Riots, Battles and Cycles

Riots, Battles and Cycles ^∗

St´ephane Auray

EQUIPPE (EA 4018), Universit´es Lille Nord de France, GREDI, Universit´e de Sherbrooke and CIRP´ EE, Canada.

Aur´elien Eyquem

GATE (UMR 5824), Universit´e de Lyon, Ecole Normale Sup´erieure LSH, France, GREDI, Universit´e de Sherbrooke, Canada.

Fr´ed´eric Jouneau-Sion

EQUIPPE (EA 4018), Universit´es Lille Nord de France.

March 2009

Abstract

This paper proposes a theoretical framework to investigate the impact of military conflicts on business cycles, as well as defense policies through enrolment mechanisms. Our framework is a variation of a Real Business Cycle model first proposed by Hercowitz and Sampson (1991) that admits explicit solutions. We extend the initial model to account for specific (potentially large) shocks that destroy the stock of capital. We consider two types of dynamics on the depreciation rate of capital: short–term shocks, that may be interpreted as riots and captured by a Moving Average specification, and mid–term shocks, that may be interpreted as wars and captured by a Markov Switching process. Using a panel data with 11 countries starting in the 19-th century we show that our model is able to reproduce the large variations of consumption to product ratios observed during major crises. Moreover the different regimes of the Markov Process are shown to be consistent with observed episodes of conflicts. Post-war US data are then used to provide estimates of the deep parameters. Our model reproduces usual business cycle facts. We also characterize the macroeconomic dynamics after shocks on the depreciation rate of capital.

Theses estimates are then used to investigate enrolment policies. We also question the goals defense policies should aim at. Finally, it provides a simple framework to quantify the welfare effects of alternative (simple) defense technologies.

Keywords: military policy, Real Business Cycle model, random coefficient autoregressive model.

JEL Classification: E13, E32, H56.

∗

We would like to thank Christian Francq for insightful comments. Corresponding author:

[email protected]

1 Introduction

Military and defense policies are a major source of central action in economics. As a first step, casual observation reveals that the budget of defense departments may reach a significant amount of the total public expenditures, and the amount of military forces may represent a significant fraction of the working–age population. For instance, according to Stockholm International Peace Research Institute, defense–related public expenditure account for 42% of the overall taxes in the USA. The figures are typically high for low income countries where most of the conflicts now take place. However, serious analysis of military expenditure cannot rely solely on accurate accounting.

First, enlarging military expenditures is by no means neutral from the political viewpoint. Major conflicts – especially World Wars I and II – have been preceded in large investment in military sector. Also, military expenditures always raise sharply during major conflicts and they are a key feature of the so–called “economics of war”. Finally any decent analysis of public policies should also questions its goals.

Various attempts have been made in the recent macroeconomic literature to treat specifically military/defense/peacekeeping policies. The question may then be raised whether this public effort should be incorporated in the total public expenditures or not (see Ramey (2008)). Another stream of literature aims at quantifying the impact of conflicts. Indeed, according to a “myth” wars may be “good for the economy” (often–quoted channels are increasing investment, less unemployment, boosted R&D ...). Most of the economic proponents to this view emphasize long–term effects.

In the short term however, war episodes destroy lives and private – as well public – belongings.

Another skeptical view toward the above “myth” comes from accurate measurements of direct costs of modern wars. Globalization and technical progress make wars more costly (Stigtlitz and Bilmes (2007) count the cost of Iraq war in trillions of dollars). Finally indirect costs may also be very high. Martin, Mayer and Thoenig (2008a, 2008b) examine the link between war episodes and international trade flows. Again, data show that major conflicts reduce international trade flows.

Conflicts are therefore a possible source of economic downturns.

However, the main part of this economic literature remains empirical. The major goal of this

paper is to propose a theoretical framework that is empirically consistent and that accounts for

some of the major features of military issues. First, conflicts destroy capital and enrolment policies

cut civilian working resources (possibly permanently if one accounts for human losses and/or severe physical damages). Moreover, the shocks related to war episodes cannot be considered as small with respect to the whole economy. Second, conflicts last for periods similar to economic cycles. Third, although several important economic mechanisms change during war episodes the ultimate needs and means of the population remain. Finally, defense expenditures are typically mostly – if not only – public. Of course, the model we propose is not able to account for other important features.

In particular we do not tackle the difficult questions raised by the causes of war (for example, escalation problems are not considered). Also, the whole range of public interventions during wars (from specific monetary policies to subsidies in R&D technologies) are not considered. Finally the main part of the paper is written for a closed economy. A supplementary section presents some extensions to account for possible imported consumption. This extension is used to discuss some issues related to embargos or sieges. It should of course not be considered as a complete economic treatment of international military issues.

Our framework is a variation of a Real Business Cycle model proposed by Hercowitz and Samp- son (1991). We choose this model because it admits explicit solutions. As an important conse- quence, shocks may be as large as desired. We extend the initial model to account for specific shocks that destroy capital stocks. These shocks may be viewed as destructions due to conflicts or possibly natural disasters. In our view, this second interpretation causes no problem as long as nation–specific considerations are absent from the model (in other words, whether destructions are caused by “enemies” or tsunami makes no difference in our model). We consider two types of dynamics : short–terms shocks are captured by a Moving Average specification and mid–term shocks by a Markov Switching process. Short–term shocks may be interpreted as riots (or natural disasters as explained above) whereas mid–term shocks are interpreted as wars. Part of the working force may be hired to limit capital destructions. This will allow us to question the goal defense policies should aim at.

Our model is able to reproduce some interesting facts. First, our model reproduces usual responses to productivity shocks (consumption, working hours and investment grow after booms).

Second, investment grows as an immediate response to conflicts but the response is negative in the

mid–term. On the contrary, production and working hours typically fall after destructions. Also

our model is able to quantify the long–term welfare gains induced by perpetual peace.

The paper is organized as follows. Section 2 proposes a presentation of the related literature.

Section 3 details the model. In particular it shown that the model is able to capture large fluc- tuations of the ratio consumption/product observed during conflicts if capital depreciation rate varies. Section 4 proposes an inference strategy to calibrate the model. Section 5 is devoted to the analysis of Impulse Response Functions as well as welfare analysis. Section 6 presents some possible extensions of the baseline model. Section 7 concludes.

2 Measuring macro-economic consequences of military activities

Using post–war data for developed countries has become customary in current macro-econometric literature. Indeed, the usefulness of the neoclassical framework during war episode is questionable.

