Raise the Geometric Mean Wage!
Many metrics have been proposed for measuring a country’s prosperity, such as GDP. I would like to suggest a slightly unusual one: the geometric mean wage.
Like how the normal average of n numbers in their sum divided by n, the geometric mean of n numbers is the n-th root of their product. Alternatively, you can take the numbers’ logarithms, find the normal average of these logarithms, then raise e to the power of this average. As an equation:
GM(a1, a2, a3, ..., an) = (a1 a2 a3 ... an)^(1/n) = exp(ln(a1) + ln(a2) + ln(a3) + ... + ln(an))
Why are the Existing Options Bad?
Let’s look at the existing metrics and see why they’re suboptimal.
- The most prominent metric for this is GDP. It’s OK, but has the problem that taking away good choices can raise GDP. For example, imagine I buy a certain pair of shoes once every 10 years. If the shoes instead lasted 5 years, but the price and quality stayed the same, this would cause me to spend more money on shoes, which would raise the GDP. I’m pretty sure most people could agree that this is an objectively bad situation though. For this reason, I prefer wages over GDP as a metric of prosperity.
- The traditional mean doesn’t take inequality into account. A society where half the people (randomly chosen) make 100 000 $/year and the other half makes 40 000 $/year seems worse than one where everyone makes 70 000 $/year.
- The median and mode handle inequality somewhat, but discard most of the data. If, for example, the bottom 40% of society suddenly lost their jobs and made no money, the median wage would not change.
- True measures of inequality, such as the Gini coefficient, have the problem that destroying people’s money (without benefiting anyone else or changing anyone’s behaviour) is considered good. We aren’t trying to recreate the world of Harrison Bergeron!
- These problems could be somewhat solved by having multiple metrics – for example the average and the Gini coefficient – but that raises the question of how to compare places where, for example, country A has a better average but country B has a better Gini. It would be best if we could combine these two things into one metric.
Deriving the Geometric Mean from Marginal Economics
In order to determine what this ideal metric should be, we can use data about how people actually value money. People do not, in fact, value every dollar the same! Instead, as people get richer, they value each additional dollar less. Data consistently shows that the value of making N dollars is proportional to the logarithm of N:
GDP vs. self-reported life satisfaction (Our World in Data)
So, instead of measuring the average wage, what if we measured the average of how much people value their wage? We can then convert this value back to a dollar amount by exponentiating it. This is the geometric mean!
Properties of the Geometric Mean
This geometric mean has two important properties:
- Generating new wealth (raising people’s wages) always increases the geometric mean, and destroying wealth always reduces it.
- Redistributing wealth from rich to poor always increases the geometric mean, and redistributing wealth from poor to rich always decreases it. None of the existing alternatives I mentioned above has both of these properties.
The geometric mean is almost always lower than the normal average; they’re the same when everyone’s equal, but the geometric mean declines as people get less equal.
I believe the inequality-adjusted HDI may use the geometric mean wage, but it’s not completely clear.
Like in the HDI, this can still be combined with things like life expectancy and amount of schooling, since money isn’t everything.
Caveats
The claim that the value of making N dollars is proportional to log(N) has only been shown for normal-sized wages. Extremely low wages completely break this system, so we would likely have to set some minimum wage, say 1 $/year, and treat all wages below the minimum as if they were the minimum. Extremely high wages don’t break the system, but may not be as valuable as the geometric mean suggests.
Also, the geometric mean simply does not work for metrics that can contain negative numbers, such as net income (income minus cost of living), a metric that I otherwise think is great. You’ll have to use some other kind of average in that case.