Exploitation Hypothesis and Numerical Calculations

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.

Appendix: Multi-public Goods and Multi-constraint Model

Cornes and Itaya (2010) considered an economy that consists of two players: one private good and two voluntarily provided public goods. The researchers showed that if the players are different in preferences, the number of unknown variables is strictly lower than the number of equations representing the Nash equilibrium in which both players simultaneously make positive contributions to both public goods. Thus, they claimed “there ‘almost surely’ does not exist a Nash equilibrium in which both players simultaneously make positive contributions to both public goods” (Cornes and Itaya 2010, Proposition 2(i), p. 369). Cornes and Itaya’s (2010) proposition is based on the relationship between the number of unknown variables and the number of equations.

The main message of this appendix is as follows. As we will see, because the number of resource constraints for each player can be equal to the number of public goods in a multi-public good economy, as in our model developed in this chapter, their claim does not necessarily hold.

The organization of this appendix is as follows. In Section “Generalized Version of Cornes and Itaya’s Proposition”, we review Cornes and Itaya’s (2010) proposition in a more general setting than their model. Their proposition consists of two parts. First, they derived a necessary condition that two agents simultaneously make positive contributions to both public goods in a voluntary contribution in a two public goods game. Second, they showed that the necessary condition is not satisfied if both agents are different in preferences. We extend their result to derive the necessary condition that H agents simultaneously contribute positive amounts to J public goods. This result is a generalized version of Cornes and Itaya’s (2010) proposition.

Moreover, we also show that Cornes and Itaya’s (2010) proposition does not necessarily hold in a variety of multi-public good models. In Section “Interior Equilibrium in our Model”, we discuss whether Cornes and Itaya’s (2010) claim applies to our model. In Section “Multi-constraint Model”, we investigate a general model in which each player faces many resource constraints in addition to those of public goods. We then derive the corresponding necessary condition, which can be satisfied in general. We provide several examples in which two players with different preferences simultaneously make positive contributions to two public goods in a Nash equilibrium.

6.1.1 Generalized Version of Cornes and Itaya’s Proposition

In this section, we extend Cornes and Itaya’s (2010) model to an economy consisting of H players. The players are indexed by $h = 1, \ldots ,H$. They consume I private goods, indexed by $i = 1, \ldots ,I.$ They voluntarily provide J public goods, indexed by $j = 1, \ldots ,J.$ The amount of private good i consumed by player h is denoted by $c_{i}^{h}$. The amount of public good j is represented by $G_{j}$. The utility of player h is given as:

$$U^{h} \left( {c_{1}^{h} , \ldots ,c_{I}^{h} ,G_{1} , \ldots ,G_{J} } \right),$$

(6.101)

where $U^{h} (.)$ is strictly increasing, strictly quasi concave, and continuously differentiable. The amount of public good j is given as the sum of contributions:

$$G_{j} = \sum\limits_{h = 1}^{H} {g_{j}^{h} } \;{\text{for}}\;j = 1, \ldots ,J,$$

(6.102)

where $g_{j}^{h}$ is player h’s contribution to public good j. The budget constraint of player h is given as:

$$Y^{h} = \sum\limits_{i = 1}^{I} {q_{i} c_{i}^{h} + \sum\limits_{j = 1}^{J} {p_{j} g_{j}^{h} } ,}$$

(6.103)

where $Y^{h}$ represents the exogenously determined income of player h, $q_{i}$ represents the price of private good i, and $p_{i}$ represents the marginal cost of the contribution to public good j. We assume that $Y^{h}$, $q_{i}$, and $p_{j}$ are positive. In general, $Y^{h}$ can be interpreted as a resource, which may be transformed into a private good and public good. In this sense, we interpret Eq. (6.103) as the resource constraint of player h. In the next two sections, we introduce additional constraints including the standard budget constraint.

