Nation-building, nationalism, and $\hbox {wars}^*$

Abstract

This paper explores how wars make nations, above and beyond their need to raise the fiscal capacity to finance warfare. As army size increases, states change the conduct of war, switching from mercenaries to mass conscript armies. In order for the population to accept fighting and enduring wars, the government elites provide public goods, reduce rent-extraction, and adopt policies to build a nation – i.e., homogenize the culture of the population. Governments can instill “positive” national sentiment, in the sense of emphasizing the benefit of the nation, but they also can instill “negative” sentiment, in terms of aggressive propaganda against the opponent. We analyze these two types of nation-building and study their implications.

Appendix

1.1 A.1. Proofs

Proof of Proposition 1:

The elite chooses $\lambda _A =0$ because the elite does not gain from costly homogenization. Plugging $\lambda _A =0$ into (4), the government budget constraint becomes $\pi _{A}t_{A}q=g_{A}.$ This allows us to write the elite’s problem as

$$\begin{aligned} \displaystyle \max _{\pi _{A}} \theta \pi _{A}t_{A}q+y_{A}+(1-\pi _{A}) \frac{t_{A}q}{s_{A}} \end{aligned}$$

(A.1)

This expression is linear in $\pi _{A}$ and is increasing when $\theta > \frac{1}{s_{A}}$. Then, public good provision is maximal when $1-s_{A}\theta \le 0$ and zero otherwise. $\square$

Proof of Lemma 1:

We proceed by steps.

Step 1. We show that effort is increasing in $NB_{A}$.

Optimal effort solves the following problem:

$$\begin{aligned} \max _{e_{A}\ge 0}\frac{1}{q}\left( \int _{0}^{q}{\small U}_{i,A}^{-}{\small di}+P_{A}(e_{A},e_{B})\int _{0}^{q}({\small U}_{i,A}^{+}{\small -U} _{i,A}^{-})di\right) -e_{A} \end{aligned}$$

(A.2)

Using (7) and (13) we obtain

$$\begin{aligned} \max _{e_{A}}\left( \int _{0}^{q}\frac{{\small U}_{i,A}^{-}}{q}{\small di}+ \frac{{\small qe}_{A}}{{\small qe}_{A}{\small +(1-q)e}_{B}}{\small NB} _{A}\right) -e_{A} \end{aligned}$$

(A.3)

If the solution is interior, the first order condition is:

$$\begin{aligned} {\small NB}_{A}\frac{{\small q[qe_{A}+(1-q)e_{B}]-q}^{2}{\small e}_{A}}{[ {\small qe}_{A}{\small +(1-q)e}_{B}]^{2}}=1 \end{aligned}$$

(A.4)

After taking the square root

$$\begin{aligned}{}[{\small q(1-q)e_{B}NB_{A}]}^{1/2}=[{\small qe}_{A}{\small +(1-q)e} _{B}] \end{aligned}$$

(A.5)

This leads to the optimal effort in country A:

$$\begin{aligned} e_{A}^{*}=max\bigg \{\frac{[{\small q(1-q)e_{B}NB_{A}]}^{1/2}}{{\small q} }-\frac{{\small (1-q)e}_{B}}{{\small q}},\ {\small 0}\bigg \} \end{aligned}$$

(A.6)

From (A.6) it is immediate that optimal effort is increasing in ${\small NB_{A}.}$ Note that for an interior solution one needs that

$$\begin{aligned} {\small e}_{B}<\frac{{\small q}}{{\small (1-q)}}{\small NB_{A}}. \end{aligned}$$

(A.7)

Step 2. We compute $NB_{A}$

First, from (8) we have:

$$\begin{aligned}&\frac{1}{q}\int _{0}^{q}U_{i,A}^{+}di \\&\quad =-\frac{1}{q}\theta g_{A}a(1-\lambda _A )\left[ \int _{0}^{C_{A}}(C_{A}-i)di+\int _{C_{A}}^{q}(i-C_{A})di\right] \\&\qquad +\theta g_{A}+y_{A}-t_{A}+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \\&\quad =-\frac{1}{q}\theta g_{A}a(1-\lambda _A )\left( C_{A}^{2}-\frac{C_{A}^{2}}{2}+\frac{ q^{2}}{2}-C_{A}q-\frac{C_{A}^{2}}{2}+C_{A}^{2}\right) \\&\qquad +\theta g_{A}+y_{A}-t_{A}+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \\&\quad =-\theta g_{A}a(1-\lambda _A )\left( \frac{C_{A}^{2}}{q}+\frac{q}{2}-C_{A}\right) \\&\qquad +\theta g_{A}+y_{A}-t_{A}+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \end{aligned}$$

Similarly, from (9)

$$\begin{aligned} \frac{1}{q}\int _{0}^{q}U_{i,A}^{-}di= & {} -\frac{1}{q}\theta g_{B}a\int _{0}^{q}\left[ (C_{B}-\lambda _A C_{A})-(1-\lambda _A )i\right] di+\frac{1}{q}\left[ \theta g_{B}-t_{A}+y_{A}\right] q \\= & {} -\frac{1}{q}\theta g_{B}a\left[ (C_{B}-\lambda _A C_{A})q-(1-\lambda _A )\frac{q^{2}}{ 2}\right] +\theta g_{B}-t_{A}+y_{A} \\= & {} -\theta g_{B}a\left[ C_{B}-\lambda _A C_{A}-(1-\lambda _A )\frac{q}{2}\right] +\theta g_{B}-t_{A}+y_{A} \end{aligned}$$

Then

$$\begin{aligned} NB_{A}= & {} \frac{1}{q}\int _{0}^{q}(U_{i,A}^{+}-U_{i,A}^{-})di \nonumber \\= & {} -\theta g_{A}a(1-\lambda _A )\left( \frac{C_{A}^{2}}{q}+\frac{q}{2}-C_{A}\right) +\theta g_{A}+y_{A}-t_{A}+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \nonumber \\&+\theta g_{B}a\left[ C_{B}-\lambda _A C_{A}-(1-\lambda _A )\frac{q}{2}\right] -\theta g_{B}+t_{A}-y_{A} \nonumber \\= & {} \theta [g_{A}-g_{B}-g_{A}a(1-\lambda _A )\left( \frac{C_{A}^{2}}{q}+\frac{q }{2}-C_{A}\right) ] \nonumber \\&+\theta g_{B}a\left[ C_{B}-\lambda C_{A}-(1-\lambda _A )\frac{q}{2}\right] +\gamma _{A} \frac{t_{B}(1-q)}{\chi q} \end{aligned}$$

(A.8)

The derivatives in Lemma 1 can be computed from the above expression. Throughout we will focus our analysis on parameters for which there exist values of $g_A$ and $\gamma _A,$ where $g_{A}\ge 0$, $\gamma _{A}\ge 0$, $g_{A}\le t_{A}q$, and $\gamma _{A}\le 1$, and such that (A.7) holds. In words: there exists some feasible policy $(g_A, \gamma _A)$ such that the elite can motivate positive war effort on the part of citizens. $\square$

Proof of Proposition 2:

Assume $\lambda _A = 0.$ Define

$$\begin{aligned} EU_{e}=NB_{e,A}\left( \frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}} \right) +U_{e,A}^{-}-e_{A} \end{aligned}$$

(A.9)

The elite chooses ${\small g}_{A}\in [0,{\small t}_{A}{\small q}]$ and ${\small \gamma }_{A}\in [0,1]$ to maximize $EU_{e}$. We denote by $\gamma _{A}^{\star }$ and $g_{A}^{\star }$ the optimal solutions. Using (10) and (11) we compute the net benefit of winning for the elite

$$\begin{aligned} NB_{e,A}=\theta g_{A}+\left( 1-\frac{g_{A}}{t_{A}q}\right) \frac{t_{A}q}{s_{A}}+\frac{ \left( 1-\gamma _{A}\right) t_{B}(1-q)}{s_{A}}-\theta g_{B}(1-a(C_{B}-C_{A})) \end{aligned}$$

(A.10)

Step 1.  We show that it is not optimal to set $\gamma _{A}^{\star }=g_{A}^{\star }=0.$

From above, we restrict our analysis to parameters for which there exists a value of $g_A$ and $\gamma _A,$ where $g_{A}\ge 0$, $\gamma _{A}\ge 0$, $g_{A}\le t_{A}q$, and $\gamma _{A}\le 1$, and such that (A.7) holds. We also assume $NB_{e,A}>NB_{A}$. Effort is strictly positive only if $g_A>0$ or $\gamma _A>0$ or both. It remains to observe that a policy that induces positive effort is strictly preferred by the elite to a policy $\gamma _{A}=g_{A}=0.$ If a policy $(g_A,\gamma _A)$ results in citizens choosing $e_A>0$ then, from (A.3), it must be that $\frac{{\small qe}_{A}}{{\small qe}_{A}{\small +(1-q)e}_{B}}{\small NB}_{A} -e_A>0,$ but since $NB_{e,A}>NB_A$ we know from (A.9) that the elite must strictly prefer this policy to one that induces zero effort.

