income is above _τ_ . Combined with the total population this yields an estimator for _λ_ .

5
and for the Theil index _T_ (see e.g. Shorrocks, 1980): where _MLDk_ and _Tk_ measure the mean-log-deviation and Theil index for population sub-group _k_ , respectively. Note that the between-group inequality components of both the mean-log-deviation and the Theil index equal the difference between the arithmetic- and the geometric mean income levels. They differ only in the weights used in the geometric mean; the mean-log-deviation weighs the subgroup means by their population shares whereas the Theil index weighs them by their incomes shares.

An inspection of the three sub-group decompositions tells us that the Theil index will be most sensitive to the top tail of the income distribution.<sup>6</sup> To illustrate the significance of the top tail to total inequality consider the limit where the population share of top income earners tends to zero ( _λ →_ 0) while their income share tends to some positive value ( _s >_ 0). It can readily be seen that the between-group inequality component of the Gini coefficient tends to _s >_ 0 in that case, while the within-group inequality among top income earners tends to zero, i.e. _G →_ (1 _− s_ ) _Gini_ 1 + _s_ . It follows that the between-group inequality component for the mean-log-deviation tends to log (1 _− s_ )<sup>_−_1</sup> , while also here (as with the Gini) the within-group inequality among top earners tends to zero (yet it does not discount the contribution of within-group inequality among non-top earners), i.e. _MLD → MLD_ 1 _−_ log (1 _− s_ ). The Theil index stands out as the only of the three inequality measures where the within-group inequality among top earners does not vanish (i.e. makes a positive contribution to total inequality) while the between-group inequality component will tend to infinity (when _μ_ 2 tends to infinity as _λ →_ 0 while _s >_ 0).

# **2.2 An alternative to top income data: Challenges**

As stated in the previous section, DB-2 (the top income database) typically takes the form of tax record data. These data have at least two advantages: (1) it directly observes realized incomes (which makes the estimation of _F_ 2( _y_ ) or any income statistics such as inequality among top earners rather straightforward), and (2) it provides a count of the number of top income earners, which makes for

> 6Hence it is expected that any efforts made to fix the top tail of the income distribution by bringing in complementary data (top income database) will be rewarded the most by the Theil index.

6
a straightforward estimation of _λ_ . A key disadvantage of tax record data is that it is often difficult to obtain access to them. Moreover, they are more likely to be available in developed countries with good quality data systems in place, and less likely to be available in developing countries.

This paper explores the feasibility of using an alternative to tax record data that are more readily available. The empirical application presented in Section 4 considers data on house prices compiled from publicly available real estate property listings as the alternative.<sup>7</sup> The advantage of these data is that their availability extends to developing countries. The flip-side is that they also introduce a number of key methodological challenges due to the fact that the alternative database (a) observes predictors of income, not actual incomes, and (b) need not constitute a proper sample, so that it is unclear what population is being represented by the data.

The following two subsections aim to provide workable solutions to these two challenges that will hopefully contribute to a wider application of this approach.

# **2.2.1 A database of predictors of top incomes**

Let us first focus on the challenge posed by observing a predictor of household income rather than actual income. Consider the following assumption.

**Assumption 1** _Suppose that household income per capita can be described by:_

_where xh denotes the predictor of household income, εh denotes a zero expectation error term, subscript h indicates the household, and where β denotes a vector of model parameters._ The assumption of a log-linear model is motivated by ease of exposition and by the fact that it fits our empirical data remarkably well. This assumption can be relaxed however by accommodating flexible functional forms for _m_ ( _xh_ ; _β_ ) if the data call for it. In our application the value of the household’s house (or rental value) will serve as the predictor _xh_ .

To obtain some intuition for the implications of Assumption 1 it may be helpful to verify what it implies for the relationship between the expenditure share on housing and household income. Note however that we are concerned with predicting household income per capita rather than household income. (In

> 7 Alternatively one could for example also look to data on mortgages or credit card statements. However, this approach may not be feasible in countries with underdeveloped or non-existing mortgage markets.