First, war may induce very large shocks in the economy. This is at odd with the usual practice to build DSGE models as linear approximations around a deterministic steady state. Second, mod- ern macro-econometric literature typically treats government spending as unanticipated “shocks”

whereas central policies may account for more than half of the GDP during major conflicts.

Despite this difficuties, several recent empirical works have emphasized the importance of a careful look at military and conflict data in macroeconomics.

McGrattan and Ohanian (2008) first showed that properly written RBC models are able to capture many economic phenomena of WWII (despite the fact that much of the empirical literature is based on post–war data). One of the reason may be the following: even during war episodes the very basic needs – consumption – and means of production – capital and labor – remain and RBC models precisely rely on these basic means and needs.

Ramey (2008) emphasize that much of the variability of the shocks in public expenditure is explained by military expenditures. The main reason is that other sources of public expenditures (education, health, ...) are much more stable over time. Hence, once trends are removed, un- expected shocks in public expenditures are mainly military. Moreover the “narrative approach”

(Ramey and Shapiro (1998)) emphasizes the role of wars/peace episodes in the identification of

unexpected shocks. Ramey (2008) argues that distinction between military and civilian public ex-

penditures induces major differences in the analysis of private response unanticipated central policy

shocks.

In two recent contributions Barro et al. (2009) and Barro and Urs` ua (2009) made clear that modern wars (especially WWII) have disastrous consequences on civilian economics (this is true both for “winners” and “loosers”). They analyze a rich panel data with 24 countries and more than 100 years. Consequences of wars on consumption are analyzed using a small neo-classical model.

They evaluate the consequences of these extremely bad events. In particular, they induce a major increase in volatility measurement capable to explain observed equity premia without relying on implausible and/or sophisticated models of risk aversion.

Much empirical effort has been devoted to precise accounting of the costs of war. As one can expect, there is a considerable disagreement among authors (see Arias and Ardila (2003) for a discussion). Most empirical contributions show that net effects of wars (including destructions and possible positive effects induced by larger public expenditures) are negative. For instance, Collier (1999) argues that civil war might cause a 2.2 percentage point loss in the annual growth rate. In a welfare analyze, Hess (2003) obtains an average cost of conflicts equal to 102.3 USD per person.

Adopting an alternative perspective, Martin et al. (2008b) identify the impact of international trade on the occurrence of armed conflicts. Using an extensive data set on bilateral trade and armed conflicts, they show that increasing bilateral trade flows (through bilateral trade agreements for example) significantly reduces the probability of armed conflict with the corresponding trade partner without increasing the probability of conflicts with other trade partners. In this sense, trade openness could be seen as a peace–promoting technology. However, they show that multilateral trade openness reduces the bilateral trade dependence and thus the cost of bilateral armed conflicts, which increases the probability of war. This mixed evidence shows that trade openness and armed conflicts are closely related in the data but that both the sign and the magnitude of the relation depend on the specific characteristics of trade flows and agreements.

Finally, many contributions study particular conflicts to understand the precise linkage between military engagements and economic activities. For instance, the massive role played by engineering and logistics problems has been made clear for World War II while specialists of World War I put forth the major impact of human losses. Many debates are still open as well on the role of economic

“repairs” due to allies by Germany and their consequences in the failure of the Weimar Republic.

3 Modeling disastrous consequences of modern wars

A single model cannot address all the above issues. Our goal is to fill the existing gap between em- pirical contributions and macroeconomic models dealing with the economic gains/costs of conflicts.

We argue that Real Business Cycle (RBC hereafter) models are the proper tool to handle some of these questions. First, these models are well–suited to analyze both the long and short–run con- sequences of macroeconomic shocks. In particular, they rely on explicit transmission mechanisms that explain propagation of these shocks in the economy. However, a careful look at Barro et al.

data set reveals that specific models are needed to handle major crises episodes.

3.1 Variations of Consumption to Product ratio

The approach followed by Barro and his co-authors focusses on the analysis of consumption or GDP. It also usefull to examine the ratio of these quantities. The following table gives the year when the ratio Consumption/Product was the lowest from 1815 onwards together with the ratio of its average to its minimal value. We also compute this ratio six years after its minimum value has been reached and compare it to its average value. Indeed, Barro et al. findings are consistent with a six years average duration of major crises.

¹

Variations of Consumption to Product ratios (data : Barro (2009))

1

The data set for consumption is incomplete. For instance reliable figures for US consumption during the Civil

War are not available (see Barro and Urs‘ua (2009) for details.

Country Minimal Year Minimum value (C as % of Y)

C/Y six years af- ter min. year (as % of average value C/Y)

USA 1944 55 109

Canada 1943 75 178

United Kingdom 1943 66 144

Netherland 1943 62 205

Belgium 1942 79 180

France 1943 51 139

Swizerland 1961 86 145

Spain 1936 80 160

Portugal 1977 83 148

Germany 1944 42 174

Italy 1943 68 155

Finland 1944 69 198

Sweden 2001 73 NA

Norway 1997 61 NA

Denmark 2001 67 NA

Austria 1944 61 184

Japan 1945 36 135

The above figures clearly shows that the worst bellicose episodes induce a major drops in civilian activities. Major participant to WWII all experienced the lowest ratio during that war.

Minor participants together with neutral states did not experienced such large drops at this time.

A quick glance at the charts for Denmark, Norway and Sweden reveals that the drop in private

consumption/product ratio is largely due to the recent (that is post-war) upward trend in budget

expenditures. Remark that Spain attains its lowest level during its civil war and recall Spain remains

neutral during WWI and WWII. The case of Portugal is also interesting: the three droughts for this

ratio happens in 1918 (a brief civil war episode after the murder of Sidonio Pa`ıs) 1973 (carnation

Revolution) and 1977 (when over a million of refugees from former Portuguese African colonies returned to Europe). Finally the last column to the above table witnesses the strength of the recovery effect. Six years after the minimum of this ratio was reached all countries experienced larger than average ratios.

Standard RBC models would typically face difficulties in explaining such large switches in the relationship between consumption and product. One reason may be that the shocks at hand are too large. We could then rely on mixtures of distributions for the shocks (similar to what is done in the empirical insurance literature). However such large shocks are incompatible with approximation devices used in the RBC literature and exact resolution might be needed (see McGrattan and Oha- nian (2008)). Also identification of large versus “ordinary” shocks could be difficult and ultimately rely on pure statistical assumptions. Finally, such model cannot allow for specific channels although public interventions often differs both quantitatively and qualitatively during major crises. We now propose a model which is able to capture the difference bewteen major crises induced by military episodes and usual technological shocks.