Step 2. We prove that it cannot be that the solution is interior for both public good and transfers. That is, it cannot be $g_{A}^{\star }\in (0,t_{A}q)$ and $\gamma _{A}^{\star }\in (0,1).$

We show that if $\chi <\frac{1-\theta s_{A}}{ q\theta (1-a\Delta )},$ then either $\gamma _{A}^{\star }\in (0,1)$ and $g_{A}^{\star }=0,$ or $g_{A}^{\star }>0$ and $\gamma _{A}^{\star }=1.$ If $\chi \ge \frac{1-\theta s_{A}}{q\theta (1-a\Delta )},$ then either $g_{A}^{\star }\in (0,t_{A}q)$ and $\gamma _{A}^{\star }=0, or$ $\gamma _{A}^{\star }>0$ and $g_{A}^{\star }=t_{A}q.$

The Lagrangian of the problem is

$$\begin{aligned} L(g_{A},\gamma _{A};\psi ,\omega )= & {} \left( U_{e,A}^{+}-U_{e,A}^{-}\right) \left( \frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}}\right) +U_{e,A}^{-}-e_{A} \nonumber \\&+\psi g_{A}+\omega \gamma _{A}+{\widehat{\psi }}(t_{A}q-g_{A})+\widehat{ \omega }(1-\gamma _{A}) \end{aligned}$$

(A.11)

where $\psi$, $\omega$, ${\widehat{\psi }}$, and ${\widehat{\omega }}$ are the multipliers of the constraints $g_{A}\ge 0$, $\gamma _{A}\ge 0$, $g_{A}\le t_{A}q$, and $\gamma _{A}\le 1.$

Taking the first-order conditions with respect to $\gamma _{A}$ and $g_{A}:$

$$\begin{aligned} \frac{\partial L(g_{A}, \gamma _{A}; \psi , \omega )}{\partial \gamma _{A}}= & {} \frac{\partial NB_{e,A}}{\partial \gamma _{A}} P(e_{A},e_{B}) + NB_{e,A} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}} \frac{\partial e_{A}}{\partial NB_{A}} \frac{\partial NB_{A}}{\partial \gamma _{A}} \nonumber \\&- \frac{\partial e_{A}}{\partial NB_{A}} \frac{\partial NB_{A}}{\partial \gamma _{A}} + \omega - {\hat{\omega }} = 0 \end{aligned}$$

(A.12)

$$\begin{aligned} \frac{\partial L(g_{A}, \gamma _{A}; \psi , \omega )}{\partial g_{A}}= & {} \frac{\partial NB_{e,A}}{\partial g_{A}} P(e_{A},e_{B}) + NB_{e,A} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}} \frac{\partial e_{A}}{\partial NB_{A}} \frac{\partial NB_{A}}{\partial g_{A}} \nonumber \\&- \frac{\partial e_{A}}{\partial NB_{A}} \frac{\partial NB_{A}}{\partial g_{A}} + \psi - {\hat{\psi }} = 0 \end{aligned}$$

(A.13)

Using the interior condition on effort $e_{A},$

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}NB_{A}=1, \end{aligned}$$

(A.14)

rearranging terms, we can write:

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{\frac{t_{B}(1-q) }{s_{A}}}{\frac{t_{B}(1-q)}{\chi q}}-\omega ^{\prime }+{\widehat{\omega }} ^{\prime } \end{aligned}$$

(A.15)

where $\omega ^{\prime } = \frac{\omega }{P(e_{A},e_{B})}$ and ${\widehat{\omega }} ^{\prime } = \frac{{\widehat{\omega }}}{P(e_{A},e_{B})},$ and

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{(\frac{1}{s_{A}} -\theta )}{\theta (1-a\Delta )}-\psi ^{\prime }+{\widehat{\psi }}^{\prime } \end{aligned}$$

(A.16)

where $\psi ^{\prime } = \frac{\psi }{P(e_{A},e_{B})}$ and ${\widehat{\psi }} ^{\prime } = \frac{{\widehat{\psi }}}{P(e_{A},e_{B})}.$

Suppose $g_{A}^{\star }\in (0,t_{A}q).$ Then, $\psi ^{\prime }={\widehat{\psi }}^{\prime }=0$ and we have

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{(\frac{1}{s_{A}} -\theta )}{\theta (1-a\Delta )} \end{aligned}$$

(A.17)

If

$$\begin{aligned} \frac{\frac{t_{B}(1-q)}{s_{A}}}{\frac{t_{B}(1-q)}{\chi q}} > \frac{(\frac{1 }{s_{A}}-\theta )}{\theta (1-a\Delta )} \end{aligned}$$

(A.18)

(equivalently $\chi >{\overline{\chi }}$), then from (A.15) it must be that $\omega '>0$ and so $\gamma ^*_A = 0.$ If instead $\chi < {\overline{\chi }}$, then from (A.15) it must be that ${\hat{\omega }}'>0$ and so $\gamma _{A}^{\star }=1.$ At the non-generic value $\chi = {\overline{\chi }}$, then we can also have an interior solution for $\gamma _A^*.$ Suppose instead $\gamma _{A}^{\star }\in (0,1).$ Following a symmetric argument, we can show that if (A.18) holds then $g_{A}^{\star }=t_{A}q$ and if instead $\chi < {\overline{\chi }}$ then $g_{A}^{\star }=0.$ At the non-generic value $\chi = {\overline{\chi }}$, then we can also have an interior solution for $g_A^*.$ Finally, observe that if neither $g_A^*$ nor $\gamma _A^*$ are interior then from step 1 we have either $(g_{A}^{\star }=t_{A}q, \gamma _A^* = 1),$ or $(g_{A}^{\star }=t_{A}q, \gamma _A^* = 0),$ or $(g_A^* = 0,\gamma _A^* = 1)$. When $(g_{A}^{\star }=t_{A}q, \gamma _A^* = 0),$ then ${\hat{\psi }}'>0,$ $\psi ' =0,$ ${\hat{\omega }}'=0,$ and $\omega ' >0.$ Then it must be that $\chi >{\overline{\chi }}.$ Symmetrically if $(g_A^* = 0,\gamma _A^* = 1)$ then it must be that $\chi < {\overline{\chi }}.$

To avoid unfruitful complications in the analysis from now on we will assume that at $\chi = {\overline{\chi }},$ when the elite is indifferent between investing in $g_A$ or in $\gamma _A,$ the elite invests first in $g_A$ and then invests in $\gamma _A$ only if $g_A$ reaches its upper limit. In the paper we consider only the case where $\gamma _A^*$ and $g_A^*$ do not reach their upper limit.

We next show uniqueness of the equilibria to be used in the proceeding results. We show that the LHS of (A.16) is strictly decreasing in $g_A$ when $\gamma _{A}=0$ and that the LHS of (A.15) is decreasing in $\gamma _A$ when $g_{A}=0.$ If the first order conditions give us a unique critical point, this guarantees that it solves the optimization problem. Assuming an interior solution, we rewrite the first-order conditions with respect to $g_{A}$ and $\gamma _{A}$:

$$\begin{aligned} \frac{q(1-q)e_{B}\left( NB_{e,A}-NB_{A}\right) }{qe_{A}(qe_{A}+(1-q)e_{B})} \theta (1-a\Delta )\frac{\sqrt{(1-q)qe_{B}}}{2q\sqrt{NB_{A}}}= & {} \left( \frac{1}{s_{A} }-\theta \right) \end{aligned}$$

(A.19)

$$\begin{aligned} \frac{q(1-q)e_{B}\left( NB_{e,A}-NB_{A}\right) }{qe_{A}(qe_{A}+(1-q)e_{B})} \frac{t_{B}(1-q)}{\chi q}\frac{\sqrt{(1-q)qe_{B}}}{2q\sqrt{NB_{A}}}= & {} \frac{ t_{B}(1-q)}{s_{A}} \end{aligned}$$

(A.20)

where

$$\begin{aligned} NB_{e,A}-NB_{A}= & {} \theta g_{A}+(1-\frac{g_{A}}{t_{A}q})\frac{t_{A}q}{s_{A}}+ \frac{(1-\gamma _{A})t_{B}(1-q)}{s_{A}} \nonumber \\&-\theta g_{B}(1-a(C_{B}-C_{A})) \nonumber \\&-\theta g_{A}(1-a\Delta )+\theta g_{B}(1-a(C_{B}-\frac{q}{2}))-\gamma _{A} \frac{t_{B}(1-q)}{\chi q}. \end{aligned}$$

