2. Stylized facts

In this section I describe the data sources and three stylized facts: (1) the military employment share of the population rises, peaks, and falls with income; (2) the ratios of military expenditure shares to employment shares are increasing in income; (3) countries’ military employment and expenditure shares are decreasing in their income relative to their opponents’.

Military expenditure and employment data in the post-Napoleonic era are from the National Military Capacity Database v4.0Footnote ⁵ which covers up to 146 countries in an unbalanced panel from 1816 to 2007. Expenditure data from 1914 is nominal US dollars, which I convert to real expenditures using the NIPA GDP deflator. Before 1914, expenditure data is nominal British pounds, which I convert to real expenditures using the Bank of England’s historical GDP deflators and historical USD/GBP exchange rates. The employment data include only troops under command of the national government which are intended for combat with foreign parties, excluding national reserves, civil defense units, and forces of feudal lords not operating with the central government. It also excludes non-military support staff that are contracted by the military.

Historical military employment data before 1816 are from Sorokin (Reference Sorokin1937), which includes estimates of historical army sizes for 12 European countries. This dataset is not perfectly comparable to the National Military Capacity’s dataset, chiefly because the Sorokin dataset only measures army sizes during wartime. Accordingly, I adjust the Sorokin estimates by a constant factor, so that average employment shares are equal for both sources during the common years 1816–1925 in which both sources have data.

Conflict data are from the Correlates of War database (Sarkees and Wayman, Reference Sarkees and Wayman2010) which catalogs the participants and years for all conflicts classified as wars between states from 1816 to 2007. Lastly, income and population data are from The Maddison Project (Reference Project2013).

The military employment share of the population rises then falls with income. Figure 1 plotted this pattern over a millennium for 12 large European countries. But the nonmonotonic relationship with income is more general. To test this relationship I run the following regression:

(1) \begin{equation} \log s_{j,t} = \beta _0 + \beta _1 \log \big(Y_{j,t}\big) + \beta _2 \log \big(Y_{j,t}\big)^2 + \alpha _j + \varepsilon _{j,t} \end{equation}

where $s_{j,t}$ is the military employment share of country $j$ in year $t$ , $Y_{j,t}$ is their real GDP per capita, and $\alpha _j$ is a country fixed effect. Table 1 reports the results in an unbalanced panel from 1816-2007. The regressions are unweighted and Huber–White standard errors are reported.

Table 1. Military shares and income

Notes: Heteroskedasticity-consistent standard errors in parentheses. The regression equations (1) and (2) also include country-specific fixed effects. Observation years range from 1816 to 2007.

The rise and fall of armies is evident in the baseline specification, which is reported in column $(1)$ : the coefficient on $\log \big(Y_{j,t}\big)$ is positive and the coefficient on the quadratic term is negative and statistically significant, confirming the hump-shaped pattern across many countries. Countries choose different military employment shares in wartime versus peacetime, so it is possible that the unconditional trends could be driven by the declining frequency of warfare. Therefore I also run the regressions only including observations for which a country was at war, which is when the theory applies. Column $(2)$ shows these results for the quadratic regressions, and confirms the theory during war years. Indeed, during war years the trend is stronger, with a larger coefficient on the linear term and a more negative coefficient on the quadratic term. Lastly, to see whether recently declining employment shares are robust to omission of the world wars, columns $(3)$ and $(4)$ omit the quadratic term and show that employment shares decline with income since 1950.

Military employment shares rise and fall, but the evidence for the long run trend of military expenditure shares is mixed. I run regression (1) for expenditure shares and the baseline estimate is reported in column $(5)$ of Table 1, which does not estimate a negative coefficient on the quadratic term. Yet, the regression using only war observations (column $(6)$ ) does find a negative coefficient, so the data are not clear on the long run trend. The baseline model described in Section 3 features a military expenditure share that is nonzero in the limit. The model also features conflict in every period, so the “war only” regressions are most comparable.

