The Moorish Wanderer

Middle Class Tax Burden

Posted in Dismal Economics, Flash News, Moroccan Politics & Economics, Morocco by Zouhair ABH on December 6, 2012

Listening to the Head of Government from time to time is entertaining and instructive at once. Whether one likes him and his politics or not, this is representative democracy at work. But overall he started to make use of statistics to get his points across, which is a marked improvement in his argument, and he is better for it. But he is still light on policy-making however.

There is this specific claim about the middle classes, and his failure to address it -or should I say, the failure of opposition caucuses to confront him on the issue- the tax hike on higher income was a sensible move, but it was not matched with an equivalent tax cut for these middle classes. Which leads me to beg the question: was this government policy to achieve some level of fiscal equity, or was it just a move to increase fiscal receipts? These are the questions I would have liked members of parliament to ask the Head of Government.

I argue here the present tax system, with or without the tax hikes on the top income-earners, is structurally unfair to everyone with an annual income below 300.000 dirhams, and specifically to the middle class (middle class as defined mathematically to be the median income per household in a defined income distribution)

First, I use 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 misses a particular aspect crucial to policy-making: though inter-decile ratios are not constant over time, they can be proven to be centred around the asymptotic value (notably the \ln(2) between the mathematical expectation and median) but there is little in the exponential distribution for the policy-maker to exercise their social preferences.

Log Normal vs Exponential sample distributions. The Log-normal allows for 'more' high income households.

Log Normal vs Exponential sample distributions. The Log-normal allows for ‘more’ high income households.

The Log-normal distribution is not that different, but it has the advantage (and from a computational point of view, an additional difficulty) of fielding two parameters in its probability density function. As indeed one can see in the following densities:

g(x)=\lambda e^{-\lambda x} the exponential, and

f(x)=\frac{1}{x{\sqrt{2\pi \sigma }}}e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}} the log-normal

Both distributions are different in form, but not so much in sample representations. Indeed, the exponential distribution is reputed to be strictly decreasing. But it can be argued households with no income (i.e. with zero or close to zero annual income) need to be taken out of the population, perhaps because they can always rely on transferred income (or because those with no income do not form a household) in any case, the sample population used to generate the exponential distribution of income does not look like it.

second, let us consider a Taylor approximation around the median point of the proposed distribution, that is:


It computes the marginal income around the median. Marginal income is the key to understand the present taxation system – as it divides up a household income into brackets, each subjected to increasing tax rates. In essence, the derivative around the median gives a fair idea of any additional income for this population (our median class) and how it would be taxed. A ‘fair’ tax structure would minimize the marginal tax burden around the median -namely, the marginal increase in tax rate for these households. In fact, the optimal tax plan would be a flat tax rate for all the median class, because then additional gains around would not be excessively taxed. A numerical example would be that of a household with an annual taxable income of 78,000 dirhams – a relatively small 4% increase (or 3,000 dirhams) is best left taxed at the same rate (or infinitesimally the same) while the present system takes away 940 dirhams from the 3,000 increase. A marginal tax burden of 32% for a 4% increase in income is not exactly fair.

So, the derivative around the median provides a generalized result that can then be compared to the present tax system, and assuming a strictly positive marginal increase in their income, the median household would observe the following result:

f(x) = \dfrac{\partial(\left | \exp \mu \right |\sigma \sqrt{2\pi})^{-1}}{\partial \exp \mu}=\frac{\dot{\exp \mu}}{(\exp \mu) \sigma\sqrt{2\pi}}

once this is plugged back into the earlier Taylor series, the net benefit for a median household is such:

f(x) = \frac{\dot{\exp \mu}}{\sigma\sqrt{2\pi}}\times\frac{(x - \exp \mu)}{\exp \mu}

And this is a pretty neat result in many aspects: the term \frac{(x - \exp \mu)}{\exp \mu} refers to the gross benefit for a median household gaining a supplement of x dirhams. But this needs to be replaced into the perspective of the whole distribution, so it is ‘discounted’ with the impact on the median itself – that’s \dot{\exp \mu} and then weighted by the measure of inequality (or income dispersion) \sigma\sqrt{2\pi}

The impact on the general welfare can then be computed by integrating (i.e. generalizing the individual boost around all median household) around the additional x dirhams to the median:

\int_{0}^{x}(\frac{\dot{\exp \mu}}{\sigma\sqrt{2\pi}}\times\frac{(u - \exp \mu)}{\exp \mu})\mathrm{d}u with an expected welfare gain of \frac{\dot{\exp \mu}}{\sigma\sqrt{2\pi}}\left(\frac{u^2}{\exp \mu}-u \right) which can be verified for u>1 is a net gain (any additional dirham contributes to generate additional welfare, that is).

So there is good evidence that suggests the total income distribution is improved when median income increases. The impact on the average household income is not as high as one would expect (up to a \sigma^2 term) but the general welfare of all individuals close enough to the median definitely improves, and those below the median can expect over time to catch up to it. Obviously, a tax rate that does not take this effect into account might indeed stifle the described effect. And this is what we are set out to demonstrate.

My third and last step involves using logged income to emphasise the effect of tax burden on middle class. This means we are back to the useful bell-shaped curve Gauss-Laplace. The log here does not denote of any particular distribution, but indeed could depict the social preferences a policy-maker needs to display in view of the results computed earlier on: since welfare gains are highest for median households, the policy-maker needs to place a larger emphasis upon them – and the Normal distribution serves this purpose pretty well – in mathematical terms, the tangent is almost flat around the median.

Plotting the densities of tax rates and income provides no particular explanation as to the transfer effect, nor the tax burden per income. For instance, there is no particular correlation between income and their theoretical tax brackets, a strange result given the progressive tax structure in place. Additionally, a supposed preference for median income household (captured by a Normal distribution centred around the median income) contradicts the present tax structure: the average tax rate is 27%, whose corresponding income coincides with income between 60,000 and 180,000 annual income. But since there is not such rate, it has to be a convex combination of the 30% and 34% rates, with the 30% rate falling on income between 60,000 and 80,000 – our median class. The convex combination puts the weight on these households at 57%. In fact, those with income between 74,313 and 77,330 dirhams per annum pay 7% more in taxes than the immediate tax bracket (those with income marginally above 80,000 a year) just because of the present tax system. In aggregate terms, this is almost 4 Bn dirhams in deadweight loss due to the present system.

The main problem with the present tax system is its ‘jumping function’ which results in disproportionately larger tax burden for those at the margins. Unfortunately for the middle class, many of them are on the margin, the closer to 80,000 a year, the higher the tax burden. A good example can be provided for the figures mentioned before: incomes of 74,313 and 77,330 dirhams pay respectively 8,293.87 and 9,198.36. And although the difference in income is merely 4%, the richer household will pay 10% more than their immediate neighbour. In fact, this fiscal injustice reaches its peak around the median.