7
fact our dependent variable denotes household expenditure per capita; we will get back to this.) Let us abstract away from this distinction, for this thought experiment only, by considering the household size fixed (so that it is absorbed in the constant _β_ 0). It can be verified that the assumed functional form implies that the expenditure share on housing is a convex declining function of income when _β_ 1 _>_ 1. The expenditure share is constant for _β_ 1 = 1 and an increasing function of income for 0 _< β_ 1 _<_ 1. More specifically, it is a concave increasing function for _β_ 1 _∈_ (<sup><u>1</u></sup> 2<sup>_,_1),alinearincreasingfunctionfor</sup><sup>_β_1=</sup><sup><u>1</u></sup> 2<sup>,andaconvexincreasing</sup> function for _β_ 1 _∈_ (0 _,_<sup><u>1</u></sup> 2<sup>).Despiteitssimplicity,thelog-linearassumptionpermits</sup> a reasonable degree of flexibility in how the expenditure share on housing varies with income. Our prior would be that _β_ 1 _>_ 1, which is consistent with the empirical evidence that is available for the Engel curve on housing expenditure, see e.g. Larsen (2014). _β_ 1 _>_ 1 also ensures that the expenditure share stays below 1 when incomes tend to extreme values. It is unclear however whether this carries over when we substitute household expenditure for household income. We could not find any empirical study that investigates how expenditure on housing varies with total household expenditure. It is conceivable that one might find values of _β_ 1 that are below 1. We would however still expect it to be above<sup><u>1</u></sup> 2<sup>.</sup> (Note that the model in that case does not rule out expenditure shares above 1 for extreme household expenditure values.) Let _Fε_ ( _e_ ; _σ_ ) denote the distribution function of _εh_ with unknown parameter vector _σ_ . We will assume that _εh_ is identically distributed across households, although this assumption can easily be relaxed. Note that the unknown parameter vectors _β_ and _σ_ both have to be estimated. In our empirical application, where the value of housing is considered as a predictor of income, the two can be estimated using the household income survey, since it includes both data on household incomes and data on the value of housing.

It will be convenient to define _n_ ( _τ, y_ ) as the number of households with income between _τ_ and _y_ , _n_ ( _τ_ ) as the number of households with income exceeding _τ_ , and _n_ as the total number of households in the population. For ease of exposition we will ignore the fact that the data may constitute a sample with sampling weights. _F_ 2( _y_ ) (= _Pr_ [ _Y ≤ y|Y > τ_ ]) and _λ_ (= _Pr_ [ _Y > τ_ ]) are seen to solve:

When DB-2 does not contain data on household incomes but data on a predictor of household incomes instead, we have that _n_ ( _τ, y_ ) and _n_ ( _τ_ ) can no longer be observed with certainty and so have to be estimated. Consider first an estimator 8
for _n_ ( _τ_ ): where 1( _a > b_ ) denotes the indicator function that equals 1 if _a > b_ and 0 otherwise. In practice of course _β_ and _σ_ will have to be replaced with their respective estimators _β_<sup>ˆ</sup> and _σ_ ˆ. Similarly, an estimator for _n_ ( _τ, y_ ) can be obtained: Given _n_ ˆ( _τ, y_ ) and _n_ ˆ( _τ_ ), we may construct the estimators _F_<sup>ˆ</sup> 2( _y_ ) = _n_ ˆ( _τ, y_ ) _/n_ ˆ( _τ_ ) ˆ and _λ_<sup>ˆ</sup> = _n_ ( _τ_ ) _/n_ . Combined with the estimator for _F_ 1( _y_ ), which is estimated using DB-1 (i.e. the household income survey), we have all we need to estimate _F_ ( _y_ ) (see eq. 1), the income distribution for the complete population. This in turn is all we need to compute any inequality measure of choice.

No assumptions have been made about the distribution of _xh_ at this point. Let us assume that the top end of the distribution of _xh_ can be described by a Pareto distribution.

**Assumption 2** _Let G_ 2( _x_ ) _denote the distribution function of x conditional on x > x_ 0 _. It is assumed that G_ 2( _x_ ) _follows a Pareto distribution with shape parameters α:_

For ease of exposition let us also assume that the income threshold _τ_ is set sufficiently high such that _Y > τ_ implies _X > x_ 0.

