defense expenditure and civilian consumption

Defense Expenditure and Civilian Consumption

Defense Expenditure and Civilian Consumption: The Dynamics of the Interrelationship

Introduction

The inter relationship between economic growth and military expenditure has interested

researchers recently. One way this inter relationship has been studied is the impact of military

expenditure on the economic growth, specially in the case of less developed countries.[Benoit

(1973, 1978), Kennedy (1974), Degar (1980), Faine, Annez and Taylor (1980, 1984), Degar and

Sen (1983)]. The debate in this line stated with the finding of Benoit (1973) that military

spending has source positive impact on economic growth in the less developed countries. Later

on his finding has been supported by Kennedy (1974). But other researchers Degar (1974),

Degar and Sen (1983), Faine, Annez and Taylor (1980, 1984) have shown that military spending

has negative impact on economic growth. Even causality between defense expenditure and

economic growth has been studied extensively in case od developing countries [ Dakurah et el,

2001; Deger and Smith, 1983; Kollias et el, 2004; Kusi, 1994 ].

A relation between military expenditure and economic growth is established easily from the

basic Kegusian framework. In an economy with excess production capacity, increased aggregate

demand from military or any other source will drive up output, capacity utilization and even the

rate of profit. Investment may increase in response to higher profits, to put the economy on a

faster long-term growth path.

But such arguments apply more to developed economies than the less developed countries. In the

latter, shortage of crucial inputs such as capital, skilled manpower are likely to affect output in a

negative way than its effects on the aggregate demand. But the explanation Benoit (1973)

offered for his finding (positive correlation between defence burden and GDP growth) was a

productivity shift – newly formed military capital could have supplementary military uses that

would contribute to overall economic growth. Again military training impact some skill which

will be used in the economy after the completion of military service. This argument through

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plausible, can also be reversed. Military spending can easily divert resources from domestic

capital formation – resources in the form of foreign exchange, skilled manpower and production

capacity in the non traded goods sector and so on.

But to the knowledge of the present research the existing literature does not discuss the impact of

economic growth on military expenditure. This hypothesis has not been rigorously tested though

it has been stated that growth appeared to exert a weak influence on defence expenditure [Benoit

(1977, p.276)].

Military spending is done to purchase defence against both external threat and internal

instability. Moreover, the concept of insecurity comes more from the perception of the citizens

about the surroundings, about the environment. Thus given the stock of arms of neighbouring

countries a better perception about the basic insecurity and/ or ease of tension may had to lower

demand for defence as a public good and hence lower military expenditure. In the less develop

countries we see another problem – and that is problem of initial security. Society often suffers

from violence, secessionist movement and the sort of tension which are sometimes the fall out of

organizational changes associated with economic growth, like asymmetry in the regional balance

within the country, change in the distribution of income and a feeling of deprivation among a

section of population.

The hypothesis we have here is the following. While economic growth increases the level of

civilian consumption, the level of civilian consumption can be taken as an index of economic

growth. Now with the increase in the level of civilian consumption, the demand for defence good

increases too. This comes from two sources: First, defence is public good and increasing sense of

security increases social welfare given the same level of civilian consumption. Again an increase

in social consumption increases the sense of insecurity of the citizens also. This happens because

of the following: As mentioned earlier a country passes through continuous social change as

economic growth occurs. This creates some strains in the society. So the problem of initial

security increases Further, with the creation of social capital, a country feels increasing need to

protect these valuable things against both internal and external threat. These are the basic of our

assumption that the consumption has a fall out in the sense that it increases insecurity too.

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OBJECTIVES OF THE PRESENT STUDY :

For the present study we propose we explore the relationship between consumption and level of

insecurity and through this the relation between economic growth and military expenditure. For

this we build up a theoretical model which captures their relationship in an optimizing

framework. Then we will test relationship by using suitable economic technique.

One aspect of this relation is the impact of military expenditure of the economic growth for the

context of the optimizing model we will study empirically the ‘spin off’ of military expenditure

on economic growth.

