deletions | additions       diff --git a/sectionComparison_of.tex b/sectionComparison_of.tex  index 0a36326..1c31540 100644  --- a/sectionComparison_of.tex  +++ b/sectionComparison_of.tex 
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\section{Comparison of two decision schemes}  Adopting the social point of view, the purpose here is to investigate the relative efficiency of the two decision schemes discussed above. We give conditions such that the Tullock contest is socially better than the cost-benefit analysis. Suggestion %Suggestion  du reviewer 2: Brennan %Brennan  and Buchanan, "Towards a Tax Constitution for Leviathan", Journal of Public Economics. 8 (1977), pp. 255-274; Brennan 255-274  %Brennan  and Buchanan. "Tax Instruments as Constraints on the Disposition of Public Revenues", Journal of Public Economics, 9 (1978), pp. 301-318;  Brennan 301-318  %Brennan  and Buchanan, The Power to Tax. (Cambridge: Cambridge University Press, 1980);  Brennan 1980)  %Brennan  and Buchanan, The Reason of Rules. (Cambridge: Cambridge University Press 1985) \bigskip{}  To better understand our investigation, it is useful to frame it in terms of the theory of the second-best. The economy considered here is subject to two irremovable constraints. On the one hand, the damage is private information. On the other hand, the liability system is undermined by judgment proofness. In this framing, our purpose is to compare two decision schemes that perform differently with respect to these two constraints: \begin{itemize}  \item Cost-benefit analysis: once the regulator's beliefs are given, this decision scheme has the advantage of being costless. However, alike any command-and-control instrument, it fails to incite the players to elicit information; \item Tullock contest: by definition, the functioning of the contest involves the expense of socially wasteful efforts. However, in association with the \textit{ex post} liability, it gives the players an incentive to reveal information about the damage. \end{itemize}  \bigskip{}  Suppose first that the regulator decides whether the new product should be marketed or not based on a cost-benefit analysis. He anticipates that the \textit{ex ante} expected social surplus will equal $b-1/2$, if it is approved, and $0$, otherwise. Accordingly, he will approve it if and only if $b>1/2$. It follows that the \textit{ex ante} expected social surplus will equal $\max\left(0,b-1/2\right)$.  \bigskip{}  Suppose now that the regulator decides whether the new product should be marketed or not using the Tullock contest (1980). Anticipating the equilibrium of the contest, the \textit{ex ante} expected social surplus will equal\footnote{The calculus can be found in the Appendix.}\[  f\left(\frac{k}{b}\right)=4\left(1-\sqrt{1-\frac{k}{b}}\right)-2\frac{k}{b}-\frac{1}{2}\left(\frac{k}{b}\right)^{2}.  \]  \begin{alignat}  {1} 
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= & \left(b-\frac{1}{2}\right)+\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\left(\frac{1}{2}-2b+k+\frac{1}{3}b^{2}f\left(\frac{k}{b}\right)\right),\label{eq:s*}  \end{alignat}  where we define for all $0\leq k/b\leq1$  $$  f\left(\frac{k}{b}\right)=  4\left(1-\sqrt{1-\frac{k}{b}}\right)-2\frac{k}{b}-\frac{1}{2}\left(\frac{k}{b}\right)^{2}.  $$  \bigskip{}