The utility maximization problem of player h is summarized as:

$$\hbox{max} U^{h} \left( {c_{1}^{h} , \ldots ,c_{I}^{h} ,\;G_{1} , \ldots ,G_{J} } \right),$$

subject to Eqs. (6.102) and (6.103) with respect to $c_{1}^{h} , \ldots ,c_{I}^{h} ,\;g_{1}^{h} , \ldots ,g_{J}^{h} .$ We then have the following proposition, which is a generalized version of Cornes and Itaya’s (2010) proposition:

Proposition 6.7

We suppose that $H > 1,\;J > 1$ and consider H players, I private goods , and J public goods economy. We also assume that each player faces one resource constraint . Except for coincidence, there is no Nash equilibrium in which all players simultaneously contribute positive amounts to all public goods .

Proof

The Lagrangian function of player h’s utility maximization problem is defined as:

$$L^{h} = U^{h} \left( {c_{1}^{h} , \ldots ,c_{I}^{h} ,G_{1} , \ldots ,G_{J} } \right) + \lambda^{h} \left\{ {Y^{h} - \sum\limits_{i = 1}^{I} {q_{i} c_{i}^{h} } - \sum\limits_{j = 1}^{J} {p_{j} } g_{j}^{h} } \right\}.$$

(6.104)

The first order conditions of an interior solution to $h's$ utility maximization problem are given as:

$$\frac{{\partial U^{h} }}{{\partial c_{i}^{h} }} - \lambda^{h} q_{i} = 0,\;{\text{for}}\;i = 1, \ldots ,I,$$

(6.105)

$$\frac{{\partial U^{h} }}{{\partial G_{j} }} - \lambda^{h} p_{j} = 0,\;{\text{for}}\;j = 1, \ldots ,J.$$

(6.106)

Solving these conditions for public good 1, we obtain:

$$\lambda^{h} = \frac{1}{{p_{1} }}\frac{{\partial U^{h} }}{{\partial G_{1} }}.$$

(6.107)

Substituting Eq. (6.107) into Eq. (6.105), we obtain I equations. Substituting Eq. (6.107) into Eq. (6.106) for public good $j = 2, \ldots ,J,$ we also obtain J − 1 equations as follows:

$$\frac{{\partial U^{h} }}{{\partial G_{j} }} - \frac{{p_{j} }}{{p_{1} }}\frac{{\partial U^{h} }}{{\partial G_{1} }} = 0\;{\text{for}}\;j = 2, \ldots ,J.$$

(6.108)

Thus, we obtain I + J − 1 equations for each player.

Aggregating the resource constraints of all players, we obtain:

$$\sum\limits_{h = 1}^{H} {Y^{h} } = \sum\limits_{h = 1}^{H} {\sum\limits_{i = 1}^{I} {q_{i} c_{i}^{h} } } + \sum\limits_{j = 1}^{J} {p_{j} G_{j} } .$$

(6.109)

Equations (6.105), (6.106), and (6.109) constitute a system of equations representing a Nash equilibrium in which all players contribute positive amounts to all public goods.

Table 6.12 summarizes the number of unknown variables and that of equations. Comparing the total numbers, we obtain:

Table 6.12 The number of unknown variables and equations in a generalized version of Cornes and Itaya’s proposition

$$\left\{ {H(I + J - 1) + 1} \right\} - \left( {HI + J} \right) = (H - 1)(J - 1).$$

If there is more than one player and more than one public good, the number of equations in the interior Nash equilibrium exceeds the number of unknown variables.

6.1.2 Interior Equilibrium in Our Model

In this subsection, we show that the number of unknown variables coincides with that of the equations in our model. The expected welfare maximization problem of country A is rewritten as follows:

$$\hbox{max} W^{A} = pU(C^{1A} ) + (1 - p)U(C^{0A} ),$$

subject to

$$\begin{aligned} & C^{1A} = Y^{A} - m_{1}^{A} - m_{2}^{A} , \\ & C^{0A} = Y^{A} - \bar{L}^{A} - m_{1}^{A} + \text{L} (s), \\ & p = p(M_{1} ), \\ & s = \frac{{pM_{2} }}{1 - p}, \\ & M_{1} = m_{1}^{A} + m_{1}^{B} , \\ & M_{2} = m_{2}^{A} + m_{2}^{B} . \\ \end{aligned}$$