# **Assumption 3**

9
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Note that _θ_ controls the thickness of the top end of the income distribution, which is a key determinant of income inequality; the smaller the value of the tail index _θ_ , the larger the proportion of high incomes, the higher the value of inequality. Under the assumption that top incomes are Pareto distributed, the mean top income level takes on the following form: This mean top income level features in the computation of the top income shares as well as the computation of the between-inequality components.<sup>8</sup>

# **2.2.2 Population underlying top income database is unclear**

Let us next address the challenge that emerges when the data underlying DB-2 are not necessarily representative of the whole population (i.e. households with incomes exceeding _τ_ ). Consider the possibility that DB-2 has “over-sampled” some and “under-sampled” other households among the top earners, such that DB-2 no longer yields a consistent estimator for _F_ 2( _y_ ) unless some corrective efforts are made. This is a rather realistic scenario as the data may constitute a series of transactions or listing prices rather than a proper sample drawn from the target population. For ease of exposition we will assume that DB-2 observes actual household incomes and not predictors of income, so that we may focus exclusively on the challenges presented in this section.

We will assume that the data are representative for selected sub-populations and that a representative “sample” can be obtained by anchoring DB-2 to some known population totals. Suppose that the target population can be sub-divided into _D_ districts with _d_ = 1 _, . . . , D_ indicating the district. The top income distribution for district _d_ will be denoted by _F_ 2 _,d_ ( _y_ ) = _Pr_ [ _Y ≤ y|Y > τ, district d_ ]. By extension, let _F_ 1 _,d_ ( _y_ ) = _Pr_ [ _Y ≤ y|Y ≤ τ, district d_ ]. Using this notation the complete income distribution for district _d_ , denoted _Fd_ ( _y_ ), satisfies:

where _λd_ = _Pr_ [ _Y > τ |district d_ ]. The density functions corresponding to _F_ 1 _,d_ ( _y_ ), _F_ 2 _,d_ ( _y_ ) and _Fd_ ( _y_ ) will be denoted by _f_ 1 _,d_ ( _y_ ), _f_ 2 _,d_ ( _y_ ) and _fd_ ( _y_ ), respectively.

> 8As an alternative to assuming a Pareto distribution for the top tail, and estimating the tail index parameter, one could also appeal to multiple imputation methods, see e.g. Douidich et al. (2015). This approach might in fact be more practical in case a more flexible functional form for _m_ ( _xh_ ; _β_ ) is being considered.

11
By definition the distribution of top incomes for the whole population solves: with _P_ 2 _,d_ = _Pr_ [ _Y > τ, district d_ ]. These mixing probabilities permit the following decomposition: where _πd_ denotes the share of the total population (regardless of income) residing in district _d_ . We make the following assumption.

# **Assumption 5** _It is assumed that:_

- _The data at hand permit consistent estimation of_ ( _F_ 2 _,d, f_ 2 _,d_ ) _and_ ( _F_ 1 _,d, f_ 1 _,d_ ) _for all d._

- _The district population shares {πd} are known._

That leaves _λd_ = _Pr_ [ _Y > τ |district d_ ] as the only unknown that needs to be estimated. One way to estimate _λd_ is to impose the assumption that _fd_ ( _y_ ) is a continuous function.

**Assumption 6** _fd_ ( _y_ ) _is a continuous function of y._

Let _f_<sup>ˆ</sup> 1 _,d_ ( _τ_ ) and _f_<sup>ˆ</sup> 2 _,d_ ( _τ_ ) denote the estimators for _f_ 1 _,d_ ( _τ_ ) and _f_ 2 _,d_ ( _τ_ ), respectively. Assumption 5 ensures that these are consistent estimators. The following proposition derives an estimator for _λd_ by appealing to Assumption 6.

**Proposition 7** _Let f_<sup>ˆ</sup> _k,d_ ( _y_ ) _denote a consistent estimator for fk,d_ ( _y_ ) _for k_ = 1 _,_ 2 _. Under Assumption 6, λ_<sup>ˆ</sup> _d presented below provides a consistent estimator for λd:_