3.2 The model

The representative household maximizes,

E

0

∞

X

t=0

β

^t

log(C

t

− h(N

t

)

^1+µ

), (1)

where β is the discount factor, C

_t

is the level of consumption chosen by the representative and N

_t

is the total amount of working time supplied by the representative agent in period t. Part of this time will be used in a military “service”. More precisely, when agents supply the N

_t

hours of work at period t and the capital installed at date t is K

_t−1

, the level of production Y

_t

is given by

Y

_t

= A

_t

K

_t−1^α

((1 − e

_t

)N

_t

)

^1−α

.

where 0 < e

_t

< 1 is the enrolment rate at date t and A

_t

is an exogenous shock. As usual, we interpret A

_t

as a technological shock.

We consider the enrolment process as the unique source of public intervention and e

t

as exoge-

nous. By doing so we implicitly assume it is the state’s privilege to enrol forces. This corresponds to the medieval duty known as “ost” in the (continental) European feodal times. A more con- temporary view may be as follows. Assume the state as the power to tax wages in order to enrol soldiers. The state’s budget balance implies

w

^m_t

e

_t

N

_t

= τ

_t

(w

^m_t

e

_t

N

_t

+ w

_t

(1 − e

_t

)N

_t

)

where w

^m_t

and w

_t

are respectively the hour wages paid to soldiers and civilians and τ

_t

is the tax rate. We then have

e

t

= τ

_t

w

_t

(1 − τ

_t

)w

^m_t

+ τ

_t

w

_t

.

In particular if soldiers and civilians are equally paid e

_t

= τ

_t

(accordingly e

_t

is less than τ

_t

if and only if soldiers are better paid than civilians).

If we set A

^′_t

= A

_t

(1 − e

_t

)

^1−α

we see that supplying time to the military sector amounts to divide the technological shock A

_t

by the same amount. We then write

Y

t

= A

^′_t

K

_t−1^α

N

_t^1−α

. The law of motion for the capital stock is given by

K

_t

= A

_K

K

_t−1^δt

I

_t^1−δt

where I

_t

= Y

_t

− C

_t

is the investment flow at date t and δ

_t

is stochastic random rate of depreciation

taking values in ]0, 1[. This law of motion is an adaptation of the model proposed by Hercowitz and

Sampson (1991) (see also Collard (1999))in which we incorporated a possible random depreciation

rate as Ambler and Paquet (1994) did. This extra source of randomness captures the effect of

increasing depreciation of capital induced by short term and/or long lasting conflicts. Notice that

with the above formula we need K

_t−1

/I

_t

> 1 to insure that ceteris paribus, K

_t

decreases when δ

_t

gets smaller.

The associated Lagragian is log(C

_t

− hN

_t^1+µ

) + λ

_t

A

_K

K

_t−1^δt

(A

_t

K

_t−1^α

((1 − e

_t

)N

_t

)

^1−α

− C

_t

)

^1−δt

− K

_t

+βE

_t

h

log(C

_t+1

− hN

_t+1^1+µ

) + λ

_t+1

A

_K

K

_t^δt+1

(A

_t+1

K

_t^α

((1 − e

_t+1

)N

_t+1

)

^1−α

− C

_t+1

)

^1−δt+1

− K

_t+1

i The first order conditions for optimality provide the following system

²

1

Ct−hN_t^1+µ

= λ

_t

K

_t

(1 − δ

_t

)/I

_t

(1)

h(1+µ)N_t^1+µ

Ct−hN_t^1+µ

= λ

_t

K

_t

(1 − δ

_t

)(1 − α)Y

_t

/I

_t

(2) λ

_t

K

_t

= βE

_t

[λ

_t+1

K

_t+1

(δ

_t+1

+ (1 − δ

_t+1

)αY

_t+1

/I

_t+1

])] (3)

K

_t

= A

_K

K

_t−1^δt

I

_t^1−δt

(4)

If we divide (2) by (1) we get

h(1 + µ)N

_t^1+µ

= (1 − α)Y

_t

Plugging this into Y

_t

= A

^′_t

K

_t−1^α

N

_t^1−α

gives

Y

_t

=

1 − α h(1 + µ)

_α+µ^1−α

(A

^′_t

)

^α+µ1+µ

K

^α

1+µ+α α+µ

t−1

(5)

Denote I

_t

/Y

_t

= S

_t

, and λ

_t

K

_t

= X

_t

for all t. Equations (1) and (3) are now equivalent to the following system

S

t

=

_(1+µ)(1+X^{Xt(1−δt)(µ+α)}

t(1−δt))

(1

^′

)

Xt

β

= α

_µ+α^1+µ

+ E

t

[X

t+1

(δ

t+1

+ α

^1+µ_µ+α

(1 − δ

t+1

))] (3

^′

)

It is easily seen that S

_t

∈ [0, 1 − (1 − α)/(1 + µ)] ⊂ [0, 1] as soon as X

_t

> 0, 1 > α > 0, and µ ≥ 0.

We consider the following stochastic process

δ

_t

= δ(η

_t

) + σ

_δ

ǫ

_δ,t

+ φ

_δ

ǫ

_δ,t−1

2

At this point we assume that the FOC provide a (local) maximum to the problem. A full analysis of the Second

Order Conditions is beyond the scope of this paper.

where (η

_t

)

_t≥0

is Markov process of order 1 with a finite number of states J and (ǫ

_δ,t

)

_t≥0

stands for an second-order stationary process (more precisely we assume E

_t

[ǫ

_δ,t+1

] = ǫ

_δ

and V ar

_t

[ǫ

_δ,t+1

] = 1).

We also assume η

_t

is independent from (ǫ

_δ,s

)

_s≤t

.

The shock ǫ

_δ,t

may then be interpreted as the outcome of a short-term problem in the storage production function. This could be caused by natural disasters (massive storms, earthquakes...) or human actions (riots for instance). Hence we shall assume P (σ

_δ

ǫ

_δ,t

+ φ

_δ

ǫ

_δ,t−1

≤ 0) = 1. As there is no reason for these shocks to last, the MA specification is convenient.

³

Also remark, we need P (δ

_t

∈ [0, 1[) = 1 a condition which is easy to obtain for MA processes.

Our model may be extended to more general MA processes (at least if (ǫ

_δ,t

)

_t∈Z

is an iid process).