The ratio of a country’s military expenditure share of GDP to its military employment share of the population increases with income. Figure 2 plotted this pattern for France, but it is evident across countries. To demonstrate, I regress the log ratio $\log (R_{j,t})$ for country $j$ in year $t$ on log GDP per capita $\log \big(Y_{j,t}\big)$ :

(2) \begin{equation} \log R_{j,t} = \beta _0 + \beta _1 \log \big(Y_{j,t}\big) + \alpha _j + \varepsilon _{j,t} \end{equation}

where $\alpha _j$ are country fixed effects. I report estimates of $\beta _1$ in column $(7)$ in Table 1. The baseline regression is statistically significant at the $1\%$ level and implies a $0.14$ elasticity of the share ratio with respect to income. Column $(8)$ reports the regression using only war observations, which is only significant at the $5\%$ level but has a larger estimate than the baseline. These estimates correspond to an elasticity of substitution for military labor and goods between $1.15$ and $1.24$ when taken to the model.

A country’s military employment and expenditure share is decreasing in its income relative to its opponents. This is true unconditionally, but it is the conditional pattern that is relevant to the theory. Conditional on a country’s own per capita income, its military employment and expenditure shares are also decreasing in their relative income. Specifically, the model predicts that the shares are decreasing in the relative aggregate income, while per capita income is what drives the time series patterns. To test this effect, I regress log employment or expenditure shares $\log (s_{j,t})$ on the log of the ratio $\log \left(\tilde{Y}_{j,t}\right)$ of country $j$ ’s GDP relative to its opponentsFootnote ⁶ in year $t$ , and a quadratic function of log GDP per capita $\log \big(Y_{j,t}\big)$ :

(3) \begin{equation} \log R_{j,t} = \beta _0 + \beta _1 \log \left(\tilde{Y}_{j,t}\right) + \beta _2 \log \big(Y_{j,t}\big) + \beta _3 \log \big(Y_{j,t}\big)^2 + \varepsilon _{j,t} \end{equation}

I report the results in Table 2.Footnote ⁷ In all cases, the shares are decreasing in relative income. While their point estimates are similar, the effect is likely larger for expenditures than employment: the log ratio of these shares is increasing in relative income. When controlling for per capita income, the elasticity of the shares is between $-0.10$ and $-0.14$ and is statistically significant. Section 4 describes why this is predicted in wars between asymmetric countries.

Table 2. Military shares and relative income

Notes: Heteroskedasticity-consistent standard errors in parentheses. Results correspond to regression equation (3). Observation years range from 1816 to 2007.

Conditioning on per capita income is crucial. Skaperdas and Syropoulos (Reference Skaperdas and Syropoulos1997) show in a general setting that more productive combatants will retain a lower share of their surplus, because returns to military power are decreasing and countries only capture a fraction of available surplus. Unconditional evidence is not strong; in Table 2, the coefficients on relative income in the unconditional regressions are close to zero. But incentives are not the whole story: productivity growth affects the supply of military inputs. As countries grow, laborers move out of agriculture and become available for military service, and as countries substitute from soldiers to goods, the growth rate of military power catches up to the growth rate of income. Conditioning on per capita income controls for these effects and supports the asymmetric predictions of the theory.

Table 2 also demonstrates that the main patterns documented in Table 1 are robust to controlling for conflict opponents’ income. As income increases, the military employment share rises, then declines. The expenditure share rises with income, and while the estimates do not reveal the predicted concavity, the true relationship must eventually be concave, because the share is bounded above by one. Finally, the log share ratio also rises with income, and while the model predicts this substitution effect, it makes no clear prediction about the concavity or convexity of this relationship.

3. Model

In the baseline model, two countries are endowed with productivity and population and always go to war. They solve a static game, which has a Nash equilibrium in pure strategies.

3.1 Preferences and technology

There are two countries, indexed by $j\in{1,2}$ . Country $j$ has population $N_j$ and productivity $Z_j$ , which are exogenous.Footnote ⁸ Each country’s decisions are made by a government, which maximizes the consumption $C_j$ of its citizens.