The first order conditions for the interior solution are given by:

$$\begin{aligned} \frac{{\partial W^{A} }}{{\partial m_{1}^{A} }} = & p^{{\prime }} (M_{1} )\left( {U(C^{1A} ) - U(C^{0A} )} \right) \\ & - \left\{ {p(M_{1} )U_{Y} (C^{1A} ) + (1 - p(M_{1} ))\left( {1 - \frac{{\partial \text{L} }}{{\partial M_{1} }}\left( {\frac{{p(M_{1} )M_{2} }}{{1 - p(M_{2} )}}} \right)} \right)U_{Y} (C^{0A} )} \right\} = 0, \\ \frac{{\partial W^{A} }}{{\partial m_{2}^{A} }} & = p(M_{1} )\left( {\text{L}^{{\prime }} \left( {\frac{{p(M_{1} )M_{2} }}{{1 - p(M_{2} )}}} \right)U_{Y} (C^{0A} ) - U_{Y} (C^{1A} )} \right) = 0 \\ \end{aligned}$$

We note that the unknown variables in the first order conditions are consumptions $(C^{0A} ,C^{1A} )$ and provisions of both public goods $(M_{1} ,M_{2} )$. Combining the budget constraints of both countries, we obtain:

$$\begin{aligned} & C^{1A} + C^{1B} = Y^{A} + Y^{B} - M_{1} - M_{2} , \\ & C^{0A} + C^{0B} = Y^{A} + Y^{B} - \bar{L}^{A} - \bar{L}^{B} - M_{1} + 2\text{L} \left( {\frac{{p(M_{1} )M_{2} }}{{1 - p(M_{2} )}}} \right). \\ \end{aligned}$$

Table 6.13 summarizes the number of unknown variables and equations of an interior Nash equilibrium. As summarized in this table, we have six unknown variables and six equations. Thus, Cornes and Itaya’s (2010) claim does not hold in our model of two public goods. Because our model includes two states of the world and two resource constraints corresponding to the respective state of the world, we have as many equations as unknown variables. In the next section, we generalize this finding.

Table 6.13 Number of unknown variables and equations in our model

6.1.3 Multi-constraint Model

In this section, we extend the model presented in Section “Generalized Version of Cornes and Itaya’s Proposition” to a multi-constraint model in which the number of resource constraints is equal to the number of public goods.

We replace the budget constraint of player h, Eq. (6.103), with the following constraints:

$$Y_{k}^{h} = \sum\limits_{i = 1}^{I} {q_{ik} c_{i}^{h} } + \sum\limits_{j = 1}^{J} {p_{jk} g_{j}^{h} } ,\;{\text{for}}\;k = 1, \ldots ,J,$$

(6.110)

where $Y_{k}^{h} ,q_{ik} ,p_{ik}$ are positive constants and their interpretations depend on the setting of the model. For example, when we consider a household production model, k is the type of the resource, $Y_{k}^{h}$ is player h’s endowment of the type k resource, $q_{ik}$ is the units of the type k resource required for one unit of production of private good $i$, and $p_{jk}$ represents the units of the type k resource required for one unit of contribution to public good j. We note that the number of h’s resource constraints, Eq. (6.110), is equal to the number of public goods, J. We define a matrix of $p_{jk}$ as P:

$$P \equiv \left( {\begin{array}{*{20}c} {p_{11} } & \ldots & {p_{1J} } \\ \vdots & \ddots & \vdots \\ {p_{J1} } & \ldots & {p_{JJ} } \\ \end{array} } \right).$$

We also assume that:

$$\left| P \right| \ne 0.$$

(6.111)

To summarize, we revise the utility maximization problem of player h as:

$$\hbox{max} U^{h} \left( {c_{1}^{h} , \ldots ,c_{I}^{h} ,G_{1} , \ldots ,G_{J} } \right),$$

subject to Eqs. (6.102) and (6.110) with respect to $c_{1}^{h} , \ldots ,c_{I}^{h} ,\;g_{1}^{h} , \ldots ,g_{J}^{h} .$ We then have the following proposition:

Proposition 6.8

We consider H players, I private goods , and J public goods economy. We also assume that the number of each player’s resource constraints is J and the matrix of coefficients of contributions in resource constraints, P, is full rank. The number of equations of the interior Nash equilibrium is then always equal to the number of unknown variables. In other words, Cornes and Itaya’s (2010) claim is not applicable to this situation.

