Atkinson–Stiglitz theorem

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

${\displaystyle {\overline {U}}_{1}\leq V_{1}(C_{1},Y_{1})\quad .}$

The second condition is that the government revenue ${\displaystyle R}$, which is equal to or more than the revenue requirement ${\displaystyle {\overline {R}}}$, is increased by a given amount:

${\displaystyle R=-(C_{1}-Y_{1})N_{1}-(C_{2}-Y_{2})N_{2}\;,}$

${\displaystyle {\overline {R}}\leq R\;,}$

where ${\displaystyle N_{1}}$ and ${\displaystyle N_{2}}$ indicate the number of individuals of each type. Under these conditions, the government needs to maximise the utility ${\displaystyle V_{2}(C_{2},Y_{2})}$ of category 2. Then writing down the Lagrange function for this problem:

${\displaystyle {\mathcal {L}}=V_{2}(C_{2},Y_{2})+\mu V_{1}(C_{1},Y_{1})+\lambda _{2}(V_{2}(C_{2},Y_{2})-V_{2}(C_{1},Y_{1}))+\lambda _{1}(V_{1}(C_{1},Y_{1})-V_{1}(C_{2},Y_{2}))+\gamma \left(-(C_{1}-Y_{1})N_{1}-(C_{2}-Y_{2})N_{2}-{\overline {R}}\right)\;,}$

which ensures satisfaction of the self-selection constraints, we obtain the first order conditions:

${\displaystyle \mu {\frac {\partial V_{1}}{\partial C_{1}}}-\lambda _{2}{\frac {\partial V_{2}}{\partial C_{1}}}+\lambda _{1}{\frac {\partial V_{1}}{\partial C_{1}}}-\gamma N_{1}=0\;,}$

${\displaystyle \mu {\frac {\partial V_{1}}{\partial Y_{1}}}-\lambda _{2}{\frac {\partial V_{2}}{\partial Y_{1}}}+\lambda _{1}{\frac {\partial V_{1}}{\partial Y_{1}}}+\gamma N_{1}=0\;,}$

${\displaystyle {\frac {\partial V_{2}}{\partial C_{2}}}+\lambda _{2}{\frac {\partial V_{2}}{\partial C_{2}}}-\lambda _{1}{\frac {\partial V_{1}}{\partial C_{2}}}-\gamma N_{2}=0\;,}$

${\displaystyle {\frac {\partial V_{2}}{\partial Y_{2}}}+\lambda _{2}{\frac {\partial V_{2}}{\partial Y_{2}}}-\lambda _{1}{\frac {\partial V_{1}}{\partial Y_{2}}}+\gamma N_{2}=0\;.}$

For the case where ${\displaystyle \lambda _{1}=0}$ and ${\displaystyle \lambda _{2}=0}$, we have

${\displaystyle {\frac {\partial V_{i}/\partial Y_{i}}{\partial V_{i}/\partial C_{i}}}+1=0\;,}$

for ${\displaystyle i=1,2}$, and therefore the government can achieve a lump-sum taxation. For the case where ${\displaystyle \lambda _{1}=0}$ and ${\displaystyle \lambda _{2}>0}$, we have

${\displaystyle {\frac {\partial V_{2}/\partial Y_{2}}{\partial V_{2}/\partial C_{2}}}+1=0\;,}$

and we find that the marginal tax rate for category 2 is zero. And as to category 1, we have

${\displaystyle {\frac {\partial V_{1}/\partial Y_{2}}{\partial V_{1}/\partial C_{1}}}=-{\frac {1-\lambda _{2}(\partial V_{2}/\partial Y_{1})/N_{1}\gamma }{1+\lambda _{2}(\partial V_{2}/\partial C_{1})/N_{1}\gamma }}\;.}$

If we put ${\displaystyle \delta _{i}={\frac {\partial V_{i}/\partial Y_{1}}{\partial V_{i}/\partial C_{1}}}\;,\quad (i=1,2)}$, then the marginal tax rate for category 1 is ${\displaystyle \delta _{1}+1}$.

Also, we have the following expression:

${\displaystyle \delta _{1}=-\left({\frac {1-\nu \delta _{2}}{1+\nu }}\right)\;,}$

where we denote ${\displaystyle \nu }$ by

${\displaystyle \nu ={\frac {\lambda _{2}(\partial V_{2}/\partial C_{1})}{N_{1}\gamma }}\;.}$

Therefore, by assumption, ${\displaystyle \delta _{1}<\delta _{2}}$, and so we can directly prove that ${\displaystyle -1<\delta _{1}<\delta _{2}}$. Accordingly, we find that the marginal tax rate for category 1 is positive.

For the case where ${\displaystyle \lambda _{1}>0}$ and ${\displaystyle \lambda _{2}=0}$, 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.