**Proof** Evaluating the first-order derivative of _Fd_ ( _y_ ) from eq. (16) with respect to _y_ yields:

By Assumption 6, _fd_ ( _y_ ) is continuous in _y_ , which imposes that (1 _− λd_ ) _f_ 1 _,d_ ( _y_ ) = _λdf_ 2 _,d_ ( _y_ ) for _y_ = _τ_ . Rearranging the terms in this equality gives us the following solution for _λd_ : The estimator for _λ_ is obtained by replacing _f_ 1 _,d_ ( _τ_ ) and _f_ 2 _,d_ ( _τ_ ) with their estimators. Provided that all terms on the right-hand side of eq. (21) are consistently 12
estimated, which is guaranteed by Assumption 5, it follows that the estimator for _λd_ will be consistent. □ Finally, note that the sub-group inequality decompositions presented in Section 2.1 can readily be extended to accommodate the sub-division of the top tail into _D_ districts. (Note that the bottom segment can in principle stay as is, i.e. need not to be sub-divided into districts.) Let us denote the income share going to the top tail from district _d_ by _sd_ = _P_ 2 _,d_ ( _μ_ 2 _,d/μ_ ), where _μ_ 2 _,d_ = _E_ [ _Y |Y > τ, district d_ ]. Note that the population- and income shares corresponding to the bottom segment now solve 1 _−_<sup>�</sup> _d_<sup>_λd_and 1</sup><sup>_−_�</sup> _d_<sup>_sd_, respectively.Similarly, let us</sup> denote the Theil index or the mean-log-deviation for the top incomes from district _d_ by _T_ 2 _,d_ and _MLD_ 2 _,d_ , respectively. Using this notation, the decomposition of the Theil index and the mean-log-deviation into the 1 + _d_ sub-groups is seen to solve:

# **3 Data**

This paper uses two different types of data-sets: (1) Household Income, Expenditure and Consumption Survey (HIECS) data, and (2) listings of homes for sale derived from (large) real-estate databases. All data used in this study are for Egypt. The HIECS is from 2008/9. The house price data are slightly more recent, covering the period early 2013 to 2015, and come from two different real-estate firms. Details are given below.

## **3.1 Egyptian Household Income, Expenditure and Consumption Survey**

The Egypt HIECS 2008/9 is conducted by the Central Agency for Public Mobilization and Statistics (CAPMAS). We were given a 50% sample of the survey (approximately 24,000 observations).<sup>9</sup> Throughout the paper, our welfare aggregate is expenditure per capita which is consistent with standard practice in most developing countries. Household expenditures have been adjusted for spatial differences in prices by deflating nominal values by a spatial price index following

> 9Hlasny and Verme (2013) were able to access the 100% sample on site at CAPMAS.

13
Belhaj Hassine (2015).<sup>10</sup> Compared to income, consumption expenditure typically produces lower estimates of inequality, especially at the top. This can be explained by a declining marginal propensity to consume and by the fact that consumption surveys tend to understate the spending on durables at the top (e.g. Aguiar and Bils (2015) for the United States). For their study of top incomes in Egypt, Hlasny and Verme (2013) used income as their welfare measure. An argument for using consumption instead of income is that data on the former are often of a higher quality in developing and emerging economies and are less vulnerable to idiosyncratic noise as households tend to smooth their consumption over time. In what follows we will be abusing terminology by often referring to income inequality and the income distribution even though our data measure expenditures, not income.

As discussed in detail in Verme et al. (2014), inequality in Egypt as assessed from household surveys is low and has even declined in the decade before the 2011 revolution. The Gini coefficient of consumption expenditure declined by around 2pp from 32.8 in 2000 to 30.8 in 2009.<sup>11</sup> Our paper tests whether the low estimate in 2009 is robust to replacing the top tail of the income distribution with an estimate that is obtained using a combination of household expenditureand house price data.

# **3.2 Real Estate Data**

In late 2014/early 2015 we obtained data on houses and apartments for sale from two Egyptian real estate firms: Betak-online and Bezaat.<sup>12</sup> The two rank among the larger real-estate firms whose listing database can be accessed online; analogous to Redfin and Zillow in the United States. The data differ in detail but a listing typically consists of the asking price, the location (the city or a further subdivision), and the date when it was listed. Interviews with the Ministry of Housing in Cairo confirmed that the listing price provides a good approximation to the actual sales price.<sup>13</sup> We keep listings classified as houses, apartments, flats or villas, since these refer to private housing. There are a number of other types of listings which we exclude, the three largest groups being land, shop, and chalet.

The model that relates the value of the house to household expenditure (per capita) is estimated using the household survey data, which report (imputed) rents not property prices. We will be assuming that rent- and sale (or listing) prices are proportional to each other, which is sufficient for our needs.

