Effects of anticipated foreign military threat on arms accumulation.

Lai, Ching-Chong

I. Introduction

The literature of competitive arms accumulation based on the intertemporal optimization model has in recent years received considerable attention. For example, some models study the strategic aspects of arms accumulation in a dynamic game [1; 11; 6; 14; 15],(1) others examine the economic effect of consumption, military spending, and arms accumulation in the presence of a foreign threat [5; 6]. However, these studies focus their attention on the stability, the steady-state effect, and the short-run adjustment of military spending and the domestic arms stock following an unanticipated permanent rise in the foreign military threat.(2) It seems that very few efforts have been made to analyze the effects of an anticipated foreign military threat on the defense spending and home weapon stock. As a consequence, the first purpose of this paper is to contribute to the literature of competitive arms accumulation through examining the impact of an anticipated foreign threat. There are several examples motivated for such a study. The Middle East is a case in point. Kuwait and Saudi Arabia face the anticipated military threat from Iraq since the outbreak of the Gulf War in 1991. Recently, the peace talk between Israel and Arabs is often held, which makes their people anticipate that the enmity between both sides may be lessened and that the military conflicts in the Middle East may be lowered in the future. The South Asia is another case in point. As India insists on developing their nuclear weapons, the public in Pakistan anticipate that the military threat will be increased and hence will engage in an arms race action.

Most studies on military spending use two alternative specifications in the utility function. Brito [1], Simaan and Cruz [11], Deger and Sen [5; 6] assume that the utility function is nonseparable between consumption and the weapon stocks, while van der Ploeg and Zeeuw [14; 15] assume that the utility function is separable between consumption and the weapon stocks. The second purpose of the paper thus tries to shed light on how alternative assumptions in the utility function can lead to different dynamic adjustments of defense spending and home weapon stock in the presence of an anticipated foreign threat.(3) It is found that, when the utility function is nonseparable between consumption and the weapon stocks, the military spending may either increase or decrease on impact as a result of an anticipated permanent rise in foreign military threat. Moreover, the defense spending will make a discontinuous rise to ensure the optimum condition at the instant of the enforcement of the foreign military threat. However, when the utility function is separable between consumption and the weapon stocks, the defense spending will definitely increase on impact, and will be continuous at the moment of foreign threat realization.

The remainder of the paper is organized as follows. The theoretical structure of the model is outlined in section II. Section III will investigate the evolution of the military spending and the domestic weapon stock following an anticipated shock of foreign military threat. Finally, the main findings of our analysis ar e presented in section IV.

II. The Model

The model we shall use is similar to Deger and Sen [5; 6]. Consider a benevolent government maximizing its intertemporal utility function subject to a resource constraint and an arms accumulation restraint. In line with Deger and Sen [5; 6] and van der Ploeg and Zeeuw [15], there is no private capital accumulation, even though the government does invest in the home weapon stock. This economy produces a single composite commodity which can be consumed and used for military defense. The government provides defense security by means of arms accumulation and maintenance.(4)

This country derives utility from consumption, c, and the home weapon stock, m, and disutility from the foreign weapon stock, [m.sup.*]. Following Brito [1], Simaan and Curz [11], Deger and Sen [5; 6], van der Ploeg and Zeeuw [15], and Zou [16], the instantaneous utility function U is specified as follows:

(1) U = U (c,m,[m.sup.*]); [U.sub.1], [U.sub.2] [is greater than] 0, [U.sub.3]

[is less than] 0; [U.sub.11], [U.sub.22] [is less than] 0, [U.sub.12]

[is greater than or equal to] 0, [U.sub.13] [is less than or equal to]

0, [U.sub.23] [is greater than] 0.(5)

It is assumed, as customary, that U is concave in c and m and [U.sub.11] [U.sub.22] - [([U.sub.12]).sup.2]) [is greater than] 0. Moreover, [U.sub.12] = [U.sub.13] = 0 implies that the utility function is separable between consumption and the weapon stocks.

The objective of a social planner is to maximize the discounted sum of future instantaneous utilities:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Rho] is a constant rate of time preference.

At each instant of time, the full-employment output, y, is allocated between consumption and military spending, g. As a consequence, we may write this constraint as:

(3) y = c + g.

The defense spending is entirely on arms accumulation ?? and weapon replacement, that is,

(4) g = ?? + [Delta], (4)

where [Delta] is a constant rate of weapon depreciation.

Using c = y - g from equation (3), the intertemporal optimization problem then can be summarized as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The optimum conditions necessary for the optimization are:

(5) [U.sub.1] (y - g,m,[m.sup.*]) = [Lambda],

(6) [U.sub.2] (y - g,m,[m.sup.*]) - [Lambda][Delta] = -?? + [Lambda][Rho],

(7) ?? = g - [Delta] m,

where [Lambda] is the costate variable which can be interpreted as the imputed value of saving, measured in utility terms.