However the computation of the explicit solution soon become difficult (in case of a MA(8) process a linear system with more than 100 equations is involved). Considering the interpretation given above, the order of the MA process is largely linked to the frequency of the data. Typically, a large order for the MA specification is needed if we use ”high” frequency data (e.g. monthly). To keep computations and estimations within a feasible range, we have chosen annual data (in this case the MA(1) specification is supported by our data set, see below).

The Markov process captures lasting conflicts where capital stocks are repeatedly destroyed.

For instance we could imagine P (η

_t

∈ { 1, 2 } ) = 1 for all t. A possible interpretation in this case is that η

_t

= 1 (resp. η

_t

= 2) in peaceful (resp. bellicose) times and δ(2) < δ(1). Also, with this kind of interpretation, there is no need for φ

_δ

or σ

_δ

to vary with η

_t

(although this may be done).

Consider X

_t

= X(η

_t

)+ σ

_X

(η

_t

)ǫ

_δ,t

as a possible solution to equation (3’), and taking into account the specification for the stochastic process (δ

_t

)

_t≥0

, we must have for all state j

X(j)+σX(j)ǫδ,t

β

=

^µ(1−α)_µ+α

J

X

j^′=1

π

_j,j^′

((X(j

^′

) + σ

^′_X

(j

^′

)ǫ

_δ

)(δ(j

^′

) + (σ

_δ

+ φ

_δ

)ǫ

_δ

+ φ

_δ

ǫ

_δ,t

) + σ

_X

(j

^′

)σ

_δ

) +α

_µ+α^1+µ

J

X

j^′=1

π

_j,j^′

(1 + X(j

^′

) + σ

_X

(j

^′

)ǫ

_δ

)

where P(η

_t+1

= j

^′

| η

_t

= j) = π

_j,j^′

.

3

Recall the AR specification for technological shocks is justified since upward or downwards moves of the production

functions will typically take time to spread and fully hit the aggregated production function.

This is equivalent to the following linear system:

X(j) =

^βµ(1−α)_µ+α

J

X

j^′=1

π

_j,j^′

((X(j

^′

) + σ

^′_X

(j

^′

)ǫ

_δ

)(δ(j

^′

) + (σ

_δ

+ φ

_δ

)ǫ

_δ

) + σ

X

(j

^′

)σ

_δ

) +αβ

_µ+α^1+µ

J

X

j^′=1

π

_j,j^′

(1 + X(j

^′

) + σ

_X

(j

^′

)ǫ

_δ

) σ

_X

(j) =

^βµ(1−α)_µ+α

φ

_δ

J

X

j^′=1

π

_j,j^′

(X(j

^′

) + σ

_X

(j

^′

)ǫ

_δ

)

We may now summarize our explicit solution.

• exogenous processes

a

^′_t

∈ R

E

_t

[ǫ

_δ,t+1

] = ǫ

_δ

, V ar

_t

[ǫ

_δ,t+1

] = 1, P(ǫ

_δ,t

< 0) = 1 P(η

_t+1

= j

^′

| η

_t

= j) = π

_j,j^′

, ∀ (j, j

^′

) ∈ { 1, . . . J }

²

δ

_t

= δ(η

_t

) + σ

_δ

ǫ

_δ,t

+ φ

_δ

ǫ

_δ,t−1

• parameters and domains α, β, ∈ ]0, 1[

µ, h, σ

_δ

, − ǫ

_δ

∈ R

⁺⁺

a

_k

, φ

_δ

∈ R

0 ≤ π

_j,j^′

≤ 1 ∀ (j, j

^′

) ∈ { 1, . . . J }

²

, P

_J

j^′=1

π

_j,j^′

= 1 ∀ j ∈ { 1, . . . , J } X(j) =

^βµ(1−α)_µ+α

J

X

j^′=1

π

_j,j^′

((X(j

^′

) + σ

^′_X

(j

^′

)ǫ

_δ

)(δ(j

^′

) + (σ

_δ

+ φ

_δ

)ǫ

_δ

) + σ

_X

(j

^′

)σ

_δ

) +α(1 + µ)

J

X

j^′=1

π

_j,j^′

(1 + X(j

^′

) + σ

X

(j

^′

)ǫ

_δ

), ∀ j ∈ { 1, . . . , J } σ

_X

(j) =

^βµ(1−α)_µ+α

φ

_δ

J

X

j^′=1

π

_j,j^′

(X(j

^′

) + σ

_X

(j

^′

)ǫ

_δ

) ∀ j ∈ { 1, . . . , J } δ

_t

∈ [0, 1[

X

_t

> 0

K

_t−1

/I

_t

> 1

• endogenous processes

X

_t

= X(η

_t

) + σ

_X

(η

_t

)ǫ

_δ,t

Y

_t

/I

_t

= S

_t

=

_(1+µ)(1+X^{Xt(1−δt)(µ+α)}

t(1−δt))

y

_t

=

_α+µ^1−α

log

1−α h(1+µ)

+ α

^1+µ+α_α+µ

k

_t−1

+

_α+µ^1+µ

a

^′_t

k

_t

= a

_k

+

δ

_t

+ (1 − δ

_t

)α

^1+µ+α_α+µ

k

_t−1

+ (1 − δ

_t

)

s

_t

+

_α+µ^1−α

log

1−α h(1+µ)

+

_α+µ^1+µ

a

^′_t

The special source of randomness introduced in this model may now be more clearly justi- fied. Consider for instance the case where K

t

= A

_k,t

K

_t−1^δ

I

_t^1−δ

and assume conflicts cause a de- crease in A

_k,t

. Straightforward computation shows that k

_t

= a

_k,t

+

δ + (1 − δ)α

^1+µ+α_α+µ

k

_t−1

+ (1 − δ)

s +

_α+µ^1−α

log

1−α h(1+µ)

+

_α+µ^1+µ

a

^′_t

. We then face a much simpler model where the ”conflict- related” shocks are simply part of a “generalized” productivity shock: a

^′′_t

= a

_k,t

+ (1 − δ)

_α+µ^1+µ

(a

^′_t

+ (1 − α) log(1 − e

_t

)). In this case, the optimal policy amounts to manage enrolment as to limit the generalized productivity shocks. But this is counterfactual: when conflict induces lower a

_k,t

, this model tell us that enrolment should also decrease. In our case, when δ

_t

drops the dynamic of capital is disturbed. More precisely the stock of capital admits a Random Coefficient Autore- gressive representation. The randomness in the process δ

t

implies that k

t

evolves according to a random autoregressive coefficient with random volatility. This particular source of randomness is the consequence of the stochastic nature of the depreciation. Ambler and Paquet (1994) obtained a similar result (see their equation (5)).

