Atkinson–Stiglitz theorem explained

The Atkinson–Stiglitz theorem is a theorem of public economics. It implies that no indirect taxes need to be employed where the utility function is separable between labor and all commodities. Non-linear income taxation can be used by the government and was in a seminal article by Joseph Stiglitz and Anthony Atkinson in 1976.^[1] The Atkinson–Stiglitz theorem is an important theoretical result in public economics, spawning a broad literature that delimited the conditions under which the theorem holds. For example, Emmanuel Saez, a French-American professor and economist demonstrated that the Atkinson–Stiglitz theorem does not hold if households have heterogeneous preferences rather than homogeneous ones.^{[2] [3]}

In practice, the Atkinson–Stiglitz theorem has often been invoked in the debate on optimal capital income taxation. As capital income taxation can be interpreted as the taxation of future consumption over the taxation of present consumption, the theorem implies that governments should abstain from capital income taxation if non-linear income taxation were 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 , the budget constraint is calculated 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 maximize the utility function, the first-order condition is:U _{j} = (( 1+ t' _{j} ) ( - U _{L} ))/(w ( 1- T')) (j =1 , 2 , ... , N). The government maximizes the social welfare function, and so \int ^{inf ty} _{0} \left[ wL - \sum _{j} x _{j} - \overline{R} \right] d F = 0 . Then a density function f is used to express the Hamiltonian: H = \left[ G(U) - λ \left\lbrace wL - \sum _{j} x _{j} - \overline{R} \right\rbrace \right]f - \mu \theta U _{L} . Taking its variation about x _{j} , the condition for its maximum is used. - λ \left[ \left( (\partial x _{1})/(\partial x _{j}) \right) _{U} + 1 \right] - (\mu \theta)/(f) \left[ (\partial ^{2} U)/(\partial x _{1} \partial L) \left( (\partial x _{1})/(\partial x _{j}) \right) _{U} + (\partial ^{2} U)/(\partial x _{j} \partial L) \right] = 0 .

Then the following relation holds:

\left( (\partial x _{1})/(\partial x _{j}) \right) _{U} = - (U _{j})/(U _{1}) = - (1+ t' _{j})/(1+ t' _{1}) . Substituting this relation into the above condition yields:λ \left[ (1+ t' _{j})/(1+ t' _{1}) - 1 \right] = (\mu \theta U _{j})/(f) \left[ (\partial ^{2} U)/(\partial L \partial x _{j}) ⋅ (1)/(U _{j}) - (\partial ^{2} U)/(\partial L \partial x _{1}) ⋅ (1)/(U _{1}) \right] = (\mu \theta U _{j})/(f) (\partial)/(\partial L) \left( ln {U _{j} } - \ln \right) \;,and the following is obtained: λ \left[ (1+ t' _{j})/(1+ t' _{1}) - 1 \right] = (\mu \theta U _{j})/(f) (\partial)/(\partial L) \left( ln { (U _{j})/(U _{1}) } \right) . Note that there is no loss of generality in setting t' _{1} zero, therefore t' _{1} = 0 is put. Since U _{j} = ( 1 + t' _{j} ) \alpha , (t' _{j})/(1+ t' _{j}) = (\mu \theta \alpha)/(λ f) (\partial)/(\partial L) \left( ln { (U _{j})/(U _{1}) } \right) . Thus, no indirect taxation needs to be employed, i.e. t _{j} = 0 , provided that the utility function is weakly separable between labor and all consumption goods.

