Thursday, January 02, 2014

Turning log-consumption into a [crude] measure of short-run human welfare

Summary: [This post is intended more as notes than for a general audience.] Many have praised log income as a good proxy for individual humans' welfare. While log income alone is useful for analyzing transfers within a fixed population, when spending can also change population (e.g. through saving lives, contraception, assistance for parents) log income must be supplemented to produce a measure that tracks welfare, e.g. with an estimate of the value of a life at a subsistence income compared to the value of a life at some higher income. I then take a first pass at global income distribution statistics in this light.

Log-income vs income as an indicator of welfare
If your annual income is $1,000,000 an extra dollar will make much less difference to your quality of life than it would if your annual income was $1,000. Thus in welfare economics it is common to assume that individuals have utility functions that are proportional to the logarithm of income, i.e. with a constant change in utility per doubling of income. This is the standard utilitarian justification for redistribution of income: with log utility functions a dollar of consumption is 1,000 times as valuable for someone with 1/1,000th the consumption level.

Toby Ord points to supporting empirical work suggesting that life satisfaction scales more closely with log-income than with absolute income:

In other words it appears that over a certain range doublings of income are associated with a fairly stable difference in welfare for the people in question.

One problem it faces is that a number of features affect both income and happiness separately, e.g. painful diseases that impair market labor. Experiments like GiveDirectly can help to tease out causation from correlation.

Also, this evidence covers a limited range of incomes, but since humans require a minimum level of income to survive, typically the only level of income below subsistence that people might have is zero (for the dead or nonexistent).

Converting between log-income and welfare
The above information may allow us to compare changes in income in a population with incomes limited to a certain range. But to evaluate cases where the population changes we would need to know the difference between the welfare at particular incomes and at zero income, i.e. the welfare level of a subsistence income (or other base income).

Given a constant s for the welfare associated with a subsistence income, we could then assign an estimated welfare value to individuals as:

welfare = s + (log income) - (log subsistence income)

Giving a good value for s would be an empirically and philosophically challenging project, so I won't do so here. Instead, I will mostly compute log income less a stand-in for log subsistence income, which can be supplemented by whatever one considers a good value for s.

One way to get intuitions about s is to compare lotteries with different chances of various incomes or death/nonexistence. For example (all logarithms henceforth are base 10):

Assume subsistence income is $120 and your income is a little less than $38,000, corresponding to log incomes of 2.08 and 4.58, a difference of 2.5. If you would be indifferent (considering solely your personal welfare) between a certainty of the $200 income and a coin flip equally likely to kill you or give you the higher income, then that maps to an s of 2.5. Indifference to a lottery with a 5/6 fatality rate would correspond to an s of 0.5

Such intuition pumps would be improved on by empirical methods along the lines of Value of Statistical Life (VSL), although all such methods do have flaws.

World log income above subsistence
To get a very crude first-pass estimate of world mean log income, I turned to Giving What We Can's "How Rich Am I?" calculator, which uses 2008 data from Branco Milanovic. I pulled incomes at various percentiles, and then assigned welfare scores using a subsistence income of $120 (there are some people with lower reported income, but I wonder about the interpretation) and various values for s.

PercentileUS international dollars incomeLog (base 10) incomeLog (base 10) income less log ($120)Welfare, s=0.5Welfare, s=2.5Welfare, s=10

This gives an average log income (less log subsistence income) a little over 1. World GDP per capita (PPP) for 2008 was around $10,000, and if everyone had exactly equal incomes at that level the corresponding value would be 1.92, more than half again as great, before accounting for s. If universal development meant that everyone had incomes at the level of the U.S. per capita GDP (about $50,000), mean welfare would be 2.62+s.

In a total utilitarian calculus, the maximum potential direct welfare gains from such changes would be easily matched by a doubling or two of global population (very easily with high values of s). This is unsurprising: total utilitarianism plus logarithmic utility functions make for a standard route to the Repugnant Conclusion. Redistribution/development to support more people, rather than just to raise incomes would do better in that framework.

