Abstract
Strict liability imposed on firms involved in an imperfect competition game (here duopoly) is not an obstacle to achieving a socially optimal care level of care. However, when one input becomes scarce (here a limited generic asset), the precedent result is compromised. Duopolistic competition leads to supply more goods at a lower price at the equilibrium, but also a lower care level compared to a monopoly placed in similar conditions. When damage are too high, capped strict liability coupled with an exante regulation leads the firms to increase their care level. The weakest one may disappear letting the other one in monopoly situation.
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Notes
European Parliament: “European Parliament resolution of 20 October 2020 with recommendations to the Commission on a civil liability regime for artificial intelligence”.
https://www.europarl.europa.eu/doceo/document/TA920200276_EN.html
Office of European Union, Ethics of connected and automated vehicles: Recommendations on road safety, privacy, fairness, explainability and responsibility, Hetzer, Muehleisen, Kousaridas and AlonsoZarate, (2019).
New York Times, 23/03/2019.
Indian software developer HCL Technologies Ltd, Bloomberg (2019).
New York Times (23/03/2019).
US Department of Justice: “Boeing Charged with 737 Max Fraud Conspiracy and Agrees to Pay over $2.5 Billion” https://www.justice.gov/opa/pr/boeingcharged737maxfraudconspiracyandagreespayover25billion
Ibidem.
Directive 2004/35/CE of the European Parliament and of the Council of 21 April 2004 on environmental liability with regard to the prevention and remedying of environmental damage.
42 U.S.C. §9601 et seq. (1980). CERCLA, beyond polluted sites concerns also “accidents, spills, and other emergency releases of pollutants and contaminants into the environment”.
US Restatement of Torts §§ 519, 520 (1938), Great Lakes Dredging & Dock Co. v. Sea Gull Operating Corp., 460 So. 2d 510, 512 (Fla. 3d DCA 1984)).
Six factors listed in § 520 of the Restatement defines them: (a) existence of a high degree of risk of some harm to the person, land or chattels of others; (b) likelihood that the harm that results from it will be great; (c) inability to eliminate the risk by the exercise of reasonable care; (d) extent to which the activity is not a matter of common usage; (e) inappropriateness of the activity to the place where it is carried on; and (f) extent to which its value to the community is outweighed by its dangerous attribute.
Exemptions exist and concern an act of God, acts of war or omissions of a third party other than an employee or agent of the defendant or one whose act or omission occurs in connection with a contractual relationship with the defendant. §9607(b); New York v. Shore Realty Corp., 759 F.2d 1032, 1042 (2d Cir. 1985).
Roughly speaking liable persons are: (a) the present owner of a facility from which there has been a release of a hazardous substance;(b) the present operator of a facility; (c) the owner of the facility at the time of disposal or release;(d) the operator of the facility at the time of disposal or release;(e) anyone who arranges for the disposal or treatment of hazardous substances or who arranges with a transporter for disposal of hazardous substances;(f) any transporter of hazardous substances; and(g) an owner of a facility with knowledge of a spill or release of hazardous substances who sells or transfers without disclosing.
For instance, for the European Environmental Liability Directive 2004/35/CE, see art.8–4, and for CERCLA, the potentially responsible parties (PRPs) may escape liability for, caused by (among act of God or war,
Acts/omissions of a third party with whom a PRP has no contractual relationship, commonly referred to as a "thirdparty defense," or State and local governments.
To see the point consider the foc for the program (7) without constraints. Then \(\frac{{\delta B}_{m}\left({q}_{m},{ x}_{m} \right)}{{\delta q}_{m}}=0\) involves that: \({q{^{\prime}}}_{m}= \frac{1}{2}\left(K\mathrm{c}{x}_{m}(1{ x}_{m})D\right)\) furthermore as \(\frac{{\delta B}_{m}\left({q}_{m},{ x}_{m} \right)}{{\delta x}_{m}}={q}_{m}{q}_{m}D=cte\), then the probability of an accident is linear, then, as the monopoly does not know any constraint on the generic asset, it is sufficient to put \({x}_{m}=1\) and then \({q{^{\prime}}}_{m}= \frac{1}{2}\left(K\mathrm{c}1\right)\). It is easy to see that then \({q{^{\prime}}}_{m}>{q}_{m}^{0}\).
Naturally, if the regulator puts it at a higher level than the one reached under competition.
Because: \(\frac{{\partial B}_{i}\left(\right)}{\partial {x}_{i}}={q}_{i}(1+p{^{\prime}}\left({ x}_{i}\right)D=0\)
Indeed, \({\uppi }^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right){q}_{j}^{1}c{q}_{j}^{1}\overline{x}{q}_{j}^{1}\overline{p}D{q}_{j}^{1}>0 \Rightarrow {\uppi }^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right)c\overline{x}\overline{p}D>0\) or,
\(\frac{1}{2}\left(K\overline{p}D\overline{x}c\right)c\overline{x}\overline{p}D>0\) i.e. \(\frac{1}{2}K \frac{3}{2}\left( \overline{p}D+\overline{x}+c\right)>0\).
