Constitutional rules, according to which latter decisions will be made, can be designed ex ante to achieve ex post social objectives (Brennnan and Buchanan, 1985). The purpose here is to investigate the relative efficiency of the two decision schemes discussed previously, from the point of view of social welfare. Our results can justify and help designing ex ante constitutional rules to frame lobbying activities.

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:

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  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;

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  Tullock contest: by definition, the functioning of the contest involves the expense of socially wasteful efforts. However, in association with the ex post liability, it gives the players an incentive to reveal information about the damage.

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 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 ex ante expected social surplus will equal \(\max\left(0,b-1/2\right)\).

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 ex ante expected social surplus will equal¹¹The calculus can be found in the Appendix.

where we define for all \(0\leq k/b\leq 1\)

Clearly, from the social point of view, the contest is preferable if and only if

The purpose of the remaining of this section is to derive conditions such that this inequality holds true.

Let us first consider the situation where the level of assets of lobby \(I\)’s is small (i.e., \(k\rightarrow 0\)). Then, we can calculate that

As condition (2) is false, the cost-benefit analysis is a better decision scheme.

Let us now consider the situation where the level of assets of lobby \(I\)’s is large relative to his benefit (i.e., \(k\rightarrow b\)). Then, we can show that

In other words, the contest costlessly implements the first-best decision rule. Of course, condition (2) is then true, meaning that the Tullock contest is a better decision scheme.²²We show that the expected social welfare resulting from the contest game equals \(\lim_{k\rightarrow b}s^{*}\left(b,k\right)=b^{2}/2\).

Intuitively, these findings suggest that the second-best decision scheme will depend on the fraction of his benefit that the industrial lobby can lose in court. Proposition 1 below rigorously confirms this.

Proposition 1. For all \(b\), there exists

such that

To rephrase it, Proposition 1 says that if the industrial lobby risks in court a sufficiently large fraction of his benefit, in the sense that the ratio of his level of assets with respect to his benefit, \(k/b\), is larger than a given threshold, \(a\left(b\right)\), then it is socially preferable, ex ante, that the government decides to approve, or to ban, the new product according to the Tullock contest, rather than according to the cost-benefit analysis (and reciprocally).

Intuitively, this result can be explained as follows. On the one hand, as the lobbyists divert resources from productive activities, the contest is a costly mechanism. This favors a decision according to the cost-benefit analysis. On the other hand, as lobby \(I\) is held liable for damage, the larger the damage is, the less effort lobby \(I\) has incentive to expend in the contest, as long as he can afford to pay for it in court (i.e., when \(\delta<k\)). In equilibrium, the probability of approving the new product is thus decreasing in the harm. This favors a decision using the contest. Proposition 1 holds true, simultaneously because increasing \(k\) reduces the lobbying efforts and enlarges the region where the probability of approving the new product is negatively correlated to the harm. The picture is complete recalling that the contest performs very badly when \(k\rightarrow 0\) and costlessly implements the first-best decision rule when \(k\rightarrow b\).

Finally, for the sake of succinctness, Proposition 1 focuses on the existence of the threshold \(a\left(b\right)\) but remains silent about its value.³³In fact, we show in the proof of Proposition 1 that \(a\left(b\right)\) is bounded below by \(\tau\left(b\right)=\left(1/b\right)\max\left\{2\sqrt{1-b}(1-\sqrt{1-b}),(2\sqrt{b}-1)\right\}\). Of course, in practice, it would be useful to be more precise, in order to better identify the situations where to use the contest. Figure \ref{Fig1} below plots the frontier \(a\left(b\right)\). It emphasizes that the use of the contest should be limited to situations where the industrial lobby risks in court a large fraction of his benefit.