> 10For a recent discussion of challenges with real consumption measurement, see e.g. Van Veelen (2002) and Van Veelen and Van der Weide (2008).

> 11Source: PovcalNet, accessed 31 October 2015.

> 12The URLs are respectively: www.betakonline.com; and www.bezaat.com.

> 13For our purposes it is sufficient that the actual price is proportional to the listing price.

14
The household survey is from 2009, while the rents derived from the real estate data refer to late 2013 - early 2015. There is no real need however to express the values in prices from the same year, i.e. to inflate the 2009 expenditures to 2014 prices or to deflate the house prices to 2009 prices. Instead we will be assuming that the Pareto tail index associated with the top tail of the income distribution is stable over the 2009-2014 period.

# **3.3 Does the household survey indeed omit the rich?**

One way of illustrating whether the household data under-represent the top part of the distribution is to compare some of the characteristics of the top 1 percent the household survey with those of senior Egyptian executives. For the purpose of this exercise, household income is imputed from household expenditures in the survey using the average savings rate in Egypt for 2009.<sup>14</sup> The data on executive pay come from Payscale, an online information company providing current information on salary, benefits, and compensation by type of job, location, and other characteristics. The numbers are presented in Table 1.<sup>15</sup>

||% surveyed population|Minimum|Median|Maximum|
|---|---|---|---|---|
|Household income|Top 1%|11,995|14,666|98,080|
|CEO total pay|Top 1.2%|23,723|68,970|168,545|
|CFO total pay|Top 0.8%|22,551|54,563|212,393|

Table 1: Annual income of top earners in Egypt (USD, nominal, 2009 prices) We focus on the total compensation of senior executives, who represent 2 percent of survey participants and have the highest reported median compensation among survey participants.<sup>16</sup> Therefore, in principle, these households should be in the top 1% of households in Egypt’s household survey. However, since the median senior executive income is closer to the maximum income than to the median income of the richest 1 percent in the household survey, and the maximum income earned by senior executives is much higher than the maximum income in 14 We assume that household income reflects mainly the income of the household head and that the top households save at the average rate. The source for the average savings rate is the World Bank’s World Development Indicators (WDI).

> 15Household income is imputed based on information on household expenditures in Egypt’s 2009 household survey and the average saving rate in Egypt in 2009. The total pay of senior executives in Egypt is obtained from a global database of salaries and compensation for 2015. If anything we expect the data on executive pay to be on the conservative side. The values in the table are deflated and converted from EGP into USD using annual average inflation and exchange rate data from the World Bank’s World Development Indicators.

> 16The senior executives surveyed by Payscale are either chief executive officers (CEOs) or chief financial officers (CFOs) in Egyptian firms.

15
the household survey, it appears that the household survey under-represents the top earning households, particularly the top earning senior executive households.

Similarly, in Vietnam the top salaries recorded in their household survey are less than half of average executive salaries obtained from corporate salary surveys (World Bank, 2014). In the case of Argentina, Alvaredo (2010) finds that while the tax data have almost 700 observations with incomes exceeding 1 million USD, there are none in the Argentine household survey. In a comparison of 16 Latin American household surveys, the ten richest households have incomes similar to a managerial wage, which is arguably substantially smaller than the incomes of top capital owners (Sz ́ekely and Hilgert, 1999).

# **4 Empirical application**

This section presents our empirical application to Egypt. As outlined in the methodology section we combine data on household expenditures with data on house prices. The household expenditures are obtained from the 2009/10 Egypt Household Income, Expenditure and Consumption Survey (HIECS), which is also used for Egypt’s official estimates of poverty and inequality. The house prices represent listing prices for houses that have been put up for sale via two large real estate firms operating in Egypt. We use the real estate database to estimate the top end, defined as the top 5 percent, of the income distribution. The “bottom” 95 percent of the income distribution is estimated using the HIECS.