References
Aghion, P., Bergeaud, A., Cette, G., Lecat, R., & Maghin, H. (2019). The invertedU relationship between credit access and productivity growth. Economica, 86, 1–31.
Ameye, E. (2010). “Channelling of nuclear third party liability towards the operator: Is it sustainable in a developing nuclear world or is there a need for liability of nuclear architects and engineers?” In: INLA/AIDN (ed) Nuclear Inter Jura 2009.
Bertolini, A., & Riccaboni, M. (2021). Grounding the case for a European approach to the regulation of automated driving: The technologyselection effect of liability rules. European Journal of Law and Economics, 51, 243–284. https://doi.org/10.1007/s1065702009671
Calabresi, G. (1970). The costs of accidents, a legal and economic analysis. Yale University Press.
Campello, M., Graham, J., & Harvey, Campbell R. (2010). The real effects of financial constraints: Evidence from a financial crisis. Journal of Financial Economics, 97(3), 470–487.
Cantú, C. E. (2001). Distinguishing the concept of strict liability for ultrahazardous activities from strict products liability under section 402a of the restatement (second) of torts: Two parallel lines of reasoning that should never meet. Akron Law Review, 35–1, 31–57.
DariMattiacci, G., & De Geest, G. (2005). « Judgment proofness under four different precaution technologies”. Journal of Institutional and Theoretical Economics, 161, 38–56.
EGL&NT (Expert Group on Liability and New Technologies New Technologies Formation) (2020). “Liability For Artificial Intelligence And Other Emerging Digital Technologies”, European Union, https://www.europarl.europa.eu/meetdocs/2014_2019/plmrep/COMMITTEES/JURI/DV/2020/0109/AIreport_EN.pdf
Esó, P., Nocke, V., & White, L. (2010). Competition for scarce resources. The RAND Journal of Economics, 41(3), 524–548.
Evans, A., & Schaefer, A. (2011). The impact of airport capacity constraints on future growth in the US air transportation system. Journal of Air Transport Management, 17(5), 288–295.
Evas, T., (2020). “Civil liability regime for artificial intelligence”, The European Parliamentary Research Service”, EPRS. https://www.europarl.europa.eu/RegData/etudes/STUD/2020/654178/EPRS_STU(2020)654178_EN.pdf
Heffron, R. J., Ashley, S. F., & Nuttall, W. J. (2016). The global nuclear liability regime post Fukushima Daiichi. Progress in Nuclear Energy, 90, 1–10.
Helland, E. A., & Tabarrok, A. (2012). Product liability and moral hazard: Evidence from general aviation. Journal of Law and Economics, University of Chicago Press, 55(3), 593–630.
Hetzer D., Muehleisen M., Kousaridas A. and AlonsoZarate J., (2019). "5G connected and automated driving: Use cases and technologies in crossborder environments," 2019 European conference on networks and communications (EuCNC), 2019, pp. 78–82, doi: https://doi.org/10.1109/EuCNC.2019.8801993.
Hiriart, Y., & Martimort, D. (2006). The benefits of extended liability. The RAND Journal of Economics, 37, 562–582.
Inderst, R., & Wambach, A. (2002). Capacity constrained firms in (labor) markets with adverse selection. Economic Theory, 19, 525–548.
Jaffee, D. M., & Modigliani, F. (1969). A theory and test of credit rationing. American Economic Review, 59, 850–872.
Kim, A. H. C. S. (2015). Internal capital markets in business groups: evidence from the asian financial crisis. The Journal of Finance, 70(6), 2539–2586.
Klein, B., & Leffler, K. B. (1981). The role of market forces in Assuring contractual performance. Journal of Political Economy, 89, 615–641.
Kolstad, C. D., Ulen, T. S., & Johnson, G. V. (1990). Ex post liability for harm vs. ex ante safety regulation: Substitutes or complements? The American Economic Review, 80, 888–901.
Lior, A., (2020) "AI Entities as AI Agents: Artificial Intelligence Liability and the AI Respondeat Superior Analogy," Mitchell Hamline Law Review: Vol. 46 : Iss. 5 , Article 2. https://open.mitchellhamline.edu/mhlr/vol46/iss5/2
Lior, A. (2021). AI strict liability VisÀVis AI monopolization. Science and Technology Law Review, 22(1), 90–126.
