Greenwald-Stiglitz theorem

Sketch

Let ${\displaystyle X}$ and ${\displaystyle q=(q_{1},q_{2},\ldots ,q_{M})}$ be the vector of goods and the associated vector of consumer prices, respectively. And ${\displaystyle p^{H}}$ is the accident probability for the case where the level of care is high, and ${\displaystyle p^{L}}$ low care. Also, the parameters regarding contract ${\displaystyle \lbrace \alpha ,\beta \rbrace }$ are given, consisting of the net benefit ${\displaystyle \alpha }$ in the event of accident and the premium ${\displaystyle \beta }$. For high care, the vector of goods is ${\displaystyle X^{H}=p^{H}X^{H1}+(1-p^{H})X^{H0}}$, where ${\displaystyle X^{H1}}$ is the consumption vector in the event of an accident and ${\displaystyle X^{H0}}$ no accident. Now the expected utility ${\displaystyle V^{H}}$ for high care is maximized, under the social revenue constraint ${\displaystyle \alpha p^{H}=\beta (1-p^{H})+(q-1)X^{H}}$ and the self-selection constraint that the expected utility for low care is less than or equal to that for high care.

As to the Lagrangian for the model,

${\displaystyle L=V^{H}+\gamma (\alpha p^{H}-\beta (1-p^{H})-(q-1)X^{H})+\lambda (V^{H}-V^{L})}$

and its partial derivatives are evaluated at q=1; this corresponds to a case where there is no differential taxation

${\displaystyle {\frac {\partial L}{\partial q_{k}}}{\Big |}_{q=1}={\frac {\partial V^{H}}{\partial q_{k}}}-\gamma {\frac {\partial X^{H}}{\partial q_{k}}}+\lambda ({\frac {\partial V^{H}}{\partial q_{k}}}-{\frac {\partial V^{L}}{\partial q_{k}}})}$

${\displaystyle {\frac {\partial L}{\partial \alpha }}{\Big |}_{q=1}={\frac {\partial V^{H}}{\partial \alpha }}+\gamma p^{H}+\lambda ({\frac {\partial V^{H}}{\partial \alpha }}-{\frac {\partial V^{L}}{\partial \alpha }})=0}$

${\displaystyle {\frac {\partial L}{\partial \beta }}{\Big |}_{q=1}={\frac {\partial V^{H}}{\partial \beta }}-\gamma (1-p^{H})+\lambda ({\frac {\partial V^{H}}{\partial \beta }}-{\frac {\partial V^{L}}{\partial \beta }})=0}$

When it comes to the marginal utility of income ${\displaystyle \mu ^{i1},\quad (\mu ^{i0})}$ with care level i in the event of an accident (no accident), Roy's identity says ${\displaystyle {\frac {\partial V^{i}}{\partial q_{k}}}=-p^{i}\mu ^{i1}{\frac {\partial X^{i1}}{\partial q_{k}}}-(1-p^{i})\mu ^{i0}{\frac {\partial X^{i0}}{\partial q_{k}}}}$. Also, ${\displaystyle {\frac {\partial V^{i}}{\partial \alpha }}=p^{i}\mu ^{i1}}$ and ${\displaystyle {\frac {\partial V^{i}}{\partial \beta }}=-(1-p^{i})\mu ^{i0}}$.

Accordingly, adding ${\displaystyle {\frac {\partial L}{\partial \alpha }}{\Big |}_{q=1}{\frac {\partial X^{H1}}{\partial q_{k}}}+{\frac {\partial L}{\partial \beta }}{\Big |}_{q=1}(-{\frac {\partial X^{H0}}{\partial q_{k}}})}$ to ${\displaystyle {\frac {\partial L}{\partial q_{k}}}{\Big |}_{q=1}}$ allows several terms to offset, resulting in

${\displaystyle {\frac {\partial L}{\partial q_{k}}}{\Big |}_{q=1}=\lambda (p^{L}\mu ^{L1}(X_{k}^{L1}-X_{k}^{H1})+(1-p^{L})\mu ^{L0}(X_{k}^{L0}-X_{k}^{H0}))}$

Then, by putting ${\displaystyle {\bar {\mu }}^{L}=p^{L}\mu ^{L1}+(1-p^{L})\mu ^{L0}}$, the equation becomes

${\displaystyle {\frac {\partial L}{\partial q_{k}}}{\Big |}_{q=1}=\lambda {\bar {\mu }}^{L}\left({\bar {X}}_{k}^{L}-{\bar {X}}_{k}^{H}\right)}$, where

${\displaystyle {\bar {X}}_{k}^{i}={\frac {1}{{\bar {\mu }}^{L}}}(p^{L}\mu ^{L1}X_{k}^{i1}+(1-p^{L})\mu ^{L0}X_{k}^{i0})}$. This equation means that the partial derivative in the left hand side of the equation is negative if ${\displaystyle {\bar {X}}_{k}>{\bar {X}}_{k}^{L}}$ and positive if ${\displaystyle {\bar {X}}_{k}<{\bar {X}}_{k}^{L}}$. It is therefore apparent that differential commodity taxation or subsidy is welfare-improving, and this result is interpreted in the following way. The individual's level of consumption of a commodity is correlated to his experience with taxation of that commodity. Therefore, in cases where he reduces his consumption of the commodity when switching from the risky activity to the safe, the impact of taxing that commodity will be mitigated if he is choosing the safe option. This change has a decisive effect on his behaviour, making him strictly prefer the safe activity to the risky. In cases where his level of consumption of the commodity is increased when switching from the risky to the safe, a subsidy on that commodity will have a greater beneficial effect if he is doing the safe activity and lead him to strictly prefer the safe option over the risky.

