Risk Management and International Security

Abstract

Economics of national security or international security is so immense a topic as to be intimidating. As in most of economics it has both positive and normative components. The goal of this chapter is to offer examples of both. Positive analyses, attempt to understand the economic origins of conflicts among nations and the economic foundations of success or failure in these conflicts.

Appendix: Mutual Insurance Model

2.1.1 Analytical Framework

Here we will expand the analysis of a country’s preparation and response to an emergency disruption when it can purchase or sell insurance from a trusted alliance partner. That is, we select one of the alternative instruments examined in the previous sections of the chapter, to focus on incentives among allies. The results are neither obvious nor expected.

There are two partner countries in the alliances, country 1 and country 2. They are identical in preferences but may be heterogeneous in income and emergency costs. They do not provide an international public good. There is an intra-alliance mutual insurance market.

Assuming additively separable utility functions, country i’s expected utility is given by

$$W_{i} = (1 - \alpha )V(c_{i}^{A} ) + \alpha V(c_{i}^{B} ),$$

(2.24)

where $W_{i}$ is expected welfare of country i, $c_{i}$ is private consumption of country $i\;(i = 1,2)$. $c_{i}$ is subject to uncertainty. In state A, which occurs at the probability of $1 - \alpha$, country i enjoys $c_{i}^{A}$. In state B, which occurs at the probability of $\alpha$, country i cannot enjoy $c_{i}^{A}$ but can enjoy $c_{i}^{B}$. $\alpha$ indicates the probability of an economically disruptive production emergency or a ‘war’. Note that subscript i refers to the country. Country 1 may buy insurance from or sell insurance to 2.

Country i’s budget constraint in each state is given by

$$c_{i}^{A} = Y_{i} - ps_{i}$$

(2.25a)

$$\begin{aligned} c_{i}^{B} & = (1 - \pi_{i} )Y_{i} - ps_{i} + s_{i} \\ & {\kern 1pt} = c_{i}^{A} - \pi Y_{i} + s_{i} \\ \end{aligned}$$

(2.25b)

where $Y_{i}$ is exogenously given (identical) national income of country i. $\pi_{i} ({>}0)$ indicates the net fraction of resources lost for each dollar of private production during the period of contingency when a war or natural disaster actually occurs—resources lost by reason of being diverted to a war effort or being cut off because of disruptions in production activities, thus leaving $(1 - \pi_{i} )Y_{i}$ of production under contingency B. p is the price of insurance, and $\pi$ is called the penalty ratio.

s is the insurance return in event of emergency with p the price of insurance. In other words, $ps$ indicates a demanding or buying country’s premium paid to a supplying or selling country during state A. If $s > 0\;( < 0)$, s represents the demand (supply) country’s return in event of state B with p the price of insurance, that is, the premium per dollar of insurance coverage. Country 1’s return $s_{1}$ may be positive or negative with country 2’s return, $s_{2}$, necessarily opposite in sign. It is also assumed that uncertainty is restricted to production so that the insurance premium paid by a buyer country to a seller country in state B is risk free and is not subject to the penalty ratio.

We assume that each government (or consumer) determines its insurance demand or supply, treating exogenous parameters $\alpha ,\pi$ and insurance price p as given.

From Eqs. (2.25a) and (2.25b) we have

$$pc_{i}^{B} + \rho c_{i}^{A} = (1 - p\pi_{i} )Y_{i}$$

(2.26)

where $\rho = 1 - p$. We assume $p < 1\;(\rho > 0)$ to investigate a meaningful realistic problem. The price of insurance, p, also means the price of consumption in state B, while $1 - p$ means the price of consumption in state A. Effective income in the left hand side of Eq. (2.26) evaluates emergency costs $\pi_{i} Y_{i}$ using p, the price of consumption in state B.

Each country maximizes its expected welfare (2.24) subject to its budget constraint (2.26). The first order condition for each country’s optimization is given by

$$\alpha \rho V_{c}^{B} = p(1 - \alpha )V_{c}^{A}$$

(2.27)

where $V_{c}^{A} \equiv dV/dc^{A} ,V_{c}^{B} \equiv dV/dc^{B}$. At the optimum, the marginal utility gain from allocation of an extra dollar to s (LHS of 2.27) just equals the expected marginal utility cost of that last dollar (RHS of 2.27). For simplicity subscript i in Eq. (2.27), referring to country i, is omitted and $c_{i}$ is rewritten as c.