(A.21)

It can be shown that the LHS of (A.19) is decreasing in $g_{A}$ because $e_{A}$ and $NB_{A}$ are increasing in $g_{A}$ and $NB_{e,A}-NB_{A}$ is decreasing in $g_{A}$ (given Assumption 1). Similarly, the LHS of (A.19) is decreasing in $\gamma _{A}$ because $e_{A}$ and $NB_{A}$ are increasing in $\gamma _{A}$ and $NB_{e,A}-NB_{A}$ is decreasing in $\gamma _{A}$. $\square$

Proof of Proposition 3:

Expression (18) is the first order condition with respect to $g_{A}$, which can be written as

$$\begin{aligned} \frac{q(1-q)e_{B}\left( NB_{e,A}-NB_{A}\right) }{qe_{A}(qe_{A}+(1-q)e_{B})} \theta (1-a\Delta )\frac{\sqrt{(1-q)qe_{B}}}{2q\sqrt{NB_{A}}}=\left( \frac{1}{s_{A} }-\theta \right) \end{aligned}$$

(A.22)

We can rewrite (A.21) as

$$\begin{aligned} NB_{e,A}-NB_{A}=\theta g_{B}a(C_{B}-C_{A})-a\theta g_{B}(C_{B}-\frac{q}{2} )+\Omega \end{aligned}$$

(A.23)

where $\Omega$ is a term that does not depend on $g_{B}.$ When $C_{A}\le \frac{q}{2}$ we have that $NB_{e,A}-NB_{A}$ increases in $g_{B}.$ By Lemma 1, $NB_{A}$ and $e_{A}$ decrease in $g_{B}$. Then, we have that the LHS of (A.22) increases in $g_{B}.$ Finally, since the LHS of (A.22) decreases in $g_{A},$ this proves Proposition 3. Note that $C_{A}\le \frac{q}{2}$ is a sufficient condition (not a necessary one). $\square$

Proof of Lemma 2:

The Lagrangian of the problem with $\lambda _A$ is

$$\begin{aligned} L(g_{A},\gamma _{A}, \lambda _A;\psi ,\omega )= & {} (U_{e,A}^{+}-U_{e,A}^{-})(\frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}})+U_{e,A}^{-}-e_{A} \nonumber \\&+\psi g_{A}+\omega \gamma _{A}+ \nu \lambda _A + \widehat{\psi }(t_{A}q-g_{A} - h \lambda )+\widehat{ \omega }(1-\gamma _{A}) + {\widehat{\nu }}(1-\lambda _{A}) \end{aligned}$$

(A.24)

where $\psi$, $\omega$, $\nu ,$ ${\widehat{\psi }}$, ${\widehat{\omega }}$ and ${\widehat{\nu }}$ are the multipliers of the constraints $g_{A}\ge 0$, $\gamma _{A}\ge 0$, $\lambda _A \ge 0,$ $g_{A}+ h\lambda _A \le t_{A}q$, $\gamma _{A}\le 1,$ and $\lambda _{A}\le 1.$

Our results present the case where equilibrium policies do not hit their upper constraints. For homogenization this implies $\lambda _A^* <1.$

When $g_A^* = 0$ it is immediate that homogenization is of no value to the elite and so $\lambda _A^* = 0$. When $g_A^*$ is interior, $g_A^* \in (0, t_Aq - h \lambda _A^*),$ then the first order condition with respect to $g_A$ is

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{(\frac{1}{s_{A}} -\theta )}{\theta (1-a(1-\lambda _A )\Delta )} . \end{aligned}$$

(A.25)

Then either $\lambda _A^*=0$ or $\lambda _A^*>0.$ If $\lambda _A^*>0$ then the first order condition with respect to $\lambda$ is

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{\frac{h}{s_{A}}}{ \theta g_{A}a\Delta } . \end{aligned}$$

(A.26)

Since the left hand sides of (A.25) and (A.26) are identical, then $\lambda _A^*$ satisfies

$$\begin{aligned} \frac{\theta (1-(1-\lambda _A^* )a\Delta )}{\frac{1}{s_{A}}-\theta }=\frac{\theta (g_{A}^*a\Delta -g_{B}a(C_{A}-\frac{q}{2}))}{\frac{h}{s_{A}}} \end{aligned}$$

(A.27)

where $C_{A}=\frac{q}{2}.$ It follows that if $\lambda _A^*>0$ then it is an increasing function of $g_{A}^*:$

$$\begin{aligned} \lambda _A^* = \frac{1-\theta s_{A}}{h}g_{A}^*-\frac{ 1-a\Delta }{a\Delta }. \end{aligned}$$

(A.28)

$\square$

Proof of Proposition 4:

We continue to consider the case when policy parameters do not hit their upper constraints. Suppose $g_{A}^{\star }\in (0,t_{A}q - h \lambda ).$ Then (A.25) holds. From Lemma 2, the optimal level of homogenization is either $\lambda _A^* = 0$ or $\lambda _A^* = \frac{1-\theta s_{A}}{h}g_{A}-\frac{ 1-a\Delta }{a\Delta }.$ Following a symmetric argument to Proposition 2, suppose $g_A^* \in (0, t_Aq - h \lambda _A)$ then if

$$\begin{aligned} \frac{\frac{t_{B}(1-q)}{s_{A}}}{\frac{t_{B}(1-q)}{\chi q}}>\frac{(\frac{1}{ s_{A}}-\theta )}{\theta (1-a(1-\lambda _A^{\star })\Delta )} \end{aligned}$$

(A.29)

it must be that $\gamma _A^* = 0.$ If the reverse inequality holds then it must be that $\gamma _A^* = 1$ (a case we do not consider). Suppose $\gamma _A^*\in (0,1).$ Then by the same argument, if the inequality in (A.29) is reversed then $g_A^* = 0$ and it follows that $\lambda _A^* = 0.$ Compared to the threshold when nation-building is not feasible, we note that the right-hand side of (A.29) is weakly lower, thus weakly increasing the set of parameters for which public good is provided. $\square$

Proof of Proposition 5:

First note that $e_A^*$ continues to be given by the expression in (A.6), but the term $NB_A$ in $e_A^*$ becomes

$$\begin{aligned} NB_A = \theta g_A \left[ 1 - a\left( \frac{C_A^2}{q} +\frac{q}{2} + C_A\right) \right] - (1 - \lambda _2)\theta g_B \left[ 1 - a\left( C_B - \frac{q}{2}\right) \right] . \end{aligned}$$

(A.30)

The expected utility of the elite continues to be given by

$$\begin{aligned} EU_{e}=NB_{e,A}\left( \frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}} \right) +U_{e,A}^{-}-e_{A} \end{aligned}$$

(A.31)

where, as with benchmark homogenization,

$$\begin{aligned} NB_{e,A}=\theta g_{A}+\left( 1-\frac{g_{A}+ h \lambda _2 }{t_{A}q}\right) \frac{t_{A}q}{s_{A}}+\frac{ (1-\gamma _{A})t_{B}(1-q)}{s_{A}}-\theta g_{B}(1-a(C_{B}-C_{A})). \end{aligned}$$

(A.32)

It continues to hold that the elite always chooses at least one of $\gamma _A^*, g_A^*, \lambda _2^*$ to be strictly positive. The Lagrangian of the problem with $\lambda _2$ is

$$\begin{aligned}&L(g_{A},\gamma _{A}, \lambda _2;\psi ,\omega ) \nonumber \\&\quad =(U_{e,A}^{+}-U_{e,A}^{-})\left( \frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}}\right) +U_{e,A}^{-}-e_{A} \end{aligned}$$

(A.33)

$$\begin{aligned}&\qquad +\psi g_{A}+\omega \gamma _{A}+ \nu \lambda _2 + \widehat{\psi }(t_{A}q-g_{A} - h \lambda _2)+\widehat{ \omega }(1-\gamma _{A}) + {\widehat{\nu }}(1-\lambda _2) \end{aligned}$$

(A.34)

where $\psi$, $\omega$, $\nu ,$ ${\widehat{\psi }}$, ${\widehat{\omega }}$ and ${\widehat{\nu }}$ are the multipliers of the constraints $g_{A}\ge 0$, $\gamma _{A}\ge 0$, $\lambda _2 \ge 0,$ $g_{A}+ h\lambda _2 \le t_{A}q$, $\gamma _{A}\le 1,$ and $\lambda _2\le 1.$ We continue to consider the case where policy choices do not hit their upper constraints. Then the first order conditions with respect to $\gamma _A,$ $g_A,$ and $\lambda _2$ are respectively

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}= & {} \frac{\frac{t_{B}(1-q) }{s_{A}}}{\frac{t_{B}(1-q)}{\chi q}} - \omega ' \end{aligned}$$

(A.35)

