Abstract
We analyze the tradeoff between safety and production. The government chooses safety effort and tax rate in the first stage, and then the company strikes a balance between safety effort and production in the second stage. The government, representing the general public, earns taxes on production. Both players’ safety efforts mitigate the negative impact of a disaster. The disaster probability is modeled as a contest between the disaster magnitude and the two players’ safety efforts. Seven propositions are developed. First, as the safety effort of one player approaches infinity, the marginal change in the other player’s safety effort, with respect to the first player’s safety effort, approaches zero. Second, an infinitely large safety effort by any player causes the disaster probability and the negative impact of the disaster to decrease toward a constant. Third, as one player’s safety effort approaches infinity, the other player’s safety effort approaches zero. Fourth, the two players’ safety efforts are strategic substitutes so that an increase in one player’s safety effort decreases the other player’s safety effort. This enables the players to free ride on each other’s safety efforts. Fifth, higher tax rate causes the company to exert higher safety effort. Sixth, with maximum tax rate the company focuses exclusively on safety effort and generates no profit. Seventh, the company’s safety effort is inverse \(U\) shaped in the disaster magnitude.
This is a preview of subscription content,
to check access.Notes
See http://www.infoplease.com/ipa/A0001451.html for major oil spills since 1967, and http://news.yahoo.com/s/ac/20110317/sc_ac/8079848_worst_oil_spills_in_history for the five worst oil spills in history. The gravity of disasters can be measured according to economic loss, human loss, and symbolic loss. The July 6, 1988 North Sea off Scotland Occidental Petroleum’s Piper Alpha rig disaster killed 167 people, http://en.wikipedia.org/wiki/Piper_Alpha#cite_note2. The North Sea off Norway March 27, 1980 capsize of the Alexander L. Kielland platform killed 123 people, http://en.wikipedia.org/wiki/Alexander_L._Kielland_(platform), all retrieved August 8, 2014.
http://en.wikipedia.org/wiki/Deepwater_Horizon_oil_spill, retrieved August 8, 2014.
http://en.wikipedia.org/wiki/Exxon_Valdez_oil_spill, retrieved August 8, 2014.
See http://www.infoplease.com/ipa/A0001451.html for major oil spills since 1967, and http://news.yahoo.com/s/ac/20110317/sc_ac/8079848_worst_oil_spills_in_history for the five worst oil spills in history. The gravity of disasters can be measured according to economic loss, human loss, and symbolic loss. The July 6, 1988 North Sea off Scotland Occidental Petroleum’s Piper Alpha rig disaster killed 167 people, http://en.wikipedia.org/wiki/Piper_Alpha#cite_note2. The North Sea off Norway March 27, 1980 capsize of the Alexander L. Kielland platform killed 123 people, http://en.wikipedia.org/wiki/Alexander_L._Kielland_(platform), all retrieved August 8, 2014.
http://www.coso.org/documents/COSOBoardsERM4pagerFINALRELEASEVERSION82409_001.pdf, retrieved August 8, 2014.
The scaling function \(F(D,S,s)\) may be estimated by extrapolating from past data. Examples of factors impacting the scaling function are lawyer’s fees required to keep court cases going to prevent a company from being financially responsible for a disaster, public relations costs to ensure a company’s reputation when suffering from a disaster, and various costs to businesses and society of such disasters that impact the company.
Presenting Proposition 7 for general \(F(D,S,s)\) is technically possible but requires a much more complicated condition than (10).
A set of sufficient but not necessary conditions for (10) to be satisfied are \(\frac{\partial ^{2}p(D,S,s)}{\partial S^{2}}\ge \)0, \(\frac{\partial ^{3}p(D,S,s)}{\partial S^{3}}\le \)0, \(\left {\frac{\partial ^{3}p(D,S,s)}{\partial S^{3}}} \right \ge \left {\frac{\partial ^{2}p(D,S,s)}{\partial S^{2}}} \right \), \(\frac{\partial ^{3}p(D,S,s)}{\partial S\partial D^{2}}\ge \)0 and \(\frac{\partial ^{3}p(D,S,s)}{\partial S^{2}\partial D} \le \)0.
Abbreviations
 \(R\) :