Citizens serve one of two roles. They can be soldiers or they can be workers. $N_{X,j}$ denotes the number of soldiers and $N_{Y,j}$ denotes the number of workers, which must add up to the population:

(4) \begin{equation} N_{X,j}+N_{Y,j}=N_j \end{equation}

Workers have productivity $Z_j$ and produce three types of goods: agriculture $A_j$ , guns $G_j$ , and surplus $Y_j$ . $Y_j$ is the “butter” in this framework - the goods that are enjoyed and that countries use guns to contest. The resource constraint for goods is:Footnote ⁹

(5) \begin{equation} Z_jN_{Y,j} = A_j + G_j +Y_j \end{equation}

Agriculture is used to feed the population. Each person consumes $\nu$ units of agriculture, so the agricultural goods constraint is:

(6) \begin{equation} A_j=\nu N_j \end{equation}

As a result, $\nu$ behaves like a subsistence constraint.

Guns and soldiers are used to produce military power, $X_j$ , with a production function:

(7) \begin{equation} X_j=f\big(N_{X,j},G_j\big) \end{equation}

Assume this production function $f(N_X,G)$ is quasi-convex and has constant returns to scale.

The substitutability of guns and soldiers in this production function will be crucial for determining the long-run behavior of the economy.

3.2 War

The two countries go to war, where military power is used to compete over a share $\theta$ of the two countries’ combined surplus of goods $Y_j$ . The contestable share of the joint surplus of the two countries is $\theta (Y_1 + Y_2)$ . The war function $\Gamma (X_1,X_2)$ determines the share of this surplus that accrues to country $1$ . $\Gamma (\cdot,\cdot )$ is increasing in the first argument, decreasing in the second argument, bounded by $\Gamma (\cdot,\cdot )\in [0,1]$ , and is symmetric so that $\Gamma (X_1,X_2)=1-\Gamma (X_2,X_1)$ .

A country uses its surplus for consumption, net of war gains or losses. Then the consumption for country $1$ is determined by:

(8) \begin{equation} C_1=\Gamma (X_1,X_2) \theta (Y_1+Y_2) + (1-\theta )Y_1 \end{equation}

The country faces a trade-off in turning goods into consumption. Spending more on $Y_1$ increases the total surplus, but decreases the resources that can be spent on $X_1$ , reducing the share of total surplus that is retained. When $\theta =1$ , all surplus is potentially contestable, and war resembles the structure in Skaperdas (Reference Skaperdas1992).

3.4 Equilibrium conditions

The country’s decision can be rewritten as an unconstrained maximization problem in two variables - guns and soldiers - here expressed from the perspective of country $1$ :

(9) \begin{align} \max _{N_{X,1},G_{1}} (\Gamma (f(N_{X,1},G_{1}),X_{2})\theta +1-\theta )(Z_{1}(N_{1}-N_{X,1})-\nu N_{1}- G_{1})+\Gamma (f(N_{X,1},G_{1}),X_{2})\theta Y_{2} \end{align}

The optimality conditions of this maximization problem are lengthy, but can be written in an intuitive form by substituting in some definitions. I also omit for readability the arguments of the function $\Gamma$ , its partial derivative with respect to the first argument $\Gamma _1$ , and marginal return to military from increasing soldiers $f_N$ and guns $f_G$ . The equilibrium condition for guns is

(10) \begin{equation} \Gamma _{1}f_{G}\theta (Y_{1}+Y_{2})=\Gamma \theta +1-\theta \end{equation}

On the left hand side is the marginal increase in consumption, from an increase in military power, due to an increase in guns. On the right hand side is the marginal consumption given up by shifting output from surplus to guns. The equilibrium condition for soldiers is

(11) \begin{equation} \Gamma _{1}f_{N}\theta (Y_{1}+Y_{2})=(\Gamma \theta +1-\theta )Z_{1} \end{equation}

On the left hand side is the marginal increase in consumption from increased military power, and on the right hand side is the marginal consumption lost from shifting soldier to workers. This marginal cost has a coefficient of $Z_1$ , the productivity level, which differs from the guns equilibrium condition. This is because at the margin, using output for guns reduces surplus one for one, but employing workers as soldiers reduces surplus by the factor $Z_1$ .

Dividing equation (11) by equation (10) shows that the marginal product of guns in the military power function must decrease with productivity, relative to the marginal product of soldiers:

(12) \begin{equation} \frac{f_{N}}{f_{G}}=Z_{1} \end{equation}

Productivity $Z_1$ is the relative price of soldiers with respect to guns. If guns and soldiers are substitutes, then rising productivity will increase guns more than one for one, relative to soldiers.

