Editing Invisible hand (section)

===Joseph E. Stiglitz===
The [[Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel|Nobel Prize]]-winning economist [[Joseph E. Stiglitz]], says: "the reason that the invisible hand often seems invisible is that it is often not there."<ref>The Roaring Nineties, 2006</ref><ref name=HAND1>[http://blogs.iht.com/tribtalk/business/globalization/?p=177 ALTMAN, Daniel. ''Managing Globalization.'' In: ''Q & Answers'' with Joseph E. Stiglitz, Columbia University and ''The International Herald Tribune'', October 11, 2006 ] {{webarchive|url=https://web.archive.org/web/20090626040606/http://blogs.iht.com/tribtalk/business/globalization/?p=177 |date=June 26, 2009 }}</ref> Stiglitz explains his position:

{{quote|Adam Smith, the father of modern economics, is often cited as arguing for the "invisible hand" and [[free market]]s: firms, in the pursuit of profits, are led, as if by an invisible hand, to do what is best for the world. But unlike his followers, Adam Smith was aware of some of the limitations of free markets, and research since then has further clarified why free markets, by themselves, often do not lead to what is best. As I put it in my new book, [[Making Globalization Work]], the reason that the invisible hand often seems invisible is that it is often not there. Whenever there are "[[externalities]]"—where the actions of an individual have impacts on others for which they do not pay, or for which they are not compensated—markets will not work well. Some of the important instances have long understood environmental externalities. Markets, by themselves, produce too much pollution. Markets, by themselves, also produce too little basic research. (The government was responsible for financing most of the important scientific breakthroughs, including the internet and the first telegraph line, and many bio-tech advances.) But recent research has shown that these externalities are pervasive, whenever there is imperfect information or imperfect risk markets—that is always.  Government plays an important role in banking and securities regulation, and a host of other areas: some regulation is required to make markets work. Government is needed, almost all would agree, at a minimum to enforce contracts and property rights. The real debate today is about finding the right balance between the market and government (and the third "sector" – governmental non-profit organizations). Both are needed. They can each complement each other. This balance differs from time to time and place to place.<ref name="HAND1"/>}}

The preceding claim is based on Stiglitz's 1986 paper, "Externalities in Economies with Imperfect Information and [[Incomplete Markets]]",<ref>{{cite journal | last2 = Stiglitz | first2 = Joseph E.| last1 = Greenwald | first1 = Bruce C. | title = Externalities in economies with imperfect information and incomplete markets | journal = [[Quarterly Journal of Economics]] | volume = 101 | issue = 2 | pages = 229–64 | publisher = [[Oxford University Press]] | date = May 1986 | jstor = 1891114 | doi = 10.2307/1891114 | doi-access = free }} ([http://socsci2.ucsd.edu/~aronatas/project/academic/Stiglitz%20Greenwald.pdf PDF; 853 kb])</ref> which describes a general methodology to deal with externalities and for calculating [[Optimal tax|optimal corrective taxes]] in a general equilibrium context. In it he considers a model with households, firms and a government.

Households maximize a utility function <math>u^{h}(x^{h}, z^{h})</math>, where <math>x^{h}</math> is the consumption vector and <math>z^{h}</math> are other variables affecting the utility of the household (e.g. pollution). The budget constraint is given by <math>x^{h}_{1}+q \cdot \bar{x}^{h}\leq I^{h}+\sum a^{hf} \cdot \pi^{f}</math>, where q is a vector of prices, a<sup>hf</sup> the fractional holding of household h in firm f, π<sup>f</sup> the profit of firm f, I<sup>h</sup> a lump sum government transfer to the household. The consumption vector can be split as <math>x^{h}=\left( x^{h}_{1}, \bar{x}^{h} \right)</math>.

Firms maximize a profit <math>\pi^{f}=y^{f}_{1}+p\cdot \bar{y}_{1}</math>, where y<sup>f</sup> is a production vector and p is vector of producer prices, subject to <math>y^{f}_{1}-G^{f}(\bar{y}^{f}, z^{f}) \leq 0</math>, G<sub>f</sub> a production function and z<sup>f</sup> are other variables affecting the firm. The production vector can be split as <math>y^{f}=\left( y^{f}_{1}, \bar{y}^{f} \right)</math>.

The government receives a net income <math>R=t \cdot\bar{x}-\sum I^{h}</math>, where <math>t=(q-p)</math> is a tax on the goods sold to households.

It can be shown that in general the resulting equilibrium is not efficient.