Finally, we propose to define and estimate equilibrium demand for military expenditure for a

country.

The Model

It is assumed that the social economic welfare is measured by a measured by a strictly concave

utility function of current consumption (C) and the state of insecurities (S) which is a stock

concept. Such a function follows closely the type of money demand function used in the

literature though objection have been raised in recent time. The marginal utilities of consumption

is positive but diminishing,. The marginal utility of insecurity is negative and decreasing. The

disutility may be due to aesthetic consideration, fear of the citizens about their security from

external threat or internal violence.

By consumption we mean civilian consumption.

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Formally the conditions on the utility function U are

U = U (c, s) U (c, s) R2

Uc > 0, Ucc < 0, c > 0 ……….(1)Us < 0, Uss < 0, s > 0

Ucc Uss – Ucs 2 0 ……….(2) Further we assume that an increase in the level of insecurities reduces the marginal utility of consumption, or

Ucs < 0 ………(3)

This is economically reasonable, as increasing perception of insecurity reduces satisfaction from the same level of consumption.

The following limit conditions are imposed on partial dervatives of U :

lim Uc (c, s) = ∞ for all S > 0 ………(4)c 0

andlim Us (c, s) = 0 for all C > 0 ………(5)s 0

Condition (4) ensures that no optional policy will lead to a zero level of consumption. This

assumption is standard in the optimal growth literature.

Condition (5) states that small deviations from a situation of perfect security dose not reduce

welfare. This condition is sufficient to ensure that no optimal trajectory moves the economy to

the fear of insecurity.

Full employment of the economy is assumed and a fixed level of output (Yo) is produced in each

time period. This may be interpreted as output net of replacement investment. This output is

allocated to consumption (C) and military expenditure (M) so that

Y o = C + M ………(6)

Thus there exists a trade off between Consumption and Utility expenditure. The stock of

insecurity increases as a result of the Consumption process. The flow increases at an increasing

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rate with respect to consumption. The stock of insecurity is subject to decay at a constant

exponential rate . Further, in the absence of military expenditure the stock of insecurity

evolves over time according to

S* = f (c) – S ………(7)

Where f (c) R2 f (O) = 0F’ 0, f” 0, c > 0

And also S* indicates rate of change of S with respect to time.

[ Henceforth, asterix * after a variable will indicate the rate of change with respect to time]

Andlim f ' (c) = 0c 0

It is possible for the economy to slow the accumulation of the level of insecurity by increasing

military expenditure. The amounts insecurity reduced will be function (g) of the amount military

expenditure E. this function satisfy the following condition :

g (M) R2 , g (O) = 0 ………(8)

g' (M) > 0, g'' ( M) < 0 , M > 0

and

lim g’ (.) = ∞

M 0

Society’s spending on military purposes either reduces the level of insecurity or transform it to

some form which can be disposed of at no cost to society other than through M.

Thus the society’s net contribution to the flow of insecurity is measured by

f (c) – g (M)

Form (6) the level of military expenditure can be determined solely by the choice of consumption level.

M = Yo – C ……….(9)

The net contribution than is a function of the consumption level and is defined 5


W ' (C) = f (C) – g (Yo – C) ………..(10)

The function W has the following properties

W’ (c) = f ' ( .) + g ' (.) > 0, as f ' > 0, g ' > o ………..(11)

AndW ' ' (c) = f ' ' – g ' ' > 0

The flow of the insecurity level increases at an increasing rate with respect to consumption. We also see from (8)

lim W ' ( C) = lim f ' ( C) + lim g ' (Yo – C) = ∞ …………(12)

CYo CYo CYo

Let C0 be the solution of W (C) = 0

ThenW ( C) < 0 for C < C0

And

W(C) > 0 for C > C0

For C< C0 the economy is reducing the level of insecurity in a net sense. For C > C0, it is increasing the level of insecurity. Thus C0 is the level of consumption which will just maintain a secure position. This is shown in following figure 1.:

Figure 1______________________________________________________________

+

0 C 0

Co

–

______________________________________________________________

W (c) may be thought of as the control function, which controls the level of insecurities.