This remark also raises the question of the existence of a stationary distribution for the endoge- nous process k

_t

.

⁴

This is a key issue for (at least) two reasons. First, part of our inferential strategy relies on asymptotic approximations which are valid only if our dynamical system accepts station- ary solutions. Second, the existence of the welfare is not guarantee if no long-term distribution may be exhibited. We obtain the following result

Proposition 1 If

E

log

δ

_t

+ (1 − δ

_t

)α 1 + µ + α α + µ

< 0 and lim

_i→+∞

1

i log( | a

^′_t−i

| ) < 0

4

This point is not discussed in Ambler and Paquet (1994).

then there exists a stationary solution to

k

_t

= a

_k,t

+ ρ

_k,t

k

_t−1

+ σ

_k,t

a

^′_t

(6)

where

a

_k,t

= a

_k

+ (1 − δ

t

)

s

t

+

^1−α_α+µ

log

1−α h(1+µ)

ρ

_k,t

= δ

_t

+ (1 − δ

_t

)α

^1+µ+α_α+µ

σ

_k,t

= (1 − δ

_t

)

^1+µ_α+µ

Proof: We derive for all t ≥ 1

k

t

= a

_k,t

+ k

0 t−1

Y

i=0

ρ

_k,t−i

+

t−1

X

i=1

(a

_k,t−i

+ σ

_k,t−i

a

^′_t−i

)

i

Y

j=1

ρ

_k,t+j−i

Now a

_k,t

is a bounded stationary process (it is almost trivial considering − 1/exp(1) < (1 − δ

t

) log(1 − δ

t

) < 0). Let us denote a

_k,t

≤ a

_k

. Since ρ

_k,t

> 0 we have

k

_t

≤ a

_k

+ (k

₀

+ a

_k

)

t−1

Y

i=0

ρ

_k,t−i

+

t−1

X

i=1

σ

_k,t−i

a

^′_t−i

i

Y

j=1

ρ

_k,t+j−i

(7)

A unique stationary solution to equation (6) exists whenever the rhs of (7) a.s. converge to a single real valued random variable as t goes to infinity. Let us first study the a.s. convergence of

t−1

X

i=1

σ

_k,t−i

a

^′_t−i

i

Y

j=1

ρ

_k,t+j−i

.

A sufficient condition is the absolute convergence. We then study the following sum

t−1

X

i=1

| σ

_k,t−i

a

^′_t−i

|

i

Y

j=1

| ρ

_k,t+j−i

| .

Using the Cauchy principle, a sufficient condition for convergence is then given by

lim

_i→+∞

σ

^1/i_k,t−i

| a

^′_t−i

|

^1/i

i

Y

j=1

| ρ

_k,t+j−i

|

^1/i

< 1

Taking logarithms, a sufficient condition is then given by

lim

_i→+∞

1

i log(σ

_k,t−i

) + lim

_i→+∞

1

i log( | a

^′_t−i

| ) + lim

_i→+∞

1 i

i

X

j=1

log( | ρ

_k,t+j−i

| ) < 0

As σ

_k,t

is also bounded from above, we must show

lim

_i→+∞

1

i log( | a

^′_t−i

| ) + lim

_i→+∞

1 i

i

X

j=1

log( | ρ

_k,t+j−i

| ) < 0 Now, if we have E[log(δ

_t

+ (1 − δ

_t

)α

^1+µ+α_α+µ

)] < 0 we get a.s.

lim

_i→+∞

1 i

i

X

j=1

log( | ρ

_k,t+j−i

| ) < 0

and the Cauchy Principle also shows that in this case we have lim

_t→+∞

Q

_t−1

i=0

ρ

_k,t−i

= 0. a.s. Q.E.D One may remark that the condition lim

_i→+∞¹_i

log( | a

^′_t−i

| ) < 0 is very weak. For instance it does not rule out the possibility that a

_t

is not a stationary process. Second, the condition E[log(δ

_t

+ (1 − δ

_t

)α

^1+µ+α_α+µ

)] < 0 is also weaker than E[δ + (1 − δ)α

^1+µ+α_α+µ

] < 1 (the stationarity condition in perpetual peaceful time with δ

t

= δ).

4 Inferential strategy

The estimation of the model raises two different difficulties. First, as usual in DSGE models, the so-called “deep” parameters are complicated functions of the observable moments. Second, the identification of regimes is difficult. We shall first rely on the study of the consumption/product ratio to identify the regimes. This will unable us to show that our model is well suited to capture major bellicose time period. In a second step, the deep parameters will be estimated using the identified regimes.

4.1 Identification of regimes

Collecting reliable macro–level data during major long–lasting conflicts is a difficult task. Never-

theless major efforts by A. Maddison (2001) completed by a recent work of Barro (2009) provide

consensual figures about private consumption and GDP for several countries on an annual basis.

The the list of countries we used together with their starting date of record in Barro’s data base is as follows USA (1834), Canada (1871), United Kingdom (1830), Netherlands (1815), France (1824), Germany (1851), Denmark (1844), Italy (1874), Spain (1874), Sweden (1861), Switzerland (1851).

Other countries present in Barro’s data set have not been included since consumption is often not observed regularly before 1945. Our selection includes all major parties of WWI and WWII except Russia (absent) and Turkey (for which the starting date, 1923 appears too recent since Turkey involvement was particularly important in WWI only). It also includes two long–standing neutral countries and one country which experienced a major civil war (namely Spain).

As the sample is not very large, it seems reasonable not the increase the number of parameters in the transition matrix. We choose a specification ` a la Chib (1998) with an (infinite) transition matrix such that P(η

_t+1

= j | η

_t

= j) = 1 − P(η

_t+1

= j + 1 | η

_t

= j) = π

_j

for all 1 ≤ j and P(η

_t+1

= J | η

_t

= J ) = 1. This specification, while compatible with our model is close to the

”structural break” literature. It captures the fact that no conflict nor peace time exactly resemble previous one. Also remark that it may be that post-war depreciation rate δ

t

is higher than before the conflict. Indeed, it may be argue that the remaining stock of capital after a conflict is of better quality. This may be due, for instance, to successful R&D investment during war (or to post-war

“repairs”, annexations,...). Of course, only part of the this transition matrix is identifiable since the part of this ”history” is censored.

We also choose a parametric specification for ǫ

_δ,t

, assuming these shocks are uniformly dis- tributed over the interval [0, − √

12] (recall we imposed P(ǫ

_δ,t

< 0) = 1 and V ar

_t

[ǫ

_δ,t+1

] = 1).