Company’s resource
 \(E\) :

Company’s productive effort
 \(S\) :

Company’s safety effort
 \(s\) :

Government’s safety effort
 \(A\) :

Company’s unit cost of production
 \(B\) :

Company’s unit cost of safety effort
 \(H(S)\) :

Production function
 \(b\) :

Government’s unit cost of safety effort
 \(D\) :

Disaster magnitude
 \(p(D,S,s)\) :

Disaster probability
 \(F(D,S,s)\) :

Scaling function for how the company is negatively affected by disaster
 \(f(D,S,s)\) :

Scaling function for how the government is negatively affected by disaster
 \(\tau \) :

Taxation percentage variable chosen by the government
 \(u\) :

Government’s expected profit
 \(U\) :

Company’s expected profit
References
Asche, F., & Aven, T. (2004). On the economic value of safety. Risk Decision and Policy, 9, 253–267.
Azaiez, N., & Bier, V. M. (2007) Optimal resource allocation for security in reliability systems. European Journal of Operational Research, 181(2), 773–786.
Birsch, D., & Fielder, J. H. (1994). The Ford Pinto case: A study in applied ethics, business, and technology. Albany, NY: State University of New York Press.
Blome, C., & Schoenherr, T. (2011). Supply chain risk management in financial crises: A multiple casestudy approach. International Journal of Production Economics, 134(1), 43–57.
Boase, J. P. (2000). Beyond government? The appeal of publicprivate partnerships. Canadian Public Administration, 43(1), 75–92.
Boyer, M., & Laffont, J. J. (1999). Toward a political theory of the emergence of environmental incentive regulation. Rand Journal of Economics, 30(1), 137–157.
Carmichael, H. L. (1986). Reputations for safety: Market performance and policy remedies. Journal of Labor Economics, 4(4), 458–472.
Cheung, M., & Zhuang, J. (2012). Regulation games between government and competing companies: Oil spills and other disasters. Decision Analysis, 9(2), 156–164.
England, R. W. (1988). Disaster–Prone technologies, environmental risks, and profit maximization. Kyklos, 41(3), 379–395.
Flinders, M. (2005). The politics of publicprivate partnerships. The British Journal of Politics & International Relations, 7(2), 215–239.
Golbe, D. L. (1986). Safety and profits in the airline industry. The Journal of Industrial Economics, 34(3), 305–318.
Hale, A. R. (2003). Safety management in production. Human Factors and Ergonomics in Manufacturing & Service Industries, 13(3), 185–201.
Hausken, K. (2005). Production and conflict models versus rent seeking models. Public Choice, 123, 59–93.
Hausken, K. (2006). Income, interdependence, and substitution effects affecting incentives for security investment. Journal of Accounting and Public Policy, 25, 629–665.
Hausken, K., Bier, V., & Zhuang, J. (2009). Defending against terrorism, natural disaster, and all hazards. In V. M. Bier & M. N. Azaiez (Eds.), Game theoretic risk analysis of security threats (pp. 65–97). New York: Springer.
Hausken, K., & Zhuang, J. (2013). The impact of disaster on the strategic interaction between company and government. European Journal of Operational Research, 225(2), 363–376.
Kunreuther, H. (2008). Reducing losses from catastrophic risks through longterm insurance and mitigation. Social Research, 75(3), 905–930.
Kunreuther, H., & Heal, G. (2003). Interdependent security. The Journal of Risk and Uncertainty, 26(2/3), 231–249.
Kunreuther, H., & Useem, M. (2010). Learning from catastrophes: Strategies for reaction and response. Wharton School Publishing, Pearson Education Inc., New Jersey.
Osmundsen, P., Aven, T., & Tomasgard, A. (2010). On incentives for assurance of petroleum supply. Reliability Engineering & System Safety, 95(2), 143–148.