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By selecting the consumption level the society uniquely determines the amount insecurity it

generates in net term.

The control function has two components: One is an active control represented by g (M). List us

now consider transferring a unit of output from the consumption sector to the military

expenditure. In terms of controlling the level of insecurity, economy gain in two ways : First,

with increase in military expenditure, level of insecurity reduces. Second, since the consumption

level is lower, less flow of insecurity is being generated.

With military expenditure, the level of insecurity is governed by the growth function:

S* = W ( C ) - S ………….(13)

THE OPTIMAL SOLUTION

The optimizing problem before society is to maximize the discounts flows of utility as follows: Max e – ρ t U (C, S) dt ρ > 0 …………(14)

where ρ is the rate of discount.

Subject to,S* = W( C ) – S, S(0) = So

Yo – 0 < U, S < U

This is an optimal control problem with one state variable S and one control variable C. S o is historically given initial level of insecurity.

Here the improper integral in 14 has a maximum because

∞ ∞ e –ρt U (c, s) dt e–ρt U(Yo,0) dt

0 0 = U(Yo,0)/p

which is finite for ρ > 0

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Using Pontryagin’s Maximum Principle the necessary condition for a solution to the

optimizing problem ( 14 ) are deduced as follows

The current value Hamiltonian is

H = U(c, s) + (W(c) - S)

When is the costate variable. Also there exist function γ (t) and q (t) such that the current value Lagrangain is

L = H + y γ [W(c) – S] + q (Yo – c)

For optimization necessary condition are :

L = 0 < = > Uc (C , S) + W’(C ) + γ W’ (C ) – q = 0c

or ( + γ) W’ (C ) = q – Uc (C , S)

or + γ = Uc (C, S) +__q______ W’ (C) W’ (C ) ……………(15)

d = ρ - L = ρ - Us (c, s) + + γ dt S

or * = (ρ + ) - Us (C , S ) + γ …………….(16)

q 0, q (Yo – C) = 0 ……………(17)

γ 0, γ S = 0 γ S* = 0 ……………(18)

A policy of all-out consumption is non-optimal since from (12) and (15) we find that:

lim [-Uc ( C,S ) + q ] = 0 {as lim W’ (c) = 0cYo W’ (c) cY o

= > lim ( + γ ) = 0 C Yo

But forC = Yo S * > 0 or S > 0

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γ = 0

and hence = 0, which can only exist for an instant since

* = – Us ( C, S ) > 0

For an interior solution (S > 0, 0 < C < Yo) the necessary condition are

= Us (C, S) < 0 …………….(19) W ' (c)

And * = ( ρ + ) - Us ( C, S) ……………(20)

Now costate variable has the interpretation of a shadow price of the corresponding state

variable if the objective function has the dimension of an economic value [intriligator (ch.11)].

Form 19 see that the shadow price of the insecurity is negative. Again 19 gives the derived

demand for consumption as an implicit function of the shadow price and the level of insecurity.

We can visualize the system solved as :

C = C ( , S)

(from implicit function theorem this is locally true)

Using implicit function rule we can write,

_c = W ' > 0 …………….(21) - (Ucc + W '' )

From Equation (19) we write

W ' (c) – Uc (C, S) > 0

c = ______ W’ (c) W” – Ucc (C, S)]

and C = – Ucs _____ < 0 …………..(21a)S Ucc + W” (c)

as Ucs < 0

The sign of Equation (21A) follows from assumption. [ Ucc < 0, W” (c) < 0, Ucs < 0, and

W’(C) > 0 ]. The level of consumption is an increasing function of and a decreasing function

of the level of insecurity.

