The Moorish Wanderer

The Economic Chronicles of the Kingdom, 1955-2011 Part.2

Posted in Dismal Economics, Moroccan Politics & Economics, Morocco, Read & Heard by Zouhair ABH on July 17, 2012

I should thank Romain Ferrali for his comment/question on some figures I have used on my Capdema presentation (quite a successful gathering, I am told. the Capdema event, obviously) about Morocco’s “1%”.

A challenge indeed, given the scarce information about income distribution from both HCP and the World Bank.

My initial -and strongest- assumption about income distribution is its stability across time: incomes evolve overtime, but the differences between the median and most percentiles relative to their respective incomes are assumed (and tested) to remain constant, or so we shall observe.

A caveat the reader would do well to consider: these incomes are computed on the basis of Gross National Income (from the World Bank Opendata) divided up by the number of households, either provided by HCP census or estimated on the basis of past annual demographic growth. Such a crude method bears several shortcomings in terms of sources of income not accounted for, the heterogeneity of different sources of income, the transfert effects between household to name but three. On the other hand, the primary source of interest remains the dynamics of income distribution, and if indeed additional information arises, I would be glad to carry on and finesse it further. This method might explain why HCP and myself disagree on the definition of “middle classes”: mine is simply the statistical definition of the income halfway between the poor and the rich.

The computations are based essentially on a set of three assumptions:

1/ Income distribution is constant: the same exponential distribution (with different parameter \lambda for each year)

2/ Parameter \lambda i.e. the inverse of GNI per capita, has its own statistical distribution.

3/ the parameter is stationary, possibly with a normal distribution whose mean and variance are estimated on the basis of time series.

An earlier blogpost assumed income distribution was exponential; an educated guess, you might say, given the fact that the decile-based cumulative density function clearly indicates a strong level of inequality – not very scientific indeed, but given the information at hand, it was the best I could come up with. I was lucky enough to find a paper than vindicates partially my assumption.

On income distribution in the United States, Dragulescu & Yakovenko note:

“The exponential Boltzmann-Gibbs distribution naturally applies to the quantities that obey a conservation law, such as energy or money [10]. However, there is no fundamental reason why the sum of incomes (unlike the sum of money) must be conserved. Indeed, income is a term in the time derivative of one’s money balance (the other term is spending). Maybe incomes obey an approximate conservation law, or somehow the distribution of income is simply proportional to the distribution of money, which is exponential [10]. Another explanation involves hierarchy.Groups of people have leaders, which have leaders of a higher order, and so on. The number of people decreases geometrically (exponentially) with the hierarchical level. If individual income increases linearly with the hierarchical level, then the income distribution is exponential.”

The authors provide two possible explanation for that particular distribution: it is either linked to the amount of money at hand, i.e. the higher the income a household earns, the higher its cash-in-hand is going to be, and because money is conserved, income benefits from that effect too. The second explanation is more sociological: income is assumed to have an institutional link to hierarchy, the social and professional status of a particular household confers a certain level of income. Here again, conservation in hierarchical statuses confers on income the same distribution.

The exponential distribution is a very useful statistical device. Its density function needs only one parameter, and is defined such:

f(x) = \lambda \exp^{-\lambda x}, x>0, \lambda>0

the restrictions on x is useless in this particular case, since income is obviously a positive amount of money, and lambda is necessarily positive, since:

\mathbb{E}(x) = \frac{1}{\lambda}

and there lies the usefulness of the said distribution: all we need is a time series of GNI per Capita or per Household to generate yearly income distribution. The idea is to use these as random vectors to check the effects of inequality over a long period of time. Once this set of distribution is generated, we test the results against empirical data from 1985, 1991, 1999, 2001, 2007 and 2010.  (available on the World Bank data nomenclature) The data at hand is the decile/quantile distribution of concentrated income.

we test whether differences in both values for available years are not statistically significant. Don’t bother with seemingly larger differences for 2007 and 2010, the sample size puts it in perspective.

