# The Cost of not Redistributing Money Part 1

1. Distributive Efficiency

I’m on a bit of a roll at the moment writing essays arguing for things that should be obvious, but for some reason often seem to be forgotten. Abba Lerner’s concept of distributive efficiency seems like a reasonable continuation. Personally I’m more of a believer in re-engineering the whole system than redistribution within the existing system — nonetheless it’s an interesting exercise to map how much the United States loses -from a utilitarian point of view- from not redistributing. In order to do the calculations I’ve enlisted the help of my good friend Kieran Latty, whose work on this very question is almost chillingly comprehensive.

If money is redistributed through payments and progressive taxation it is generally thought that there will be a cost, called the deadweight loss- the economy as a whole will shrink if people have less incentive to work due to progressive taxation. What is often forgotten is that in another sense there is also a cost to not redistributing money.

A dollar is worth a lot more to someone with an income of 10,000 a year than to someone with an income of 100,000 a year, and it is worth even less to someone on one million dollars a year. Thus there’s a certain inefficiency in giving a dollar to someone who already has many dollars, equally then there is a certain inefficiency to some having many dollars while others have few- those dollars could be more effectively contributing to the happiness of humanity if they were given to those who had less. Using certain mathematical techniques it is possible to quantify how much better a society would be in which everyone had an equal portion of income, yet the total pool of income was the same.

2. The technical aspects

To briefly run through the technical side, it’s possible to estimate the rate at which people’s marginal utility of income declines through a number of methods. For example, you can ask them questions about which gambles they would accept and you can look at how much they are willing to pay for different forms of insurance. These will all give an estimate of the personal income elasticity of the marginal utility of income (Latty 2011) which we will call eta (η).

The Atkinson index is a popular normative measure of inequality which gives the proportion of income in a society which could be sacrificed, without reducing ‘social welfare’ — if inequality was also reduced to zero.

The Index has a single parameter, (ε) , which regulates the degree of inequality aversion. For larger values of epsilon, the Atkinson index is more sensitive to income inequality. It turns out that if we are utilitarians, and the utility function has a certain form [the constant elasticity of substitution form (CES)] then epsilon (ε) is simply equal to the elasticity of marginal utility of income, or eta (η).

Latty (2011) on the basis of a literature review commends a value of the personal elasticity of the marginal utility of income (η) of 1.5, and data from the Luxemburg income study for the U.S. gives a value for the Atkinson index for ε=1 of 0.241.

Fortunately, if income distributions are approximately lognormal, then we can obtain somewhat accurate estimates for the Atkinson index for alternative values of epsilon. If we apply this procedure, and set ε=1.5, we obtain an estimate for the Atkinson index of 0.339.

3. Headline figure: The cost of inequality in the United States (sans relative income effects).

If the above mentioned assumptions are correct (and we will discuss later how they are not), then the Atkinson index for the United States is about 0.339. It turns out that once we adopt a more nuanced view, which models relative income and status effects explicitly, then this value should be considered a lower bound.

Thus, this analysis suggests that, from a utilitarian point of view, the United States is ‘wasting’ (again, not factoring in the costs of redistribution), at least a third and a bit of its income through the distributive inefficiency of its income allocation.

Another way to look at this is that an economy with the same proportional distribution of income as the United States would have to be approximately 50% bigger in size per capita than a hypothetical economy with a perfectly even distribution of income in order to be ‘as good’ from a Utilitarian perspective (if we are also holding population constant).

Of course this isn’t a complete case for extensive redistribution- it only deals with one side of the ledger, and doesn’t consider the other side- the deadweight loss associated with redistribution (or the possibility that some forms of redistribution in the US at the moment might actually be economically beneficial, for that matter). Since the deadweight loss from drastically equalising income is likely large, considering this side of the ledger in isolation, without being aware of the limitations, would be misleading. Nonetheless, we’ve at least seen that one side of the ledger is pretty costly — a lot of utility is lost through the unevenness of distribution.

4. Coming soon in part 2! Relative income effects

Thus far our approach has been very individualistic. We assume that each agent’s welfare is only affected by their own income, but in truth the income of others affects us in numerous ways- even if it doesn’t affect our income. In particular, people often experience stress and dissatisfaction in an environment where many people are much richer than them, and the very structure of social institutions starts to warp. In part 2 we’ll see what happens when the effects of others income on our wellbeing- relative income effects- are included. It turns out that estimates of the social welfare losses from income inequality can rise very appreciably when we include analysis of relative income effects.

(1): Latty, K. (2011) Income distribution, growth and social-welfare: towards an economic solution to the growth-equality trade-off problem URL= https://ses.library.usyd.edu.au/handle/2123/8260

(2): Latty, K (2015) A five parameter Atkinson like index featuring relative income effects, with a seven parameter extension for nonlinear (prioritarian) social welfare functions URL=https://www.academia.edu/6099318/A_five_parameter_Atkinson_like_index_featuring_relative_income_effects_with_a_seven_parameter_extension_for_nonlinear_prioritarian_social_welfare_functions

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