MacLeod, B. (2007). “Reputations, relationships and contract enforcement. Journal of Economic Literature, 45, 595–628.
Nie, P.Y., & Chen, Y.H. (2012). Duopoly competitions with capacity constrained input. Economic Modelling, 29(2012), 1715–1721.
Polinsky, A. M., (1980). “Strict Liability vs. Negligence in a Market Setting”, The American Economic Review, Vol. 70, No. 2, Papers and Proceedings of the ninetysecond annual meeting of the american economic association (May 1980), pp. 363–367.
Polinsky, M., & Shavell, S. (2010). The uneasy case for product liability. Harvard Law Review, 123, 1437–1492.
Schafer, HB. and Schonenberger, A., (2000). “Strict Liability versus Negligence” In: Bouckaert, BoudewijnJDe Geest, Gerrit (Eds.): Encyclopedia of Law and Economics (Vol. II: Civil Law and Economics): 597–624
Shavell, S., (2003). “Economic Analysis of Accident law”, NBER, Working Paper 9483 https://www.nber.org/system/files/working_papers/w9483/w9483.pdf
Shavell, S., (2018). “The mistaken restriction of strict liability to uncommon activities” The Journal of Legal Analysis, Forthcoming, dec. 12. Available at SSRN: https://ssrn.com/abstract=3298135
Shavell, S. (1984). Liability for harm versus regulation of safety. Journal of Legal Studies, 13, 357–374.
Shavell, S. (1984). A model of the optimal use of liability and safety regulation. RAND Journal of Economics, 15(271), 280.
Shavell, S. (1986). The judgmentproof problem. International Review of Law and Economics, 6, 45–58.
Shavell, S. (1987). Economic analysis of accident law. Harvard University Press.
Yu, S., Townsend R.M., Wu, Z., (2019). “Internal capital markets in business groups and the propagation of credit supply shocks”, IMF Working Papers, https://www.imf.org/en/Publications/WP/Issues/2019/05/21/InternalCapitalMarketsinBusinessGroupsandthePropagationofCreditSupplyShocks46882
Stiglitz, J., & Weiss, A. (1981). Credit rationing in markets with imperfect information. American Economic Review, 71(3), 393–410.
Tirole, J. (1988). The theory of industrial organization. MIT Press.
Veit, D. J., Weidlich, A., & Krafft, J. A. (2011). An agentbased analysis of the German electricity market with transmission capacity constraints. Energy Policy, 37(10), 4132–4144.
Vergès, E. (2014). Risks and Uncertainties of Scientific Innovations in French liability law: Between radical departure and continuity. McGill Law Journal / Revue De Droit De McGill, 59(4), 889–912. https://doi.org/10.7202/1026132ar
Viscusi, W. K. (2012). “Does Product Liability Make Us Safer? “(April 4, 2012). Regulation, Vol. 35, No. 1, 2012, Vanderbilt Public Law Research Paper No. 12–20, Available at SSRN: https://ssrn.com/abstract=2034546
Viscusi, W. K., & Zeckhauser, R. J. (2011). Deterring and compensating oil spill catastrophes: The need for strict and twotier liability. HKS Working Paper No. RWP11025. Available at SSRN: https://ssrn.com/abstract=1928134 or https://doi.org/10.2139/ssrn.1928134
Widmer, P. (2014). « La responsabilité pour choses et activités dangereuse dans les Projets européens », https://doczz.fr/doc/5677956/1laresponsabilit%C3%A9pourchosesetactivit%C3%A9sdangereuse
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Appendices
Appendix 1
1.1 1.1 Strict liability equilibrium for firms without restriction on inputs
In this economy, both firms use the generic asset without restriction. This assumption means that the firms disconnect their market activities (production and selling) from accident prevention. Under duopolistic competition this leads to the following program:
From the firstorder conditions, we get:
Calculating the best response functions requires determining the care level because otherwise, they would interfere with the expression of quantity \({q}_{i}\) in relation to quantity \({q}_{j}, i\ne j, i,j=\mathrm{1,2}\). Similar to free competition, this determination is independent of the knowledge of \({q}_{i}\) for firm \(i\). Then it is preferable to fix the firms’ optimum care level (here \({x}_{i}^{0}>0\)), such that^{Footnote 20}:
Solving the above equations for \({q}_{1}\) and \({q}_{2}\) we can define the game Nash equilibrium \(\left({q}_{1}^{*},{q}_{2}^{*}\right)\):
And,
1.2 1.2 The constrained monopoly choice
Let us assume that the monopoly is constrained in such a way that the program is the following
Under the constraints

1)
We define the condition for which the manager may decide to comply or not with the firstbest care level. Understanding this condition will involve deducing the condition for not complying to it. Using (7) and the results in 3.1, then, without constraint, the firm determines the firstbest care level.