Other Approaches

Joseph Stiglitz explains why indirect taxation is unnecessary, viewing the Atkinson–Stiglitz theorem from a different perspective.^[4]

Basic concepts

Suppose that those who are in category 2 are the more able. Then, two conditions are imposed for Pareto efficient taxation at which a government aims. The first condition is that the utility of category 1 is equal to or more than a given level:

\overline{U} _{1} \le V _{1} (C _{1} , Y _{1} ) . The second condition is that the government revenue R , which is equal to or more than the revenue requirement \overline{R} , is increased by a given amount: R = - (C _{1} - Y _{1} ) N _{1} - (C _{2} - Y _{2} ) N _{2} , \overline{R} \le R , where N _{1} and N _{2} indicate the number of individuals of each type. Under these conditions, the government needs to maximize the utility V _{2} (C _{2} , Y _{2} ) of category 2. Then writing down the Lagrange function for this problem:l{L} = V _{2} (C _{2} ,Y _{2} ) + \mu V _{1} (C _{1} ,Y _{1} ) + λ _{2} ( V _{2} (C _{2} , Y _{2} ) - V _{2} (C _{1} ,Y _{1} ) ) + λ _{1} ( V _{1} (C _{1} , Y _{1} ) - V _{1} (C _{2} ,Y _{2} ) ) + \gamma \left( - (C _{1} - Y _{1} ) N _{1} - (C _{2} - Y _{2} ) N _{2} - \overline{R} \right) , ensuring the satisfaction of the self-selection constraints, the first-order conditions are:\mu (\partial V _{1})/(\partial C _{1}) - λ _{2} (\partial V _{2})/(\partial C _{1}) + λ _{1} (\partial V _{1})/(\partial C _{1}) - \gamma N _{1} = 0 , \mu (\partial V _{1})/(\partial Y _{1}) - λ _{2} (\partial V _{2})/(\partial Y _{1}) + λ _{1} (\partial V _{1})/(\partial Y _{1}) + \gamma N _{1} = 0 , (\partial V _{2})/(\partial C _{2}) + λ _{2} (\partial V _{2})/(\partial C _{2}) - λ _{1} (\partial V _{1})/(\partial C _{2}) - \gamma N _{2} = 0 , (\partial V _{2})/(\partial Y _{2}) + λ _{2} (\partial V _{2})/(\partial Y _{2}) - λ _{1} (\partial V _{1})/(\partial Y _{2}) + \gamma N _{2} = 0 .

For the case where

λ _{1} =0 and λ _{2} =0 :(\partial V _{i} / \partial Y _{i})/(\partial V _{i} / \partial C _{i}) + 1 = 0 , for i =1 , 2 , therefore the government can achieve a lump-sum taxation. For the case where λ _{1} =0 and λ _{2} > 0 :(\partial V _{2} / \partial Y _{2})/(\partial V _{2} / \partial C _{2}) + 1 = 0 , the marginal tax rate for category 2 is zero. As to category 1:(\partial V _{1} / \partial Y _{2})/(\partial V _{1} / \partial C _{1}) = - (1-λ _{2} (\partial V _{2} / \partial Y _{1} ) / N _{1} \gamma)/(1+λ _{2} (\partial V _{2} / \partial C _{1} ) / N _{1} \gamma) . If \delta _{i} = (\partial V _{i} / \partial Y _{1})/(\partial V _{i} / \partial C _{1}) , (i =1 , 2 ) , the marginal tax rate for category 1 is \delta _{1} + 1 .

Also, note the following equation:

\delta _{1} = - \left( (1- \nu \delta _{2})/(1+ \nu) \right) , where \nu is denoted by:\nu = (λ _{2} (\partial V _{2} / \partial C _{1} ))/(N _{1} \gamma) . Therefore, by assumption, \delta _{1} < \delta _{2} , and so -1 < \delta _{1} < \delta _{2} can be directly proven. Accordingly, the marginal tax rate for category 1 is positive.

For the case where

λ _{1} > 0 , and λ _{2} = 0 , the marginal tax rate for category 2 is negative. The lump-sum tax imposed on an individual of category 1 would become larger than that for category 2 if the lump-sum tax were feasible.