At the long-run equilibrium, the economy is characterized by ?? = ?? = 0 and g, m, and [Lambda] are at their stationary level, namely ??, ??, and ??. From equations (5)-(7) we can easily derive the following steady-state relationships:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Delta] = ([Delta] + [Rho])([Delta][U.sub.11] - [U.sub.12]) + [U.sub.22] - [Delta][U.sub.21] [is less than] 0. These results tell us that a rise in the foreign threat leads to more military spending (less consumption), more arms accumulation, and uncertain movements in the shadow price of home weapon stock. However, if the marginal utility from consumption is independent of the weapon stocks ([U.sub.12] = [U.sub.13] = 0), equation (10) definitely indicates that the shadow price will increase in response to a rise in [m.sup.*].

Following Deger and Sen, the analysis can be simplified by transforming the differential equations (6) and (7) into a system involving only g and m. Differentiating (5) with respect to time and substituting (6) and (7) into the resulting equation, we have:

(11) ?? = [[U.sub.2] + [U.sub.12] (g - [Delta] m) - ([Delta] + [Rho])

[U.sub.1]/[U.sub.11].

Equation (11) states how the military spending will change over time.

Expanding equations (11) and (7) around the stationary values of ?? and ??, we have:

(12) ?? = H (g,m,[m.sup.*]),

(13) ?? = J (g,m,[m.sup.*]),

where [H.sub.g] = ([Delta] + [Rho]) [is greater than] 0, [H.sub.m] = [U.sub.22] - [Delta] [U.sub.12] - ([Delta] + [Rho]) [U.sub.12]/[U.sub.11] [is greater than] 0, [[H.sub.m][.sup.*]] = [U.sub.23] - ([Delta] + [Rho]) [U.sub.13]]/[U.sub.11] [is less than] 0, [J.sub.g] = 1, [J.sub.m] = - [Delta] [is less than] 0, and [[J.sub.m][.sup.*]] = 0.

Let [[micro].sub.1] and [[micro].sub.2] be the two characteristic roots that satisfy dynamic equations (12) and (13), we then have:

(14) [[micro].sub.1][[micro].sub.2] = - [[Delta]([Delta] + [Rho]) +

[H.sub.m]] [is greater than] 0.

Obviously, the two characteristic roots of the system are of opposite signs. This implies that the system displays the saddlepoint stability, which is common to perfect foresight models.(6)

The phase diagram is illustrated in Figure 1. It is clear from equations (12) and (13) that the slopes of loci ?? = 0 and ?? = 0 are:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Figure 1 ILLUSTRATION OMITTED]

As indicated by the direction of the arrows in Figure 1, the lines SS and UU represent the stable and unstable branches respectively. Evidently, the convergent saddle path SS is always downward sloping and is flatter than the ?? = 0 locus, while the divergent branch UU is always upward sloping and is steeper than the ?? = 0 schedule.

In the next section, we will use the graphical apparatus like Figure 1 to illustrate the possible adjustment patterns of g and m in the presence of an anticipated permanent shock in [m.sup.*].

III. Dynamics of a Shock in Foreign Military Threat

Assume that initially, at time t = 0, the economy is in a steady state with [m.sup.*] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Meanwhile, the public perfectly anticipate that the country will suffer from a permanent rise in the foreign military threat from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at t = T in the future. It should be noted that the special situation where T = 0 implies an unanticipated permanent shock in [m.sup.*].

In Figure 2, the initial equilibrium, where ?? = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersects ?? = 0, is established at [E.sub.0]; the initial military spending and home weapon stock are [g.sub.0] and [m.sub.0] respectively. Upon the shock of an anticipated permanent rise from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the ?? = 0 ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) locus will shift upward to ?? = 0 ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), while ?? = 0 remains intact.(7) g = 0 ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) intersects ?? = 0 at point [E.sub.*], with g and m being ?? and ?? respectively. Obviously, the new stationary values of military spending and home weapon stock are at higher levels.