Proof

The Lagrangian function can be defined as:

$$\tilde{L}^{h} = U^{h} \left( {c_{1}^{h} , \ldots ,c_{I}^{h} ,G_{1} , \ldots ,G_{J} } \right) + \sum\limits_{k = 1}^{J} {\lambda_{k}^{h} \left( {Y_{k}^{h} - \sum\limits_{i = 1}^{I} {q_{ik} c_{i}^{h} } - \sum\limits_{j = 1}^{J} {p_{jk} g_{j}^{h} } } \right)} .$$

(6.112)

The first order conditions of an interior solution are given as:

$$\frac{{\partial \tilde{L}^{h} }}{{\partial c_{i}^{h} }} = \frac{\partial U}{{\partial c_{i}^{h} }} - \sum\limits_{k = 1}^{J} {\lambda_{k}^{h} q_{ik} } = 0,\;{\text{for}}\;i = 1, \ldots ,I,$$

(6.113)

$$\frac{{\partial \tilde{L}^{h} }}{{\partial g_{j}^{h} }} = \frac{{\partial U^{h} }}{{\partial G_{j} }} - \sum\limits_{k = 1}^{J} {\lambda_{k}^{h} p_{jk} } = 0,\;{\text{for}}\;j = 1, \ldots ,J.$$

(6.114)

Equation (6.114) is rewritten as the following matrix form:

$$P\left( {\begin{array}{*{20}c} {\lambda_{1}^{h} } \\ \vdots \\ {\lambda_{J}^{h} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{\partial U^{h} }}{{\partial G_{1} }}} \\ \vdots \\ {\frac{{\partial U^{h} }}{{\partial G_{J} }}} \\ \end{array} } \right).$$

(6.115)

Because we assume that matrix P is full rank, we have:

$$\left( {\begin{array}{*{20}c} {\lambda_{1}^{h} } \\ \vdots \\ {\lambda_{J}^{h} } \\ \end{array} } \right) = P^{ - 1} \left( {\begin{array}{*{20}c} {\frac{{\partial U^{h} }}{{\partial G_{1} }}} \\ \vdots \\ {\frac{{\partial U^{h} }}{{\partial G_{J} }}} \\ \end{array} } \right).$$

(6.116)

Substituting Eq. (6.116) into Eq. (6.113), we obtain:

$$\frac{\partial U}{{\partial c_{i}^{h} }} - \left( {q_{i1} , \ldots ,q_{iJ} } \right)P^{ - 1} \left( {\begin{array}{*{20}c} {\frac{{\partial U^{h} }}{{\partial G_{1} }}} \\ \vdots \\ {\frac{{\partial U^{h} }}{{\partial G_{J} }}} \\ \end{array} } \right) = 0,\;{\text{for}}\;i = 1, \cdots ,I.$$

(6.117)

Thus, we have $H \times I$ first order conditions for all players.

Finally, we aggregate the resource constraints, Eq. (6.110), and obtain J resource constraints:

$$\sum\limits_{h = 1}^{H} {Y_{k}^{h} } = \sum\limits_{h = 1}^{H} {\sum\limits_{i = 1}^{I} {q_{ik} c_{i}^{h} } } + \sum\limits_{j = 1}^{J} {p_{jk} G_{j} } ,\;j = 1, \ldots ,J.$$

(6.118)

Table 6.14 summarizes the number of variables and equations. As shown in this table, when each player faces as many resource constraints as the number of public goods, the number of unknown variables is equal to that of the equations.