3.5 Functional forms

Some functional forms are useful to further characterize the baseline economy. The production function for military power is a CES aggregator of guns and soldiers:

(13) \begin{equation} f(N_X,G)\equiv \left (\alpha N_{X,j}^{\frac{\epsilon -1}{\epsilon }}+(1-\alpha )G_j^{\frac{\epsilon -1}{\epsilon }}\right )^{\frac{\epsilon }{\epsilon -1}} \end{equation}

This functional form is unusual in the conflict literature, where power typically only depends on expenditures. Instead there are two inputs, and the elasticity of substitution $\epsilon$ will be a crucial parameter determining the relationship between income growth and warfare.Footnote ¹⁰

Lastly, a functional form is needed for the war function $\Gamma (X_1,X_2)$ , satisfying the conditions in section 3.2. One such function is that countries receive a share of surplus equal to their share of military power:

(14) \begin{equation} \Gamma (X_1,X_2)=\frac{X_1}{X_1+X_2} \end{equation}

This has the convenient property of homogeneity of degree zero, and is the linear form of the Tullock (Reference Tullock and Tullock1980) rent-seeking function, which is often used in the conflict literature (Garfinkel and Skaperdas, Reference Garfinkel and Skaperdas2007).

3.6 Symmetric equilibrium

I first examine symmetric equilibria, where $Z_1=Z_2$ and $N_1=N_2$ . Section 4 considers the implications of asymmetric equilibria.

With the assumed functional forms, the equilibrium condition (10) becomes

(15) \begin{equation} (1-\alpha )\frac{c}{g}=\frac{\theta +2(1-\theta )}{\theta }\left(\frac{x}{g}\right)^{\frac{\epsilon -1}{\epsilon }} \end{equation}

where lower case letters denote per capita variables, e.g. $c\equiv \frac{C}{N}$ . Substituting in $x=f(n_X,g)$ , equation (15) becomes

(16) \begin{equation} (1-\alpha )\frac{c}{g}=\frac{2-\theta }{\theta }\left (\alpha \left(\frac{n_{X}}{g}\right)^{\frac{\epsilon -1}{\epsilon }}+1-\alpha \right ) \end{equation}

and substituting for the marginal products, equation (12) becomes

(17) \begin{equation} \frac{\alpha }{1-\alpha }\left(\frac{g}{n_X}\right)^{\frac{1}{\epsilon }}=Z \end{equation}

Then given productivity $Z$ , the symmetric equilibrium is characterized by three variables $(n_X,g,c)$ and three equations: (15), (17), and the per capita symmetric budget constraint,

(18) \begin{equation} Z (1-n_X)= \nu +g + c \end{equation}

3.7 Equilibrium in the limit

To understand how productivity growth affects the economy, it is useful to start by considering equilibrium in the limit as $Z$ becomes large. Define shares of total income $s_C\equiv \frac{c}{Z}$ and $s_G\equiv \frac{g}{Z}$ . Shares $s_C$ , $s_G$ , and $n_X$ are all bounded by $0$ and $1$ . Let $(\overline{s_{C}},\overline{s_G},\overline{n_X})$ denote their limits, if they exist.

Express the condition (17) in terms of shares and rearrange to get

(19) \begin{equation} \frac{s_G}{n_X}=Z^{\epsilon -1}\left(\frac{1-\alpha }{\alpha }\right)^\epsilon \end{equation}

If $\epsilon \gt 1$ , then the right hand side of this equation grows as productivity $Z$ grows. On the left hand side, $s_G$ is positive and bounded above by one, so as $Z$ becomes large, it must be that $n_X$ becomes small: $\overline{n_X}=0$ . This is why it central that guns and soldiers are substitutes to imply a declining share of soldiers as income grows. Guns become cheap relative to soldiers as productivity grows. If the two inputs are substitutes in the military function, then this price change implies a quantity shift from soldiers toward guns. Equation (19) maps directly to the regressions estimated in columns (5) and (6) of Table 1. This corresponds to an elasticity of substitution $\epsilon$ between $1.15$ and $1.24$ .