[Figure 2 ILLUSTRATION OMITTED]

Before we proceed with the analysis, four points should be noted. First, [0.sup.+] denotes the instant after the foreign policy-switch announcement; [T.sup.-] and [T.sup.+] denote the instant before and after the implementation of foreign military threat, respectively. Second, during the dates between [0.sup.+] and [T.sup.-], the foreign threat remains unchanged and the point [E.sub.0] should be treated as the reference point to govern the dynamic adjustment. Third, since the public become aware that the foreign threat will increase from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the moment of [T.sup.+], the transversality condition requires the economy to move to a point exactly on the stable arm SS associated with at that instant of time. Fourth, because [m.sup.*] will rise from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at time [T.sup.+] and both the domestic weapon stock and the shadow price ([Lambda]) are not allowed to have an anticipated discontinuity, along the optimal condition [U.sub.1] (y - g, m, [m.sup.*]) = [Lambda] the optimal path of military spending exhibits a discontinuous jump to rise at time [T.sup.+]. However, the optimal path of g will be continuous at [T.sup.+] if the utility function is additively separable between consumption and the weapon stocks ([U.sub.12] = [U.sub.13] = 0).(8) Based on this information, Figure 2 illustrates the case that the utility function is nonseparable between consumption and the weapon stocks. Under such a situation, two adjustment patterns possibly occur.(9) Firstly, at the instant [0.sup.+], g will immediately rise from [g.sub.0] to [g.sub.[0.sup.+]], while m is fixed at [m.sub.0] since it is predetermined. In consequence, the economy will jump from the point [E.sub.0] to [B.sub.1] on impact. Since the point [B.sub.1] lies vertically above the point [E.sub.0], from [0.sup.+] to [T.sup.-], as the arrows indicate, both g and m continue to increase, and the economy moves from [B.sub.1] to [B.sub.2]. At time [T.sup.+], as the foreign military threat has been enacted, a sudden rise in the military spending exactly places the economy at the point [B.sub.T] on the convergent stable path SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Thereafter, from [T.sup.+] onwards, g turns to decrease and m continues to accumulate as the economy moves along the SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) curve towards its stationary equilibrium [E.sub.*]. Secondly, at the instant [0.sup.+], g will at once fall from [g.sub.0] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], while m is fixed at [m.sub.0] since it is predetermined. Consequently, the economy will jump from the point [E.sub.0] to [C.sub.1] on impact. Since the point [B.sub.1] lies vertically below the point [E.sub.0], from [0.sub.+] to [T.sup.-], as the arrows indicate, both g and m continue to decrease, and the economy moves from [C.sub.1] to [C.sub.2]. At time [T.sub.+], as the foreign military threat has been implemented, a sudden rise in the defense spending will place the economy exactly at the point [C.sub.T] on the stable arm SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Thereafter, from [T.sub.+] onwards, g continues to fall and m turns to accumulate as the economy moves along the SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) curve towards its long-run equilibrium [E.sub.*]. Obviously, these results are not identified in the literature of a dynamic model of arms accumulation.

On the other hand, Figure 3 depicts the case that the utility function is additively separable in consumption and the weapon stocks as proposed by van der Ploeg and Zeeuw [15]. Following the similar description as that in Figure 2, at the instant [0.sup.+], g will instantaneously rise from [g.sub.0] to [g.sub.[0.sup.+]], while m still remains at [m.sub.0] since it is predetermined. In association with the discrete adjustment in g, the economy will jump from the point [E.sub.0] to [B.sub.1] on impact. Since the point [B.sub.1] lies vertically above the point [E.sub.0], from [0.sup.+] to [T.sup.-], as the arrows indicate, both g and m continue to increase, and the economy moves from [B.sub.1] to [B.sub.T]. At time [T.sup.+], as the foreign military threat has been carried out, the economy exactly arrives at the point [B.sub.T] on the convergent stable path SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Thereafter, from [T.sup.+] onwards, g turns to decrease and m continues to accumulate as the economy moves along the SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) curve towards its steady-state equilibrium [E.sub.*].(10)

[Figure 3 ILLUSTRATION OMITTED]

Why does the feature of the utility function lead to so different adjustment patterns of the military spending and the domestic weapon stock prior to the implementation of the foreign military threat? This question can be answered through considering alternative scenarios. First, when the utility function is nonseparable between consumption and the weapon stocks, the marginal utility of consumption is affected by the foreign military threat ([U.sub.13] [is less than] 0). As the public perfectly know that the defense spending will make a discontinuous increase when the foreign threat comes into force at time [T.sup.+], the forward-looking agent will react in advance to satisfy the transversality condition at [T.sup.+]. If the marginal utility of consumption will be reduced significantly as a result of an anticipated rise of foreign threat (the absolute value of [U.sub.13] is large), the defense spending will be increased to a great extent so as to take the economy exactly to the stable locus at the instant [T.sup.+]. Thus, the optimizing agent will in advance decrease the defense spending prior to the realization of foreign threat, so as to ensure a greater increase in the defense spending at [T.sup.+]. However, if the marginal utility of consumption affected by the foreign military threat is very weak (the absolute value of [U.sub.13] is small), the defense spending will not be increased very much at the moment of [T.sup.+]. As a result, the defense spending will increase in advance so as to ensure a smaller increase in the defense spending at time [T.sup.+]. Second, when the marginal utility of consumption is independent of the foreign threat and domestic arms stock ([U.sub.13] = [U.sub.12] = 0), the defense spending is continuous at the moment of [T.sup.+]. Since anticipating more foreign threat in the future leads to more defense spending in the future, the defense spending will increase in advance so as to satisfy the continuity requirement when the foreign military threat is announced by the foreign country.