$$\frac{{\delta B}_{m}\left({q}_{m},{ x}_{m} \right)}{{\delta x}_{m}}=1p\mathrm{^{\prime}}\left({ x}_{m}\right)D=0$$Let \({x}_{m}^{0}\) be this level that meet this condition, and the probability of an accident is known \({p(x}_{m}^{0})\). It follows that, once the optimum care level determined, the monopoly’s program becomes:
\(\underset{{q}_{m},{x}_{m}}{\mathrm{max}}{B}_{m}\left({q}_{m},{ x}_{m} \right)={q}_{m}\left(K{q}_{m}\right){q}_{m}(\tilde{c }{p(x}_{m}^{0})D)\) under the constraint: \({q}_{m}\left(\tilde{c }\right)\le { Y}_{m}\) (where \(\tilde{c }=c+{x}_{m}^{0}\)).
As \(\left\{\tilde{c }, {p(x}_{m}^{0}),D\right\}\) are given, the program is a classical monopoly program that expresses as:
$$\underset{{q}_{m}\ge 0,{x}_{m}\ge 0}{\mathrm{max}}{B}_{m}\left({q}_{m} \right)\mathrm{under the constraint }{q}_{m}\left(\tilde{c }\right)\le { Y}_{m}$$(7')We determine the equilibrium quantity \({q}_{m}^{0}\) and, consequently, let \({B}_{m}\left({q}_{m}^{0}\right)\) be the equilibrium payoff of the monopoly.

(2)
If the monopolist chooses to lessen the equilibrium care level, the program is (7), (7a), (7b) and the solution is\({B}_{m}\left({q}_{m}^{1} ,{x}_{m}^{1}\right)\), where \(\left({q}_{m}^{1},{x}_{m}^{1}\right)\) is the equilibrium solution with\({x}_{m}^{1}<{x}_{m}^{0}\).
Proposition A
Considering 1) and 2) above, the program (7), (7a, b) and the program (7’), then:

i)
If \({B}_{m}\left({q}_{m}^{0} \right)>{B}_{m}\left({q}_{m}^{1},{x}_{m}^{1}\right)\) , then, the monopolist keeps the firstbest care level corresponding to the unrestricted program.

ii)
If \({B}_{m}\left({q}_{m}^{0} \right)<{B}_{m}\left({q}_{m}^{1},{x}_{m}^{1}\right)\) , then, the monopolist chooses to lessen the care level.
Proof
The proof results from the comparison of 1) and 2).
Appendix 2
2.1 2.1 Strict liability equilibrium for firms with restricted inputs
Each firm disposes of a limited but generic asset (a “generic input” or a “generic asset”) \(({Y}_{i}\ne {Y}_{j}, i,j=\mathrm{1,2}, i\ne j)\) and makes a strategic tradeoff between production and care according to the other firms’ choices. The program corresponds to the Eqs. (6) to (6c) in the maintext argument. Solving this program proves the following proposition:
Proposition 1
Under the assumption that the coefficients \(K\),\(D\), \({c}_{i}\) combine to make the payoff function quasiconcave and since it is twice derivable, then the program (6) to (6’c) admits at least one Nash equilibrium \(\left({q}_{\mathrm{i}}^{0},{x}_{\mathrm{i}}^{0}\right)\) which maximizes the firm’'s (respectively firm j’s) payofffunctions\({B}_{i}\left({{ q}_{i,}x}_{i} \right), i=\mathrm{1,2}\).
Proof of proposition 1
To simplify we solve the program for firm 1; the proof is the same for firm 2. The object is to set up the bestresponse functions, and then to determine the game’s Nash equilibrium. The results are derived for firm 2 given the supply i.e., \({\overline{q}}_{2}\). We consider that the coefficients \(D\),\(K,\) \({c}_{1}\le 1\) are such that the function \({B}_{1}\left({q}_{1},{ x}_{1}\right)\) is quasiconcave. This is analyzed in part a) of the proof. Part b) defines the best response functions of each firm while part c) describes the Nash equilibrium of the game. However, first we must write the Karush–Kuhn–Tucker conditions which are:

(a)
Quasiconcavity of the program
To simplify the process of obtaining a solution we consider strictly positive \({x}_{1}\) and \({q}_{1}\), discarding \({x}_{1}=0\) (no prevention) and \({q}_{1}=0\) (no production), and negative values (i.e., we study the “normal” state of affairs). We also consider the particular case where the constraint is binding (\({\mathrm{i}.\mathrm{e}. \lambda }_{1}>0\)). Thus, the firm is limited by its asset. We reject \({\lambda }_{2}>0\mathrm{ and }{\lambda }_{3}>0\) because if they are not equal to zero this means that \({x}_{1}=0\) and/or \(p\left({ x}_{1}\right)=0\) which would violate our assumptions that \({x}_{1}\) is strictly positive. It is easy to verify this when \({\lambda }_{1}={\lambda }_{2}={\lambda }_{3}=0\), then, the system becomes similar to the situation without rationed asset.