For example, consider the log-linear utility function

$$W = (1 - \alpha )\log c_{i}^{A} + \alpha \log c_{i}^{B}$$

(2.28)

Then the ordinary demand functions for $c_{i}^{A} ,c_{i}^{B}$, and $s_{i}$ are respectively given by

$$c_{i}^{A} = \frac{1 - \alpha }{1 - p}(1 - p\pi_{i} )Y$$

(2.29a)

$$c_{i}^{B} = \frac{\alpha }{p}(1 - p\pi_{i} )Y$$

(2.29b)

$$s_{i} = \frac{{1 - p - (1 - \alpha )(1 - p\pi_{i} )}}{p(1 - p)}Y$$

(2.29c)

Eq. (2.29c) implies that if

$$\frac{{p(1 - \pi_{2} )}}{{1 - p\pi_{2} }} > \alpha > \frac{{p(1 - \pi_{1} )}}{{1 - p\pi_{1} }}$$

(2.30)

then, $s_{1} > 0 > s_{2}$; which is compatible with or consistent with mutual insurance.

From Eqs. (2.29a) and (2.29b), we also have

$$c_{i}^{B} - c_{i}^{A} = \frac{{(1 - p\pi_{i} )(\alpha - p)}}{p(1 - p)}Y$$

(2.31)

which is negative if $p > \alpha$.

2.1.2 Compensated Demand Function

Let us define the following expenditure function:

$$\text{Min} E_{i} \equiv pc_{i}^{B} + \rho c_{i}^{A} \quad {\text{subject}}\;\;{\text{to }}W_{i} \ge \overline{W}_{i}$$

Since preferences are identical between countries, the functional form is also the same between them. Considering Eq. (2.26), we have as the expenditure function

$$E(W_{i} ,\alpha ,1 - p,p) = \tilde{E}(W_{i} ,\alpha ,\rho ,p) = (1 - p\pi_{i} )Y_{i}$$

(2.32)

From Eq. (2.32) expenditure function E will determine $W_{i}$ as a function of income $Y_{i}$, the probability of “war” $\alpha$, the penalty ratio $\pi_{i}$, and the price of insurance p.

By a variant of Shephard’s Lemma we know

$$\tilde{E}_{ip} \equiv \partial \tilde{E}(W_{i} ,\alpha ,\rho ,p)/\partial p = c^{B} (W_{i} ,\alpha ,\rho ,p)$$

(2.33)

$$\tilde{E}_{i\rho } \equiv \partial \tilde{E}(W_{i} ,\alpha ,\rho ,p)/\partial \rho = c^{A} (W_{i} ,\alpha ,\rho ,p)$$

(2.34)

$$E_{ip} \equiv \partial E(W_{i} ,\alpha ,1 - p,p)/\partial p = c_{i}^{B} - c_{i}^{A} = s(W_{i} ,\alpha ,1 - p,p,\pi ,Y_{i} ) - \pi_{i} Y_{i}$$

(2.35)

where $s(\;)$ is the compensated demand (or supply) function for insurance and $c^{j} (\;)$ is the compensated demand function for private consumption in state $j\;(j = A,B)$. It follows from Eq. (2.35) that $s_{\pi } \equiv \partial s/\partial \pi Y = 1$.

Alternatively, from Eqs. (2.24) and (2.26), we solve for $s,c^{A}$ and $c^{B}$ respectively as functions of W and p, which give the compensated demand (or supply) functions for s, $c^{A}$ and $c^{B}$; Eqs. (2.35), (2.33) and (2.34), respectively.