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}= & {} \frac{(\frac{1}{s_{A}} -\theta )}{\theta (1-a\Delta )} - \psi ' \end{aligned}$$

(A.36)

$$\begin{aligned} \frac{\partial P(e_{A},e_{B})}{\partial e_{A}}\frac{NB_{e,A}-NB_{A}}{ P(e_{A},e_{B})}\frac{\partial e_{A}}{\partial NB_{A}}= & {} \frac{\frac{h}{s_{A}}}{ \theta g_{B}(1-a(C_{B}-\frac{q}{2}))} - \nu ' \end{aligned}$$

(A.37)

where $\omega ',$ $\psi ',$ and $\nu '$ are the values of $\omega ,$ $\psi ,$ and $\nu$ scaled by positive constants. Note that $\Delta$ in A.36 is not a function of $\lambda _2.$

We follow the same strategy as previous proofs. Suppose $\gamma _{A}^{\star }\in (0,1).$ When

$$\begin{aligned} \frac{\frac{t_{B}(1-q)}{s_{A}}}{\frac{t_{B}(1-q)}{\chi q}}<\min \left\{ \frac{(\frac{1}{s_{A}}-\theta )}{\theta (1-a\Delta )},\frac{\frac{h}{s_{A}}}{ \theta g_{B}(1-a(C_{B}-\frac{q}{2}))}\right\} \end{aligned}$$

(A.38)

then it must be that $\psi '>0$ and $\nu '>0$ and hence $g_{A}=0$ and $\lambda _{2}=0.$ When the inequality in (A.38) is reversed, the only way the first order conditions can be satisfied is if $\omega ' >0.$ This implies $\gamma _A^* = 0.$

Condition (A.38) is equivalent to $\chi <\min \left\{ {\overline{\chi }},{\widetilde{\chi }}\right\} .$ Thus when $\chi > \min \left\{ {\overline{\chi }},{\widetilde{\chi }}\right\} ,$ then $\gamma _A^* = 0.$ The choice between using $g_{A}$or $\lambda _{2}$ is driven by the inequality

$$\begin{aligned} \frac{(\frac{1}{s_{A}}-\theta )}{\theta (1-a\Delta )} > \frac{\frac{h}{s_{A}}}{ \theta g_{B}(1-a(C_{B}-\frac{q}{2}))}. \end{aligned}$$

(A.39)

If (A.39) holds, since we cannot have both $\psi '>0$ and $\nu '>0$ (otherwise all policy choices would be zero). Then it must be that $\psi '>0$ and so $g_A^* = 0$ and $\lambda _2^* \in (0,1).$ A symmetric argument holds to show that when the inequality in (A.39) is reversed then $g_A^* \in (0, q t_A)$ and $\lambda _2^* =0.$ The inequality in (A.39) gives us the sign of $\varphi .$ When $\chi = \min \left\{ {\overline{\chi }},\widetilde{ \chi }\right\} ,$ then the elite is indifferent between using either $\gamma _A$ or one of the other instruments. For simplicity of statement, we assume they invest in one of the other instruments. When $\varphi =0$ then the elite is similarly indifferent between using $g_A$ or $\lambda _2.$ For simplicity of statement, we assume they invest in $\lambda _2.$ $\square$

Proof of Proposition 6:

The elite choose between two forms of nation-building: negative (denoted $\lambda _{2}$) and positive (denoted $\lambda _A )$. First, note that when $\lambda _A =\lambda _{2}=0$ the net benefit of winning is the same for both types of nation-building. The net benefit in case of negative indoctrination can be written as

$$\begin{aligned} {\widehat{NB}}_{A}=\theta \left( g_{A}-g_{A}a\left( \frac{C_{A}^{2}}{q}+\frac{q}{2} -C_{A}\right) \right) -\theta (1-\lambda _{2})g_{B}\left( 1-a\left( C_{B}-\frac{q}{2}\right) \right) \end{aligned}$$

(A.40)

The derivative of the average net benefit with respect to $\lambda _{2}$ is

$$\begin{aligned} \frac{\partial {\widehat{NB}}_{A}}{\partial \lambda _{2}}=\theta g_{B}\left( 1-a\left( C_{B}-\frac{q}{2}\right) \right) \end{aligned}$$

(A.41)

From (A.8) the derivative of the net benefit with respect to $\lambda _A$ is

$$\begin{aligned} \frac{\partial NB_{A}}{\partial \lambda _A }=\theta g_{A}a\Delta \end{aligned}$$

(A.42)

For the equilibrium value of $g_A^*,$ if the following holds

$$\begin{aligned} \theta g_{A}a\Delta <\theta g_{B}\left( 1-a\left( C_{B}-\frac{q}{2}\right) \right) , \end{aligned}$$

(A.43)

then ${\widehat{NB}}_{A}\ge NB_{A}.$ This implies positive homogenization is not used since the elite value homogenization only through its impact on the net benefit of winning the war. Using the fact that fiscal capacity puts an upper bound on spending, that is $g_{A}<qt_{A}+\lambda h \le qt_A,$ if

$$\begin{aligned} t_{A}<\frac{\theta g_{B}(1-a(C_{B}-\frac{q}{2}))}{q\theta a\Delta } \end{aligned}$$

(A.44)

then homogenization, if used, will be negative. $\square$

Proof of Proposition 7:

To compute war effort, we write down the utility of an ordinary citizen $i\in [0,q]$ of country A in the last period in case of victory and defeat. The two utilities are respectively given by:

$$\begin{aligned} U_{i,A}^{+}= & {} \theta g_{3}-\theta g_{3}a\left| (1-\lambda _1 )i+\lambda _1 C_{A}-C_{A}\right| +y-\tau _3+\gamma \frac{t_{B}(1-q)}{\chi q}. \end{aligned}$$

(A.45)

$$\begin{aligned} U_{i,A}^{-}= & {} \theta g_{B}-\theta g_{B}a[C_{B}-(1-\lambda _{1})i-\lambda _{1}C_{A}]+y-\tau _{3}. \end{aligned}$$

(A.46)

Some comments are in order. First, payoff (A.45) depends on $g_3$, public spending in the post-war period. When the elites commit to provide high $g_3$, soldiers exert more effort. Second, these payoffs are evaluated using the utility at war time, when effort is chosen. This is why $\lambda _{1}$ (not $\lambda _{3}$) enters $U_{i,A}^{-}$ and $U_{i,A}^{+}$. Finally, we assume $\tau _3=t_3$: the income tax is the maximum possible tax that is feasible at time 3. Note, in fact, from Lemma 1 that the income tax does not affect war effort. It is then immediate that the elites will select the highest possible income tax.

The difference between (A.45) and (A.46) is the net-benefit of winning of soldier i. Then, using (13), we can compute the average net benefit of winning in country A is $NB_A$. We now write down the elites’ payoffs in case of victory and defeat:

$$\begin{aligned} U_{e,A}^{+}= & {} \theta g_{3}+y+(1-\pi _{3})\frac{\tau _{3}q}{s_{A}}+(1-\gamma _{A})\frac{t_{B}(1-q)}{s_{A}} . \end{aligned}$$

(A.47)

$$\begin{aligned} U_{e,A}^{-}= & {} \theta g_{B}-\theta g_{B}a(C_{B}-C_{A})+y. \end{aligned}$$

(A.48)

The difference between (A.47) and (A.48) gives $NB_{e,A}$, the net benefit of winning for the elites. We now write down how effort is computed. Given public policies, war effort maximizes the average expected payoff of all citizens:

$$\begin{aligned} \max _{e_{A}}\ \ \frac{1}{q}\left( \int _{0}^{q}{\small U}_{i,A}^{-}{\small di} +P_{A}(e_{A},e_{B})\int _{0}^{q}({\small U}_{i,A}^{+}{\small -U} _{i,A}^{-})di\right) -c(m) e_{A}. \end{aligned}$$

(A.49)

Following similar steps as in the proof of Lemma 1, we obtain:

$$\begin{aligned} e_{A}^{*}=max\bigg \{\frac{[{\small q(1-q)e_{B}NB_{A}]}^{1/2}}{{\small q c(m)} }-\frac{{\small (1-q)e}_{B}}{{\small q}},\ {\small 0}\bigg \} \end{aligned}$$

(A.50)

Lemma 1 holds under this new specification. In addition, war effort is increasing in military equipment, which reduces the cost of effort.