We can study the concavity or the quasiconcavity of \({B}_{1}\left({q}_{1},{ x}_{1}\right)\).
Since the constraint is binding, we replace \({x}_{1}\) by \({x}_{1}=\) \(\frac{{Y}_{1}}{{q}_{1}}{c}_{1}\) in the payoff function:
$${B}_{1}\left({q}_{1}\right){=q}_{1}\left(K{q}_{1}{\overline{q}}_{2}\right){ Y}_{1}{q}_{1}p\left(\frac{{ Y}_{1}}{{q}_{1}}{c}_{1}\right)D$$It is easy to see that \({q}_{1}p\left(\frac{{ Y}_{1}}{{q}_{1}}{c}_{1}\right)D\) increases and \({q}_{1}\left(K{q}_{1}{\overline{q}}_{2}\right)\) decreases as \({q}_{1}\) increases in the interval \([0,{Y}_{1}]\). Then on this interval, both functions \({q}_{1}p\left(\frac{{ Y}_{1}}{{q}_{1}}{c}_{1}\right)D\) and \({q}_{1}\left(K{q}_{1}{\overline{q}}_{2}\right)\) cross once. Thus, there exists \({\tilde{q }}_{1}\in [0,{Y}_{1}]\) such that:
$${\tilde{q }}_{1}\left(K{\tilde{q }}_{1}{\overline{q}}_{2}\right){\tilde{q }}_{1}p\left(\frac{{ Y}_{1}}{{\tilde{q }}_{1}}{c}_{1}\right)D=0$$It follows that: \({B}_{1}\left({\tilde{q }}_{1}\right)={B}_{1}\left(0\right)={Y}_{1}\) and for \({\tilde{q }}_{1}\) and 0, \({B}_{1}()\) is negative. To be sustainable, there must be an interval \([{q}_{a},{q}_{b}]\subset [0,{Y}_{1}]\) such that for all \({q}_{1}^{a}\in [{q}_{a},{q}_{b}]\), \({B}_{1}\left({q}_{1}^{a}\right)>0\), with \({B}_{1}\left({q}_{a}\right)={B}_{1}\left({q}_{b}\right)=0\). If \({q}_{1}^{a}\) does not exist that fulfills this condition, and the program is not viable because, \({B}_{1}\left({q}_{a}\right)<0\).
Then the condition for having a viable program (i.e., a positive payoff) is that:
Condition A:
\(K>\frac{{ Y}_{1}}{{q}_{1}^{a}}+{q}_{1}^{a}+{\overline{q}}_{2}+p\left({q}_{1}^{a}\right)D, {q}_{1}^{a}\in [{q}_{a},{q}_{b}]\)
With this assumption, from \({B}_{1}\left({\tilde{q }}_{1}\right)={B}_{1}\left(0\right)={Y}_{1}\), and with \({q}_{1}^{a}\in [{q}_{a},{q}_{b}]\subset [0,{Y}_{1}]\), there exists \({q}_{1}^{a}\), such that \({B}_{1}\left({q}_{1}^{a}\right)>0\), and from Rolle's lemma there exists at last one \({q}_{1}^{*}\), \({q}_{1}^{*}\in [{q}_{a},{q}_{b}]\) such that \({B}_{1}\mathrm{^{\prime}}\left({q}_{1}^{*}\right)=0\). It follows that for \([0,{Y}_{1}]\), the program admits an interval \([{q}_{a},{q}_{b}]\subset [0,{Y}_{1}]\), where for all \({q}_{1}^{a}\in [{q}_{a},{q}_{b}]\), \({B}_{1}\left({q}_{1}^{a}\right)>0\) and \({B}_{1}\left(q\right)\le 0,\) for all \(q\notin [{q}_{a},{q}_{b}]\). Then the program is quasiconcave on \([{q}_{a},{q}_{b}]\).