Now we need to consider a case where income level and several commodities are observals. Each individual's consumption function is expressed in a vector form as

${\displaystyle {\textbf {C}}_{1}=\sum _{j}C_{1j}{\textbf {e}}_{j}}$

${\displaystyle {\textbf {C}}_{2}=\sum _{j}C_{2j}{\textbf {e}}_{j}\;.}$

In this case, the government's budget constraint is

${\displaystyle R\leq \sum _{k=1}^{2}(Y_{k}N_{k})-N_{1}\sum _{j}C_{1j}-N_{2}\sum _{j}C_{2j}\;.}$

Then we have

${\displaystyle \mu {\frac {\partial V_{1}}{\partial C_{1j}}}-\lambda _{2}{\frac {\partial V_{2}}{\partial C_{1j}}}+\lambda _{1}{\frac {\partial V_{1}}{\partial C_{1j}}}-\gamma N_{1}=0\;,}$

${\displaystyle \mu {\frac {\partial V_{1}}{\partial Y_{1}}}-\lambda _{2}{\frac {\partial V_{2}}{\partial Y_{1}}}+\lambda _{1}{\frac {\partial V_{1}}{\partial Y_{1}}}+\gamma N_{1}=0\;,}$

${\displaystyle {\frac {\partial V_{2}}{\partial C_{2j}}}+\lambda _{2}{\frac {\partial V_{2}}{\partial C_{2j}}}-\lambda _{1}{\frac {\partial V_{1}}{\partial C_{2j}}}-\gamma N_{2}=0\;,}$

${\displaystyle {\frac {\partial V_{2}}{\partial Y_{2}}}+\lambda _{2}{\frac {\partial V_{2}}{\partial Y_{2}}}-\lambda _{1}{\frac {\partial V_{1}}{\partial Y_{2}}}+\gamma N_{2}=0\;.}$

Here we restricting ourselves to the case where ${\displaystyle \lambda _{1}=0}$ and ${\displaystyle \lambda _{2}>0}$. It follows that

${\displaystyle {\frac {\frac {\partial V_{2}}{\partial C_{2j}}}{\frac {\partial V_{2}}{\partial C_{2n}}}}=1\;,\quad {\frac {\frac {\partial V_{2}}{\partial C_{2j}}}{\frac {\partial V_{2}}{\partial Y_{2}}}}=1\;.}$

Suppose all individuals have the same indifference curve in C-L plane. The separability between leisure and consumption enables us to have ${\displaystyle {\frac {\partial ^{2}U_{k}}{\partial C_{kj}\partial L_{k}}}=0\;,}$ which yields

${\displaystyle {\frac {\partial V_{1}}{\partial C_{1j}}}={\frac {\partial V_{2}}{\partial C_{1j}}}\;.}$

As a result, we obtain

${\displaystyle {\frac {\frac {\partial V_{1}}{\partial C_{1j}}}{\frac {\partial V_{1}}{\partial C_{1n}}}}=1\;.}$

Thus we find that it is unnecessary to impose taxes on commodities.[4]

We need to consider a case where high ability individuals (who usually earn more money to show their ability) pretend to be like they are not more able. In this case, it could be argued that the government needs to randomize the taxes imposed on the low ability individuals, for the purpose of increasing the effectiveness of screening. It is possible that under certain conditions we can do the randomization of the taxes without damaging the low ability individuals, and therefore we discuss the conditions. For the case where an individual chooses to show his ability, we see a tax schedule be related to ${\displaystyle \lbrace C_{2}^{*},Y_{2}^{*}\rbrace }$. For the case where an individual chooses to hide his ability, we see one of two tax schedules: ${\displaystyle \lbrace C_{1}^{*},Y_{1}^{*}\rbrace }$ and ${\displaystyle \lbrace C_{1}^{**},Y_{1}^{**}\rbrace }$. 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 ${\displaystyle Y}$. As the comsumption is maximized, a higher ${\displaystyle {\overline {C}}_{1}}$ is set for a higher ${\displaystyle {\overline {Y}}_{1}}$. Then the relations between those variables are

${\displaystyle C_{1}^{*}={\overline {C}}_{1}+h\;,\quad Y_{1}^{*}={\overline {Y}}_{1}+\lambda h}$

${\displaystyle C_{1}^{**}={\overline {C}}_{1}-h\;,\quad Y_{1}^{**}={\overline {Y}}_{1}-\lambda h\;.}$

The utility function is ${\displaystyle V_{2}(C_{1}^{*},Y_{1}^{*})}$ and ${\displaystyle V_{2}(C_{1}^{**},Y_{1}^{**})}$, and we have the condition for the optimum:

${\displaystyle V_{2C^{*}}(d{\overline {C}}_{1}+dh)+V_{2Y^{*}}(d{\overline {Y}}_{1}+\lambda dh)+V_{2C^{**}}(d{\overline {C}}_{1}-dh)+V_{2Y^{**}}(d{\overline {Y}}_{1}-\lambda dh)=0\;,}$

and likewise

${\displaystyle V_{1C^{*}}(d{\overline {C}}_{1}+dh)+V_{1Y^{*}}(d{\overline {Y}}_{1}+\lambda dh)+V_{1C^{**}}(d{\overline {C}}_{1}-dh)+V_{1Y^{**}}(d{\overline {Y}}_{1}-\lambda dh)=0\;.}$

And accordingly we have

${\displaystyle {\begin{bmatrix}SV_{2C}&SV_{2Y}\\SV_{1C}&SV_{1Y}\end{bmatrix}}{\begin{bmatrix}d{\overline {C}}\\d{\overline {Y}}\end{bmatrix}}=-{\begin{bmatrix}DV_{2C}+\lambda DV_{2Y}\\DV_{1C}+\lambda DV_{1C}\end{bmatrix}}dh\;,}$

where ${\displaystyle SV_{kC}=V_{kC^{*}}+V_{kC^{**}}}$ and ${\displaystyle SV_{kY}=V_{kY^{*}}+V_{kY^{**}}}$ and ${\displaystyle k=1,2}$. Similarly ${\displaystyle DV_{kC}=V_{kC^{*}}-V_{kC^{**}}}$ and ${\displaystyle DV_{kY}=V_{kY^{*}}-V_{kY^{**}}}$.

Then we have

${\displaystyle \lim _{h\rightarrow 0}{\frac {d({\overline {Y}}-{\overline {C}})}{dh}}={\frac {F_{1}-F_{2}}{(-2)(MRS_{1}-MRS_{2})}}\;,}$

where ${\displaystyle MRS_{k}=-({\frac {\partial V_{k}}{\partial C_{1}}})^{-1}{\frac {\partial V_{k}}{\partial Y_{1}}}}$. As to ${\displaystyle F_{1},F_{2}}$ we denote them by ${\displaystyle F_{1}=({\frac {\partial V_{2}}{\partial C_{1}}})^{-1}M_{2}(1-MRS_{1})}$ and ${\displaystyle F_{2}=({\frac {\partial V_{1}}{\partial C_{1}}})^{-1}M_{1}(1-MRS_{2})}$. Also we define ${\displaystyle M_{k}}$ by ${\displaystyle M_{k}=DV_{kC}+\lambda DV_{kY}}$. But the first derivative of ${\displaystyle {\overline {Y}}-{\overline {C}}}$ with regard to ${\displaystyle h}$, at ${\displaystyle h=0}$, is zero (because ${\displaystyle M_{k}=0}$), and so we need to calculate its second derivative.

${\displaystyle {\frac {d^{2}({\overline {Y}}-{\overline {C}})}{dh^{2}}}=H_{1}+H_{2}\;,}$

where ${\displaystyle H_{1}={\frac {d(F_{1}-F_{2})}{dh}}{\frac {1}{-2(MRS_{1}-MRS_{2})}}}$ and ${\displaystyle H_{2}=(-1){\frac {d({\overline {Y}}-{\overline {C}})}{dh}}{\frac {d\ln {(-2)(MRS_{1}-MRS_{2})}}{dh}}}$. And so ${\displaystyle H_{2}}$ vanishes at ${\displaystyle h=0}$. Then we have

${\displaystyle {\frac {d^{2}({\overline {Y}}-{\overline {C}})}{dh^{2}}}={\frac {I_{1}+I_{2}}{(-1)(MRS_{1}-MRS_{2})}}\;\;.}$

${\displaystyle I_{1}=(V_{2CC}+2\lambda V_{2CY}+\lambda ^{2}V_{2YY})({\frac {\partial V_{2}}{\partial C_{1}}})^{-1}(1-MRS_{1})}$

${\displaystyle I_{2}=(-1)(V_{1CC}+2\lambda V_{1CY}+\lambda ^{2}V_{1YY})({\frac {\partial V_{1}}{\partial C_{1}}})^{-1}(1-MRS_{2})}$

Since ${\displaystyle MRS_{2}<MRS_{1}<1}$, we obtain the condition under which randomization is desirable:[4]

${\displaystyle (V_{2CC}+2\lambda V_{2CY}+\lambda ^{2}V_{2YY})(V_{1C_{1}}+V_{2Y_{1}})-(V_{1CC}+2\lambda V_{1CY}+\lambda ^{2}V_{2YY})(V_{2C_{1}}+V_{2Y_{1}})<0\;.}$