economics taxes

# The Atkinson-Stiglitz Theorem

## Atkinson-Stiglitz

The Atkinson-Stiglitz theorem (AST) Atkinson-Stiglitz theorem assumes utilitarian logic and was later generalized a bit Laroque. It proves

where the utility function is separable between labor and all commodities, no indirect taxes need be employed

Put in English: given the authors' assumptions, the government need only tax labor income, though this tax can be negative (i.e. welfare) and can be based on other characteristics that predict one's ability or need (i.e. tagging). The main point is that there is no need to tax consumption. This and the Diamond-Mirrless production efficiency result Diamond together strongly suggest that taxing labor income should be the primary way governments raise revenue.

The obvious question, of course, is what assumptions Atkinson and Stiglitz made. Let me list them:

• Everyone has the same preferences ("homogeneity").
• People are rational.
• People have utility functions.
• Wages are fixed.
• With regards to labor and consumption, the utility function is linearly separable. To put this another way, it assumes everyone's utility functions can be represented as $f(L) + g(c)$.

This work was later expanded into showing that even if you have a commodity tax and a non-optimal income tax, you can eliminate the former to generate a Pareto improvement Kaplow (see also Deaton for related work). Later, this was also generalized to utility functions in which consumption and labor are not linearly separable Hellwig.

This is all cool, but the assumptions this edifice is built on are obviously false, and there has been significant work examining what violations of these assumptions end up implying about optimal taxes and transfers. In principle, then, all non-excise non-labor-income taxes should be justified by this later work (or at least justifiable).

We'll cover these issues in the next.

## Exceptions to Atkinson-Stiglitz

The best overview I've found of the limitations of the AST is sections 3-5 of Indirect Taxation and Redistribution by Broadway and Pestieau Boadway. They point to a number of issues that pop up when you relax the assumption that everyone has the same preferences.

In particular, if people with high wages have different intrinsic preferences than people with low wages, consumption taxes on the goods/services preferred by the high-wage people can be justified in addition to an optimal income tax Saez.

Just to be clear, this is not the claim that we should tax things that rich people buy more of. This is the claim that we should tax things that rich people buy more of even if we gave the poor people the same incomes. An example of such a difference is savings, which is one argument in favor of a capital gains tax Saez.

Another obvious exception to the AST is if these preferences can be observed directly - i.e. if the government can observe a household has particularly high needs, then that households should pay less in taxes - the general principle behind "tagging". This could, for instance, be used to justify subsidizing respirators or childcare.

An alternative reason to tax a good even with optimal income taxing is if that consumption of that good positively correlates with leisure Saez.

Yet another way the AST breaks down is if households can obtain consumption resources through labor not observable by the government, e.g. black market labor. In this case, a consumption tax is probably appropriate so as to capture revenue from a household when they earn money on the black market but spend it in the legal market place.

Conversely, different preferences for leisure complicate the theorem's proof, but don't change the overall result, since a consumption tax can't distinguish lazy high-wage people from hard-working poor people any better than an income tax.

Finally, Naito examines how the AST works when we assume that high ability people can achieve higher returns from accumulating human capital than low ability people. This model, prescribes a tax on goods/services that positively correlate with human capital development (e.g. education) Naito.

## Conclusions

Where this all leaves us is paradoxically disappointedly murky and also crystal clear

The nominal interpretation of AST is clearly refuted: even ignoring externalities, there are many cases where consumption taxes can improve social welfare. The above can be summarized as saying "tax income for most of your redistributive and revenue needs, but/and also tax/subsidize consumption in these specific cases."

If this seems obvious/useless to you, I think it can be helpful to see how this varies a great deal from typical arguments surrounding consumption taxes.

For instance, people often want to tax yachts more and food less, but the above makes justifying such policies quite non-obvious. Conversely, the idea that we should tax college tuition isn't exactly obvious (well, mostly).

Just as economics as a whole is (in some sense) a cataloging of all the problems and solutions with what is overall a reasonably good idea (competitive markets), this subfield is (in some sense) a cataloging of the all the problems and solutions of what is overall a reasonably good starting point (income taxes > consumption taxes).

A common aphorism in statistics goes "all models are wrong, but some are useful". If so, then the Atkinson-Stiglitz Theorem is a model of the highest degree - an extremely useful device for investigating what optimal taxation would actually look like even as it falls short of perfection itself.

Wikipedia contributors. (2019, October 20). Atkinson-Stiglitz theorem. In Wikipedia, The Free Encyclopedia. Retrieved 01:12, April 18, 2020, from https://en.wikipedia.org/w/index.php?title=Atkinson%E2%80%93Stiglitz_theorem&oldid=922185884 Kaplow, L., 2006. On the undesirability of commodity taxation even when income taxation is not optimal. Journal of Public Economics 90, 1235–1250. https://doi.org/10.1016/j.jpubeco.2005.07.001. Laroque, G., 2005. Indirect taxation is superfluous under separability and taste homogeneity: a simple proof. Economics Letters 87, 141–144. https://doi.org/10.1016/j.econlet.2004.10.010. Hellwig, M. F. (2010). A generalization of the Atkinson–Stiglitz (1976) theorem on the undesirability of nonuniform excise taxation. Economics Letters, 108(2), 156–158. https://doi.org/10.1016/j.econlet.2010.04.035 Boadway, R., & Pestieau, P. (2003). 21 Indirect Taxation and Redistribution: The Scope of the Atkinson-Stiglitz Theorem. Economics for an imperfect world: Essays in honor of Joseph E. Stiglitz, 387. https://www.econstor.eu/bitstream/10419/189292/1/qed_wp_1005.pdf Naito, H. (2007). Atkinson-Stiglitz Theorem with Endogenous Human Capital Accumulation. The BE Journal of Economic Analysis & Policy, 7(1). https://doi.org/10.2202/1935-1682.1516. Saez, E. (2002). The desirability of commodity taxation under non-linear income taxation and heterogeneous tastes. Journal of Public Economics, 83(2), 217-230. https://doi.org/10.1016/S0047-2727(00)00159-6. Diamond, Peter A.; Mirrlees, James A. (1971). "Optimal Taxation and Public Production I: Production Efficiency". The American Economic Review. 61 (1): 8–27. Deaton, A. (1979). Optimally uniform commodity taxes. Economics Letters, 2(4), 357-361. https://doi.org/10.1016/0165-1765(79)90050-8