Atkinson–Stiglitz theorem

The Atkinson–Stiglitz theorem is a theorem of public economics which states "that, where the utility function is separable between labor and all commodities, no indirect taxes need be employed" if non-linear income taxation can be used by the government and was developed in a seminal article by Joseph Stiglitz and Anthony Atkinson in 1976. The Atkinson–Stiglitz theorem is generally considered to be one of the most important theoretical results in public economics and spawned a broad literature which delimited the conditions under which the theorem holds, e.g. Saez (2002) which showed that the Atkinson–Stiglitz theorem does not hold if households have heterogeneous rather than homogeneous preferences. In practice the Atkinson–Stiglitz theorem has often been invoked in the debate on optimal capital income taxation: Because capital income taxation can be interpreted as the taxation of future consumption in excess of the taxation of present consumption, the theorem implies that governments should abstain from capital income taxation if non-linear income taxation is an option since capital income taxation would not improve equity by comparison to the non-linear income tax, while additionally distorting savings.

Optimal taxation

For an individual whose wage is $w$, its budget constraint is given by

$\sum _{j}q_{j}x_{j}=\sum _{j}(x_{j}+t_{j}(x_{j}))=wL-T(wL)\;,$

where $q_{i}$ and $x_{i}$ are the price and the purchase of the i-th commodity, respectively.

To maximise the utility function, the first order condition is:

$U_{j}={\frac {(1+t'_{j})(-U_{L})}{w(1-T')}}\;(j=1,2,...,N).$

The government maximises the social welfare function, and so

$\int _{0}^{\infty }\left[wL-\sum _{j}x_{j}-{\overline {R}}\right]dF=0\;.$

Then we use a density function $f$ to express the Hamiltonian:

$H=\left[G(U)-\lambda \left\lbrace wL-\sum _{j}x_{j}-{\overline {R}}\right\rbrace \right]f-\mu \theta U_{L}\;.$

Taking its variation with regard to $x_{j}$, we use the condition for its maximum.

$-\lambda \left[\left({\frac {\partial x_{1}}{\partial x_{j}}}\right)_{U}+1\right]-{\frac {\mu \theta }{f}}\left[{\frac {\partial ^{2}U}{\partial x_{1}\partial L}}\left({\frac {\partial x_{1}}{\partial x_{j}}}\right)_{U}+{\frac {\partial ^{2}U}{\partial x_{j}\partial L}}\right]=0\;.$

Then the following relation holds:

$\left({\frac {\partial x_{1}}{\partial x_{j}}}\right)_{U}=-{\frac {U_{j}}{U_{1}}}=-{\frac {1+t'_{j}}{1+t'_{1}}}\;.$

Substituting this relation into the above condition yields:

$\lambda \left[{\frac {1+t'_{j}}{1+t'_{1}}}-1\right]={\frac {\mu \theta U_{j}}{f}}\left[{\frac {\partial ^{2}U}{\partial L\partial x_{j}}}\cdot {\frac {1}{U_{j}}}-{\frac {\partial ^{2}U}{\partial L\partial x_{1}}}\cdot {\frac {1}{U_{1}}}\right]={\frac {\mu \theta U_{j}}{f}}{\frac {\partial }{\partial L}}\left(\ln {U_{j}}-\ln {U_{1}}\right)\;,$

and we obtain

$\lambda \left[{\frac {1+t'_{j}}{1+t'_{1}}}-1\right]={\frac {\mu \theta U_{j}}{f}}{\frac {\partial }{\partial L}}\left(\ln {\frac {U_{j}}{U_{1}}}\right)\;.$

Note that there is no loss of generality in setting $t'_{1}$ zero, therefore we put $t'_{1}=0$. Since $U_{j}=(1+t'_{j})\alpha$, we have

${\frac {t'_{j}}{1+t'_{j}}}={\frac {\mu \theta \alpha }{\lambda f}}{\frac {\partial }{\partial L}}\left(\ln {\frac {U_{j}}{U_{1}}}\right)\;.$

Thus it turns out that no indirect taxation need to be employed, i.e. $t_{j}=0$, provided that the utility function is weakly separable between labour and all consumption goods.

See also

• Redistribution of wealth
• Pareto optimality