Totally differentiating (2.24) and (2.26), yields

$$\left[ {\begin{array}{*{20}c} {\alpha \rho V_{c}^{B} /p,} & {\alpha V_{c}^{B} } \\ { - p(1 - \alpha )V_{cc}^{A} ,} & {\rho \alpha V_{cc}^{B} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {dc^{A} } \\ {dc^{B} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right]dW + \left[ {\begin{array}{*{20}c} 0 \\ {\alpha V_{c}^{B} /p} \\ \end{array} } \right]dp$$

(2.36)

where $V_{cc}^{A} \equiv d^{2} V^{A} /dc^{A} dc^{A} ,V_{cc}^{B} \equiv d^{2} V^{B} /dc^{B} dc^{B} ,V^{A} \equiv V(c^{A} ),V^{B} \equiv V(c^{B} )$.

Hence we have

$$c_{W}^{A} \equiv \partial c^{A} /\partial W = \rho \alpha V_{cc}^{B} /\gamma > 0$$

(2.37)

$$c_{W}^{B} \equiv \partial c^{B} /\partial W = p(1 - \alpha )V_{cc}^{A} /\gamma > 0$$

(2.38)

$$s_{W} \equiv \partial s/\partial W = c_{W}^{B} - c_{W}^{A} = \left[ {p(1 - \alpha )V_{cc}^{A} - \alpha \rho V_{cc}^{B} } \right]/\gamma$$

(2.39)

$$\begin{aligned} E_{W} & \equiv \partial E/\partial W = p\partial c^{B} /\partial W + \rho \,\partial c^{A} /\partial W \\ & = \left[ {p^{2} (1 - \alpha )V_{cc}^{A} + \rho^{2} \alpha V_{cc}^{B} } \right]/\gamma = \frac{p}{{\alpha V_{c}^{B} }} > 0 \\ \end{aligned}$$

(2.40)

$$c_{p}^{A} \equiv \partial c^{A} /\partial p = - \alpha^{2} V_{c}^{B} V_{c}^{B} /p\gamma > 0$$

(2.41)

$$c_{p}^{B} \equiv \partial c^{B} /\partial p = \alpha^{2} V_{c}^{B} V_{c}^{B} \rho /p^{2} \gamma < 0$$

(2.42)

$$s_{p} \equiv \partial s/\partial p = c_{p}^{B} - c_{p}^{A} = \alpha^{2} V_{c}^{B} V_{c}^{B} /p^{2} \gamma < 0$$

(2.43)

where $\gamma \equiv \alpha V_{c}^{B} \left[ {p^{2} (1 - \alpha )V_{cc}^{A} + \rho^{2} \alpha V_{cc}^{B} } \right]/p < 0$.

Greater values of W require higher values of $c^{A} ,c^{B}$, and E. An increase in p raises $c^{A}$ but reduces $c^{B}$ and s—results that are qualitatively plausible. However, the sign of Eq. (2.43) is ambiguous. To see this, suppose relative risk aversion is constant $\left( {\frac{{cV_{cc} }}{{V_{c} }} = - \lambda } \right)$. Then considering the first-order condition (2.26), (2.43) may be rewritten as

$$s_{W} = \alpha \rho \lambda V_{c}^{B} (c^{A} - c^{B} )/c^{A} c^{B} \gamma$$

(2.43′)

Thus, if $c^{A} > c^{B}$, as we will assume then it follows that $s_{W} < 0$ and $E_{p} \equiv \partial E/\partial p = c^{B} - c^{A} < 0$. In summary, in so far as $\pi_{1} Y_{1} \ge \pi_{2} Y_{2}$ and $Y_{1} \le Y_{2}$ (and strict inequality obtains for one or both), then $W_{1} < W_{2}$ and $s_{1} > s_{2}$.

2.1.3 Two Country Model

The two country model will then be summarized by

$$E(W_{1} ,p) = (1 - p\pi_{1} )Y_{1}$$

(2.44)

$$E(W_{2} ,p) = (1 - p\pi_{2} )Y_{2}$$

(2.45)

$$s(W_{1} ,p,\pi_{1} Y_{1} ) + s(W_{2} ,p,\pi_{2} Y_{2} ) = 0$$

(2.46)

Equation (2.44) gives expected welfare of country 1 as a function of insurance price, p, and its own effective income, $(1 - p\pi_{1} )Y_{1}$. Similarly, Eq. (2.45) shows expected welfare of country 2. Equation (2.46) gives the equilibrium condition for the private mutual insurance market and shows the equilibrium value of insurance price, p. Since there are no externalities, this equilibrium is Pareto efficient.