Problem settings

The model is constructed with three main parts: households, firms, and the government. Let ${\displaystyle x^{h}=(x_{1}^{h},{\bar {x}}^{h})}$ be an N-dimensional consumption vector of household h; ${\displaystyle x_{1}^{h}}$ is consumption for the numeraire good and ${\displaystyle {\bar {x}}^{h}}$ is for the N-1 nonnumeraire goods. Likewise ${\displaystyle y^{f}=(y_{1}^{f},{\bar {y}}^{f})}$ is an N-dimensional production function of firm f.

Household utilities ${\displaystyle u^{h}}$ are maximized, placing financial constraints on spending ${\displaystyle E^{h}}$

${\displaystyle x_{1}^{h}+q\cdot {\bar {x}}^{h}\leq I^{h}+\sum _{F}a^{hf}\cdots \pi ^{f}}$

where ${\displaystyle q}$ is an (N-1)-dimensional vector representing the prices of N-1 nonnumeraire goods, and ${\displaystyle \pi ^{f}}$ is the profits of firm f. Also, ${\displaystyle I^{h}}$ is a lump sum transfer from the government to household h. Then, the compensated demand for good ${\displaystyle k}$ keeping ${\displaystyle z^{h},u^{h}}$ unchanged is ${\displaystyle {\hat {x}}_{k}^{h}\equiv {\frac {\partial E^{h}}{\partial q}}{\Big |}_{z^{h},u^{h}}}$.

Under the condition that production of the numeraire good ${\displaystyle y_{1}^{f}}$ is equal to or less than production function ${\displaystyle G^{f}}$, profit maximization for firms is done:

${\displaystyle \pi ^{f}=y_{1}^{f}+p\cdot {\bar {y}}^{f}}$

where ${\displaystyle p}$ is an (N-1)-dimensional vector representing produces' prices for the N-1 nonnumeraire goods. This implies that ${\displaystyle {\frac {\partial \pi _{*}^{f}}{\partial p_{k}}}{\Big |}_{z^{f}}=y_{k}^{f},\quad (k=1,2,\ldots ,N)}$ for the firm's maximum profit function ${\displaystyle \pi _{*}^{f}}$.

For the government, its net income is the difference between revenue and expenditure:

${\displaystyle R\equiv t\cdot {\bar {x}}-\sum _{H}I^{h}}$

where ${\displaystyle t=q-p}$ and ${\displaystyle {\bar {x}}=\sum _{H}{\bar {x}}^{h}}$.

Initial equilibrium

If the original equilibrium is Pareto optimal, there exists a solution that would increase the government's net income without affecting household utilities. This means that the objective function ${\displaystyle R}$ is maximized at ${\displaystyle t=0}$ under the constraint for the government: ${\displaystyle I^{h}+\sum a^{hf}\pi ^{f}=E^{h}}$.

Then considering the toral derivative of the constraint, together with the fact that ${\displaystyle {\frac {dq}{dt}}=I_{N-1}+{\frac {dp}{dt}}}$, yields

${\displaystyle {\frac {\partial E^{h}}{\partial q}}+({\frac {\partial E^{h}}{\partial q}}-\sum _{f}a^{hf}{\frac {\partial \pi ^{f}}{\partial p}}){\frac {dp}{dt}}={\frac {dI^{h}}{dt}}+(\sum _{F}a^{hf}{\frac {\partial \pi ^{f}}{\partial z}}{\frac {dz^{f}}{dt}}-{\frac {\partial E^{h}}{\partial z}}{\frac {dz^{h}}{dt}})}$

Given that ${\displaystyle \sum _{h}a^{hf}=1}$, the first derivative of the constraint is summed over all households

${\displaystyle {\bar {x}}+({\bar {x}}-{\bar {y}}){\frac {dp}{dt}}=\sum _{H}{\frac {dI^{h}}{dt}}+(\Pi -B)}$

with technological externalities ${\displaystyle \Pi =\sum _{F}{\frac {\partial \pi _{*}^{f}}{\partial z^{f}}}{\frac {dz^{f}}{dt}}}$ and ${\displaystyle B=\sum _{H}{\frac {\partial E^{h}}{\partial z^{h}}}{\frac {dz^{h}}{dt}}}$. Furthermore, the bracket term becomes zero as ${\displaystyle {\bar {x}}={\bar {y}}}$ in any market equilibrium, and it is presented as

${\displaystyle \sum _{H}{\frac {dI^{h}}{dt}}={\bar {x}}-(\Pi -B)}$

Then, the first derivative of the objective function is

${\displaystyle {\frac {dR}{dt}}={\frac {d{\bar {x}}}{dt}}\cdot t+\Pi -B}$

The necessary condition for the system to be optimal at the initial equilibrium is:

${\displaystyle {\frac {dR}{dt}}=\Pi -B=0}$

It is therefore apparent that the initial equilibrium is Pareto improved by welfare-improving tax measures unless externalities ${\displaystyle \Pi }$ and ${\displaystyle B}$ are compleltely offsetting.