The following practical decisions and assumptions are made: (a) we restrict the analysis to urban Egypt only (this can be extended to apply to all of Egypt under the assumption that rural households do not rank in the top of the income distribution in Egypt), (b) it is assumed that house price quotes are proportional to (imputed) rental values (as the household expenditure survey contains data on rents only, and we rely on the survey to identify the relationship between house value and household income),<sup>17</sup> (c) it is assumed that the Pareto tail index of the income distribution has been stable between 2009/10 (the time of the survey) and 2013/14 (the time of the house price database), (d) it is assumed that one house constitutes one household (the fact that top income households could be associated with multiple houses may lead us to under-estimate inequality) and that all houses are domestically owned, and (e) we will only be using house price data for Cairo and Alexandria to estimate the top tails of their respective income distributions. For the rest of urban Egypt the entire income distribution will be estimated using the HIECS. The latter decision is motivated by the fact that: (i)

> 17Under a non-arbitrage condition, house prices and rental values are expected to move in parallel, see e.g. Himmelberg et al. (2005).

16
the lion-share of the “rich” that are missing or whose incomes are understated in the HIECS arguably reside in either Cairo or Alexandria, and (ii) the real estate markets are most developed in Cairo and Alexandria such that the coverage and the quality of the house price data are highest for these two cities (henceforward we will refer to these as districts).

Table 2 provides some basic statistics on the number of observations available to us. For the house price databases we only counted observations above the median house price value (which practically coincides with the mode of the house price density). Since we are interested in the top tail behavior of the house price distribution, we do not use the lower house price values.

|sub-group|Database<br>Betak-online<br>Bezaat|HIECS|
|---|---|---|
|Cairo|5772<br>8475|1289|
|Alexandria|1293<br>2012|767|
|Urban Egypt||6935|

Table 2: Number of observations used The following sections proceed with the empirical application that combines the household expenditure survey and the house price data. A validation of our methodology in a controlled setting where only the survey data are used can be found in the Annex.

# **4.1 Pareto tail index estimated on income survey data**

This subsection presents first estimates of the Pareto tail index of Cairo’s and Alexandria’s income distributions by using household survey data only. These estimates will serve as a reference point. Under the assumption of Pareto distributed top tails we have that: 1 _− F_ 2( _y_ ) = � _τy_ � _−θ_ . Rearranging terms yields: If this assumption holds true, a plot of log ( _y_ ) against _−_ log (1 _− F_ 2( _y_ )) should reveal a linear relationship with a slope parameter equal to<sup><u>1</u></sup> _θ_<sup>.Figure1provides</sup> this plot using the top 10 percent of the household expenditure data from the HIECS. For the majority of data points a linear relationship seems to provide a reasonable fit. A deviation from linearity can be observed however toward the far end of the income spectrum, where the slope appears to fall. Consequently, we should expect estimates of _θ_ to come out higher if we were to increase the income threshold above which observations are included.

17
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estimates of _Pr_ [ _Y > τ, district d_ ] for _d_ = _Cairo, Alexandria_ . We estimate these by: _Pr_ [ _Y > τ, district d_ ] = _Pr_ [ _Y > τ |district d_ ] _Pr_ [ _district d_ ], where _Pr_ [ _district d_ ] (the share of the urban population residing in district _d_ ) is obtained from the most recent population census and where _Pr_ [ _Y > τ |district d_ ] is estimated using Proposition 7. For comparison the latter is also estimated using data from the HIECS only. The two different estimators are denoted by _λ_<sup>ˆ</sup> _prop_ 7 and _λ_<sup>ˆ</sup> _svy_ , respectively. _Pr_ [ _Y > τ, district d_ ] and _Pr_ [ _district d_ ] are denoted by _P_ and _π_ , respectively, such that _P_<sup>ˆ</sup> _prop_ 7 = _πλ_<sup>ˆ</sup> _prop_ 7 and _P_<sup>ˆ</sup> _svy_ = _πλ_<sup>ˆ</sup> _svy_ (where we suppressed the subscript _d_ for ease of notation). The estimates are presented in Table 4.

|sub-group|_π_|ˆ_λprop_7|ˆ_λsvy_|ˆ_Pprop_7|ˆ_Psvy_|
|---|---|---|---|---|---|
|Cairo|0.251|0.116|0.101|0.029|0.025|
|Alexandria|0.130|0.079|0.048|0.010|0.006|
|Other urban|0.619|0.028|0.028|0.017|0.017|