Table 6.14 The number of unknown variables and equations in the model developed in Sect. 6.2

Proposition 6.8 is intuitively explained as follows. On one hand, the system of equations representing an interior Nash equilibrium consists of the following three types of equations: (1) the first order conditions with respect to private goods, (2) the first order conditions with respect to contributions, and (3) the resource constraints. On the other hand, this system of equations reduces to a system of equations of unknown variables representing private goods and public goods. In general, the number of equations can exceed the number of unknown variables because the first order conditions with respect to contributions do not have corresponding variables in the system of equations, which consists of variables representing private goods and public goods. However, when the number of players’ resource constraints is equal to the number of public goods, the first order conditions with respect to contributions are solved with respect to Lagrange multipliers. These equations are then substituted in the first order conditions of private goods. Thus, the number of equations becomes equal to that of unknown variables.

This result establishes the inapplicability of Cornes and Itaya’s (2010) claim—when agents are different in preferences, there almost surely does not exist a Nash equilibrium in which all players simultaneously make positive contributions to all public goods—to a situation in which each of the players faces as many resource constraints as the number of public goods.

6.1.4 Examples of Multi-resource Constraint Models

In this section, we present several examples in which each player faces as many resource constraints as the number of public goods. These include:

• (1) States-of-the-world model: The players face several contingent states of the world. The public goods in different states are treated as different public goods.¹¹

• (2) International public goods provided by fragmented national government model: Each national government is divided into as many administrative departments as international public goods and the departments are so bureaucratic that they have virtually independent budget constraints.

• (3) Household production model: The players are endowed with several factors of production. They produce private goods and public goods using these factors. The player has as many resource constraints as the number of production factors.¹²

• (4) Dynamic model with liquidity constraint: The players voluntarily provide a public good in each period. The public goods provided in different periods are treated as different public goods. Due to the liquidity constraint, the budget constraints of each player do not reduce to a single inter-temporal budget constraint.

6.1.4.1 Example of State-of-the-World Model: Clear Water Supply

Let us consider two contingent states: D and W. State D has a dry climate and state W has a wet climate. The probability of state D is p and the probability of state W is $1 - p.$

There are two households: $h = 1,\;2.$ They consume private goods and clear water. We assume that clear water is provided as a public good and the marginal cost of clear water depends on the state of the world. The utility of household $h$ is given as:

$$u^{h} = pU^{h} (c_{D}^{h} ,G_{D} ) + (1 - p)U^{h} (c_{W}^{h} ,G_{W} ),$$

where $c_{D}^{h}$ and $c_{W}^{h}$ are consumptions, $G_{D}$ and $G_{W}$ are provisions of clear water, and the subscripts represent the state of the world. The amounts of public goods supplied are given as:

$$G_{D} = g_{D}^{1} + g_{D}^{2} ,$$

(6.119)

$$G_{W} = g_{W}^{1} + g_{W}^{2} ,$$

(6.120)

where $g_{D}^{h} ,g_{W}^{h}$$(h = 1,\;2)$ are household h’s contributions to the provision of clear water in the two states. The budget constraint of household h is given as:

$$y^{h} = c_{D}^{h} + p_{D} g_{D}^{h} ,$$

(6.121)

$$y^{h} = c_{W}^{h} + p_{W} g_{W}^{h} ,$$

(6.122)

where $y^{h}$ is fixed income and $p_{D}$ and $p_{W}$ are the marginal costs of clear water ($p_{D} > p_{W} > 0$). The number of resource constraints for each player is equal to that of the public goods.

6.1.4.2 Example of International Public Goods Provided by Fragmented National Governments Model

We assume that there is one private good, one national public good, and two international public goods. For example, the national public good is national defense and the international public goods are scientific knowledge and international peacekeeping operations.