Given that $\overline{n_X}=0$ , condition (16) implies $\overline{s_{C}}=\overline{s_{G}}\frac{2-\theta }{\theta }$ in the limit. And the budget constraint (18) becomes $1 - n_X =\frac{\nu }{Z}+ s_G + s_C$ , so in the limit as $Z\rightarrow \infty$ ,

(20) \begin{equation} 1= \overline{s_G} + \overline{s_C} \end{equation}

This also implies that as productivity rises, the share of total income spent on agriculture, $\frac{\nu N}{ZN}$ goes to zero. Solving for each share yields $\overline{s_{C}}=1-\frac{\theta }{2}$ and $\overline{s_{G}}=\frac{\theta }{2}$ .

Figure 3 plots the symmetric equilibrium for various levels of productivity, to show how the equilibrium evolves as productivity grows from $Z=1$ to high levels. In this example $\nu =.9$ , so initially agriculture is $90\%$ of total income $Z$ . As income grows and a lower share needs to be paid to agriculture, more is spent on consumption, which rises to its’ long run share $\overline{s_{C}}$ , which in this example is $.5$ because $\theta =1$ . Spending on both military inputs rise as a share of income initially as the economy transitions out of agriculture. But eventually the substitution effect starts to dominate, so the soldier share falls to $\overline{n_X}=0$ while the guns share rises to $\overline{s_{G}}$ .

Figure 3. Example symmetric equilibria as income grows.

4. Asymmetric equilibrium

In this section, I relax the assumption of symmetry to explain the third stylized fact: countries’ military employment and expenditure shares are decreasing in their income relative to their opponents’.

Countries’ incentives are quantitatively different when they are richer or poorer than their neighbors. When conditioning on their productivity level, a country’s military employment and expenditure share is decreasing in its income relative to its opponents. The game-theoretic nature of the model generates cross-sectional implications that differ observably from the time series implications. In the time series, the model predicts that the employment share rises and falls with income and the expenditure share rises and asymptotes. Whereas in the cross section when controlling for productivity, the model predicts that both the employment share and expenditure share are decreasing in relative income.

When a country’s income rise in the asymmetric case, then the country’s incentive to spend on war falls. The available surplus rises less than one for one with the country’s income, because it only retain a fraction of the total available surplus, yet the marginal share retained decreases more than one for one, for a given guns expenditure share share. A lower guns share is required to increase the marginal share retained, so guns expenditure shares are decreasing in neighbors’ incomes. The optimal allocation of workers implies that solider are proportional to guns by a factor that depends on the level of productivity. So conditional on the productivity level, employment shares are also decreasing in relative income.

To characterize some properties of asymmetric equilibria, I approximate the model in the long run when incomes are large. In general, the asymmetric equilibrium must be calculated numerically, but I can analytically characterize asymmetric behavior for the approximation. The first implication of the long run approximation is that for large $G$ , $f(N_X,G)\approx \left (1-\alpha \right )^{\frac{\epsilon }{\epsilon -1}}G$ . The second implication is for large $Z$ , $\frac{\nu }{Z}\approx 0$ . Then the equilibrium condition for guns (10) becomes

(21) \begin{equation} \frac{G_{2}}{(G_{1}+G_{2})^{2}}(Z_{1}N_1-G_{1}+Z_{2}N_2-G_{2})=\frac{G_{1}}{G_{1}+G_{2}}+\frac{1-\theta }{\theta } \end{equation}

Rewrite the guns expenditures as shares of total income $G_1=s_{G,1}Z_1N_1$ , and define relative aggregate income of country $1$ $z_1\equiv \frac{Z_1 N_1}{Z_2 N_2}$ , to get an expression for country $1$ ’s best response function $g\big(s_{G,2},z_1\big)$ to country $2$ ’s guns share and relative income:

(22) \begin{equation} s_{G,1}=g\big(s_{G,2},z_1\big)=\frac{s_{G,2}}{z_1}\left(\sqrt{\theta \frac{z_{1}+1}{s_{G,2}}}-1\right) \end{equation}

The asymmetric Nash Equilibrium satisfies two equations: $s_{G,1}=g(s_{G,2},z_2;\,\theta )$ and $s_{G,2}=g\left(s_{G,1},\frac{1}{z_2};\,\theta \right)$ . For $z_2$ near $1$ , this solution is close to $\left(s_{G,1},s_{G,2}\right)=\left(\frac{1}{\theta },\frac{1}{\theta }\right)$ , the symmetric outcome from Section 3.3. But when incomes are asymmetric, the guns shares diverge.