Table 4: Estimates of _π_ , _λ_ and _P_ Notice that our estimate of _λ_ finds that the percentage of households residing in Cairo and Alexandria with incomes exceeding _τ_ is larger than what the HIECS alone would have us believe. This combined with the earlier observation that _θ_ ˆ _mix < θ_ ˆ _svy_ leads us to believe that relying on survey data alone will arguably under-estimate both the number of households with high incomes as well as the size of their incomes (either because top income earners are missing in the survey or because they under-report their incomes, or both). Table 5 compares estimates of top income shares _S_ obtained using the HIECS to those obtained using both the HIECS and the house price data. The additional columns compare estimates of inequality among top income households (i.e. only including households whose income exceeds _τ_ ) for three different measures of inequality.

|sub-group|_Smix_|_Ssvy_|_Ginimix_|_Ginisvy_|_MLDmix_|_MLDsvy_|_Theilmix_|_Theilsvy_|
|---|---|---|---|---|---|---|---|---|
|Cairo|0.159|0.118|0.414|0.227|0.295|0.087|0.532|0.107|
|Alexandria|0.042|0.036|0.283|0.156|0.141|0.038|0.208|0.041|
|Other|0.063|0.066|0.223|0.223|0.082|0.082|0.097|0.097|

Table 5: Estimates of _S_ , _Gini_ , _MLD_ and _Theil_ (for the top tail) Estimates of total inequality for (urban) Egypt are obtained by adding estimates of bottom- and between inequality to the estimates of top inequality reported in Table 5. Bottom inequality (i.e. inequality among households with income below _τ_ ) is estimated using the HIECS only. The between inequality component is estimated using data from both sources as it is a function of average income among top earners (which is a function of _θ_ ; see eq. 15) as well as a 23
function of _λ_ (in the case of MLD) and of the top income share _S_ (in the case of the Theil index), see equations (4) and (7). In the case of the Gini coefficient we implement the approximate decomposition that is also used by Alvaredo (2011): _Gini ≈_ (1 _−_<sup>�</sup> _d_<sup>_λd_)(1</sup><sup>_−_�</sup> _d_<sup>_sd_)</sup><sup>_Gini_1 + �</sup> _d_<sup>_sd_.</sup>

||Survey and House|prices<br>Survey only|
|---|---|---|
|Gini|0.470|0.364|
|MLD|0.278|0.217|
|Theil|0.420|0.258|

Table 6: Estimates of inequality for (urban) Egypt in 2009/10: Survey-only versus Survey+House prices The total inequality estimates are presented in Table 6. The survey-only estimate of the Gini coefficient for (urban) Egypt in 2009/10 stands at 36 _._ 4. This is relatively low by international standards and hence would suggest that Egypt ranks among lower inequality countries. Our estimate of the Gini coefficient is 47 _._ 0 which is considerably higher than the official estimate. The level of top incomes recorded in the HIECS is found to be at odds with house prices observed toward the top end of the market in Cairo and Alexandria. Our estimates represent an attempt to correct for this. We repeated the analysis for other choices of inequality measures, specifically for the MLD and Theil measures. Noticeable increases in inequality can be observed for all measures considered. The magnitude of the adjustment is largest for the Theil index which is consistent with the fact that the Theil index is most sensitive to the top tail of the income distribution when compared to the other two choices of inequality measures.

The precision of our estimate of inequality is largely determined by the precision with which we are able to estimate _α_ and _β_ 1 (provided that the assumptions under which the estimators have been derived reasonably apply to the data at hand). It is instructive to verify what level of inequality would be obtained using rather conservative values for _θ_ . Note that a most conservative estimate of _θ_ can be obtained by combining a value of _α_ from the top end of the estimated range with a value of _β_ 1 from the low end of the estimated range (but taking _β_ 1 _≥_ 2<sup><u>1</u></sup> which rules out housing expenditure shares that are convex increasing functions of household expenditure; see the discussion following Assumption 1). For Cairo this gives us a value of around 2 _._ 4 (1 _._ 2 _/_ 0 _._ 50; see Figures 4 and 6). For Alexandria we obtain a value that is just over 2 _._ 5 (1 _._ 25 _/_ 0 _._ 5; see Figures 4 and 6). Note that these values are in the range of the respective survey-only estimates of _θ_ (see Figure 2). In other words, it would take a very conservative estimate for _θ_<sup>ˆ</sup> _mix_ to reproduce the survey-only estimate of inequality. The estimate we consider most reasonable finds a Gini coefficient for (urban) Egypt of 47 _._ 0, which is roughly 24
10 points higher than the survey-only estimate. Of course, by the same token, we may also be under-estimating inequality. Working with values of _θ_ toward the lower end of our estimated range yields estimates of inequality that are noticeably higher than the Gini coefficient of 47 _._ 0. Using our estimates for _θ_ (and _λ_ ) in a back-of-the-envelope calculation, we find that there are approximately 170 households in Cairo whose household expenditure exceeds 1 million USD per year. Although no other information on the number of millionaires in Egypt is currently available, this estimate seems rather conservative.