(b)
Formation of the bestresponse function\({x}_{1}>0\) in (6.g) implies:\({q}_{1}{q}_{1}{p}^{\mathrm{^{\prime}}}\left({ x}_{1}\right)D+{\lambda }_{1}{q}_{1}=0\) and consequently, \({x}_{1}^{0}>0\) such that:
$${p}^{\mathrm{^{\prime}}}\left({x}_{1}^{0}\right)=\left(\frac{1+{\lambda }_{1}}{D}\right)$$Concerning \({q}_{1}>0\), by (6.i):
$$K2 {q}_{1}{\overline{q}}_{2}{ x}_{1}{c}_{1}p\left({ x}_{1}\right)D{\lambda }_{1}\left({c}_{1}+{x}_{1}\right)=0$$But \({\lambda }_{1}>0\), implies \({q}_{1}\left({c}_{1}+{x}_{1}\right){ Y}_{1}=0\).It follows that we determine a system with three unknowns \({q}_{1},{x}_{1},\) \({\lambda }_{1}\) and, three equations:
$${q}_{1}\left({c}_{1}+{x}_{1}\right){ Y}_{1}=0$$(a1)$${p}^{\mathrm{^{\prime}}}\left({x}_{1}\right)=\left(\frac{1+{\lambda }_{1}}{D}\right)$$(a2)$$K2 {q}_{1}{\overline{q}}_{2}{ x}_{1}{c}_{1}p\left({ x}_{1}\right)D{\lambda }_{1}\left({c}_{1}+{x}_{1}\right)=0$$(a3)To define the best response function for firm 1, it is sufficient to extract \({\lambda }_{1}\) from (a2), \({\lambda }_{1}=1D{p}^{\mathrm{^{\prime}}}\left({x}_{1}\right),\) and report this value in (a3) where, previously, \({q}_{1}\), from (a1) has been expressed in terms of \({x}_{1}\). Then considering these changes, from (a3) we get an equation in terms only of \({x}_{1}\):
$$KD p\left({x}_{1}\right)\frac{2 {Y}_{1}}{{c}_{1}+{x}_{1}}{\overline{q}}_{2}+D\left({c}_{1}+{x}_{1}\right){p}^{\mathrm{^{\prime}}}\left({x}_{1}\right)=0$$Obviously, solving this equation depends on the probability density \(p\left({x}_{1}\right)\). The solution obtained depends on \({\overline{q}}_{2}\), then \({x}_{1}^{0}={x}_{1}\left( {\overline{q}}_{2}\right), {x}_{2}^{0}={x}_{2}( {\overline{q}}_{1})\). We replace all expressions in \({x}_{1}\) in (a3) to determine the best response functions or reaction functions. Then by the implicit function theorem the first order condition for firm 1 alone defines what is its best strategy given \({\overline{q}}_{2}\). A similar argument holds for firm 2:
$${q}_{1}\left( {\overline{q}}_{2}\right)=\frac{1}{3}\left[K2Dp\left( {x}_{1}^{0}\right)+p\left({x}_{2}^{0}\right)+2{c}_{1}D{p}^{\mathrm{^{\prime}}}{(x}_{1}^{0})+2D{x}_{1}^{0}{p}^{\mathrm{^{\prime}}}({x}_{1}^{0}){c}_{2}Dp{^{\prime}}\left({x}_{2}^{0}\right)D{x}_{2}^{0}{p}^{\mathrm{^{\prime}}}({x}_{2}^{0})\right]$$$${q}_{2}\left( {\overline{q}}_{1}\right)=\frac{1}{3}\left[K2Dp\left({x}_{2}^{0}\right)+p\left({x}_{1}^{0}\right)+2{c}_{2}D{p}^{\mathrm{^{\prime}}}{(x}_{2}^{0})+2D{x}_{2}^{0}{p}^{\mathrm{^{\prime}}}({x}_{2}^{0}){c}_{1}Dp{^{\prime}}\left({x}_{1}^{0}\right)D{x}_{1}^{0}{p}^{\mathrm{^{\prime}}}({x}_{1}^{0})\right]$$ 
(c)
Nash equilibrium determination
From the reaction functions we can define the Nash equilibrium of the game (see definition 1 above) which consists of determining the quantities \(\left({q}_{1}^{0},{q}_{2}^{0}\right)\) which the firms will supply at equilibrium. The Nash equilibrium can be seen as occurring when each firm 1 strategy is the best response to the other firm’s strategical choice (similarly for firm 2). Substituting firm 2’s best response function into firm 1 gives:
$${q}_{1}^{0}{=q}_{1}\left({q}_{2}\left( {\overline{q}}_{1}\right)\right)$$And for firm 2.
$${q}_{2}^{0}{=q}_{2}\left({q}_{1}\left( {\overline{q}}_{2}\right)\right)$$Having determined \({q}_{1}^{0}\) and \({q}_{2}^{0}\), we deduce the game’s equilibrium level of care for both firms (\({x}_{1}^{0}\) and \({x}_{2}^{0}\))and then \({B}_{1}\left(\left({x}_{1}^{0},{q}_{1}^{0}\right),\left({x}_{2}^{0},{q}_{2}^{0}\right)\right),{B}_{2}\left(\left({x}_{2}^{0},{q}_{2}^{0}\right),\left({x}_{1}^{0},{q}_{1}^{0}\right)\right)\) is the Nash game equilibrium.