The identification of the regimes is greatly facilitated if we only use part of the information.

More precisely recall we have

X

t

(1 − δ

t

) = 1 − C

t

/Y

t

C

_t

/Y

_t

−

^1−α_1+µ

Hence if

^1−α_1+µ

is known the process (1 − C

_r

/Y

_r

)(C

_r

/Y

_r

−

^1−α_1+µ⁻¹

, for r = 1, 3, . . . admits an explicit Markov Chain representation. Now as 0 <

^1−α_1+µ

< 1/2 we propose a grid search over this parameter and identify all the dates when the regime changes. Finally as we aims at capturing the major conflicts we tried to identify up to five sub-periods (pre WWI, WWI, inter-war, WWII, post-war).

Using this technique, the regimes are as follows

Country end of regime 1 end of regime 2 end of regime 3 end of regime 4

USA 1915 1940 1945 1981

Canada 1904 1906 1945 1978

United Kingdom 1913 1919 1939 1945

Netherlands 1941 1950 1962 1988

France 1909 1940 1943 1957

Switzerland 1959 1971 1974 1988

Spain 1909 1935 1959 1981

Germany 1913 1939 1949 1969

Italy 1912 1950 1964 1993

Finland 1938 1946 1955 1996

Sweden 1915 1963 1979 1994

Norway 1915 1963 1979 1994

Denmark 1913 1958 1979 1994

Japan 1932 1945 1952 1967

Of course, as we did not use prior information about the “true” dates of conflicts we cannot expect perfect matching but the results are in reasonable accordance with actual facts for most countries. In particular remark that neutral countries have very different dates than the others.

4.2 Deep parameters identification

If we consider the regimes as given, then η

_t

is known. In this case, likelihood may be computed for the series C

t

/Y

t

and Y

t

. More precisely ǫ

_δ,t

may be extracted from

X

_t

(1 − δ

_t

) = 1 − C

_t

/Y

_t

C

_t

/Y

_t

−

^1−α_1+µ

if µ, α, β, σ

_δ

, φ

_δ

and X(j), σ

X

(j), δ(j), for all j are known. Moreover X(j), σ

X

(j) may be computed using the Rational Expectation relationship

X

_t

β = α 1 + µ

µ + α + E

_t

[X

_t+1

(δ

_t+1

+ α 1 + µ

µ + α (1 − δ

_t+1

))].

Now when ǫ

_δ,t

is extracted, δ

_t

may be computed. Now as

ln((1 − α)/(h(1 + µ))) + y

_t

= (1 + µ)n

_t

we have using y

_t

− a

^′_t

− (1 − α)n

_t

= αk

_t−1

αk

t−1

= α + µ

1 + µ y

t

− a

^′_t

+ 1 − α 1 + µ ln

h(1 + µ) 1 − α)

.

taken this into account in k

_t

= a

_k

+ δ

_t

k

_t−1

+ (1 − δ

_t

)(s

_t

+ y

_t

) we get

z

t+1

= a

^′_t+1

− δ

t

a

^′_t

where

z

_t

= α + µ

1 + µ y

_t+1

+ 1 − α 1 + µ ln

h(1 + µ) 1 − α)

− δ

_t

α + µ

1 + µ y

_t

+ 1 − α 1 + µ ln

h(1 + µ) 1 − α)

− α (a

_k

+ (1 − δ

_t

)(s

_t

+ y

_t

)) ,

which allows us to extract a

^′_t

. As usual we choose a gaussian stationary AR(1) process for a

^′_t

= ρ

_a

a

^′_t−1

+ σ

_a

ǫ

_a,t

and maximize the likelihood of the sequence ǫ

_δ,t

, ǫ

_a,t

wrt to all the deep parameters.

We performed this estimation technique for UK data. We first consider the estimation of the deep parameters related to preferences and production function

β µ α

0.883 3.84 0.498

We remark that the estimations of the parameters governing preferences and productions func- tions are in accordance with other available results (although α is much larger than usual post- WWII estimates). Notice h cannot be identified with available data since y

_t

is observed up to some constant. The parameters associated with the augmented TFP process are as follows (again they are similar to usual figures with an autoregressive coefficient close to 1).

ρ

_a

σ

_a

0.974 0.069

The parameters associated with the δ

_t

process

δ

₁

δ

₂

δ

₃

δ

₄

δ

₅

δ

₆

0.908 0.574 0.687 0.772 0.942 0.462

π

1

π

2

π

3

π

4

π

5

0.977 0.656 0.975 0.202 0.947 σ

_δ

φ

_δ

0.155 6.477E-07

Clearly UK experienced a significant drop in the average value of δ

_t

during both wars. During the first World the drop is almost 40%. The complete recovery only appears after WWII. Also notice a significative difference between the average quality of capital in the XIX-th and XX-th centuries. As 1/π

_j

is an estimate of the anticipated duration of regime j one may notice that the duration of WWII seems slightly underestimated as 1/pi

₄

= 1.29. It is however difficult to say whether this is due to the quality of data and/or the ability of the model to reproduce such trouble times.

5 IRFs and welfare analysis

We now use the estimated model to analyze the short–term impact of shocks. Welfare computation is then used to question the cost/advantages of defense and peacekeeping policies.

5.1 Impulse Response Functions

As we derived an explicit dynamic system as a possible solution of our RBC model, Impulse Response Functions may be obtained without simulations.

First consider (generalized) productivity shocks affecting the process a

^′_t

. Responses to tempo- rary upward movement of the process a

^′_t

display the usual hump–shaped pattern with the usual directions. More precisely, investment, consumption and working hours increase along with a

^′_t

. Recall that increasing enrolment is equivalent to a negative productivity shocks. As a consequence, enrolment in perpetual peaceful times (that is when δ

_t

is fixed) destroys civilian working hours both directly (since (1 − e

_t

)N

_t

decreases) and indirectly since N

_t

also decreases.

We now consider responses after a negative shock to the depreciation rate, i.e. δ

_t

drops. The

evolution of the capital stock K

_t

is driven by two effects. On the one hand, par of the accumulated

stock of capital is destroyed, but also investment increases. Hence the net effect is ambiguous.