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Let us now turn to an analysis of the phase diagram in the (S, ) plane. First, consider the

behaviour of the level of insecurity (stock). We define:

Q(, S) = W [C (, S)] - S = 0 …………..(22)

Q W’ (c) C ____ = ------------------ > 0 …………..(23)

Q = W’ (c) C – θ < 0 …………..(24)S S

As C / S < 0

There fore,d = Q / S = - W’ (c) c + > 0 …..(25)dS S*= o Q / W’ (c) c /

The locus of Q (, S) = 0 is upward sloping to the right in the phase diagram with

lim = lim – Uc (C, S) = – Uc (C0 , 0) …..(26)S0 so = o S0 W’ (c) W’ (C0 )

CCo

For fixed level of , So is a decreasing function of the level of insecurity. This is evident if we examine (24).

Therefore So < 0 to the right of So = 0 and So > 0 to the left.

We consider the behaviour of shadow price . We define:

N (, S) = ( ρ + ) - Us[C(, S), S] = 0 ……………(26a)

N = ( ρ + ) – Ucs c > 0 ……………(26b)

Ns = - Uss – Ucs c ……………(26c) S

Now

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d / dS | = - Uss + Usc c / S o = 0 (ρ + ) – Usc c /

= Uss + Usc [ - Ucs / Ucc + W” ]

_____________________________ (ρ +) – Usc c /

by substituting form 21a

- (Ucc Uss – U 2 cs ) - Uss W” = _______________________________ < 0 ………(26d)

- (Ucc + W”) (ρ + - Ucs . c / )

The negative sign of (26d) comes from the sign of respective terms.

Thus * = 0 is downward sloping to the right with

lim lim Us (C, S) / (ρ = )S0 = S0

* = 0 = 0 from (5)

For constant S,

* = (ρ + ) – Us [C(, S), S] is an increasing function of ,

Hence * > 0 is true above o= 0 and * < 0 is below o= 0 , [from equation 26b]

To determine the stability properties of the equilibrium we consider the Jacobian Matrix of the

system

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S* = W(c) - S* = ( ρ +) - Us(C,S)

Evaluated at the equilibrium (*, S*). The Jacobian will be as follows:

- + W ' c / S W ' C /

J = …………(27)- Uss - Ucs c / S (p + ) – Ucs c /

[Chiang, Ch.11]

The Phase diagram will look like the following:

Figure 2 _________________________________________________________________________

S* 0 S

I S* = 0

II *

IV

III - Uc (C 0 ,0) W’ (C0)

* = 0

Therefore,

det J = ( - + W ' c ) (p+ - Ucs c ) – W ' c ( - Uss – Ucs c )12


S S ……..(28)

Since c = – UcsS Ucc+W '' ……..(21a)

Substituting the value of c in the term we have S

W ΄ c [Uss + Ucs ( – Ucs )] Ucc + W '

= W ΄ c [ Uss Ucc + Uss W ΄΄ – Ucs 2 ] (Ucc + W '' ) Therefore the value of the Jacobian will be

| J | = (- + W’ c ) (p + -Ucs c ) S

+ W ' c / [ – Uss Ucc – Ucs 2 ) – W ΄΄ Uss ] < 0 (Ucc + W '' ) ……….(29)

Since the determinant of J is negative it follows that equilibrium is a saddle point. From the figure we find then equilibrium is unique.

Regions I and II are traps in the sense that if any path enters in these regions, it remains forever.

We show that the trajectory which lies along the stable branch of the saddle point is the optimal solution to the inter- temporal maximization problem of Equation (14).

Path in region I or which enters I can be dismissed immediately as non optimal since S* > 0 and * > 0.

Eventually = 0 while * = Us > 0 and becomes positive violating the conditions as in Equation ( 19).

To show that the equilibrium path is the optimal path, we proceed as follows.