Sadka, E. (2007). Publicprivate partnerships—a public economics perspective. CESifo Economic Studies, 53(3), 466–490.
Skiba, A. K. (1978). Optimal growth with a convexconcave production function. Econometrica, 46(3), 527–539.
Skaperdas, S. (1996). Contest success functions. Economic Theory, 7, 283–290.
Tirole, J. (1988). The theory of industrial organization. Boston: MIT Press.
Tullock, G. (1980) Efficient rentseeking. In Buchanan, J. M., Tollison, R. D., & Tullock, G. (Eds.), Toward a theory of the rentseeking society, Texas A. & M. University Press, College Station, pp. 97–112.
Viscusi, W. K. (1993). The value of risk to life and health. Journal of Economic Literature, 31, 1912–1946.
Wu, D. D., & Olson, D. L. (2009). Risk issues in operations: Methods and tools. Production Planning & Control, 20(4), 293–294.
Wu, D. D., Olson, D. L., & Birge, J. R. (2011a). Guest editorial. Computers and Operations Research, 39(4), 751–752.
Wu, D. D., Olson, D. L., & Birge, J. R. (2011b). Introduction to special issue on “Enterprise risk management in operations”. Journal of Production Economics, 134(1), 1–2.
Zhuang, J., & Bier, V. M. (2007). Balancing terrorism and natural disastersdefensive strategy with endogenous attack effort. Operations Research, 55(5), 976–991.
Zohar, D. (1980). Safety climate in industrial organizations: Theoretical and applied implications. Journal of Applied Psychology, 65(1), 96–102.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the United States Department of Homeland Security (DHS) through the National Center for Risk and Economic Analysis of Terrorism Events (CREATE) under award number 2010ST061RE0001. This research was also partitially supported by the United States National Science Foundation (NSF) award numbers 1334930 and 1200899. However, any opinions, findings, and conclusions or recommendations in this document are those of the authors and do not necessarily reflect views of the DHS, CREATE, or NSF. Author names were listed alphabatically by last name.
Appendix: Proof of Propositions 1–7
Appendix: Proof of Propositions 1–7
Proof of Proposition 1
When \(S = 0\), the limit \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{dS(s)}{ds}\) is always zero so we only consider \(S >0\). Differentiating the first equation in (5) gives
Since \(p\) and \(F\) are bounded and monotonic in s it follows that \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial p(D,S,s)}{\partial s}=0\) and \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial F(D,S,s)}{\partial s}=0\), and thus \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial ^{2}p(D,S,s)}{\partial S\partial s}=0\) and \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial ^{2}F(D,S,s)}{\partial S\partial s}=0\). Hence (11) implies \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{dS}{ds}=0\). Although \(s\) is determined before \(S\), in an overall sense the players are interested in \(\mathop {\lim }\nolimits _{S\rightarrow \infty } \frac{ds}{dS}=0\). In contrast to (9) we thus consider the government’s first order condition
which is analogous to (5) for the company. Hence analogously, \(\mathop {\lim }\nolimits _{S\rightarrow \infty } \frac{ds}{dS}=0\). \(\square \)
Proof of Proposition 2
Follows since \(\frac{\partial p}{\partial s}\le 0\) ,\(\frac{\partial p}{\partial S}\le 0\) , \(\frac{\partial f}{\partial s}\le \)0, \(\frac{\partial f}{\partial S}\le \)0, \(\frac{\partial F}{\partial s}\le \)0, \(\frac{\partial F}{\partial S}\le \)0 as stated in Assumption 1, and since \(p, f, F\) have lower bounds 0. \(\square \)
Proof of Proposition 3