Various commodities

Consider a case where income level and several commodities are observable. Each individual's consumption function is expressed in a vector form as:

bf{C} _{1} = \sum _{j} C _{1j} bf{e} _{j} bf{C} _{2} = \sum _{j} C _{2j} bf{e} _{j} . In this case, the government's budget constraint is:R \leq \sum ^{2} _{k =1} ( Y _{k} N _{k} ) - N _{1} \sum _{j} C _{1j} - N _{2} \sum _{j} C _{2j} . Then: \mu (\partial V _{1})/(\partial C _{1j}) - λ _{2} (\partial V _{2})/(\partial C _{1j}) + λ _{1} (\partial V _{1})/(\partial C _{1j}) - \gamma N _{1} = 0 , \mu (\partial V _{1})/(\partial Y _{1}) - λ _{2} (\partial V _{2})/(\partial Y _{1}) + λ _{1} (\partial V _{1})/(\partial Y _{1}) + \gamma N _{1} = 0 , (\partial V _{2})/(\partial C _{2j}) + λ _{2} (\partial V _{2})/(\partial C _{2j}) - λ _{1} (\partial V _{1})/(\partial C _{2j}) - \gamma N _{2} = 0 , (\partial V _{2})/(\partial Y _{2}) + λ _{2} (\partial V _{2})/(\partial Y _{2}) - λ _{1} (\partial V _{1})/(\partial Y _{2}) + \gamma N _{2} = 0 . Here, λ _{1} =0 and λ _{2} > 0 . Therefore, it follows that: ((\partial V _{2})/(\partial C _{2j}))/((\partial V _{2})/(\partial C _{2n})) = 1 , ((\partial V _{2})/(\partial C _{2j}))/((\partial V _{2})/(\partial Y _{2})) = 1 . Suppose all individuals have the same indifference curve in C-L plane. The separability between leisure and consumption can be expressed as: (\partial ^{2} U _{k})/(\partial C _{kj} \partial L _{k}) = 0 , yielding (\partial V _{1})/(\partial C _{1j}) = (\partial V _{2})/(\partial C _{1j}) . As a result:((\partial V _{1})/(\partial C _{1j}))/((\partial V _{1})/(\partial C _{1n})) = 1 . Thus, Stiglitz stated it is unnecessary to impose taxes on commodities.

Conditions for randomization

Consider a scenario in which individuals with high abilities, who typically earn higher incomes as a reflection of their skills, downplay their abilities. In this case, it could be argued that the government needs to randomize the taxes imposed on the low ability individuals, to increase the effectiveness of screening. It is possible that under certain conditions the taxes can be randomized without damaging the low-ability individuals. For the case where an individual chooses to show their ability, a tax schedule is related to

\lbrace C ^{*} _{2} , Y ^{*} _{2} \rbrace . For the case where an individual chooses to hide their ability, there are two tax schedule possibilities: \lbrace C ^{*} _{1} , Y ^{*} _{1} \rbrace and \lbrace C ^{**} _{1} , Y ^{**} _{1} \rbrace . The randomization is done so that the risk of the former case should differ from that of the latter.

To avoid hitting the low ability group, the mean consumption must be shifted upwards at each

Y . As the consumption is maximized, a higher \overline{C} _{1} is set for a higher \overline{Y} _{1} . Then the relations between those variables are: C ^{*} _{1} = \overline{C} _{1} + h , Y ^{*} _{1} = \overline{Y} _{1} + λ h C ^{**} _{1} = \overline{C} _{1} - h , Y ^{**} _{1} = \overline{Y} _{1} - λ h .

The utility function is

V _{2} ( C ^{*} _{1} , Y ^{*} _{1} ) and V _{2} (C ^{**} _{1} , Y ^{**} _{1} ) , therefore the condition for the optimum is:V _{2 C ^{*}} (d \overline{C} _{1} + d h ) + V _{2 Y ^{*}} (d \overline{Y} _{1} + λ d h) + V _{2 C ^{**}} (d \overline{C} _{1} - d h ) + V _{2 Y ^{**}} (d \overline{Y} _{1} - λ d h ) = 0 , and likewise: V _{1 C ^{*}} (d \overline{C} _{1} + d h ) + V _{1 Y ^{*}} (d \overline{Y} _{1} + λ d h) + V _{1 C ^{**}} (d \overline{C} _{1} - d h ) + V _{1 Y ^{**}} (d \overline{Y} _{1} - λ d h ) = 0 . And accordingly:\begin{bmatrix} SV _{2} & SV _{2} \\ SV _{1} & SV _{1} \end{bmatrix} \begin{bmatrix} d \overline{C} \\ d \overline{Y} \end{bmatrix} = - \begin{bmatrix} DV _{2C} + λ DV _{2} \\ DV _{1C} + λ DV _{1} \end{bmatrix} d h , where SV _{k} = V _{k C ^{*}} + V _{k C ^{**}} and SV _{k} = V _{k Y ^{*}} + V _{k Y ^{**}} and k =1 , 2 . Similarly, DV _{k} = V _{k C ^{*}} - V _{k C ^{**}} and DV _{k} = V _{k Y ^{*}} - V _{k Y ^{**}} .

