We're all familiar with the concept, but for those who weren't aware that it had been formalised, given a title and a Wikipedia entry, the Politician's Syllogism goes like this:
- We must do something.
- This is something.
- Therefore, we must do this.
And so it is with the minimum wage. Poverty and inequality are problems that demand some sort of policy response, and if you're not familiar with the file, then increasing the minimum wage sounds like a halfway-plausible response. At least it's something, right?
If you are a politician, that hardly matters: Something Was Done. And until recently, my view was that this exercise had essentially no effect. Increasing the minimum wage amounted to a random redistribution, so there wasn't much point in spending much energy on a policy that wouldn't affect poverty and inequality.
But the study I blogged about in this post got me to thinking about the question of how and why a random redistribution of income could be welfare-reducing. The story in the study went something like this:
[M]ost minimum wage earners are young, and the income of these workers is a significant contribution to low-income households. For families in which the minimum wage worker keeps her job, the increase in the minimum wage can increase family income above the LICO so long as hours worked are not significantly affected. For families in which the minimum wage worker loses her job and/or loses too many hours, the lost income may bring family income below the LICO. Canadian data suggest that the latter effect dominates: the net effect of an increase in minimum wages on poverty rates is positive
This sounds plausible, but I was wondering if it could be generalised somehow. And I think it can. It's actually pretty simple.
I shall follow Marshall's instructions and do the math first. Suppose that
- Individual base incomes xi are distributed across the population according to a certain distribution.
- Governments redistribute income so that net incomes are yi = xi + zi, where zi has mean zero and is independent of original incomes xi
- Individual utility functions u(•) are concave
Then if we use expected utility as a measure for social welfare, we get
(I keep meaning to figure out how to do equations in the typepad editor one day. Today is apparently not that day.)
Given the above hypotheses - which populate a fair number of economic models - purely random redistributions of income are welfare-reducing. If you know the math, I don't need to explain it to you (except perhaps my notation). If you don't know the math, I probably can't. So I'm going to move on to the next stage of Marshall's recommended methodology by translating it to English.
Suppose that the economy is populated by economic twins: everyone has a partner with identical characteristics. And now suppose that we take $1000 from one twin and give it to her counterpart. The effect on total income is zero - it's a pure redistribution - but the effect on total welfare is negative.
Consider one representative pair. If they have preferences with declining marginal utility, the gain from receiving an extra $1000 will be less than the loss from being deprived of $1000. So even if the total income of a given pair stays the same, the gains of the winner are outweighed by the losses of the loser. This will be the case across all couples, and hence for the population. And that's the basic story. In a pure redistribution, there will be winners and losers. But if marginal utility is decreasing, total losses will outweigh total gains. (Okay, you can burn the math now.)
This raises the stakes a bit for those tempted by the Politician's Syllogism when it comes to poverty and inequality. Random redistributions make things worse.