Figure 4. Asymmetric best response functions.

The guns share best response is decreasing in relative income $z_1$ . When country $1$ is richer, country $1$ chooses a lower guns share for any given foreign guns share. When $z_1\gt 1$ (i.e. country 1 is richer), then $s_{G,1}\lt s_{G,2}$ . This case is plotted in Figure 4, for $\theta =.8$ and a 20% income difference. The best response lines cross above the 45 degree line, so the richer country chooses a lower guns share in equilibrium. Figure 5 plots the equilibrium guns share of country $1$ against its relative income $\frac{Z_1N_1}{Z_2N_2}$ for several values of $\theta$ . As country $1$ becomes richer relative to country $2$ , it chooses a lower guns share.Footnote ¹¹

Figure 5. Equilibrium guns expenditure shares and relative incomes.

Military employment shares must be decreasing in relative income too. The military employment share is always related to the guns share by equation (19), so that for the shares are always proportional for a given productivity level. Foreign endowments do not enter this relationship. So if relative populations change, or if the foreign country becomes more productive, the employment share will change one for one with the guns share. If domestic productivity increases, this reduces the guns share, but reduces the employment share more than one for one, because the substitution effect shifts soldiers into production. In both cases, the military employment share is decreasing in relative income.

5. The repeated game

The baseline model in Section 3 considers a static Nash equilibrium, which is a good approximation when wars are infrequent, so that time discounting is significant between periods. In this section, I generalize this setup to a repeated game environment. I show that when the discount factor is sufficiently small, the baseline results are robust: military employment shares rise and fall, and countries substitute from soldiers to guns. Furthermore, as discount factors or productivity growth rise in the repeated game, the long run military expenditure share falls. When discount factors or productivity growth are sufficiently high, countries can play a strategy that eliminates warfare entirely. I begin by considering grim trigger strategies which are easily characterizable, and follow by considering punishment strategies that are subgame perfect.

Each country’s decisions are made by a government, which now maximizes the present discounted utility of its citizens. The period utility function is power utility with parameter $\sigma$ over consumption per capita, so the present discounted utility in period $t=0$ is

(23) \begin{equation} \sum _{t=0}^{\infty }\beta ^t\left(\frac{C_{j,t}}{N_{j,t}}\right)^\sigma \end{equation}

When $\beta =0$ , the country’s objective is the same as in the baseline model.

When $\beta \gt 0$ , countries can support a better equilibrium than the static Nash equilibrium. In general, repeated games admit many possible equilibria. First consider grim trigger strategies: countries play one allocation as long as the other country does the same, but if one country deviates from this allocation, the other country plays the grim trigger allocation forever after. Within this set of strategies, the best sustainable Nash equilibrium is achieved by choosing the minmax punishment: the allocation that gives the other country the least utility, given that the other country is playing its best response. In this game, the minmax allocation is the one that maximizes military power $f(N_X,G)$ . So country $j$ ’s minmax allocation solves the following maximization problem:

\begin{equation*} \max _{N_{X,j},G_j} f\big(N_{X,j},G_j\big) \end{equation*}

\begin{equation*} \text {s.t. }Z_j(1-N_{X,j})=\nu N_{X,j}+G_j \end{equation*}

which shares an optimality condition with the baseline model: $\frac{\alpha }{1-\alpha }\left(\frac{G_j}{N_{X,j}}\right)^{\frac{1}{\epsilon }}=Z_j$ .

An equilibrium is sustainable when playing a grim trigger strategy if countries prefer the equilibrium in perpetuity, over the best one period deviation followed by the best response to the minmax strategy. This implies the following condition, where $C_t$ denote period $t$ consumption in a sustainable equilibrium, $\hat{C}_t$ denotes the best one period deviation, and $C_t^{MM}$ denotes the best response to the minmax strategy:

(24) \begin{equation} \sum _{t=0}^{\infty }\beta ^t\left(\frac{C_{t}}{N_{t}}\right)^\sigma \geq \hat{C}_0+\sum _{t=1}^{\infty }\beta ^t\left(\frac{C_{t}^{MM}}{N_{t}}\right)^\sigma \end{equation}

The best sustainable grim trigger equilibrium is the one where (24) holds with equality.