If the three assumptions turn out to be correct, we should observe generated results close enough to empirical percentages from these years, and thus conclude to the robustness of the estimated income distribution. The policy implications of these results are infinite: fiscal policy, among others, would gain a lot from addressing issues of truth-telling and other institutional dysfunctions. But for now, I am focused on trying to describe as explicitly as possible statistical properties of income households in Morocco since its independence.

A first test to check whether income distribution is exponential, is to compare synthetic and empirical median income per household. The exponential distribution has the following property:

\int_{0}^{x}\left(\lambda \exp^{-\lambda t}\right)dt= 0.5 = median

which means the difference (or ratio) between average and median income per household is a constant commensurate to ln(2) the null hypothesis in this case is to check whether observed discrepancies between both datasets are statistically insignificant. At 95% confidence interval, we get most empirical values lay between the 45-52% percentiles, which, given the size of the selected samples, is a pretty robust evidence these differences amount to very little. We therefore retain at high levels of confidence the assumption of the exponential distribution.

For instance, median income in synthetic data for 1985 was 23,293 dirhams, vs empirical median income of 24,210 dirhams, which falls within the 51%-52% percentile (which is more than enough to test at 95% confidence). It is worth pointing out however that these discrepancies, for all their statistical irrelevance, are systematically in favour of empirical data, which points out to an income distribution marginally more unequal than the exponential distribution suggests. However, because the fit is robust enough, we shall settle for the synthetic model. A final caveat perhaps: the test was carried on 6 particular dates, which still does not preclude significantly different results for the 51 remaining years. The likelihood of such event nonetheless is very low in view of the levels of confidence used earlier.

The fact the distribution has been the same (with its parameter \lambda evolving with GNI per household) since 1955 might lead to think that income inequality has remained constant since 1955 (recall the ratio Median/Mean is \log(2)) and for some inter-quartile ratios,  results are stationary, which means inter-quartile income inequality has not change significantly over the past half a century.

large discrepancies between the top 1% and the median incomes starting from the 1970s

The picture is not all that clear, though: first off, the upper bound evolves frequently, a properties that has to do with the elusive nature of high income households (the 1% more affluent) the synthetic income distribution. If anything, there seem to be no particular link that growth since 1955 has contributed to influence income inequality one way or the other. What looks to be painfully clear however, is that the richest 1% have enjoyed a distribution of income growth whose trend is undoubtedly in their favour: between 1955 and 2010, the richest 1% have improved their income relative to all percentiles below the median by 6%, even as GNI per Household grew an average 6.91% over the same period of time. It might look like jumping to conclusions, but unequal distribution of income growth seem to contribute a lot, if indeed 86% of it goes to the top 1%.

I would say the graph and these computations understate the discrepancies between top and ‘regular’ earners: the sample size goes only as far as list maximum values of an annual income of 1,077,000 dirhams per annum, even as rarefied incomes are larger by far. If anything, these computations would instead minimize the reality of income inequality, because extreme values on the right hand-side tail are bounded.

So there it is: a quick look at the relationship between growth and inequality indices point to the lack of correlation: growth in Morocco does not necessarily bring about better quality of life to households below the median line. If anything (but the statistics gets blurry there) income inequality abated during the 1990s (a period of recession as well as structural reforms) and increased with the early 2000s (an economic expansion by many measures)

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  1. […] هذا الإقتراح ينبني أساس على نظام جبائي يختلف تمام و النظام الحالي: فهو يعتمد على تقديم مداخيل الأسر في المغرب على عبارة متغيرة عشوائية (من طراز أسي) تتطابق نسبة الضريبة على الدخل و إياه. مما يعني أن إدارة الضرائب لها معرفة دقيقة لتوزيع المداخيل، شيء ممكن لأنني بمعلومات عامة أستطعت أن أثبت ذلك […]

  2. gold price said, on August 14, 2012 at 13:06

    but do not include data for the highest-income households where most of change in income distribution has occurred.

  3. […] both Exponential and Log-Normal distributions to prove a couple of nice (and useful) properties; I referred earlier on to the exponential distribution as a possible way to model household income distribution. Yet it […]


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