The path to the equilibrium denoted by (**, C,** S** ) is compared with any other feasible path (, C, ρ )

Let (t) = e –ρt (t)………..(30)

= > * = * e –ρt – ρ e –ρt ………..(31)

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From the concavity condition of the function U(C,S), we have e –ρt [U(C**, S** ) – U (C,S)]dt0

e –ρt {Uc** (C** - C) + Us** (S** -S )} dt = e –ρt {[ - ** W’ (C** _] (C** - C) +[**( α + ρ) -0* ] ( S** - S )}dt 0 = [ - ** e –ρt ]{W’ (C **) ( C** - C ) – α (S* - S)}dt

0 + ( ρ e –ρt * - e –ρt 0*) ( S* - S)dt

0 = [ - ** e –ρt ] {W ' ( C**) ( C** - C ) – α (S* - S)dt + 0* (S* -S)dt 0 0

Since the equilibrium path is derived from the optimising condition, the equilibrium path is the

optimal path. Thus given S (0) , the economy selects that value of (0), which corresponds to

the stable branch of the saddle point. The shadow price is then changes over time according to

Equation ( 20 ), which ensures convergence to the equilibrium (S**,** ).

We see that the equilibrium is a saddle point in the (S, ) phase space. This means that if is set

at any value other than the one corresponding to the stable branch of the saddle, and then there

is change over time according to Equation ( 20) , then the chosen trajectory diverges form

optimal trajectory.

However there is no reason for the economy to get off the optimal trajectory in such a controlled

system is strictly stable subject to the choice of optimal control.

For the initial level of the insecurity lying above the equilibrium solution, the level of insecurity

must fall and as it falls over time, resources are released from military expenditure to

consumption. For initial level of insecurity below the equilibrium point, the stock of insecurity

rises over time. As a result more resources are diverted from civilian to military uses and level of

consumption falls over time.

Stationary Solution

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Let us turn to a discussion of the stationary solution. The condition on C and S are:

W (C** ) = α S* …………(32)

Uc (C*, S*) = - Us(C**, S** ) W' (C** ) …………(33) ρ +

Equation (32 ) is the condition that in equilibrium the stock of insecurity is constant. The

amount of low insecurity generated W(C** ) is just equal to the amount which reduced as a

result of decay θ S*.

In equation (33), the left hand side is the marginal utility of consumption. The numerator of the

right is the loss in utility caused by an increment in consumption. This utility is lost because

additional insecurity has been generated as a result of momentary increase in consumption. If the

stock of insecurity does not decay then the loss of utility must be permanent. The present value

of this loss is.

-Us (C, S) W ' (C) ρ

But the state of insecurity decays over time at some positive rate and loss of utility is less. The present value will be lower and this is as follows:

Present value = -Us (C, S) W ' (C) ρ + α

The present value may be interpreted as the marginal psychic cost of consumption. And equation (33) explains that the marginal utility of consumption equals the marginal psychic cost of consumption.

Variation in Parameters:

Any change of fiscal monetary policies of the government will induce changes in

the values of the parameters important in this study like ρ and α . Their changes will

have impact on the values of both civilian and military consumption of the economy.

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Above we have the stationary value of C and S for given value of the parameters ρ , α and

Y(0). We now turn to a consideration of the effects on equilibrium value of C and S of different

values of these parameters.

Differentiating the system (32), (33) totally we get

W ' dC – α dS = S d α + 0. d ρ + f '. dYo …………(34)

{ (ρ + α ) Ucc + UsW ' ' + UcsW '} dC + { ((ρ + α ) Ucs +Uss W '} dS

= - Uc (d α + d ρ) – Us f ' ' dY0

[ Note : Since W (C) = f (C) – g (Y0 –c), and C = Y0 –M ]

We define the determinant

W’ - α =

(ρ + α) Ucc + UsW” + Ucs W’ (ρ + α) Ucs + Uss W’

= W’ { (ρ + α) Ucc + Uss W’} + a { (ρ + α) Ucc + UsW” + UcsW’} < 0

as Ucs < 0 by assumptionUss < 0Ucc < 0W” < 0

We now solve (34) for the desired partial derivation using Cramer’s rule.

First consider variation in ρ and other two parameters are unchanged as

dY0 = 0 = da

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Then system (34) reduces to

W’ dc – α dS = 0

{( ρ + α ) Ucc + Us W '' + Ucs W '}dc + {{( ρ + α ) Ucs + UssW '}dp

= - Uc d ρ

Therefore C = - Uc α > 0 as Δ < 0, ………….(35) ρ Δ

and S = - UcW ' > 0 …………(36) ρ

Thus both consumption and level of insecurity move in the same direction as the discount rate.