Of course, population over the next few centuries is small compared to all the beings that have existed in Earth's past, and potentially vastly smaller relative to populations in the long run, so total utilitarian-minded types would be more concerned with long run impacts than short term human welfare effects of population size.

Postscript: log incomes of spherical cows
Logarithmic utility functions, or functions with constant relative risk aversion,  are convenient for economists in making analytically tractable models, but clearly break down at extremes. At the high end, there are very likely diminishing returns, and it should be noted that log utility functions are unbounded: utility increases along with log income without bound, so the usual infinitarian problems arise, although we can ignore these in most of the applications these metrics are used for.

At the low end, if each halving of income delivers a constant decrease in utility then utility decreases without bound as income approaches zero, and is undefined for an income of zero. But having one person with a income extremely close to zero is not vastly worse than the person having an income of $1, and an income of 0, or death, is not undefined or infinitely bad. Climate economist Martin Weitzman has published work discussing how welfare economic models of the impacts of climate change used such logarithmic utility functions and therefore suffered from these pathologies (being unable to make sensible tradeoffs about cases in which income collapses to near zero).

However, since humans cannot survive on sub-subsistence income, we can generally ignore the region between zero income and subsistence, assigning (death or nonexistence) a value of zero.


Toby Ord said...

Thanks for the nice clear presentation of this and the crunching of the numbers. Here are some scattered comments:

1) It is great to have a stab at this and get it on (virtual) paper. There are many hidden assumptions in your model, but putting them down in the light of day is the way to find them, and to challenge people to do better. In general, I really admire how you do this so often.

2) The self-reported happiness numbers in that chart are pretty bizarre, methodologically speaking. It is not at all clear that people's wellbeing is linear in them. That said, they are a reasonable starting point.

3) Regardless of the self-reported numbers, there is a literature on people's risk aversion about income. Assuming they are risk neutral about wellbeing, this lets us work out the diminishing utility of income function. People use the one-dimensional family of isoelastic utility functions, (which are those with constant relative risk aversion). Log is a special case in this family, but the usual measures are more steeply diminishing than log. You could use the parameter of this family as something else for your sensitivity analysis.

4) Consumption measures are better than income if we have them (and I think the GWWC calculator data might use consumption).

5) The GWWC data is in international dollars, which are good for the mapping into utility/wellbeing, but are confusing for distribution. I presume that when equalising world income money is moved from places with worse purchasing power to ones with more purchasing power, so *more* international dollars are created and your calculation will be an under-estimate of the good created. I'm not sure by how much. I can't imagine the amount of international dollars going up by more than a factor of 4 (roughly the PPP ration between mainstreet USA and India), so the log value can't go up by more than about 0.5.

6) When changing the population, you will change the total income/consumption, as well as changing the function that maps this to wellbeing. I'm not sure how big a deal this is or how to deal with it, but you might want to think about it.

Toby Ord said...

Also, note that an increase of 0.1 in this measure is about a 25% increase in income, and an increase of 0.01 is about a 2.5% increase. The world read GDP growth rate is about 3% on average, so we can think of that as a global increase of about 0.01 each year due to growth (if the relative inequality stays fixed). Or about 1.00 per century at historic rates.

Nick Beckstead said...

Thanks, Carl. Cool post.


> However, since humans cannot survive on sub-subsistence income, so we can generally ignore the region between zero income and subsistence, assigning (death or nonexistence) a value of zero

so we can --> we can
missing period

Carl said...

Thanks for many good points Toby, definitely worth using in a followup. I took the chart from your post.

I wonder if doing such a global human welfare estimation might be a good research project for an undergraduate looking for an EA-relevant topic.

Thanks Nick, done.

Vaniver said...

The model as you've stated it treats income as a multiple of the subsistence income- one way to get around this (and deal with what utility value you assign to death, and so on) is to consider s+log(income+x). The behavior around incomes of 0 is more reasonable, and for small x it barely impacts the behavior at high incomes.