We now study the elite’s problem. Recall that the policies chosen by the elites are: investment in fiscal capacity i, the soldiers’ pay $\gamma$, rents $\pi _1, \pi _3$, public spending $g_1, g_3$ and “positive” nation-building $\lambda _1, \lambda _3$. We abstract from commitment problems and assume that in the first period the elites chooses all policies. The elites’ problem can be simplified by noting that $g_1=0$ and $\lambda _3=0$. The fact that there is no public good provision in the first period follows from two considerations. First, only public goods in the final period affect war effort, not $g_1$.³⁹ Second, abstracting from war effort considerations, the elites prefer rents to public goods (Assumption 1). Finally, it is immediate to see that (as discussed above) in all periods taxes will be at the upper bound, established by fiscal capacity. To see this recall taxes do not discourage war effort (Lemma 1) and benefit the elites.

The elite selects the vector of policies to maximize its inter-temporal utility from period 1 to period 3 (time is not discounted):

$$\begin{aligned}&\max _{ \{i, \gamma ,g_{3}, \pi _1, \pi _3, \lambda _1\}}\ \left[ y+ \frac{\tau _1 q -h\lambda _1-0.5i^2 -ai -g_1}{s_A}\right] +\left[ y- c(m) e_{A}\right] \nonumber \\&\qquad + \left[ (U_{e,A}^{+}-U_{e,A}^{-})\left( \frac{\chi qe_{A}}{\chi qe_{A}+\chi (1-q)e_{B}}\right) +U_{e,A}^{-}\right] . \end{aligned}$$

(A.51)

subject to the government budget constraints (26), (27), and (28). Using (26) the first term in square brackets is the elites’ payoff in the first period: income plus political rents. The second term is the payoff during war. To understand this term, recall that the elites internalize the effort exerted by the average soldier, that in the second period rents are not distributed, and that taxes are entirely used to buy equipment. The final terms are the expected utility in the last period, which can be computed from (A.47) and (A.48), using the winning probability (7).

In what follows, we assume that equilibrium policies ($i, \gamma ,g_{3}, \lambda _1$) do not hit the upper constraint; this implies that rents in periods 1 and 3 are strictly positive. We now take the first-order condition with respect to fiscal capacity investment. Fiscal capacity investment enters the first-period utility directly. Moreover, it raises military equipment $m=t_1+i$, which reduces the cost of effort and increases war effort. Finally, it affects tax revenue in the last period. Recall from the last period’s budget constraint that $\pi _3 t_3q=g_3$. Hence rents in the final period are $t_3 q -g_3$. Since $g_3$ does not depend on fiscal capacity, an increase of fiscal capacity (hence higher $t_3$) increases by q the elites’ rents in case of victory. War effort $e_A$ is a function of c and, indirectly, of i. We have $\frac{\partial e_A}{\partial c} \le 0$ and $\frac{\partial c}{\partial i} \le 0$. Assuming an interior solution, and using the optimal condition for effort, the first-order condition with respect to i is given by

$$\begin{aligned} -\frac{i+a}{s_A} - \frac{ \partial c}{\partial i} e_A + P_A(e_A,e_B) \frac{\partial U_{e,A}^{+}}{\partial i} + \frac{\partial e_A}{\partial c} \frac{ \partial c }{\partial i} \left[ \left( \frac{NB_{e,A}}{NB_A}-1\right) c\right] =0 \end{aligned}$$

(A.52)

Note that $NB_{e,A}>NB_A$ as the elites have more to lose from a defeat. When $\xi =0$ we have $\frac{ \partial c }{\partial i}=0$ and that war effort does not depend on i. Then, given that $\frac{\partial U_{e,A}^{+}}{\partial i} >0$, we have that optimal investment i when $P_A(e_A,e_B)$ is close to zero. This prove the first part of the statement: when $\xi =0$, we need that the probability of victory must be sufficiently large in order to have a strictly positive investment in fiscal capacity. When $\xi >0$, note that as $\tau _1$ gets smaller, the derivative $\frac{\partial c }{\partial i}$ goes to $-\infty$ when evaluated at $m=t_1+i=0$. This implies that for sufficiently low level of $\tau _1$, investment in fiscal capacity will be strictly positive.

Finally, assuming that equilibrium policies do not hit the upper constraint, the elites maximization is identical to the problem with exogenous fiscal capacity. To see this, notice that the first term in the elites’ objective cancels out when taking derivatives with respect to $g_3$ and $\gamma$. This is because military equipment does not affect the relative effectiveness of public good and monetary payoffs in affecting war effort. Exactly as in Sect. 5, monetary payoffs are distributed to the soldiers when army size is smaller than

$$\begin{aligned} \frac{1-\theta s_{A}}{q\theta (1-(1-\lambda _A^*)a\Delta )}, \end{aligned}$$

(A.53)

where $\lambda _A^*$ is the equilibrium level of homogenization. $\square$

1.2 A.2. Binding fiscal-capacity and spoils of war

Assume $\lambda =0$. Suppose that equilibrium policies are not bounded away from their maximal levels –i.e., either $\gamma _{A}^{\star }=1$ or $g_{A}^{\star }=t_{A}q.$ Simulations show that public spending might be provided before the cutoff ${\overline{\chi }}.$ This occurs when $\gamma _{A}^{\star }$ hits the upper constraint and the elite are left only with the less efficient instrument (public good) to further boost effort. In fact, note from Figures 6 and 7 that when $\chi \le {\overline{\chi }}$, spending is strictly positive precisely when $\gamma _{A}^{\star }=1$. Similarly, from Fig. 7 we observe that soldiers’ pay is positive when $\chi >{\overline{\chi }}$. This occurs because the elite is already using public spending, the most efficient instrument, at full capacity. The graphs below show that qualitatively results are similar to those stated in Proposition 2. It bears stressing that the cutoff is the same one derived in Proposition 2.

Fig. 6

Public goods

Fig. 7

Soldiers’ pay

Fig. 8

Quasi linear utility

1.3 A.3. Quasi-linear utility

Assume the following quasi-linear utility function for all $i\in [0,q]$

$$\begin{aligned} U_{i,A}=\ln (g_{A})\theta (1-a\left| i-C_{A}\right| )+c_{i,A} \end{aligned}$$

(A.54)

Under peace, the elite maximizes

$$\begin{aligned} U_{e,A}=\theta \ln (g_{A})+y_{A}+\frac{(1-\pi _{A})t_{A}q}{s_{A}}. \end{aligned}$$

(A.55)

subject to the government’s budget constraint. It is immediate to compute that under peace, if the solution is interior (i.e., fiscal capacity is not too low), optimal spending is

$$\begin{aligned} g_{A}^{\star }=\theta s_{A} \end{aligned}$$

(A.56)

Compared to Proposition 1, there is public good provision under peace as well and public spending increases in $\theta$ and $s_{A}.$ Under war (assume $\lambda =0)$, if the solutions for $g_{A}^{\star }$ and $\gamma _{A}^{\star }$ are both interior, we have

$$\begin{aligned} g_{A}^{\star }=\theta s_{A}+\chi q\theta (1-a\Delta ), \end{aligned}$$

(A.57)

This implies that an increase in army size raises spending, as in the model in the main text, but in a continuous way. We can simulate a path for spending and soldiers’ pay as a function of army size. When army size is small, the solution is interior and public spending increases in $\chi$ according to (A.57). As army size gets sufficiently large, soldiers are not paid anymore and public spending is constant thereafter (Fig. 8).

1.4 A.4. Enemy neutral indoctrination

Assume $C_A = q/2$ and any form of indoctrination has a unitary cost h. We consider a form of indoctrination called “enemy-neutral” which does not affect citizens’ utility in case country B wins the war; it only raises the value of the public good provided in A. The utility if A wins is

$$\begin{aligned} {\widetilde{U}}_{i,A}^{+}=\theta g_{A}\left[ 1-a(1-\lambda _{1})\left| i-C_{A}\right| \right] +y_{A}-t_{A}+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \end{aligned}$$

(A.58)

where $\lambda _{1}\in [0,1].$ In case of defeat, the utility of A’s citizens is unchanged and equal to

$$\begin{aligned} {\widetilde{U}}_{i,A}^{-}=\theta g_{B}\left[ 1-a\left| i-C_{B}\right| \right] +y_{A}-t_{A}. \end{aligned}$$

(A.59)

Language policies might be considered in this type of homogenization. It is reasonable to suppose that making, say, Bretons learn French improves their ability to feel “French” and enjoy the public goods provided in Paris, but should have little or no consequence on the way they would enjoy the German public good in case of a defeat in a Franco-German war. There are two ways of considering the effect of this alternative form of homogenization on war effort. On one hand, relative to the benchmark form of homogenization in the paper, citizens located to the left of $C_{A}$, far from the border with country B, have stronger incentives to fight. On the other hand, there is a negative effect on the desired war effort of citizens located to the right of $C_{A},$ because it is not the case anymore that homogenization worsens the utility of these citizens in defeat. It can be shown that when Assumption 2 holds, the two effects exactly balance out (Lemma 3 below). Choices made by the elite and choice of effort by soldiers are the same under either form of indoctrination. This equivalence result hinges crucially on the assumption that the capital is in the middle. If the capital of country A were close to “zero,” the benchmark form of homogenization would be more effective, because bringing the population closer to the capital of A would also bring most of the citizens further away from B’s capital. Conversely, if the capital were close to the border with country B, enemy-neutral indoctrination would be more effective.