Figure 6. Soldier shares for various grim trigger equilibria.

Better equilibria can be sustained when countries are more patient, or when wars are less frequent. When $\beta =0$ , the best sustainable equilibrium must also be the best one period response, yielding the static Nash equilibrium. To see how the equilibrium changes when $\beta$ increases, consider condition (24) in the limit when expenditure shares are constant. Let $s_C$ denote the sustainable limiting consumption share, let $\hat{s}_C$ denote the best one period deviation consumption share, and let $s_C^{MM}$ denote the best response consumption share to the minmax. The best sustainable grim trigger equilibrium in the limit satisfies:

(25) \begin{equation} s_C^\sigma \sum _{t=0}^{\infty }\beta ^tZ_t^\sigma = \left(Z_0\hat{s}_{C,0}\right)^\sigma +\left(s_C^{MM}\right)^\sigma \sum _{t=1}^{\infty }\beta ^tZ_t^\sigma \end{equation}

When productivity growth is constant at rate $\mu$ , equation simplifies to:

(26) \begin{equation} s_C^\sigma = \hat{s}_{C,0}^\sigma (1-\beta (1+\mu )^\sigma )+\left(s_C^{MM}\right)^\sigma \beta (1+\mu )^{\sigma } \end{equation}

The best grim trigger equilibrium is a weighted average of the best response and the minmax response. When $\beta (1+\mu )^{\sigma }$ is higher, more weight is placed on the punishment, allowing for a better achievable best response $\hat{s}_{C,0}^\sigma$ and a better sustainable equilibrium.

Faster productivity growth $\mu$ also improves the sustainable equilibrium. With high productivity growth, countries act as if they are more patient, knowing that more consumption will be lost if they are punished by the grim trigger. Either higher patience or faster growth improve the long run outcome by reducing the guns share and increasing the consumption share. Furthermore, they improve the equilibrium at every point in transition. Figure 6 plots the transition path for grim trigger equilibria for several values of $\beta$ . All cases exhibit the rise and fall of armies. When $\beta =0$ , the best equilibrium is static Nash. As countries become more patient, better equilibria are sustainable, and the soldier share falls at every level of income.

Figure 7. Guns shares for various grim trigger equilibria.

Higher discount factors or economic growth rates result in lower long run guns expenditure shares. In the baseline model, the contestable share of surpluses $\theta$ controlled the long run guns share. When dynamics are introduced, this ingredient is unnecessary to match empirical expenditure shares, which are much lower than $\frac{1}{2}$ . The grim trigger equilibria in Figure 7 all have $\theta =1$ . When $\beta =0$ and the economy is in the static Nash equilibrium, the guns share converges to $\frac{1}{2}$ . As $\beta$ increases, the long run guns share decreases.

Sufficiently high patience or economic growth can eliminate war entirely. Grim trigger strategies do not generally produce the best possible equilibrium, nor are they subgame perfect, so I next consider sustainable subgame perfect equilibria with punishment, as in Abreu (Reference Abreu1988). Countries play a strategy where, were either country to deviate from the sustainable allocation, both countries play the worst possible allocation for $k$ periods, then revert to the sustainable allocation.

This equilibrium has two dynamic conditions. Countries must prefer not to deviate from the sustainable allocation:

(27) \begin{equation} \sum _{t=0}^{\infty }\beta ^t\left(\frac{C_{t}}{N_{t}}\right)^\sigma \geq \hat{C}_0+\sum _{t=1}^{k}\beta ^t\left(\frac{C_{t}^{P}}{N_{t}}\right)^\sigma +\sum _{t=k+1}^{\infty }\beta ^t\left(\frac{C_{t}}{N_{t}}\right)^\sigma \end{equation}

where $C_t^P$ denotes the punishment level of consumption in period $t$ . The punishment must also be credible for the equilibrium to be subgame perfect, so that a country prefers to accept its punishment for $k$ periods, rather than deviating:

(28) \begin{equation} \sum _{t=0}^{k-1}\beta ^t\left(\frac{C^P_{t}}{N_{t}}\right)^\sigma +\sum _{t=k}^{\infty }\beta ^t\left(\frac{C_{t}}{N_{t}}\right)^\sigma \geq \hat{C}_0^P+\sum _{t=1}^{k}\beta ^t\left(\frac{C_{t}^{P}}{N_{t}}\right)^\sigma +\sum _{t=k+1}^{\infty }\beta ^t\left(\frac{C_{t}}{N_{t}}\right)^\sigma \end{equation}

where $\hat{C}_t^P$ denotes the one period best response to the punishment level of consumption in period $t$ .