The benefits of insurance-alliance formation depend crucially on the nature of relative risks of emergency. Equation (2.46) implies that $s_{1}$ and $s_{2}$ must have opposite signs. Without loss of generality, assume $s_{1} > 0$ and $s_{2} = - s_{1} < 0$. Then, country 1 is the buyer, while country 2 is the seller or writer of insurance. If $Y_{1} = Y_{2}$ and $\pi_{1} = \pi_{2}$, both countries are identical whence $W_{1} = W_{2}$ and $s_{1} = s_{2} = 0$. Accordingly, heterogeneous income and/or penalty ratios are essential for the mutual insurance. We now briefly explore how such a situation would occur.

Suppose that Y is the same between two countries. Nevertheless, if $\pi$ is high in country 1 and low 2, it is still possible to have mutual insurance. With $s_{\pi } = 1$, $s_{i}$ is necessarily increasing with $\pi_{i}$. Therefore, when $\pi_{1} > \pi_{2}$, it is possible for $s_{1} = - s_{2} > 0$. From Eqs. (2.44) and (2.45) we then would know $W_{1} < W_{2}$ when $\pi_{1} > \pi_{2}$.

2.1.4 Comparative Statics

Next we investigate some comparative statics results in this insurance model. Totally differentiating Eqs. (2.44), (2.45) and (2.46), gives

$$\left[ {\begin{array}{*{20}c} {E_{1W} } & 0 & {s_{1} } \\ 0 & {E_{2W} } & {s_{2} } \\ {s_{1W} } & {s_{2W} } & {s_{1p} + s_{2p} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {dW_{1} } \\ {dW_{2} } \\ {dp} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - pY_{1} } \\ 0 \\ { - Y_{1} } \\ \end{array} } \right]d\pi_{1}$$

(2.47)

Hence, we have

$$\frac{{dW_{1} }}{{d\pi_{1} }} = - \frac{{Y_{1} }}{\Delta }\left\{ {p\left[ {E_{2W} (s_{1p} + s_{2p} ) - s_{2} s_{2W} } \right] - s_{1} E_{2W} } \right\}$$

(2.48)

$$\frac{{dW_{2} }}{{d\pi_{1} }} = - \frac{{Y_{1} }}{\Delta }s_{2} (ps_{1W} - E_{1W} )$$

(2.49)

$$\frac{dp}{{d\pi_{1} }} = \frac{Y}{\Delta }E_{2W} (ps_{1W} - E_{1W} )$$

(2.50)

where $\Delta \equiv E_{1W} E_{2W} (s_{1p} + s_{2p} ) - s_{1} E_{2W} s_{1W} - s_{2} E_{1W} s_{2W}$ and the first subscript on the partial derivatives refers to the country. As shown in Eqs. (2.40) and (2.43), $E_{W} > 0,s_{p} < 0$. Moreover, $\Delta$ is negative from the stability condition.

Therefore, it is easy to see that Eq. (2.50) is positive. To see this note that an increase in $\pi_{1}$ raises the price of insurance, p; an increase in the penalty ratio will raise the price of insurance. Since $s_{2} < 0$, Eq. (2.49) is positive. An increase in p, of course, is desirable for the seller country; and an increase in $\pi_{1}$ has a positive welfare spillover into country 2—this being the ordinary price effect. Since $s_{1} > 0$, the sign of Eq. (2.48) is negative. An increase in $\pi_{1}$, therefore, directly reduces the expected income of country 1—this being the ordinary the income effect. As country 1 is a buyer, an increase in p implies an unfavorable price effect, which will strengthen the unfavorable income effect of the increase in the penalty ratio. In sum then, an increase in $\pi_{1}$ hurts country 1, while it benefits country 2.