Then:

\lim _{h} (d (\overline{Y)/(-) \overline{C} ) }{d h} = (F _{1} - F _{2})/(( -2 ) ( MRS _{1} - MRS _{2} )) , where MRS _{k} = - ( (\partial V _{k})/(\partial C _{1}) ) ^{-1} (\partial V _{k})/(\partial Y _{1}) . As to F _{1} , F _{2} are denoted by F _{1} = ( (\partial V _{2})/(\partial C _{1}) ) ^{-1} M _{2} ( 1 - MRS _{1} ) and F _{2} = ( (\partial V _{1})/(\partial C _{1}) ) ^{-1} M _{1} ( 1 - MRS _{2} ) . Also, M _{k} is defined by M _{k} = DV _{k} + λ D V _{k} . The first derivative of \overline{Y} - \overline{C} with regard to h , at h =0 , is zero because M _{k} =0 , and so its second derivative needs to be calculated.(d ^{2} ( \overline{Y)/(-) \overline{C} )}{d h ^{2} } = H_ + H_ \;, where H _{1} = (d(F _{1} - F _{2} ))/(dh) (1)/(-2 ( MRS _{1} - MRS _{2} )) and H _{2} = ( -1 ) (d( \overline{Y)/(-) \overline{C} ) }{ d h } (d ln {( -2 )( MRS _{1} - MRS _{2} ))/(}{d) h} . And so H _{2} disappears at h = 0 . Then: (d ^{2} ( \overline{Y)/(-) \overline{C} )}{d h ^{2} } = \frac \; \; . I _{1} = ( V _{2CC} + 2 λ V _{2} + λ ^{2} V _{2} ) ( (\partial V _{2})/(\partial C _{1}) ) ^{-1} ( 1 - MRS _{1} ) I _{2} = ( -1 ) ( V _{1CC} + 2 λ V _{1} + λ ^{2} V _{1} ) ( (\partial V _{1})/(\partial C _{1}) ) ^{-1} ( 1 - MRS _{2} )

Since

MRS _{2} < MRS _{1} < 1 , the condition under which randomization is desirable is calculated: (V _{2CC} + 2 λ V _{2} + λ ^{2} V _{2} ) ( V _{1 C _{1}} + V _{2 Y _{1}} ) - (V _{1CC} + 2 λ V _{1} + λ ^{2} V _{2} ) ( V _{2 C _{1}} + V _{2 Y _{1}} ) < 0 .

See also

• Redistribution of wealth
• Pareto optimality

Notes and References

1. Atkinson . A. B. . Stiglitz . J. E. . 1976 . The Design of Tax Structure: Direct versus Indirect Taxation . . 6 . 1–2 . 55–75 [p. 74] . 10.1016/0047-2727(76)90041-4 .
2. Saez . E. . 2002 . The Desirability of Commodity Taxation under Non-linear Income Taxation and Heterogeneous Tastes . Journal of Public Economics . 83 . 2 . 217–230 . 10.1016/S0047-2727(00)00159-6 .
3. Book: Boadway . R. W. . Pestieau . P. . 2003 . Indirect Taxation and Redistribution: The Scope of the Atkinson-Stiglitz Theorem . Economics for an Imperfect World: Essays in Honor of Joseph E. Stiglitz . MIT Press . 0-262-01205-7 . 387–403 .
4. J.E. Stiglitz, Journal of Public Economics, 17 (1982) 213-124, North-Holland