# What type of an income distribution would allow the fastest growth and/or highest sustainability

Is there a theoretical framework to explore the question of what could be the ideal distribution of income among a population that yields growth and/or sustainability?

Let $$P(m)$$ be a density function of the number of people having $$m$$ dollars. How do I find the function $$P_{\text{optimal}}(m)$$ that maximizes the ability of the market to "grow". I would like to get some help figuring out the basic concepts and definitions in order to address these type of questions. Namely, how to distribute income among population in the best way for the collective.

• related concept: Ralwsian, veil of ignorance, utilitarianism. – 0x90 Sep 18 '20 at 1:38

## 1 Answer

I would like to get some help figuring out the basic concepts and definitions in order to address these type of questions. Namely, how to distribute income among population in the best way for the collective.

There is actually no single best way of distributing income for the 'collective'. This is because what is best depends on normative judgement. What is best for utilitarian might not be best for Ralwsian or libertarian.

This being said there are actually ways of how to study what an optimal (re)distribution and taxation should look like depending on income distribution. To be more specific these models try to answer the question that given what your moral/political beliefs are how you can redistribute income in most optimal way.

If you are looking for general models then Mirrlees (1971), Diamond (1998) and Saez (2001) are good place to start. These papers consider optimal income taxation in static case but they are probably good starting point, even though due to being static models they are not really good at answering growth related question but given that the dynamic models of taxation are infinitely more complex you would want to probably start here. A simplified version of the optimal income tax formula is given:

$$\frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_{n0})}$$,

with $$b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho}$$.

This formula tells you how the marginal tax rates $$\frac{T'(z_n)}{1-T'(z_n)}$$ that govern both how much people are taxed and how much redistribution transfers people receive (since a transfer is just negative tax) will depend on the efficiency parameter $$\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$$ which is given by elasticity of labor supply to income taxes and which tells you how much taxation 'hurts' the economy.

The second part $$\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $$b_n$$ which depends on both utility of consumers and the societal utility function (i.e. is our society Ralwsian, Utilitarian, Libertarian ...?- depending on different philosophy people are assigned different welfare weights - which you can interpret as telling you how much society ‘values’ given person e.g. under Ralwsian min-max principle poorest person gets weight 1 and anyone else 0). Furthermore $$f(z)$$ and $$F(z)$$ are the density and cumulative distribution function of the income distribution respectively.

Finally $$\frac{1-F(z_n)}{z_nf(z_{n0})}$$ is the part that captures the relative magnitude of distortions created by this taxation. It gives us indication about how 'broad' tax base is at given point in income distribution and on margin how many people are affected.

Again if growth is an integral part of your question you will have to go beyond static models into dynamic ones (see short literature review on those in Stantcheva 2020) but I recommend starting with the static case mentioned above.

Furthermore, if by sustainability you mean environmental sustainability that is a question that is actually, generally speaking, across economic literature divorced from any distributional question. The reason for this is that the main point of government intervention in this area is to solve some environmental issue such as $$CO_2$$ over-pollution due to lack of private property for clean air and these problems usually exist regardless of any income distribution considerations. Usually these sustainability issues involve Pigouvian taxes or subsidies so that is a key word you can use in search for those kind of models.

There are also few papers that look at interplay between Pigouvian taxes and redistribution systems. On of the best papers I read on this issue is probably Jacobs and De Mooij (2015) and this issue was also explored by Lange, & Requate (2000), but this being said there are not that many papers on this topic especially when you want to also add growth into the mix.

Per request in comments I am adding also my response comments to question if Pareto distribution is good for the economy.

The answer depends on how do you exactly define good and in what context. For example, empirically most ex ante (ex ante redistribution that is) income distributions around the world can be characterized as a log-normal distribution with Pareto tail. This is almost universally true. Now whether that’s good or bad depends on definition of what is good, for a Ralwsian such distribution is not good as a Ralwsian would in ideal world where there are no costs to redistribution prefer uniform distribution. This being said since in real life redistribution generally introduces inefficiency we can’t just choose arbitrary distribution of income we want. When you plug some real life distribution in the optimal redistribution models usually the ex-post distribution of income does not have different shape altogether rather it’s compressed or stretch here and there for example a Ralwsian social welfare function would stretch the log-normal distribution with Pareto tail at its left tail and compress it at the right Pareto tail but the shape does not change drastically. For other commonly used welfare functions same would hold but I suppose there could be some exotic one for which it would not hold.

• Is Pareto distribution good for the economy? – 0x90 Sep 17 '20 at 14:36
• @0x90 The answer here depends how do you exactly define good and in what context. For example, empirically most ex ante (ex ante redistribution that is) income distribution around the world can be characterized as a log normal distribution with Pareto tail. This is almost universally true. Now whether that’s good or bad depends on definition of what is good, for a Ralwsian such distribution is not good as a Ralwsian would in ideal world where there are no costs to redistribution prefer uniform distribution. This being said since in real life redistribution generally introduces inefficiency – 1muflon1 Sep 17 '20 at 15:24
• we can’t just choose arbitrary distribution of income we want. When you plug some real life distribution in the optimal redistribution models usually the ex-post distribution of income does not have different shape altogether rather it’s compressed or stretch here and there for example a Ralwsian social welfare function would stretch the lognormal distribution with Pareto tail at its left tail and compress it at the right Pareto tail but the shape does not change drastically. For other commonly used welfare functions same would hold but I suppose there could be some exotic one for which it – 1muflon1 Sep 17 '20 at 15:31
• would not hold. – 1muflon1 Sep 17 '20 at 15:33
• Do you mind to include this useful information about Pareto in your original answer? – 0x90 Sep 17 '20 at 17:27