Lemma 3

Equilibrium war effort, elite’s payoffs and public policies under enemy-neutral homogenization coincide with the ones obtained under the “benchmark” form of homogenization.

Proof of Lemma 3:

Under the benchmark utility, the average net benefit of winning in the country is

$$\begin{aligned} NB_{A}= & {} \theta \left( g_{A}-g_{B}-g_{A}a\left( 1-\lambda \right) \left( \frac{C_{A}^{2}}{q}+\frac{q }{2}-C_{A}\right) \right) \nonumber \\&+\theta g_{B}a\left( C_{B}-\lambda C_{A}-(1-\lambda )\frac{q}{2}\right) +\gamma _{A} \frac{t_{B}(1-q)}{\chi q} \end{aligned}$$

(A.60)

Under “enemy neutral” nation-building the average net benefit of winning in the country is

$$\begin{aligned} {\widetilde{NB}}_{A}= & {} \theta \left( g_{A}-g_{B}-g_{A}a(1-\lambda _{1})\left( \frac{ C_{A}^{2}}{q}+\frac{q}{2}-C_{A}\right) \right) \nonumber \\&+\theta g_{B}a(C_{B}-\frac{q}{2})+\gamma _{A}\frac{t_{B}(1-q)}{\chi q} \end{aligned}$$

(A.61)

Both net benefits are identical when $C_{A}=\frac{q}{2}.$ It also follows that if $C_{A}>q/2,$ “enemy neutral” would be preferable for the elite to the “benchmark” one, and vice versa when $C_{A}<q/2.$ When $C_{A}=\frac{q}{2},$ since the two forms of nation-building affect the elite utility only through the probability of winning, and since the elite’s payoffs do not depend on nation-building, we have that economic outcomes under the two forms of nation-building are identical. $\square$

1.5 A.5. Simplified model: free-riding and heterogenous effort

We simplify the main model by assuming a discrete number of soldiers. This will allow us to study free-riding in war effort. We will assume that each soldier in country A chooses his effort by taking others’ effort as given (including the effort of his fellow citizens).

To make the analysis tractable, we reduce preference heterogeneity in the home country. Country A is composed of only two groups: group 1 and group 2 with population size $P_1$ and $P_2$, respectively. The elite of country A is given by one individual. Suppose that the number of soldiers in the two groups are, respectively, $N_1<P_1$ and $N_2<P_2$. Country B has population $P_B$. The number of soldiers in B is $N_B$.

After defining $T_A\equiv t_A P_1$ and $T_B\equiv t_BP_B$ we write contraint (4) as:

$$\begin{aligned} \pi _A T_A =g_A + h\lambda \end{aligned}$$

(A.62)

As in the main model, citizens receive utility from consumption (equal to disposable income), public goods and from the spoils of war. The elite also receives rents from office. In the case of victory soldiers receive a share $\gamma _A$ of the spoils of war, while the elite keeps the remaining part. The victory payoffs of the citizens of groups 1 and 2 and of the elites are:

$$\begin{aligned} U^+_{1,A}= & {} \theta _1 g_A + \frac{\gamma _A T_B}{N_1+N_2} +y_A-t_A \end{aligned}$$

(A.63)

$$\begin{aligned} U^+_{2,A}= & {} \theta _2 g_A + \frac{\gamma _A T_B}{N_1+N_2} +y_A-t_A \end{aligned}$$

(A.64)

$$\begin{aligned} U^+_{e,A}= & {} \theta _e g_A + (1-\pi _A)T_A + (1-\gamma _A) T_B \end{aligned}$$

(A.65)

Similarly to Assumption 1 in the main text, we assume $\theta _e<1$ so that the elite would not provide public goods in peacetime. Because the elite and group 1 live in the same location, $\theta _e=\theta _1$. In addition, assume $\theta _e=\theta _1>\theta _2$, meaning that members of group 2 enjoy less the public goods than members of group 1. For example, assume that group 1 lives in the capital $C_A$ while group 2 lives closer to country B. Note that individuals obtain a share of the spoils of war regardless of effort. This is precisely what drives free-riding.

In case of defeat, the elite of country A loses power and foreign public goods are provided. The payoffs in the case of defeat are given by

$$\begin{aligned} U^-_{e,A}= & {} (1-\theta _e) g_B +y_A \end{aligned}$$

(A.66)

$$\begin{aligned} U^-_{1,A}= & {} \varphi _A (1-\theta _1) g_B +y_A-t_A \end{aligned}$$

(A.67)

$$\begin{aligned} U^-_{2,A}= & {} \varphi _A (1-\theta _2) g_B +y_A-t_A \end{aligned}$$

(A.68)

where $1 \ge \varphi _A \ge 0$, where $\varphi _A \in [0,1]$. We model homogenization as in the main model. More specifically, positive homogenization changes how much group 2 values the national public goods. The new $\theta _2$ after homogenization is $\theta '_2=\theta _1 \lambda +(1-\lambda ) \theta _2$. The utilities of citizens/ soldiers of country B can be written in a symmetric way. When $\varphi _A=0$, citizens in A do not value the foreign public good. This would correspond to negative indoctrination by A’s elite.

We compute the Nash-equilibrium. We look at symmetric equilibria in which all individuals in the same group exert the same effort. We suppose that the cost of effort is linear, which simplifies the analysis. Let $e_1\ge 0$ and $e_2\ge 0$ be the chosen effort by the two groups in country A. Total effort in A is then $E_A=e_1N_1+e_2N_2$. Effort by country B is taken as given and equal to $E_B$ (We will explore an extension with endogenous effort in the foreign country at the end of this section). The probability that country A wins is

$$\begin{aligned} P_A(E_A,E_B)= \frac{N_1e_1+e_2 N_2}{E_B+N_1e_1+e_2 N_2}\end{aligned}$$

(A.69)

We write down the net benefit of winning for the two groups:

$$\begin{aligned} NB_{1,A}\equiv & {} U^+_{1,A}-U^-_{1,A}= \left( \theta _1 (g_A +\varphi _A g_B)-\varphi _A g_B + \frac{\gamma _A T_B}{N_1+N_2}\right) \end{aligned}$$

(A.70)

$$\begin{aligned} NB_{2,A}\equiv & {} U^+_{2,A}-U^-_{2,A}= \left( \theta _1 (g_A +\varphi _A g_B)-\varphi _A g_B + \frac{\gamma _A T_B}{N_1+N_2}\right) \end{aligned}$$

(A.71)

If effort is strictly positive, optimal effort by each soldier solves the first-order conditions:

$$\begin{aligned} \frac{E_B}{(N_1e_1+N_2e_2+E_B)^2} NB_{1,A}= & {} 1 \end{aligned}$$

(A.72)

$$\begin{aligned} \frac{E_B}{(N_1e_1+N_2e_2+E_B)^2} NB_{2,A}= & {} 1 \end{aligned}$$

(A.73)

When homogenization is not total (i.e., $\lambda <1$) we have $NB_{1,A} >NB_{2,A}$. Thus, we cannot have that both first-order equations are satisfied with equality. As a result, for group 2 the solution is at the corner: optimal effort is minimum, $e_2=0$ and we have complete free riding of group 2. If the cost of effort is not linear, we have that free-riding is less extreme. When $\lambda \in [0,1)$ effort $e_1$ is implicitly given by

$$\begin{aligned} \frac{E_B}{(N_1e_1+E_B)^2} NB_{1,A} =1 \end{aligned}$$

(A.74)

Thus, we compute the optimal effort chosen by members of group 1 when there is no (complete) homogenization, denoted $e^{NH}$:

$$\begin{aligned} e^{NH} = \frac{ \sqrt{NB_{1,A} E_B}-E_B}{N_1 }, \end{aligned}$$

(A.75)

which is positive provided that $E_B$ is small enough. When $\lambda =1$ (perfect homogenization and the two groups have the same preferences) total effort is the same in both groups. Thus we obtain that after full homogenization, both groups exert the same effort, denoted $e^{H}$:

$$\begin{aligned} e^{H} =\frac{ \sqrt{NB_{1,A} E_B}-E_B}{N_1+N_2} \end{aligned}$$

(A.76)

Total effort is unchanged in country A for all $\lambda$ If the cost of effort is not linear, increasing $\lambda$ reduces free-riding (in a more continuous way) but also increases total effort. That is, $N_1 e^{NH}=(N_1+N_2)e^{H}=E_A$.