Appendix 3
Proof of proposition 2
To demonstrate a) it is sufficient to show that \({q}_{m}^{0}\) must be such that \({{q}_{m}^{0}<q}_{1}^{0}+{q}_{2}^{0}\).
Then assume that \({q}_{m}^{0}\ge {q}_{1}^{0}+{q}_{2}^{0}\). As \({q}_{1}^{0}+{q}_{2}^{0}=\frac{2}{3}\left(KD\left(1c\right)\right)\) and \({q}_{m}^{0}=\frac{1}{2}\left(KD\left(1+c\right)\right)\) (by assumption \({c}_{1}={c}_{2}=c\)), with respectively \(K>D\left(1c\right)\) and \(K>D(1+c)\) to guarantee positive quantities, and develop:
This inequality is true if \(D\left(17c\right)>K\) but this contradicts the condition that
Then it follows that:
From this result, b) it follows that the duopoly equilibrium price \({\pi }_{D}^{0}\) is lower than the monopoly price \({\pi }_{m}^{0}\) then:
It remains to prove c). For this, as previously, we compare the respective care levels namely: \({x}_{i}^{0}\) et \({x}_{m}^{0}\). Recall that these values are proportion that corresponds to “intensity” they are comprised between 0 and 1. We analyze successively \({x}_{i}^{0}\) and \({x}_{j}^{0}, i\) \(i\ne j.\)
Let us assume that \({x}_{m}^{0}>{x}_{i}^{0}\), and by taking their respective value:
After our simplifications and as \(a{Y}_{m}={Y}_{i}\)
Then:
\(K>D\left(1c\left(\frac{2+3a}{23a}\right)\right)\)
For \(a\ne \frac{2}{3}\), this expression is true because by hypothesis \(K>D(1+c)\) and,\(D\left( {1 + c} \right) > D\left( {1  c\left( {\frac{{2 + 3a}}{{2  3a}}} \right)} \right) \Rightarrow 1 >  \left( {\frac{{2 + 3a}}{{2  3a}}} \right)i.e.2 >  2\)
Consequently, \({x}_{m}^{0}>{x}_{i}^{0}\) i.e., the care level of the monopoly is higher than the level chosen by the duopoly firms.
Then a similar analysis is made for \({x}_{j}^{0}, i\ne j\). As by assumption \({Y}_{1}+{Y}_{2}={Y}_{m}\) with \(0<a,b<1:\) \(a{Y}_{m}={Y}_{i}\) et \(b {Y}_{m}={Y}_{j}\) (then obviously \(b {Y}_{m}=(1a){{Y}_{m}=Y}_{j}\). By the same argument than previously:
Let us assume that \({x}_{m}^{0}>{x}_{j}^{0}\), this involves that:
\(D\left( {1 + c} \right) > D\left( {1  c\left( {\frac{{2 + 3b}}{{2  3b}}} \right)} \right) \Rightarrow 1 >  \left( {\frac{{2 + 3b}}{{2  3b}}} \right)\)
Or, by replacing \(b {Y}_{m}=(1a){{Y}_{m}=Y}_{j}\)
\(D\left(1+c\right)>D\left(1c\left(\frac{5 3a}{3\mathrm{a}1}\right)\right) \Rightarrow 11>5\)
for \(a\ne \frac{1}{3}\)
Then, \({x}_{m}^{0}>{x}_{j}^{0}\).
Consequently,
We have simultaneously, \({{q}_{m}^{0}<q}_{1}^{0}+{q}_{2}^{0}\) and \({x}_{m}^{0}>{x}_{j}^{0}\) and \({x}_{m}^{0}>{x}_{i}^{0}\)
Appendix 4
Proof of proposition 3
General feature and proof of i)
The regulator introduces a standard corresponding to a minimum probability threshold \(\overline{p}\) which needs \({\overline{x}}_{i}, i=\mathrm{1,2}\) as the care effort needed to achieve it. It follows that the firm uses \({q}_{i}{\overline{x}}_{i}\) as the input for prevention, then \({\overline{x}}_{i}=\overline{x} , i=\mathrm{1,2}\) (by assumption the firms use the same technology). Firm \(i{^{\prime}}\) s program depends on\({q}_{i}\), given\({\overline{q}}_{j}\), we look for an interior solution\({q}_{i}>0\):
With \({q}_{i}\in \left[0,\frac{{ Y}_{i}}{\left({c}_{i}+\overline{x}\right)}\right]\). given the asset constraint \({q}_{i}\left({c}_{i}+\overline{x}\right)\le { Y}_{1}\). Then, \({q}_{i}^{0}\left({\overline{q}}_{j}\right)\) the interior solution is such that:
With:
Then the above program is concave and admits a solution. Does this solution lie within \(\left[0,\frac{{ Y}_{1}}{\left({c}_{i}+\overline{x}\right)}\right]\)? To address this, we build the best response functions and determine the Nash equilibrium of the system:
And for:
Function \({B}_{i}\left({q}_{i}^{0},{q}_{j}^{0}\right)\) reaches a maximum at \({q}_{i}^{0}\le \frac{{ Y}_{i}}{\left({c}_{i}+\overline{x}\right)}\). As \({q}_{i}^{0}\) is determined independently of \(\frac{{Y}_{i}}{\left({c}_{i}+\overline{x}\right)}\), which results in \({q}_{i}^{0}\) being higher or lower than this value. If it is lower (unsaturated constraint) the firm \(i\) (respectively \(j\)) maximizes its profit for this value. Then, it will not be able to produce beyond \(\frac{{Y}_{i}}{\left({c}_{i}+\overline{x}\right)}\) (respectively \(\frac{{Y}_{j}}{{c}_{j}+\overline{x}}\)). Thus, the production level is reduced.
