Atkinson–Stiglitz theorem

Other approaches

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

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:

${\displaystyle {\overline{U}}_1\le V_1(C_1,Y_1).}$

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:

${\displaystyle R=-(C_1-Y_1)N_1-(C_2-Y_2)N_2,}$

${\displaystyle \overline{R}\le R,}$

where ${\displaystyle N_1}$ and ${\displaystyle N_2}$ indicate the number of individuals of each type. Under these conditions, the government needs to maximize the utility ${\displaystyle V_2(C_2,Y_2)}$ of category 2. Then writing down the Lagrange function for this problem:

${\displaystyle \mathcal{L}=V_2(C_2,Y_2)+\mu V_1(C_1,Y_1)+{\lambda }_2(V_2(C_2,Y_2)-V_2(C_1,Y_1))+{\lambda }_1(V_1(C_1,Y_1)-V_1(C_2,Y_2))+\gamma \left(-(C_1-Y_1)N_1-(C_2-Y_2)N_2-\overline{R}\right),}$

ensuring the satisfaction of the self-selection constraints, the first-order conditions are:

${\displaystyle \mu \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_1}-{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_1}+{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_1}-\gamma N_1=0,}$

${\displaystyle \mu \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_1}-{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_1}+{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_1}+\gamma N_1=0,}$

${\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_2}+{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_2}-{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_2}-\gamma N_2=0,}$

${\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_2}+{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_2}-{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_2}+\gamma N_2=0.}$

For the case where ${\displaystyle {\lambda }_1=0}$ and ${\displaystyle {\lambda }_2=0}$:

${\displaystyle \frac{\mathrm{\partial }{V}_i/\mathrm{\partial }{Y}_i}{\mathrm{\partial }{V}_i/\mathrm{\partial }{C}_i}+1=0,}$

for ${\displaystyle i=1,2}$, therefore the government can achieve a lump-sum taxation. For the case where ${\displaystyle {\lambda }_1=0}$ and ${\displaystyle {\lambda }_2>0}$:

${\displaystyle \frac{\mathrm{\partial }{V}_2/\mathrm{\partial }{Y}_2}{\mathrm{\partial }{V}_2/\mathrm{\partial }{C}_2}+1=0,}$

the marginal tax rate for category 2 is zero. As to category 1:

${\displaystyle \frac{\mathrm{\partial }{V}_1/\mathrm{\partial }{Y}_2}{\mathrm{\partial }{V}_1/\mathrm{\partial }{C}_1}=-\frac{1-{\lambda }_2(\mathrm{\partial }{V}_2/\mathrm{\partial }{Y}_1)/N_1\gamma }{1+{\lambda }_2(\mathrm{\partial }{V}_2/\mathrm{\partial }{C}_1)/N_1\gamma }.}$

If ${\displaystyle {\delta }_i=\frac{\mathrm{\partial }{V}_i/\mathrm{\partial }{Y}_1}{\mathrm{\partial }{V}_i/\mathrm{\partial }{C}_1},(i=1,2)}$, the marginal tax rate for category 1 is ${\displaystyle {\delta }_1+1}$.

Also, note the following equation:

${\displaystyle {\delta }_1=-\left(\frac{1-\nu {\delta }_2}{1+\nu }\right),}$

where ${\displaystyle \nu }$ is denoted by:

${\displaystyle \nu =\frac{{\lambda }_2(\mathrm{\partial }{V}_2/\mathrm{\partial }{C}_1)}{N_1\gamma }.}$

Therefore, by assumption, ${\displaystyle {\delta }_1<{\delta }_2}$, and so ${\displaystyle -1<{\delta }_1<{\delta }_2}$ can be directly proven. Accordingly, 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.

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:

${\displaystyle {\mathbf{\text{C}}}_1=\sum _j{C}_{1j}{\mathbf{\text{e}}}_j}$

${\displaystyle {\mathbf{\text{C}}}_2=\sum _j{C}_{2j}{\mathbf{\text{e}}}_j.}$

In this case, the government's budget constraint is:

${\displaystyle R\le \sum _{k=1}^2(Y_k{N}_k)-N_1\sum _j{C}_{1j}-N_2\sum _j{C}_{2j}.}$

Then:

${\displaystyle \mu \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{1j}}-{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{1j}}+{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{1j}}-\gamma N_1=0,}$

${\displaystyle \mu \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_1}-{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_1}+{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_1}+\gamma N_1=0,}$

${\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{2j}}+{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{2j}}-{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{2j}}-\gamma N_2=0,}$

${\displaystyle \frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_2}+{\lambda }_2\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_2}-{\lambda }_1\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{Y}_2}+\gamma N_2=0.}$

Here, ${\displaystyle {\lambda }_1=0}$ and ${\displaystyle {\lambda }_2>0}$. Therefore, it follows that:

${\displaystyle \frac{\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{2j}}}{\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{2n}}}=1,\frac{\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{2j}}}{\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{Y}_2}}=1.}$

Suppose all individuals have the same indifference curve in C-L plane. The separability between leisure and consumption can be expressed as:

${\displaystyle \frac{{\mathrm{\partial }}^2{U}_k}{\mathrm{\partial }{C}_{kj}\mathrm{\partial }{L}_k}=0,}$ yielding

${\displaystyle \frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{1j}}=\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_{1j}}.}$

As a result:

${\displaystyle \frac{\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{1j}}}{\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_{1n}}}=1.}$