Let us suppose that preference for the present time increases (or discount rate is higher ).

Now given the level of insecurity, the economy will consume more.

Let us now investigate the effect of change in the decay rate α . we find that

__C = 1 {S {( ρ + α ) Ucs + W’ Uss] – α Uc} > 0 ………..(37) α Δ

Because we have Ucs < 0, Uss < 0 , Δ < 0., and

S/ c = 1 {- W’ Uc – S [{( ρ + α ) Ucc + Us W ' ' + Ucs W ' ]} 0 ………..(38) Δ <

As W ' > 0, Uc > 0, Ucc < 0, W '' < 0

Thus level of consumption moves in the same direction as decay rate. But the level of insecurity

may or may not move in the same direction as decay rate. One thing is important. If for some

reasons, tension reduces so that decay rate of insecurity become higher, society as a whole

benefited by higher level of consumption.

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Let us now consider variation in full employment output Y (0). This may be caused by

economic growth caused by an increase in capital stock or the incidence of technological

progress. Form (34 ) we have

C / Y0 = 1 [ {f ' ( ρ + α )Ucs + UssW ' } – α Uss f ' ] > 0 ……… (39) Δ

as W’ > 0, Uc > 0, Ucc < 0, W” < 0

andS / Y0 = 1 {- Us f '' W ' – f ' [( ρ + α) Ucc + Us W '' + Ucs W ']} 0

Δ <as Us < 0 …………(40)

Thus when economic growth occurs, in equilibrium the society has a higher level of

consumption. But from Equation (40) we find that there may or may not be any relation in

between economic growth and the level of insecurity.

Conclusion

We can summaries the major points of the optimal control model. First the steady state

equilibrium is stable. Thus the society can achieve efficient allocation of resources in between

civilian consumption and military expenditure. Secondly, an increase in the decay rate of the

stock of insecurity (resulting from ease of tension) will increase the equilibrium of consumption.

Thus the society is benefited. Thirdly, we have a desired demand for military expenditure

deduced from optimizing framework. Fourthly, the relationship between economic growth and

military expenditure is not so simple as suggested in the literature. We are to make empirical

investigation to see whether it can throw some light on this.

References

Benoit, E., Defense and Economic Growth in Developing Countries, Lexington Books, Lexington, 1973.

__________, Growth and Defense in Developing Countries, Economic Development and Cultural Change, 26, 271 – 280 , 1978

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Dakurah, A.H., S. P. Davis and R.K. Sampath, Defense Spending and Economic Growth in Developing Countries: a Causality Analysis, Journal of policy Modeling, 23, 651 – 658 , 2001.

Deger, S., Military Expenditure in Third World Countries: The Economic Effects, London, Routledge, 1986.

Deger, S. and S. Sen, Military Expenditure , Spin-off and Economic Development, Journal of Development Economics, 13, 1983.

Deger ,S. and R. Smith, Military Expenditure and Growth in Developing Countries, Journal of Conflict Resolution, 27, 335 – 353 , 1983.

Faine, R., P. Arnez, and L. Taylor, Defense Spending, Economic Structure and Growth: Evidence among Countries and Over Time, Economic Development and Cultural Change. 1984.

Kennedy, G., The Military in the Third world, London, 1974.

Kollias, C., G. Monolas, and S. Paleologou, Defense Expenditure and Economic in the European Union: A Causality analysis, Journal of Policy Modeling, 26, 553 – 569 , 2004.

Kusi, N.K., Economic Growth and Defense Spending in Developing Countries: A Causal Analysis, Journal of Conflict Resolution, 38, 152 – 159 , 1994.

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defense expenditure and civilian consumption

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marginal psychic cost

economic growth occurs

uss usc

ucc

public good

optimal path

optimizing framework

control function