Top

Published in:

2019 | OriginalPaper | Chapter

6. Exploitation Hypothesis and Numerical Calculations

Authors : Prof. Toshihiro Ihori, Prof. Martin C. McGuire, Shintaro Nakagawa

Published in: International Governance and Risk Management

Publisher: Springer Singapore

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The purpose of this chapter is twofold. First, we theoretically analyze simultaneous optimization of both self-protection and self-insurance by two countries facing the common risk of a disastrous event to derive simple analytical rules in the distribution of contributions. In this analysis, we provide a new perspective from the theory of collective risk management to reveal how allies share the burden of self-insurance and self-protection public goods. Second, we utilize our model to conduct numerical simulations of burden sharing in NATO from 1970 to 2010. In this simulation, we show that whether the conventional exploitation hypothesis holds depends on the risk profile that NATO faces. Our calculated results closely simulate the actual development of the military spending to GDP ratio.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Self-protection and Self-insurance Against Adversity

next chapter Threat Misestimations and the Role of NGOs

Available only for authorised users

Please refer to Buchholz and Sandler (2016) for a comprehensive review of the conventional exploitation hypothesis.

As shown in the next section, the optimal self-insurance allocation implies that if \(\text{L}^{{\prime }} > 1\), the consumption in the good state is lower than that in the bad state, \(C^{1A} < C^{0A}\). To preclude this counter-intuitive case, we assume \(\text{L}^{{\prime }} \le 1\).

Cornes and Itaya (2010) argued that if two public goods are voluntarily provided in a two-player economy and both players have different preferences, there almost surely does not exist a Nash equilibrium in which both players simultaneously contribute to both public goods. Dasgupta and Kanbur (2005) independently found the theoretically identical result to that of Cornes and Itaya (2010). Their claim is not applied to our model because players in our model face two contingent budget constraints. We provide a detailed discussion on the applicability of Cornes and Itaya (2010)’s claim in the appendix of this chapter.

Using Eq. (6.39), we obtain:

\(\frac{{\partial e(C,M_{1} ,M_{2} )}}{\partial C} < 1\) if and only if \(- \frac{{U_{YY} (C)}}{{U_{Y} (C)}} < - \frac{{U_{YY} (e(C,M_{1} ,M_{2} ))}}{{U_{Y} (e(C,M_{1} ,M_{2} ))}}\).

Thus, the slope of the curve of function \(e(.)\) in \(C^{0A} - C^{1A}\) space is steeper (more gradual) than the unity if the absolute risk aversion increases (decreases).

From Eq. (6.43), the first term on the RHS of Eq. (6.48) is non-negative. Substituting Eq. (6.45) into the second term and using our assumption, Eq. (6.35), we obtain:

\(U_{Y} (e)\frac{\partial e}{{\partial C^{0A} }} - U_{Y}^{0A} = U_{Y} (e)\frac{{\text{L}^{{\prime }} U_{YY}^{0A} }}{{U_{YY} (e)}} - U_{Y}^{0A} > 0\).

From Eq. (6.39), we obtain \(C^{1A} /C^{0A} = (L^{{\prime }} )^{ - 1/\theta }\), which implies \({\rm L}^{{\prime }} R^{0A} /R^{1A} = ({\rm L}^{{\prime }} )^{1 - (1/\theta )} > 1\) because \({\rm L}^{{\prime }} < 1\) and \(\theta < 1.\)

From Eq. (6.59), we obtain \(m_{1}^{A*} \ge m_{1}^{B*} + Y^{A} - \bar{L}^{A} - Y^{B} + \bar{L}^{B}\). Substituting this inequality in the RHS of Eq. (6.15), we obtain \(C^{1A} \le Y^{B} - m_{1}^{B*} + \bar{L}^{A} - \bar{L}^{B} - m_{2}^{A*}\). Remembering \(m_{2}^{B*} = 0\) and using Eq. (6.22), we have Eq. (6.68).

We derive the second inequality of Eq. (6.87) from our assumption that \(m_{2}^{B*} > 0\).

The derivation of the Nash equilibrium is as follows. We construct a system of equations consisting of a first order condition and corner condition. Next, we check if the inequality constraints are satisfied. For example, we numerically solve a system of equations, \(\frac{{\partial W^{A} }}{{\partial m_{1}^{A} }} = 0,\;\frac{{\partial W^{A} }}{{\partial m_{2}^{A} }} = 0,\;\frac{{\partial W^{B} }}{{\partial m_{1}^{B} }} = 0,\;\frac{{\partial W^{B} }}{{\partial m_{2}^{B} }} = 0\), and check whether the solution satisfies \(m_{1}^{A*} > 0,m_{2}^{A*} > 0,m_{1}^{B*} > 0,m_{2}^{B*} > 0\). If the solution satisfies the condition, the solution is considered an interior equilibrium. We conduct this procedure for all 16 types of Nash equilibria.