The world consists of H countries, indexed by h. The welfare of country h is given as:

$$u^{h} = U^{h} (c_{1}^{h} ,c_{2}^{h} ,G_{1} ,G_{2} ),$$

where $c_{1}^{h}$ is private good consumption, $c_{2}^{h}$ is national defense, $G_{1}$ is scientific knowledge, and $G_{2}$ is the international peacekeeping operation. The amounts of international public goods are given as:

$$G_{1} = g_{1}^{1} + g_{1}^{2} ,$$

(6.123)

$$G_{2} = g_{2}^{1} + g_{2}^{2} ,$$

(6.124)

where $g_{1}^{h} ,(h = 1,\;2)$ is country h’s contribution to scientific knowledge and $g_{2}^{h} ,(h = 1,\;2)$ is country h’s contribution to international peacekeeping operations.

Country h’s national defense and contribution to peacekeeping operations are carried out by its ministry of defense. We assume that the military budget is fixed for a political reason. Country h’s resource constraints are therefore given as:

$$Y^{h} = c_{1}^{h} + g_{1}^{h} + \bar{B}^{h} ,$$

(6.125)

$$\bar{B}^{h} = c_{2}^{h} + g_{2}^{h} ,$$

(6.126)

where $Y^{h}$ is the fixed national income and $\bar{B}^{h}$ is the fixed military budget. The number of resource constraints for each player is equal to that of public goods.

6.1.4.3 Example of Household Production Model: Capital and Labor

Assume an economy consists of two countries, one private good, two public goods, and two factors of production. The countries are indexed by $h = 1,\;2.$ The welfare of country h is given as:

$$u^{h} = U\left( {c^{h} ,G_{1} ,G_{2} } \right),$$

where $c^{h}$ is h’s private good consumption and $G_{1}$ and $G_{2}$ are public goods. The amounts of public goods are given as:

$$G_{1} = g_{1}^{1} + g_{1}^{2} ,$$

(6.127)

$$G_{2} = g_{2}^{1} + g_{2}^{2} ,$$

(6.128)

where $g_{j}^{h}$ ($h,\;j = 1,\;2$) represents h’s contribution to public good j. The country produces private goods and public goods with two factors—capital and labor—with linear production technology. The resource constraints of country h are given as:

$$Y_{K}^{h} = q_{K} c^{h} + p_{K1} g_{1}^{h} + p_{K2} g_{2}^{h} ,$$

(6.129)

$$Y_{L}^{h} = q_{L} c^{h} + p_{L1} g_{1}^{h} + p_{L2} g_{2}^{h} ,$$

(6.130)

where $Y_{K}^{h}$ is h’s capital endowment, $Y_{L}^{h}$ is h’s labor endowment, and $q_{K} ,p_{K1} ,p_{K2} ,q_{L} ,p_{L1} ,p_{L2}$ are parameters representing the production technology.

In general, if every public good requires one intrinsic factor of production and all public goods are different in their intrinsic factors, the number of resource constraints for each player is equal to that of public goods.

6.1.4.4 Example of Dynamic Model

Let us consider the T-period model. Periods are index by $t = 1, \ldots ,T$. This economy consists of H households, indexed by $h = 1, \ldots ,H$. Households consume a private good and voluntarily contribute to a public good in every period. The current utility of household h in period t is given as:

$$U^{h} (c_{t}^{h} ,G_{t} ),$$

where $c_{t}^{h}$ is h’s private good consumption and $G_{t}$ is the public good provision. The amount of the public good supplied at period t is given as:

$$G_{t} = \sum\nolimits_{h = 1}^{H} {g_{t}^{h} ,}$$

(6.131)

where $g_{t}^{h}$ is h’s contribution to the public good. The discount utility of household h is given as:

$$u^{h} = \sum\limits_{t = 1}^{T} {\delta^{t - 1} U^{h} \left( {c_{t}^{h} ,G_{t} } \right)} ,$$

where $\delta$ is the discount factor. We assume that households can neither lend nor borrow. The budget constraint at period t is:

$$Y_{t}^{h} = c_{t}^{h} + g_{t}^{h} .$$

(6.132)

The number of resource constraints of each household is T, which is equal to the number of public goods. We should remember that the condition we derived here is a necessary condition of the existence of the Nash equilibrium in which all players make positive contributions to all public goods. Even if each player faces as many resource constraints as the number of public goods, some players might not make positive contributions to some public goods in corner solutions.