Inserting \(\mathop {\lim }\nolimits _{S\rightarrow \infty } \frac{\partial p(D,S,s)}{\partial s}=0\) and \(\mathop {\lim }\nolimits _{S\rightarrow \infty } \frac{\partial f(D,S,s)}{\partial s}=0\) into (12) implies \(s=0\Leftrightarrow 0>b\). Inserting \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial p(D,S,s)}{\partial S}=0\) and \(\mathop {\lim }\nolimits _{s\rightarrow \infty } \frac{\partial F(D,S,s)}{\partial S}=0\) into the second equation in (5) implies \(S=0\Leftrightarrow 0>(1\tau )\frac{\partial H(S)}{\partial S}\). \(\square \)
Proof of Proposition 4
When \(S = 0\), \(\frac{dS(s)}{ds}\) is always zero so we only consider \(S >0\). Similar to the proof of Proposition 1, we have
Since we assume \(\frac{\partial p}{\partial S}\le 0\),\(\frac{\partial p}{\partial s}\le 0\), \(\frac{\partial F}{\partial S}\le 0\), \(\frac{\partial F}{\partial s}\le 0\), \(\frac{\partial ^{2}p}{\partial S^{2}}\ge 0\), and \(\frac{\partial ^{2}F}{\partial S^{2}}\ge 0\), the denominator of (13) is always positive. So we have \(\frac{dS}{ds}\le 0\) if and only if the numerator of (13) is positive. One sufficient condition to ensure this is: \(\frac{\partial ^{2}p(D,S,s)}{\partial S\partial s}\ge 0\) and \(\frac{\partial ^{2}F(D,S,s)}{\partial S\partial s}\ge 0\). In summary we have shown \(\frac{dS}{ds}\le 0\) if \(\frac{\partial ^{2}p(D,S,s)}{\partial S\partial s}\ge 0\) and \(\frac{\partial ^{2}F(D,S,s)}{\partial S\partial s}\ge 0\). Analogously, similar to the proof of Proposition 1 and using the first order condition (11), we can show \(\frac{ds}{dS}\le 0\) if \(\frac{\partial ^{2}p(D,S,s)}{\partial S\partial s}\ge 0\) and \(\frac{\partial ^{2}f(D,S,s)}{\partial S\partial s}\ge 0\).
Proof of Proposition 5
When \(S = 0\), \(\frac{dS(\tau )}{d\tau }=0\) so we only consider \(S > 0\). Differentiating the first equation in (5) gives
Because of \(\frac{\partial H}{\partial S}<0, \frac{\partial F}{\partial S}\le 0,\frac{\partial ^{2}p(D,S,s)}{\partial ^{2}S}\ge 0, \frac{\partial ^{2}F(D,S,s)}{\partial S^{2}}\ge 0\), and \(\frac{\partial ^{2}H(S)}{\partial S^{2}}\le 0\) as stated in Assumptions 1 and 2, we have \(\frac{dS(\tau )}{d\tau }\ge 0\). \(\square \)
Proof of Proposition 6
Inserting tax rate \(\tau =1\) into (2) gives \(U(S,s,\tau =1)=F(D,S,s)p(D,S,s)\). Since we assume \(\frac{\partial p}{\partial s}\le 0\) and \(\frac{\partial F}{\partial S}\le \)0 in Assumption 1, S needs to be maximum to maximize \(U\). Hence from (1) we have \(S(\tau =1,s)=\frac{R}{B}\). \(\square \)
Proof of Proposition 7
When \(S = 0\), \(\frac{dS}{dD}=\frac{d^{2}S}{dD^{2}}=0\) is always zero so we only consider \(S > 0\). From the first equation in (5) we have
where \(H(S)\) does not depend on \(D\) according to (1). Since we assume \(\frac{\partial ^{2}p(D,S,s)}{\partial S^{2}}\ge 0\) and \(\frac{\partial ^{2}H(S)}{\partial S^{2}}\le 0\) in Assumption 2 and \(F\ge 0\) in (2), we have \(\frac{dS}{dD}\ge 0\) if and only if \(\frac{\partial ^{2}p(D,S,s)}{\partial S\partial D}\le 0\).
From (15) we have
For the special case \(\frac{\partial ^{2}H(S)}{\partial S^{2}}=0\) (linear production), (15) and (16) become
Substituting (17) into (18), we have
So we have \(\frac{d^{2}S}{dD^{2}}\le 0\) if and only if (10) is satisfied. For the second part, when \(D\) goes to infinity, we must have \(\frac{dS}{dD}\le 0\). Since \(S\) must be nonnegative, the limit exists. \(\square \)
Rights and permissions
About this article
Cite this article
Hausken, K., Zhuang, J. The strategic interaction between a company and the government surrounding disasters. Ann Oper Res 237, 27–40 (2016). https://doi.org/10.1007/s1047901416845
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1047901416845