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!Proof
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|It is worth keeping in mind that an equilibrium for the model may not necessarily exist. If it exists and there are no taxes (I<sup>h</sup>=0, ∀h), then demand equals supply, and the equilibrium is found by:

<math>\sum \bar{x} ^{h} (q,I,z) - \sum \bar{y}^{f}(p,z)=\bar{ x } (q,I,z) - \sum \bar{y}^{f}(p,z)=0</math>

Let us use <math>\frac{\partial E^{h}}{\partial q}=E^{h}_{q}</math> as a simplifying notation, where <math>E^{h}\left( q, z^{h}, u^{h} \right)</math> is the expenditure function that allows the minimization of household expenditure for a certain level of utility. If there is a set of taxes, subsidies, and lump sum transfers that leaves household utilities unchanged and increase government revenues, then the above equilibrium is not Pareto optimal. On the other hand, if the above non taxed equilibrium is Pareto optimal, then the following maximization problem has a solution for t=0:

: <math>\begin{align}
&\underset{t,I}{\operatorname{maximize}}& & R = t \cdot \bar{ x } - \sum I^{h} \\
&\operatorname{subject\;to}
& & I^{h}+\sum a^{hf} \pi ^{f} =E^{h} (q,z^{h}; \bar{u}^{h}) \\
\end{align}</math>

This is a necessary condition for Pareto optimality. Taking the derivative of the constraint with respect to t yields:

<math>\frac{dI^{h}}{dt}+\sum a^{hf}\left( \pi^{f}_{z} \frac{dz^{f}}{dt}+\pi^{f}_{P} \frac{dp}{dt} \right)=E^{h}_{q} \frac{dq}{dt}+E^{h}_{z} \frac{dz^{h}}{dt}</math>

Where <math>\pi^{f}_{z}=\frac{\partial \pi^{f}_{*}}{\partial z^{f}}</math> and <math>\pi^{f}_{*}(p,z^{f})</math> is the firm's maximum profit function. But since q=t+p, we have that dq/dt=I<sub>N-1</sub>+dp/dt. Therefore, substituting dq/dt in the equation above and rearranging terms gives:

<math>E^{h}_{q}+\left( E^{h}_{q} - \sum a^{hf} \pi^{f}_{P} \right)\frac{dp}{dt}=\frac{dI^{h}}{dt}+\left\{ \sum a^{hf} \pi^{f}_{z} \frac{dz^{f}}{dt} -E^{h}_{z} \frac{dz^{h}}{dt} \right\}</math>

Summing over all households and keeping in mind that <math>\sum a^{hf}=1</math> yields:

<math>\sum E^{h}_{q}+\left(\sum E^{h}_{q} - \sum \pi^{f}_{P} \right)\frac{dp}{dt}=\sum \frac{dI^{h}}{dt}+\left\{\sum \pi^{f}_{z} \frac{dz^{f}}{dt} -\sum E^{h}_{z} \frac{dz^{h}}{dt} \right\}</math>

By the [[envelope theorem]] we have:

<math>\widehat{ x }^{h}_{k}(q;z^{h},u^{h})  = \left. \frac{\partial E^{h}}{\partial q}\right|_{z^{h},u^{h}}</math>

<math>\left. \frac{\partial \pi^{f}_{*}}{\partial p_{k_{1}}}\right|_{z^{f}}=y^{f}_{k}</math>;∀k

This allows the constraint to be rewritten as:

<math>\bar{x} + \left( \bar{ x } - \bar{ y } \right)\frac{dp}{dt}=\sum\frac{dI^{h}}{dt}+\left( \sum\pi^{f}_{z}\frac{dz^{f}}{dt} - \sum E^{h}_{z} \frac{dz^{h}}{dt} \right)</math>

Since <math>\bar{x}=\bar{y}</math>:

<math>\sum \frac{dI^{h}}{dt}= \bar{ x } - \left( \sum \pi^{f}_{z} \frac{dz^{f}}{dt} - \sum E^{h}_{z} \frac{dz^{h}}{dt} \right)</math>

Differentiating the objective function of the maximization problem gives:

<math>\frac{dR}{dt}= \bar{ x } + \frac{d\bar{x}}{dt} \cdot t - \sum \frac{dI^{h}}{dt}</math>

Substituting <math>\sum \frac{dI^{h}}{dt}</math> from the former equation in to latter equation results in:

<math>\frac{dR}{dt}= \frac{d\bar{x}}{dt} \cdot t +(\sum \pi ^{f}_{z} \frac{dz^{f}}{dt} - \sum E^{h}_{z} \frac{dz^{h}}{dt}) =\frac{d\bar{x}}{dt} \cdot t +(\Pi^{t} - B^{t})</math>

Recall that for the maximization problem to have a solution a t=0:

<math>\frac{dR}{dt} = \left( \Pi^{t} - B^{t} \right) = 0</math>

In conclusion, for the equilibrium to be Pareto optimal dR/dt must be zero. Except for the special case where Π and B are equal, in general the equilibrium will not be Pareto optimal, therefore inefficient.
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