For example, the average growth rate of \(Y^{A}\) in 1970–1979 is given as \((Y_{1980}^{A} /Y_{1970}^{A} )^{1/10} - 1\). The average growth rates of \(Y^{A}\) and \(Y^{B}\) reported in the table are rounded to two decimal spaces.

This chapter and Ihori et al. (2014) investigated another type of a states-of-the-world model, which consists of two states of the world and two public goods.

Cornes and Schweinberger (1996) assumed that public goods are produced by households with several production factors. However, they also assumed that households buy private goods in the market with the income they make by selling the factors. Thus, each household in their model faces only one budget constraint except for corner solutions.

Bergstrom, T., Blume, L., & Varian, H. (1986). On the private provision of public goods. Journal of Public Economics, 29, 25–49.CrossRef

Boadway, R., & Hayashi, M. (1999). Country size and the voluntary provision of international public goods. European Journal of Political Economy, 15, 619–638.CrossRef

Buchholz, W., & Sandler, T. (2016). Olson’s exploitation hypothesis in a public good economy: A reconsideration. Public Choice, 168(1–2), 103–114.CrossRef

Cornes, R., & Itaya, J. (2010). On the private provision of two or more public goods. Journal of Public Economic Theory, 12(2), 363–385.CrossRef

Cornes, R. C., & Schweinberger, A. G. (1996). Free riding and the inefficiency of the private production of pure public goods. Canadian Journal of Economics, 29(1), 70–91.CrossRef

Dasgupta, I., & Kanbur, R. (2005). Bridging communal divides: Separation, patronage and integration. In C. B. Barrett (Ed.), The social economics of poverty: On identities, groups, communities and networks. Abingdon: Routledge.

Ehrlich, I., & Becker, G. (1972). Market insurance, self-insurance, and self-protection. Journal of Political Economy, 80, 623–648.CrossRef

Ihori, T., McGuire, M. C., & Nakagawa, S. (2014). International security, multiple public good provisions, and the exploitation hypothesis. Defence and Peace Economics, 25(3), 213–229.CrossRef

Kemp, M. C. (1984). A note of the theory of international transfers. Economics Letters, 14, 259–262.CrossRef

Lohse, T., Robledo, J. R., & Schmidt, U. (2012). Self-insurance and self-protection as public goods. Journal of Risk and Uncertainty, 79(1), 57–76.

Muermann, A., & Kunreuther, H. (2008). Self-protection and insurance with interdependencies. Journal of Risk and Uncertainty, 6, 10–123.

Olson, M. (1965). The logic of collective action. Cambridge, MA: Harvard University Press.

Olson, M., & Zeckhauser, R. (1966). An economic theory of alliances. Review of Economics and Statistics, 48, 266–279.CrossRef

Sandler, T. (1992). Collective action: The theory and application. Ann Arbor, MI: University of Michigan Press.

Sandler, T. (1997). Global challenges: An approach to environmental, political, and economic problems. New York: Cambridge University Press.CrossRef

Sandler, T. (2005). Collective versus unilateral responses to terrorism. Public Choice, 124(1–2), 75–93.CrossRef

Sandler, T., & Hartley, K. (1999). The political economy of NATO: Past, present and into the 21st century. Cambridge, UK: Cambridge University Press.CrossRef

Shibata, H. (1971). A bargaining model of the pure theory of public expenditure. Journal of Political Economy, 79, 1–29.CrossRef

The World Bank. (2018). World development indicators. http://databank.worldbank.org/data/reports.aspx?source=world-development-indicators. Accessed on June 7, 2018.

Warr, P. G. (1983). The private provision of a public good is independent of the distribution of income. Economics Letters, 13, 207–211.CrossRef

Title: Exploitation Hypothesis and Numerical Calculations
Authors: Prof. Toshihiro Ihori
Prof. Martin C. McGuire
Shintaro Nakagawa
Publisher: Springer Singapore
Book: International Governance and Risk Management
Print ISBN: 978-981-13-8874-3

Electronic ISBN: 978-981-13-8875-0

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-981-13-8875-0_6