Let us first show that K

_t

must then decrease. The condition P (σ

_δ

ǫ

_δ,t

+φ

_δ

ǫ

_δ,t−1

≤ 0) = 1 implies P (δ

_t

≤ δ) = 1 where δ = max { δ(1), . . . , δ(J) } . Now, it may be shown that,

(1 − δ) log

X

_t

(1 − δ)(µ + α) (1 + µ)(1 + X

_t

(1 − δ))

,

is a concave function of δ if δ ∈ [0, 1[. As a consequence, a necessary and sufficient condition for k

_t

to decrease when δ

_t

decreases is then,

µ(1 − α) − α

²

µ + α k

_t−1

> log

X

_t

(1 − δ)(µ + α) (1 + µ)(1 + X

t

(1 − δ))

+

(1 − α) log

1−α h(1+µ)

(1 + µ)a

^′_t

α + µ − 1 − δ

1 + X

t

(1 − δ) . This condition is equivalent to K

_t−1

/I

_t

> exp

−

_1+X^1−δ_t(1−δ)

when I

_t

and X

_t

are computed for δ

t

= δ. Notice this condition is always verified if K

t−1

/I

t

> 1 (as we assume X

t

> 0).

Now, as N

_t

is an increasing function of K

_t−1

only, we obtain that working hours also decrease with one–period delay, just as production. By the same argument, the marginal productivity of labor – hence real wage – also decreases one period after the shock. Of course, the effect is reinforced if the government chooses to enlarge enrolment. However, the marginal productivity of capital – hence real interest rates – grows if µ > α

²

/(1 − α). This condition is certainly verified since the hypothesis α < 1/2 < µ is clearly not rejected.

The response of investment is positive during a few periods – as S

_t

increases. The result after two periods depends on φ. If this parameter is positive enough, then two periods after the initial shock, the investment may still be higher than its long–term level. After three periods, the investment is lower than its long–term level. Finally, as production decreases and investment increases, private consumption decreases. Consumption’s decrease is immediate (since investment rises whereas production is constant in the first period). The fall in consumption will typically be the largest two periods after the depreciation shocks if φ is large enough (since production decreases while investment may still rise). Notice that the marginal propensity to consume (1 − S

_t

) also decreases.

The chart below displays our IRFs after a depreciation rate shock using the estimates presented

in the previous section.

5 10 15 20 25 30

−0.2

−0.1 0 0.1 0.2 0.3 0.4

% deviation

Periods

IRFs to negative depreciation shocks

Capital Output Consumption Investment Hours

The “myth” according to which “war may be good for the economy” is then highly partial. The only positive effect our model produces is an increase of gross private investment. Notice however that this effect is entirely due to the need to rebuilt destroyed capital. The net and mid–term effects are both clearly negative. Also, the claim according to which employment may be larger during war episodes does not show up. Finally, reversing the previous comparative exercises, our model is able to account for economic “booms” after long periods of conflicts, as this is may be analyzed as a positive shock on δ

_t

.

5.2 Welfare analysis

The conclusions to be drawn from our IRFs are limited, as they relate to the linkage between destructive episodes and business cycles. The fact that destruction by itself cannot be good for the economy goes back to the famous “broken window fallacy” controversy. It clearly cannot be used to justify massive disarmament. A more careful study of the benefits induced by enrolment must be derived from welfare analysis.

As mentioned in the introduction, the two kinds of risks we consider (temporary and long

lasting shocks) are usually handled by two different peacekeeping forces: police and army. We first consider the optimal short–run policy, in which enrolment is entirely devoted to police forces, whereas enrolment in the army is consider in a second step.

5.2.1 Police forces and the short–run enrolment policy

Since we consider only short–run policy, we consider here that J = 1. For the sake of illustration, assume that enrolment in the police force divides the size of the temporary shocks (that is ǫ

_δ,t

/(θe

_t

)).

Enrolment at date t must then be chosen as to maximize, (1 − α)

_α+µ^1+µ

log(1 − e

_t

) + βα

^1+α+µ_α+µ

1 + βE

_t

h

δ

_t+1

+ (1 − δ

_t+1

)α

^1+α+µ_α+µ

i k

_t

− log(1 + X

_t

(1 − δ

_t

)) − E

_t

[log(1 + X

_t+1

(1 − δ

_t+1

))] .

Calibrating θ such that the average amount of police forces in the US correspond to the optimal level for average shocks, we get the following chart:

⁵

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Percieved shock

Initial size of the shock

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Enrolment rate

Percieved shock 45° line

Enrolment rate (as % of act. pop.)

Optimal enrolment is an increasing function of the size of shocks with a moderate convexity

5

The data are from the Bureau of Labor Statistics, see

http

:

//www.bls.gov/oes/current/oesnat.htm#b33−0000.

Enrolment is calculated as the sum of occupations in the protective service sector plus active military forces inside

the U.S., see

http

:

//siadapp.dior.whs.mil/personnel/M ILIT ARY /ms0.pdf

.

effect. Notice also the evolution is not very large (keeping police forces within +/-15% of its actual value covers optimally more than 3/4 of the shocks). Also, the “perceived shocks” (that is ǫ

_δ,t

/(θe

_t

)) decrease more sharply for larger shocks. The optimal enrolment absorbs more than 50%

of the largest possible shock.

Finally, recall,

(1 − δ) log

X(1 − δ)(µ + α) (1 + µ)(1 + X(1 − δ))

,

is a concave function of δ. We conclude that the police technology, ceteris paribus, should aim at increasing the distribution of δ

_t

in the second–order sense. In other words, the welfare displays risk–adversity towards ˜ δ, and there is a positive propensity to pay for lower–magnitude risks of riots and natural disasters.

5.2.2 Military forces and the long–run enrolment policy

We now consider long–run policy. Consequently, the government’s objective is to maximize the whole welfare stream. If a stationary distribution exists, the welfare may be explicitly derived.

Indeed, we have,

E

₀

"

_+∞

X

t=0

β

^t

log(C

_t

− hN

_t^1+µ

)

#

= E

₀

"

_+∞

X

t=0

β

^t

y

_t

# + E

₀

"

_+∞

X

t=0

β

^t

log

1 − S

_t

− 1 − α 1 + µ

# .

The last sum of the above equation is,

log(µ + α) − log(1 + µ) − E[log(1 + ˜ X(1 − δ))] ˜

1 − β ,

where for any asymptotically stationary process x

_t

, we denote, ˜ x = lim

^d _t→+∞

x

_t

. Now recall we have,

y

_t

= 1 − α α + µ log

1 − α h(1 + µ)

+ α 1 + µ + α

α + µ k

_t−1

+ 1 + µ

α + µ a

^′_t

.