# **5 Concluding remarks**

A growing literature has shown that household surveys provide only limited information about top incomes and therefore underestimate income inequality. This paper presents a method that corrects for this underestimation. We use the household survey for the bottom part of the distribution and combine it with another data source that provides a better coverage of the top tail. The existing literature has restricted itself to the use of tax record data to capture the top tail. Unfortunately income tax records are unavailable in many countries, including most of the developing world. Our method permits a much larger set of data for the top tail; the only requirements are that the data (i) contain a good predictor of household income, and (ii) provide a good coverage of the top tail.

We apply this method to Egypt, where estimates of inequality based on household surveys alone are low by international standards. Using publicly available data from real estate listings to estimate the top tail of the income distribution, we find strong evidence that inequality in Egypt is being underestimated. The Gini index for urban Egypt is found to increase from 36 to 47 after correcting for the missing top tail. A natural next step would be to use data on house prices to estimate the top tail of the wealth distribution, and extend the analysis to other countries.

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<!-- Start of picture text --> <2<br>2<br>a<br>SS<br>N<br>oO<br>6 8 10 12<br>Inype<br><!-- End of picture text -->
income distribution for the sub-sample (not reported here) does not show any signs of truncation. Less than 3 percent of the observations are dropped which is sufficient to produce a sizable decline in inequality (see Table 8): the estimated Gini drops from 36 _._ 4 (full sample) to 31 _._ 9 (sub-sample).<sup>19</sup> Table 7 presents the estimates of _β_ 1, _α_ , _θ_ , _λ_ and _P_ . Note that _β_ 1 is estimated using the sub-sample of households, _α_ is estimated using the imputed rent data for the full sample (which now stand in for the house price database), while _θ_ , _λ_ and _P_ are estimated using a combination of the two data sources. Observe that the estimates of the Pareto tail index _α_ obtained using the imputed rent data from the survey are markedly higher than the estimates obtained using the house price database (see Table 3 in the main text). This is consistent with the hypothesis that top income households are under-represented in the original survey. The estimates for _β_ 1 are in the same range as those obtained using the full sample.

|sub-group|ˆ_β_1|ˆ_α_|ˆ_θmix_|ˆ_θsvy_|ˆ_λprop_7|ˆ_Pprop_7|
|---|---|---|---|---|---|---|
|Cairo|0.692|1.385|2.001|2.910|0.058|0.015|
|Alexandria|0.676|2.567|3.795|3.900|0.021|0.003|

Table 7: Estimates of _β_ 1, _α_ , _θ_ , _λ_ and _P_ The corresponding estimates of total inequality for urban Egypt are shown in Table 8. While we do not perfectly reproduce the original estimates of inequality, we find the results encouraging. All measures of inequality considered show a clear upward adjustment. We slightly over-estimate the Gini, and slightly underestimate the MLD and Theil indices. This may be due to the fact that we are using an approximation to the sub-group decomposition for the Gini (see Alvaredo, 2011), while the sub-group decompositions for the MLD and Theil indices are exact.<sup>20</sup>

||Survey only (full-sample)|Survey only (sub-sample)|Survey + imputed rents|
|---|---|---|---|
|Gini|0.364|0.319|0.387|
|MLD|0.217|0.163|0.201|
|Theil|0.258|0.175|0.253|

Table 8: Estimates of inequality for (urban) Egypt in 2009/10: Survey-only versus Survey+imputed rents

> 19Note that we have not adjusted the sampling weights after dropping the selected households.

> 20There is no additive sub-group decomposition for the Gini that is exact.

29