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 ${\displaystyle \{C_2^{\ast },Y_2^{\ast }\}}$. For the case where an individual chooses to hide their ability, there are two tax schedule possibilities: ${\displaystyle \{C_1^{\ast },Y_1^{\ast }\}}$ and ${\displaystyle \{C_1^{\ast \ast },Y_1^{\ast \ast }\}}$. 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 consumption is maximized, a higher ${\displaystyle {\overline{C}}_1}$ is set for a higher ${\displaystyle {\overline{Y}}_1}$. Then the relations between those variables are:

${\displaystyle C_1^{\ast }={\overline{C}}_1+h,Y_1^{\ast }={\overline{Y}}_1+\lambda h}$

${\displaystyle C_1^{\ast \ast }={\overline{C}}_1-h,Y_1^{\ast \ast }={\overline{Y}}_1-\lambda h.}$

The utility function is ${\displaystyle V_2(C_1^{\ast },Y_1^{\ast })}$ and ${\displaystyle V_2(C_1^{\ast \ast },Y_1^{\ast \ast })}$, therefore the condition for the optimum is:

${\displaystyle V_{2C^{\ast }}(d{\overline{C}}_1+dh)+V_{2Y^{\ast }}(d{\overline{Y}}_1+\lambda dh)+V_{2C^{\ast \ast }}(d{\overline{C}}_1-dh)+V_{2Y^{\ast \ast }}(d{\overline{Y}}_1-\lambda dh)=0,}$

and likewise:

${\displaystyle V_{1C^{\ast }}(d{\overline{C}}_1+dh)+V_{1Y^{\ast }}(d{\overline{Y}}_1+\lambda dh)+V_{1C^{\ast \ast }}(d{\overline{C}}_1-dh)+V_{1Y^{\ast \ast }}(d{\overline{Y}}_1-\lambda dh)=0.}$

And accordingly:

${\displaystyle \left[\begin{array}{cc}S{V}_{2C}& S{V}_{2Y}\\ S{V}_{1C}& S{V}_{1Y}\end{array}\right]\left[\begin{array}{c}d\overline{C}\\ d\overline{Y}\end{array}\right]=-\left[\begin{array}{c}D{V}_{2C}+\lambda D{V}_{2Y}\\ D{V}_{1C}+\lambda D{V}_{1C}\end{array}\right]dh,}$

where ${\displaystyle S{V}_{kC}=V_{k{C}^{\ast }}+V_{k{C}^{\ast \ast }}}$ and ${\displaystyle S{V}_{kY}=V_{k{Y}^{\ast }}+V_{k{Y}^{\ast \ast }}}$ and ${\displaystyle k=1,2}$. Similarly, ${\displaystyle D{V}_{kC}=V_{k{C}^{\ast }}-V_{k{C}^{\ast \ast }}}$ and ${\displaystyle D{V}_{kY}=V_{k{Y}^{\ast }}-V_{k{Y}^{\ast \ast }}}$.

Then:

${\displaystyle \underset{h\to 0}{lim}\frac{d(\overline{Y}-\overline{C})}{dh}=\frac{F_1-F_2}{(-2)(MR{S}_1-MR{S}_2)},}$

where ${\displaystyle MR{S}_k=-(\frac{\mathrm{\partial }{V}_k}{\mathrm{\partial }{C}_1}{)}^{-1}\frac{\mathrm{\partial }{V}_k}{\mathrm{\partial }{Y}_1}}$. As to ${\displaystyle F_1,F_2}$ are denoted by ${\displaystyle F_1=(\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_1}{)}^{-1}{M}_2(1-MR{S}_1)}$ and ${\displaystyle F_2=(\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_1}{)}^{-1}{M}_1(1-MR{S}_2)}$. Also, ${\displaystyle M_k}$ is defined by ${\displaystyle M_k=D{V}_{kC}+\lambda D{V}_{kY}}$. 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 its second derivative needs to be calculated.

${\displaystyle \frac{d^2(\overline{Y}-\overline{C})}{d{h}^2}=H_1+H_2,}$

where ${\displaystyle H_1=\frac{d(F_1-F_2)}{dh}\frac{1}{-2(MR{S}_1-MR{S}_2)}}$ and ${\displaystyle H_2=(-1)\frac{d(\overline{Y}-\overline{C})}{dh}\frac{d\ln (-2)(MR{S}_1-MR{S}_2)}{dh}}$. And so ${\displaystyle H_2}$ disappears at ${\displaystyle h=0}$. Then:

${\displaystyle \frac{d^2(\overline{Y}-\overline{C})}{d{h}^2}=\frac{I_1+I_2}{(-1)(MR{S}_1-MR{S}_2)}.}$

${\displaystyle I_1=(V_{2CC}+2\lambda V_{2CY}+{\lambda }^2{V}_{2YY})(\frac{\mathrm{\partial }{V}_2}{\mathrm{\partial }{C}_1}{)}^{-1}(1-MR{S}_1)}$

${\displaystyle I_2=(-1)(V_{1CC}+2\lambda V_{1CY}+{\lambda }^2{V}_{1YY})(\frac{\mathrm{\partial }{V}_1}{\mathrm{\partial }{C}_1}{)}^{-1}(1-MR{S}_2)}$

Since ${\displaystyle MR{S}_2<MR{S}_1<1}$, the condition under which randomization is desirable is calculated:

${\displaystyle (V_{2CC}+2\lambda V_{2CY}+{\lambda }^2{V}_{2YY})(V_{1C_1}+V_{2Y_1})-(V_{1CC}+2\lambda V_{1CY}+{\lambda }^2{V}_{2YY})(V_{2C_1}+V_{2Y_1})<0.}$