Hence we derive,

E

₀

P

_+∞

t=0

β

^t

y

_t

= y

₀

+

^{β(1−α) log}

1−α h(1+µ)

(α+µ)(1−β)

+ α

^1+µ+α_α+µ

E

₀

P

_+∞

t=1

β

^t

k

_t−1

+

_{(1−β)(α+µ)}^β(1+µ)a∞

,

= y

₀

+

^β

(1−α) log

1−α h(1+µ)

+α(1+µ+α)E[˜k]+(1+µ)(a^∞+(1−α)E[log(1−˜e)])

(1−β)(α+µ)

.

The computation of E[˜ k] in the general case is cumbersome. To avoid complications, we consider ǫ

_δ,t

= 0. Again, we consider that enrolment is entirely devoted to lower destructive effects of possible long–term conflicts.

We may then restrict the welfare analysis to, β

α(1 + µ + α)E[˜ k] + (1 + µ)(1 − α)E[log(1 − e)] ˜

α + µ − E[log(1 + ˜ X(1 − δ))]. ˜

Let P

_j

= P(η

_t

= j), we have (following, for instance, Francq and Zakoian (2001), section 4):

P

_j

E[k

_t

| η

_t

= j] = a

_k

+ (1 − δ(j))

1−α α+µ

log

1−α h(1+µ)

+ s

∞

(j) +

_α+µ^1+µ

(a

∞

+ (1 − α)log(1 − e(j)) , +

α

^1+µ+α_α+µ

+

^{µ(1−α)−α}_α+µ ²

δ(j) P

J

j^′=1

π

_jj^′

P

_j^′

E[k

t−1

| η

t−1

= j

^′

], where s

∞

(j) = log

µ+α 1+µ

+ x(j) + E[log(1 − δ(j))] − E[log(1 + X(j)(1 − δ(j))].

We therefore conclude that the vector (P

₁

E[k

_t

| η

_t

= 1], . . . , P

_J

E[k

_t

| η

_t

= J])

^′

is the solution of a linear system. Provided the matrix Id

_J

− [

α

^1+µ+α_α+µ

+

^{µ(1−α)−α}_α+µ ²

(δ(j) + σǫ)

π

_jj^′

] is invertible, there is only one solution and E[˜ k] = P

_J

j=1

P

j

E[k

t

| η

t

= j].

It is easy to show that any positive shift in the distribution of δ increases the welfare. In other words, any defense technology that changes the initial distribution of δ to another one which stochastically dominates the first one at the first order, induces welfare gains.

Again, for concreteness, assume δ

_t

= δ + (δ − δ(η

_t

))/(θe

_t

). As in the previous “police case”, enrolment increases at the beginning of a war, grows up or down as war becomes more or less destructive and returns to its lowest level if we go back to peaceful time. Also remark that because of the term (1 − δ(j))(1 − α)

_α+µ^1+µ

log(1 − e(j)), a more efficient defense technology is also more costly.

We finally use our model to address questions about the goal a long–run defense policy should

aim at – beside trying to reduce the consequences of negative shocks on δ. Consider the case

J = 2 (recall by convention we assume η

₁

= 1 if and only if peace prevails at date t). We would

like to know how the welfare depends on the sequence of war and peace. For instance, assume we must choose among two military technologies leading to the same average value of δ at the same enrolment cost. The first one aims at preventing long wars, at the cost of more frequent war episodes. The second tries to keep peace as long as possible, at the cost of more destructive war episodes. What is the best one?

If J = 2 the welfare may be written as,

βα(1+µ+α)

(1−β)(α+µ)

E[k

_t

| η

_t

= 2] −

^log(1+X_1−β^(1−δ(2)))

+P

₁

E[k

_t

| η

_t

= 1] − E[k

_t

| η

_t

= 2] −

log(1+X(1−δ(1)))−log(1+X(1−δ(2))) 1−β

,

where P

₁

=

_2−π^1−π22

11−π22

is the (marginal, stationary) probability of peace. Moreover, we have, P

₁

E[k

_t

| η

_t

= 1] = a

₁

+ b

₁

(π

₁₁

P

₁

E[k

_t

| η

_t

= 1] + (1 − π

₁₁

)(1 − P

₁

)E[k

_t

| η

_t

= 2]),

(1 − P

₁

)E[k

_t

| η

_t

= 2] = a

₂

+ b

₂

((1 − π

₂₂

)P

₁

E[k

_t

| η

_t

= 1] + π

₂₂

(1 − P

₁

)E[k

_t

| η

_t

= 2]), E[˜ k] = P

₁

E[k

_t

| η

_t

= 1] + (1 − P

₁

)E[k

_t

| η

_t

= 2],

a

_j

= a

_k

+ (1 − δ(j))

1−α α+µ

log

1−α h(1+µ)

+ s

∞

(j) +

_α+µ^1+µ

(a

∞

− (1 − α) log(1 − e(j)) , b

j

= α

^1+µ+α_α+µ

+

^{µ(1−α)−α}_α+µ ²

δ(j).

Let us denote Ω =







b

₁

π

₁₁

b

₁

(1 − π

₁₁

) b

₂

(1 − π

₂₂

) b

₂

π

₂₂





 and w =





 a

₁

a

₂





 . We have E[˜ k] = (1, 1)Ω

⁻¹

w, and then, dE[˜ k] = − (1, 1)Ω

⁻¹

dΩΩ

⁻¹

w, or,

sgn(dE[˜ k]) = − sgn





 (1, 1)







b

₂

π

₂₂

− b

₂

(1 − π

₂₂

)

− b

1

(1 − π

11

) b

1

π

11





 dΩ







b

₂

π

₂₂

− b

₂

(1 − π

₂₂

)

− b

1

(1 − π

11

) b

1

π

11





 w





 ,

= − sgn













b

₂

π

₂₂

− b

₁

(1 − π

₁₁

) b

₁

π

₁₁

− b

₂

(1 − π

₂₂

)





 dΩ







b

₂

π

₂₂

w

₁

− b

₂

(1 − π

₂₂

)w

₂

− b

₁

(1 − π

₁₁

)w

₁

+ b

₁

π

₁₁

w

₂











 ,

= − sgn













b

₂

π

₂₂

− b

₁

(1 − π

₁₁

) b

₁

π

₁₁

− b

₂

(1 − π

₂₂

)













b

₁

dπ

₁₁

− b

₁

dπ

₁₁

− b

₂

dπ

₂₂

b

₂

dπ

₂₂













b

₂

π

₂₂

w

₁

− b

₂

(1 − π

₂₂

)w

₂

− b

₁

(1 − π

₁₁

)w

₁

+ b

₁

π

₁₁

w

₂











 .

If we consider the case δ(1) ≃ δ(2), that is to say war episodes are not too destructive. We get