Lemma A1

Total effort in country A, $E_A$, is strictly increasing in $g_A$ and increasing in $\gamma _A$, and decreasing in $g_B$. Total effort $E_A$ does not depend on homogenization $\lambda$. Homogenization affects, however, how total effort is shared. When $\lambda \in [0,1)$ soldiers of group 2 completely free-ride and exert zero effort, while members of group 1 exert a positive effort equal to (A.75). When $\lambda =1$ (perfect homogenization), in a symmetric equilibrium individual effort is the same in both groups and equal to (A.76).

Proof

Note from (A.75) and (A.76) that effort is increasing in $NB_{1,A}$, which is increasing in $g_A$ and $\gamma _A$ and decreasing in $g_B$ (given that $\theta _1<1$ and $\varphi _A\le 1$). The fact that homogenization does not change total effort by A follows from the above discussion and a comparison of (A.75) and (A.76). $\square$

We now discuss how the elite choose policies (public goods and/or monetary payoffs) to motivate the population. We assume that the elite internalize the average war effort of group 1. This assumption can be justified by the fact that both the elite and members of group 1 live in the same location, the capital. The problem of the elite is:

$$\begin{aligned} \displaystyle \max _{g_{A},\gamma _{A}} P_A(E_A, E_B) \bigg [ \theta _e (g_A +g_B)- g_B + (1-\gamma _A) T_B+ (1-\pi _A) T_A)\bigg ]-U^-_{e,A}-e_1 \end{aligned}$$

(A.77)

subject to the budget constraint (A.62). In Proposition A1, we assume that $\lambda =0$ and abstract from homogenization. In Proposition A2, we will treat $\lambda$ as a choice variable and discuss the incentives of the elite to homogenize.

Proposition A1

Let $\lambda =0.$ When army size is small so that $N_1+N_2< {\tilde{\chi }}$ , where

$$\begin{aligned} {\tilde{\chi }}\equiv \frac{1-\theta _1}{\theta _1} \end{aligned}$$

(A.78)

we have $\gamma _{A}^{\star }>0$ and $g_{A}^{\star }=0$ . When instead $N_1+N_2 \ge {\tilde{\chi }}$, we have $g_{A}^{\star }>0$ and $\gamma _{A}^{\star }=0$.

Proof

The proof is virtually identical to the one of Proposition 2. We refer to that proof for more details. Recall that $N_1$ is the number of active soldiers in country A when $\lambda =0$. Taking the first-order conditions with respect to $\gamma _{A}$ and $g_{A}:$

$$\begin{aligned}&\frac{\partial NB_{e,A}}{\partial \gamma _{A}} P(E_{A},E_{B}) + NB_{e,A} \frac{\partial P(E_{A},E_{B})}{\partial E_{A}} \frac{\partial E_{A}}{\partial NB_{1,A}} \frac{\partial NB_{1,A}}{\partial \gamma _{A}} \nonumber \\&\quad - \frac{\partial e_{A}}{\partial NB_{1,A}} \frac{\partial NB_{1,A}}{\partial \gamma _{A}} + \omega - {\hat{\omega }} = 0 \end{aligned}$$

(A.79)

$$\begin{aligned}&\frac{\partial NB_{e,A}}{\partial g_{A}} P(E_{A},E_{B}) + NB_{e,A} \frac{\partial P(E_{A},E_{B})}{\partial E_{A}} \frac{\partial E_{A}}{\partial NB_{1,A}} \frac{\partial NB_{1,A}}{\partial g_{A}} \nonumber \\&\quad - \frac{\partial e_{A}}{\partial NB_{1,A}} \frac{\partial NB_{1,A}}{\partial g_{A}} + \psi - {\hat{\psi }} = 0 \end{aligned}$$

(A.80)

where $\omega , {\hat{\omega }}, \psi$ and ${\hat{\psi }}$ are the Lagrange multipliers (see proof of Proposition 2). Recall the interior condition on individual effort $e_{A}:$

$$\begin{aligned} \frac{\partial P(E_{A},E_{B})}{\partial E_{A}}NB_{A}=1. \end{aligned}$$

(A.81)

Note that the elite internalizes that public policies will change the effort of all active soldiers, while each individual internalizes how his individual effort will affect the winning probability. Recalling $E_A=N_1e_A$, we can write the two equalities above as:

$$\begin{aligned} \frac{\partial P(E_{A},E_{B})}{\partial e_{A}}\frac{N_1 NB_{e,A}-NB_{A}}{ P(E_{A},E_{B})}\frac{\partial e_{A}}{\partial NB_{A}}= & {} \frac{T_{B} }{\frac{T_{B}}{{N_1+N_2}}}-\omega ^{\prime }+{\widehat{\omega }} ^{\prime } \end{aligned}$$

(A.82)

$$\begin{aligned} \frac{\partial P(E_{A},E_{B})}{\partial e_{A}}\frac{N_1NB_{e,A}-NB_{A}}{ P(E_{A},E_{B})}\frac{\partial e_{A}}{\partial NB_{A}}= & {} \frac{1-\theta _1}{\theta _1}-\psi ^{\prime }+{\widehat{\psi }}^{\prime } \end{aligned}$$

(A.83)

Suppose $g_{A}^{\star }\in (0,T_{A}).$ Then, $\psi ^{\prime }={\widehat{\psi }}^{\prime }=0$ and we have

$$\begin{aligned} \frac{\partial P(E_{A},E_{B})}{\partial e_{A}}\frac{N_1 NB_{e,A}-NB_{A}}{ P(E_{A},E_{B})}\frac{\partial e_{A}}{\partial NB_{A}}=\frac{1-\theta _1}{\theta _1 } \end{aligned}$$

(A.84)

If

$$\begin{aligned} \frac{T_{B}}{\frac{T_{B}}{{N_1+N_2}}}> \frac{1-\theta _1}{\theta _1 } \end{aligned}$$

(A.85)

(equivalently $N_1+N_2>{\tilde{\chi }}$), then from (A.82) it must be that $\omega '>0$ and so $\gamma ^*_A = 0.$ If instead $N_1+N_2 < {\tilde{\chi }}$, then from (A.82) it must be that ${\hat{\omega }}'>0$ and so $\gamma _{A}^{\star }=1.$ Similarly, and following the proof of Proposition 2, one can show that when $N_1+N_2<{\tilde{\chi }}$, public spending is zero, unless $\gamma ^*_A = 1.$ $\square$

Proposition A1 is in line with the main findings of Proposition 2 in the main text. When army size is small, the elite has no incentive to provide public goods.

We now treat $\lambda$ as an endogenous parameter and discuss the incentives of the elite to choose $\lambda$. The problem of the elite is

$$\begin{aligned} \displaystyle \max _{\lambda , g_{A},\gamma _{A}} P_A(E_A, E_B) \bigg [ \theta _e (g_A +g_B)- g_B + (1-\gamma _A) T_B+ (1-\pi _A) T_A)\bigg ]-U^-_{e,A}-e_1 \end{aligned}$$

(A.86)

subject to the budget constraint (A.62). From Lemma A1, note that nation-building does not change aggregate effort and the probability of winning. The fact that $\lambda$ affects only the last term of the above expression greatly simplifies the analysis. Recall that the benefit of nation-building is to decrease effort by group 1. This is valuable for the elite because the elite internalizes the effort cost of the group that lives in the capital. Comparing (A.75) and (A.76), note that when $\lambda$ goes to 1, effort by group 1 will decline more when $N_2$ is larger. Intuitively, the larger $N_2$, the stronger the incentives to reduce free-riding of group 2. Further, it is also intuitive that homogenization is more likely to be chosen when its cost is small. Finally, it is immediate from the previous discussion that partial homogenization ($\lambda <1$) is not effective to reduce free-riding. Hence, it is never chosen. We state without proof the following Proposition.

Proposition A2

(nation-building) Full homogenization is chosen only if $N_2$ is large enough and when its cost h is small enough.

Next we show that, by making an additional assumption (namely, $\varphi _B=0$), the result of Proposition A2 does not depend on the assumption that B’s effort is exogenous.

Proposition A3

(endogenous B’s effort) Let $\lambda =0$. Suppose effort by B is endogenous. Under the assumption that citizens of country B do not enjoy A’s public goods ($\varphi _B=0$) we have that the threshold at which public goods are provided is given by ${\tilde{\chi }}$, as defined in Proposition A2.