The structure of the game has to be modified to allow for an equilibrium without war. Assume that when both countries choose zero guns $G_t=0$ , then the country spends its entire income on agriculture and consumption, so that $Z_t N_t=C_t$ . Better equilibria are supported by worse punishments, so assume the worst possible punishment: $C_t^P=0$ . When supporting no war, and with productivity growth rate $\mu$ , Condition (27) becomes:

(29) \begin{equation} Z_0\sum _{t=0}^{k}\beta ^t(1+\mu )^{\sigma t}\geq \hat{C}_0 \end{equation}

$k$ periods of no war must be preferable to one period of deviation. Condition (28) becomes:

(30) \begin{equation} Z_0\beta ^k(1+\mu )^{\sigma k}\geq \hat{C}_0^P \end{equation}

which implies that one period of deviation form punishment must not be worth delaying a return to no war for an additional period. If there is some $k$ such that these conditions are satisfied, no war is sustainable.

The conditions to sustain no war are characterizable analytically in the long run symmetric case, where the set of feasible punishment lengths $k$ is increasing in the discount factor $\beta$ and the growth rate $\mu$ :

Proof. The two countries are symmetric, so the best one period deviation to the no war allocation is $\hat{C}_t=2Z_tN_t$ , because one country can choose an arbitrarily tiny military power greater than zero, and capture the entire surplus.

In the long run where $\nu =0$ and $X=(1-\alpha )^{\frac{\epsilon }{\epsilon +1}}G$ , and when $\theta =1$ , country $1$ ’s best response condition for guns (10) becomes

(31) \begin{equation} \frac{G_1}{(G_1+G_2)^{2}}(2Z-G_1-G_2)=\frac{G_2}{G_1+G_2} \end{equation}

so the best one period deviation to the punishment $G=Z$ has guns choice $\hat{G}^P$ that satisfies

\begin{equation*} \frac {\hat {G}^P}{\left(\hat {G}^P+Z\right)^{2}}\left(Z-\hat {G}^P\right)=\frac {Z}{\hat {G}^P+Z} \end{equation*}

Applying the quadratic formula and taking the unique feasible solution, the best one period deviation to the punishment is given by

(32) \begin{equation} \hat{G}^P=Z\left(\sqrt{2}-1\right) \end{equation}

And the associated consumption $\hat{C}^P=\Gamma (\hat{G}^P,Z)(Z-\hat{G}^P)$ is

(33) \begin{equation} \hat{C}^P=Z\left(\sqrt{2}-1\right)^{2} \end{equation}

With these responses $\hat{C}$ and $\hat{C}^P$ determined, the optimality condition (29) becomes

(34) \begin{equation} \sum _{t=0}^{k}\beta ^t(1+\mu )^{\sigma t}\geq 2^\sigma \end{equation}

and the credibility condition (30) becomes

(35) \begin{equation} \beta ^k(1+\mu )^{\sigma k}\geq \left(\sqrt{2}-1\right)^{2\sigma } \end{equation}

Faster productivity growth and higher patience increase the set of feasible strategies that sustain a subgame perfect equilibrium without war. The left-hand sides of both the optimality condition (34) and the credibility condition (35) are increasing in $\beta (1+\mu )^\sigma$ . Figure 8 plots the lower bound of feasible values of $\beta (1+\mu )^\sigma$ for both conditions, for a variety of punishment lengths. When $\beta (1+\mu )^\sigma$ is above both lower bounds, no war is sustainable for the given punishment length $k$ . If $\beta (1+\mu )^\sigma$ is sufficiently small, then these conditions cannot be satisfied, and war will occur.

Figure 8. Parameter lower bounds supporting equilibrium conditions without war.