An increase in $\pi_{2}$ may be analyzed in the similar way. That is, it also increases the price of insurance, hurting country 1 due to the price effect. On the other hand, while greater $\pi_{2}$ directly reduces country 2’s own effective income, it indirectly raises 2’s welfare due to the price effect. Since the latter price effect offsets the former income effect, the overall welfare effect on country 2 is ambiguous.

These results suggest that an increase in the emergency cost has different spillover effects, depending on where it occurs. If the penalty ratio rises in the demand country, it has a positive spillover effect on the supply country. On the other hand, if the penalty ratio rises in the supply country, it has a negative spillover effect on the demand/buying country. The reason here is that an increase in the penalty ratio in either country will raise the price of insurance. All these theoretical results seem intuitively and practically plausible.

Note particularly that a decrease in $\pi_{i}$ has the same qualitative effect as an increase in $Y_{i}$, national income of country i, by reducing the price of insurance. Although the penalty ratio does not directly affect the price of insurance changes in that ratio will affect the economy by changing emergency costs. Therefore, an increase in $Y_{1}$ (income of the insurance buyer) will hurt country 2, while an increase in $Y_{2}$ (income of the seller of insurance) benefits country 1. Each country directly gains from its own economic growth due to the income effect but country 2 gets smaller benefits of its own economic growth than country 1 due to a reduction in the price of the insurance that it sells. A surprising and underappreciated result that follows: World-wide economic growth is more beneficial to the country that buys insurance than it may be to the country that provides or sells it.

This structure of benefits and costs next leads us to analyze the effect of transferring income between the countries within our alliance. An income transfer like this is equivalent to a change in the penalty ratios: where the penalty ratio rises in one country but it decreases in another country. To capture this idea let the constraint of $dY_{1} + dY_{2} = 0$. Then from Eqs. (2.44), (2.45) and (2.46) we have

$$\frac{{dW_{1} }}{{dY_{1} }} = \frac{1}{\Delta }\left[ {(1 - p\pi_{1} )E_{2W} (s_{1p} + s_{2p} ) + (\pi_{2} - \pi_{1} )s_{1} (ps_{2W} - E_{2W} )} \right]$$

(2.51)

$$\frac{{dW_{2} }}{{dY_{2} }} = - \frac{1}{\Delta }\left[ {(1 - p\pi_{2} )E_{1W} (s_{1p} + s_{2p} ) - (\pi_{2} - \pi_{1} )s_{2} (ps_{1W} - E_{1W} )} \right]$$

(2.52)

$$\frac{dp}{{dY_{1} }} = \frac{1}{\Delta }\left[ { - (1 - p\pi_{1} )s_{1W} E_{2W} + (1 - p\pi_{2} )s_{2W} E_{1W} + (\pi_{2} - \pi_{1} )E_{1W} E_{2W} } \right]$$

(2.53)

These equations lay bare the relationships among differences between countries in loss from emergency, $\pi$, value of p the price of insurance, and welfare consequences of income/wealth transfer. Equation (2.53) implies that as $\pi_{2} > \pi_{1}$, then p must be of a lower value and vice versa. That is, if the penalty ratio is higher in country 1, a transfer from country 2 to country 1 will raise the price of insurance, hurting the country that buys and benefiting the supplier country. This impact represents the price effect. As for income effect from greater $\pi$, the buyer country gains and the supplying country loses. Thus, Eq. (2.51) is positive if $\pi_{2} > \pi_{1}$. In this case both income and price effects benefit buyer country 1. However, if $\pi_{1} > \pi_{2}$, the price effect of higher “p” hurts country 1, while the income effect benefits that country. Equation (2.51) becomes positive when the income effect dominates the price effect.

Equation (A29) is negative if $\pi_{2} > \pi_{1}$. In that case both the income and price effects hurt country 2 the seller or writer of insurance. However, if $\pi_{1} > \pi_{2}$, the sign of Eq. (A29) becomes ambiguous since the income and price effects offset each other. To sum up, when $\pi_{2} > \pi_{1}$, the country which receives a transfer becomes better off while the country which pays out the transfer becomes worse off. However, if $\pi_{2} < \pi_{1}$, we could have a transfer paradox: namely that a giving country gains and a receiving country loses.