IV. Concluding Remarks

This paper has made an attempt to examine the dynamic effects of an anticipated foreign military shock with which the existing literature of competitive arms accumulation does not deal explicitly. Based on our analysis, we find that, as the country faces an anticipated foreign military threat, alternative forms of the utility function are crucial for determining both the impact responses and the adjustment patterns of the military spending and the domestic arms stock. When the utility function is separable between consumption and the weapon stocks, the military spending will definitely increase on impact, and the military spending and the home weapon stock will continue to increase prior to the implementation of the foreign military threat. At the moment of foreign threat realization, the military spending will be continuous. However, when the utility function is nonseparable between consumption and the weapon stocks, the defense spending may be either increased or decreased on impact in response to an anticipated rise in the foreign military threat. Prior to the enforcement of the foreign military threat, the military spending and the home weapon stock may either continue to rise or continue to fall. At the moment of foreign threat realization, the military spending will make a discontinuous jump to rise so as to ensure the instantaneous optimum condition.

Before ending our analysis, we may note that the framework we have adopted in this paper is well suited to extend the following two directions. First, if two countries are independent and of equal size, we can set up a dynamic game model of strategic arms race between two countries. Second, if two countries are of unequal size, the problem can be handled as a dynamic Stackelberg game. In this game, we first deal with the optimization of small size country given the action of large country. After considering the reaction function of small size country as a constraint, we then tackle the optimization problem of large country.

(*) The authors are grateful to an anonymous referee for his stimulating encouragement and valuable comments on an earlier version of the paper. Any remaining errors are our responsibility.

(1.) Intriligator [7] and Intriligator and Brito [8] base on a dynamic heuristic model rather than an explicit optimizing model to examine the strategic implication of an arm race.

(2.) Isard and Anderton [9] provide a comprehensive survey of arms race models. In addition, Strauss [12] estimates and simulates the model for the NATO and Warsaw Pact alliances by using an econometric model of market processes.

(3.) Zou [16] sets up a more complicated optimization model including both capital and arms accumulation. He then studies the implications of alternative assumptions in the utility function on both long-run and short-run responses of defense spending and investment on competitive arms accumulation. However, since its dynamic system involves three differential equations, he cannot explicitly illuminate the relation between the assumptions in the utility function and the adjustment patterns of military spending and home arms stock.

(4.) Following van der Ploeg and Zeeuw [15], our model can be restated by a decentralized market economy with a representative household and a government.

(5.) Young's theorem indicates that [U.sub.12] = [U.sub.21], [U.sub.13] = [U.sub.31], and [U.sub.23] = [U.sub.32]. For a detailed description, see Chiang [4], Silberberg [10], and Takayama [13].

(6.) Since there is one predetermined variable (m) being equal to the number of stable roots, a unique perfect foresight equilibrium will exist [3; 2].

(7.) It is clear from equations (12) and (13) that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(8.) From the instantaneous optimum condition of equation (5), it can be inferred that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [Lambda] and m are continuous ([[Lambda].sub.T][.sup.-]] = [[Lambda].sub.T][.sup.+] and [[m.sub.T][.sup.-]] = [[m.sub.T][.sup.+]]), the above equation can be reduced to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, [U.sub.13] = 0 implies that [[g.sub.T][.sup.-]] = [[g.sub.T][.sup.+]] at the instant of foreign threat enforcement.

(9.) The detailed mathematical derivations for dynamic adjustment are available upon request.

(10.) When the shock of foreign military threat is unanticipated (T = 0), at the instant [0.sup.+], the defense spending will at once rise from [g.sub.0] to [[g.sub.0][.sup.+]] so as to place the economy exactly at a point like D on the convergent stable branch SS ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) in Figures 2 and 3. This result is consistent with Deger and Sen [5; 6]. Obviously, the military spending definitely increases on impact regardless of whether the utility function is separable between consumption and the weapon stocks.

References

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[14.] van der Ploeg, Frederick and Aart J. de Zeeuw. "Conflict over Arms Accumulation in Market and Command Economies," in Dynamic Policy Games in Economics, edited by Frederick van der Ploeg and Aart J. de Zeeuw. Amsterdam: Elsevier Science Publishers B. V. (North-Holland), 1989, pp. 73-90.

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