Proof

When both armies choose effort simultanously, the following first order conditions need to be satisfied:

$$\begin{aligned} \frac{E_B}{(E_A+E_B)^2} NB_{1,A}= & {} 1 \end{aligned}$$

(A.87)

$$\begin{aligned} \frac{E_A}{(E_A+E_B)^2} NB_{1,B}= & {} 1 \end{aligned}$$

(A.88)

where $NB_{1,B}$ is the net benefit of the representative soldier in B. Note that when $\varphi _B=0$, we have that $NB_{1,B}$ is not affected by public goods in A. This assumption simplifies the analysis. The two equations above lead to

$$\begin{aligned} \frac{NB_{1,B}}{E_B} =\frac{NB_{1,A}}{E_A} \end{aligned}$$

(A.89)

Solve for $E_B$ and plugging the solution into (A.87), we obtain

$$\begin{aligned} E_A=\frac{NB_{1,B} (NB_{1,A})^2 }{(NB_{1,B}+NB_{1,A})^2}. \end{aligned}$$

(A.90)

Similarly, we obtain:

$$\begin{aligned} E_B=\frac{NB_{1,A} (NB_{1,B})^2 }{(NB_{1,B}+NB_{1,A})^2}. \end{aligned}$$

(A.91)

Then, the probability of victory of A is

$$\begin{aligned} P_A(E_A,E_B)=\frac{NB_{1,A} }{(NB_{1,B}+NB_{1,A})} \end{aligned}$$

(A.92)

On this, see also Nti (1999). We can put this expression into (A.77). Since the proof of Proposition A2 does not depend on the functional form of $P_A(E_A,E_B)$, we can proceed as we did there and show that Proposition A2 also holds when B’s effort is endogenous. Notice the role of the assumption that $\varphi _B=0$. If $\varphi _B>0$ public spending would affect both $E_A$ and $E_B$, while changing $\gamma _A$ would only affect $E_A$. Therefore, when $\varphi _B>0$ we cannot proceed as in Proposition A2. $\square$

To sum up, when we allow for free-riding and endogenous effort in B, the thrust of our results remains unchanged. The additional insight that we obtain when free-riding is modelled is the following. When the cost of effort is linear, the purpose of homogenization is not to increase total effort, but to decrease free-riding and share the burden of war. If the cost of effort were not linear, homogenization would also increase total effort, giving the elites an additional incentive to homogenize.

1.6 A.6. Incentives to initiate a war

The timeline is as follows. First, public policies $(g_{A},\lambda _{A},\gamma _{A}$) are determined in country A. We maintain the assumption that public policies in country B are exogenously given. Next, a war between A and B occurs with probability $\phi$. We suppose that $\phi$ is an endogenous parameter. For tractability, we assume that the probability that a war occurs is increasing in the number of countries that wish to go to war. Finally, if a war occurs, given $e_B$, soldiers in A choose war effort.

In this section, we will not provide a full-fledged analysis and solve for all endogenous variables. The goal of this section is more limited: to investigate whether the elites of a given country wish to initiate a war. We will obtain two main results. First, we show that there exists a range of parameters for which one (or even two) countries wish to go to war. This result provides a rationale for the assumption made in the body of the paper that a war occurs for sure. Second, we show when public goods are provided, the elites have stronger incentives to initiate a war.

In the absence of transfers, the elites of country $i=A,B$ wish to go to war if and only if the expected value of a war is greater than the expected value of not going to war (Jackson and Morelli, 2011). To make our point, we will consider two extreme cases. First, we will suppose that the elite do not provide public goods: domestic taxes are entirely appropriated as rents (i.e., we are in the ‘ancien-regime” equilibrium). Second, we will suppose that domestic taxes are entirely spent for public goods. In this second case, the only rents captured by the elites are the spoils of war, which they will obtain in case of victory.

Consider the first case (no public goods in both countries). Given public policies, one can compute the probability of winning by both countries. These probabilities are needed to determine the incentives to initiate a war.

The elite of country A initiates a war if

$$\begin{aligned} P_A \left( y_{A}+\frac{t_{A}q}{s_{A}}+(1-\gamma _{A})\frac{t_{B}(1-q)}{s_{A}} \right) + (1-P_A) y_{A} -\rho e_A > y_{A}+\frac{t_{A}q}{s_{A}} \end{aligned}$$

(A.93)

The LHS is the expected value of a war, which is won by A with probability $P_A$. The RHS is the payoff of not going to war and keeping the political rents for sure. The parameter $\rho$ is the extent to which the elites internalize average war effort exerted in the country. In the main body of the paper, we assumed $\rho =1$.

Similarly, the elite of country B initiates a war if

$$\begin{aligned} (1-P_A) \left( y_{B}+\frac{t_{B}(1-q)}{s_{B}}+(1-\gamma _{B})\frac{t_{A}q}{s_{B}} \right) + P_A y_{B} -\rho e_B > y_{B}+\frac{t_{B}(1-q)}{s_{B}} \end{aligned}$$

(A.94)

After some algebra, recalling that $P_A=(qe_A)/E$ where $E\equiv qe_A+(1-q)e_B$, we can write the above expressions as

$$\begin{aligned} \frac{1}{s_A} \left[ (1-\gamma _A)(1-q)t_B -\frac{1-P_A}{P_A} t_Aq\right] > \rho \frac{E}{q} \end{aligned}$$

(A.95)

and

$$\begin{aligned} \frac{1}{s_B} \left[ (1-\gamma _B)qt_A -\frac{P_A}{1-P_A} t_B(1-q)\right] > \rho \frac{E}{1-q} \end{aligned}$$

(A.96)

We now show that it is not possible that both countries wish to go to war. A necessary condition for this would be that the expressions in the bracket parenthesis in (A.95) and (A.96) are both strictly positive. That is,

$$\begin{aligned}&(1-\gamma _A)(1-q)t_B >\frac{1-P_A}{P_A} t_Aq \end{aligned}$$

(A.97)

$$\begin{aligned}&(1-\gamma _B)qt_A -\frac{P_A}{1-P_A} t_B(1-q) \end{aligned}$$

(A.98)

After some algebra, this would require:

$$\begin{aligned} (1-\gamma _A)(1-q)t_B >\frac{ t_B(1-q)}{1-\gamma _B} \end{aligned}$$

(A.99)

which is not possible provided that $\gamma _B < 1$ and $\gamma _A < 1$. Obviously, if $\gamma _B =\gamma _A = 1$, the elites would not want to go to war.

Next, we show that there exists a range of parameters for which at least one country wishes to go to war. It is easy to show the expressions in the bracket parenthesis in (A.95) and (A.96) cannot both be strictly negative. Then, when public goods are not provided and total effort and/or $\rho$ are not too high, at least one country will want to go to war. If transfers cannot credibly made between the two countries, a war occurs for sure.

The second case is when public goods are provided and $\pi _i=1$ (no rents) in both countries $i=A,B$. Then, public spending $g_i$ is equal to $t_i$, for $i=A,B$. In case of victory, the value for the elites of each unit of public spending is $\theta$. In case of defeat, public spending is provided by the opponent, giving a smaller payoff to the elites.

The elite of country A initiates a war if

$$\begin{aligned} P_A \left( y_{A}+\theta t_A+(1-\gamma _{A})\frac{t_{B}(1-q)}{s_{A}} \right) + (1-P_A) \left( y_{A} +\theta t_B (1-(C_B-C_A)) \right) -\rho e_A > y_{A}+ \theta t_A \end{aligned}$$

(A.100)

The elite of country B initiates a war if

$$\begin{aligned} (1-P_A) \left( y_{B}+\theta t_B+(1-\gamma _{B})\frac{t_{A}q}{s_{B}} \right) + P_A \left( y_{B} +\theta t_A (1-(C_B-C_A)) \right) -\rho e_B > y_{B}+ \theta t_B \end{aligned}$$

(A.101)

Write the above inequalities as:

$$\begin{aligned} \frac{1}{s_A} (1-\gamma _A)(1-q)t_B -\frac{1-P_A}{P_A} \left[ t_A\theta - \theta t_B (1-(C_B-C_A)) \right] > \rho \frac{E}{q} \end{aligned}$$

(A.102)

and

$$\begin{aligned} \frac{1}{s_B} (1-\gamma _B)qt_A -\frac{P_A}{1-P_A} \left[ t_B\theta - \theta t_ A(1-(C_B-C_A)) \right] > \rho \frac{E}{1-q} \end{aligned}$$

(A.103)

It is relatively simple to observe that when $s_A$ and $s_B$ are sufficiently small (and either E or $\rho$ are low) both inequalities (A.102) and (A.103) hold, implying that both countries wish to go to war. As a result, wars cannot be avoided. This would also be true if transfers were allowed. Since the incidence of war is assumed to be increasing in the number of countries that wish to initiate war, the likelihood of a conflict will be higher when public goods are provided. The intuition for this result is as follows. By providing concessions to the population in the form of public spending, the elites make it worthwhile for citizens to fight in order to keep their sovereignty. At the same time, since by Assumption 1 the elites value public spending less than monetary payoffs, these concessions make peace less worthwhile for the elites. The perspective of obtaining the spoils of war, which will be shared among the elites, provides strong incentives to start a war for the elites of both countries.