a) Proof of ii) The equilibrium price is higher due to a lower supply,
To simplify we assume that (\({c}_{i}={c}_{j}\equiv c)\mathrm{ and that}{ Y}_{j}<{ Y}_{i}.\) The following relationships result from these assumptions:
\({q}_{i}^{0}={q}_{j}^{0}=\frac{1}{3}(K\overline{p}D\overline{x}c)\) or also:
Hence, firm \(j\) does not own enough resource to provide \({q}_{j}^{0}\) when firm \(i\) supplies \({q}_{i}^{0}>\frac{{ Y}_{i}}{\left(c+\overline{x}\right)}\). The firm must then forgo this value. The question then is whether \({q}_{j}^{1}=\frac{{ Y}_{j}}{\left(c+\overline{x}\right)}\), a Nashequilibrium may exist. As the constraint is binding for firm j, then:
We deduce:
Then (\({q}_{i}^{1}, {q}_{j}^{1}\)) is a Nash equilibrium for a higher price compared to if firms were not facing the limited asset. This corresponding price is \({\uppi }^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right)=K{q}_{i}^{1}{q}_{j}^{1}=\frac{1}{2}\left(K\overline{p}D\overline{x}c\right)\) and may be compared to the price that would result from a firm j without restricted assets, i.e.
It turns out that \({\pi }^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right)>{\pi }^{0}\left({q}_{i}^{0}, {q}_{j}^{0}\right)\) as firm \(j\) restrict its offer to a maximum of \(\frac{{Y}_{j}}{\left(c+\overline{x}\right)}\). Consequently, as expected, the equilibrium price is higher.
Proof of iii):The potential exit of the weakest firm
How does firm \(j\) achieve a positive profit? This involves knowing the conditions for which \({B}_{j}^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right)>0\). This is true for \(\frac{K}{3}> \left( \overline{p}D+\overline{x}+c\right)\).^{Footnote 21} It should be emphasized that this condition is general and applies to both firms because of their symmetric costs. If, instead, we consider differentiated costs, (including unit accident costs) then the following relationship makes sense:
Then, the equilibrium price per unit must be twice the value of the total marginal costs for the profit of firm j to be positive. For firm \(i\), if the costs are lower than those of \(j\), the price level at which it will make positive profits will be lower. For example, if \(\left(\overline{p}{D}_{j}+{\overline{x}}_{j}+{c}_{j}\right)=2\left(\overline{p}D\overline{x}c\right)\)(where the second member of equality designates the costs of \(i\)), then:
Firm i, for the same price level, makes a profit if:
\(\frac{1}{4}\left(K\overline{p}D\overline{x}c\right)>\left(\overline{p}D+\overline{x}+c\right)\), i.e. for a lower price.
Here, if \({\uppi }^{1}\left({q}_{i}^{1}, {q}_{j}^{1}\right)\in \left] \frac{1}{4}\left(K\overline{p}D\overline{x}c\right),\frac{1}{2}\left(K\overline{p}D\overline{x}c\right)\right[\) then, firm i achieves a positive profit while firm j makes a loss and firm \(i\) remains alone as a monopoly.
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Mondello, G. Strict liability, scarce generic input and duopoly competition. Eur J Law Econ 54, 369–404 (2022). https://doi.org/10.1007/s10657022097385
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DOI: https://doi.org/10.1007/s10657022097385
Keywords
 Tort law
 Strict liability
 Exante regulation
 Capped strict liability
 Imperfect competition
 Oligopoly
 Cournot competition
JEL Classification
 D43
 L13
 L52
 K13