Tuesday, 26 June 2012

Steve Landsburg getting it wrong about Ramsey's rule.

Frank Ramsey, a brilliant mathematician, supplied an answer to this question in 1927-

'The problem I propose to tackle is this: a given revenue is to be raised by proportionate taxes on some of or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be a minimum?”

The Generalised Ramsey rule is about optimal taxation policy and  concerned with promoting efficiency- that is, minimizing distortion to the economy.  If you tax goods which are inelastic in demand (i.e. if people still buy them when their price goes up) then, provided people spend only a minute portion of their income on them, then it hardly makes a difference- ergo no distortion, no new inefficiency.
However the necessities of the poor as well as the status symbols of the mega-rich have inelastic demand. But the poor spend a lot of their income on staples like bread. So if you tax bread they will have very little money left over to buy other things. There will be a big distortion to the Economy.  Not so if you tax Caviar. The super-rich won't notice.
Equity- the question of the fairness of the tax burden different classes bear- can conflict with Efficiency and this gives rise to the notion that, since  there is a trade off between them, there must be an optimal trade-off point.

Keeping this background in mind, let us look at what Steve Landsburg  has  to say-
The correct statement of the Ramsey rule places heavier taxes not on the goods that are inelastically demanded, = this is true. Ramsey's rule says don't tax inelastic things like Salt in India in the 1930's because there is a big income effect (if its price goes up very poor people feel a lot worse off) and small substitution effect (there is no substitute for Salt)-  but on those goods whose prices lead to the fewest distortions (in a sense that can be made precise) not just in their own markets but in others as well. What's a distortion? It's when people do something quite different after the tax. But, that happens if their standard of living is affected by the tax- this is called the 'income effect'- or if there is a close substitute whose price is not affected. Suppose there is a perfect un-taxed substitute to a taxed good. People would buy that and feel no worse off. If the price of root beer affects your peanut consumption, that goes into the calculation. If the price of butter affects your bread consumption, so does that.
If you’ll forgive a little jargon, then, the correct statement is that the optimal mix of taxes depends on a very complicated formula involving not just own-price elasticities but cross-price elasticities. Only in very special circumstances does this reduce to the cartoon version that says “inelastically demanded goods should be taxed more heavily”.
Second, once you write down the correct (quite complicated) version of the Ramsey rule, you discover that as long as all goods can be taxed, the Ramsey rule does tell you to tax them all equally. This is not true. What Landsburg has shown is that assuming you have no say in the matter and have no choice but to pay a given amount in  tax (as though it were a lump-sum tax) you'd prefer to do it under the pre-tax relative price ratios. BUT THIS IS BLINDINGLY OBVIOUS! Think about it. Any price change affects us in two ways- it changes our Real Income (how well off we are) and it changes Relative prices causing us to prefer a substitute- e.g. margarine instead of butter. Landsburg 'proves' that if we have to suffer the Income effect we wouldn't also want to suffer the Substitution effect. In other words, if I say I will  either punch you or both punch and kick you, then you would prefer me to punch you rather than that I should both punch and kick you. What a great discovery! But is it relevant to Ramsay's rule? No. Why? Ramsay's rule is about who Society should punch and slap. You yourself don't want to be both punched and slapped.  If the Income effect has already made you cry,  you don't want to also be hit by the Substitution effect. So you want the post tax price ratios to remain the same- i.e. you want a flat rate proportional tax.   But this is not  to say you want to be punched or slapped. It does not make it right for Society to punch or kick you. If the income effect is exogenously determined- i.e. the power of the punch has already been decided- you don't want to suffer the substitution effect as well- i.e. you don't want to be kicked on top of being punched. But this preference of yours tells us nothing about Allocative Efficiency. Thus, this great discovery of Landsburg is not the same thing as proving that  Society should maintain the pre-tax price ratio by taxing everything at the same rate.  Society has to decide who should be clobbered with the tax.  Society can say- 'poor working people shouldn't be taxed on the sorts of things they need to survive, while wealthy people should pay more tax on items of ostentatious consumption'. Why? If I punch Mike Tyson- he would hardly notice whereas if I deliver that same punch to a starving old woman- she may die.
Third, if only some goods can be taxed, then it’s not in general optimal to tax them all equally. The Ramsey rule (correctly stated) tells you how to tax them. It does no such thing because the thing simply can't be done. There is a complicated way of looking at 'shadow prices' and so on but its Garbage In Garbage Out.
Fourth, leisure counts as a good. If you can’t tax leisure (or equivalently subsidize labor) then it’s not in general optimal to tax everything else equally. However, if labor is supplied inelastically (as the labor economists tell us is more or less the case) then the tax-everything-equally result is restored, even when leisure can’t be taxed.  Nonsense. Landsburg assumed that the elasticity of supply of everything was infinite. Suddenly he's changed the rules and he expects his equation to still hold up. Why? Do Equations  watch Fox News?

And fifth: A great number of elementary textbooks either get this wrong or present it so misleadingly that it might as well be wrong. Students beware.
Compare this from an actual textbook

What is Landsburg rebuttal of my criticism of him?

Vik: You seem to have missed the following points:
1) This is a question about efficiency, not redistribution.
2) We are talking, in any event, about an economy with identical consumers.
3) Lump sum redistribution together with efficient taxation beats inefficient taxation every time BY THE DEFINITION of efficiency.
Putting 2) and 3) aside, 1) is the key to where you went off the rails.
Actually Steve, 
1) a question about optimal taxation is a question about the Social Welfare Function. Society isn't indifferent between an extra dollar earned by a rich man as opposed to a poor man. In the formula quoted above, the rule for optimality states that, at the margin, the distortionary effect should be equated to the redistributive impact. In other words both income and substitution effects are important. Your great discovery- 'once you've been punched by the 'Income effect' you don't want to be kicked by the 'Substitution effect' as well'-  adds nothing to the debate. 
2) An economy with identical consumers either has
a) Income homogeneity. Everybody has the same Income.  In this case just divide the Tax bill by the number of people in the country and impose a lump sum tax. Why fiddle around with Expenditure taxes? What's the point? 
b) Income heterogeneity. Some are richer than others. In this case there is a redistributive impact of a flat rate proportional tax unless the Income elasticity of all goods is the same. Here the Socially optimal tax (assuming Society ranks a poor man's dollar higher than a rich man's, in accordance with the principle of diminishing marginal utility) taxes luxuries more heavily than necessities.
3) If you can do lump sum redistribution, why can't you do lump sum taxes? By the definition of Efficiency, it is inefficient to futz around with expenditure taxes in the manner you have done. 
Conclusion- Landsburg great discovery re. tax - 'if you've been punched you don't also want to be slapped'- if the income effect has clobbered you, you ask to be spared the substitution effect-  isn't really a great discovery at all- no one, not even an Economist has ever said anything to the contrary. True, they haven't affirmed Landsburg's discovery but only because it isn't relevant to anything in any possible world. 

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