## Abstract

This paper develops a two-level model of internal and external conflict in which the paradox of power holds for internal conflict, but not for external conflict. In the model, internal conflict is imbedded in a situation of external conflict. Agents in a group fight over the distribution of resources within the group, but they cooperate to fight against other groups. Agents with low economic productivity have an advantage in the internal conflict game because they face a lower opportunity cost for investments in weapons. However, it is easier for more productive groups to mobilize resources for external conflict, and as a result they have an advantage over less productive groups. The model helps to explain why economically unproductive individuals may enjoy high living standards relative to more productive ones, but more developed groups usually defeat and conquer less developed ones. An extension of the model shows that groups with more unequal distribution of productivity might have an advantage in external conflict. The model can also be extended to study the effects of trade on the intensity of the paradox of power and income distribution within and across groups.

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## Notes

- 1.
Vasquez and Henehan (2001) document that territorial disagreements are behind 54.6% of all the wars occurred between 1816 and 1997. Using a different methodology, Sample (2002) arrives to similar conclusions. Furthermore, a common feature across the long list of territorial conflicts in human history is that, with few exceptions, Tacitus’ maxim “The gods are on the side of the stronger” prevails (see, Vasquez and Valeriano 2010).

- 2.
The existence of a government capable of financing external conflict through taxation is also important as it prevents free riding.

- 3.
The violence trap refers to a lack of incentives to produce and/or invest in environments in which coercion can be used to expropriate production.

- 4.
See Putnam (1988) for an early contribution to the analysis of conflict within and among groups.

- 5.
Durham et al. (1998) conducted the first laboratory experiment on conflict (see also Abbink 2012). Using Hirshleifer’s (1989; 1991) model of conflict, they find strong evidence supporting theoretical predictions. In particular, when the effectiveness of conflict is high, the POP always holds, while the opposite happens when the effectiveness of conflict is low. Lacomba et al. (2017) arrive at the same conclusions employing a model of conflict that allows for extreme degrees of effectiveness of conflict in their model, the winner of the conflict is the player investing more in guns.

- 6.
In the rest of the paper, aggregate quantities refer to the specification with a continuum of agents (groups). Thus, we write \(X^{j}=\int _{0}^{N^{j}}x_{h}^{j}\hbox {d}h\) and \(\int _{0}^{J}\left( G^{k}\right) ^{m}\hbox {d}k\). For the case of a finite number of agents (groups) integrals must be replaced by sums. For example, \(X^{j}=\sum \nolimits _{h=1}^{N^{j}}x_{h}^{j}\) and \(\sum \nolimits _{k=1}^{J}\left( G^{k}\right) ^{m}\).

- 7.
- 8.
For example, if the government’s goal is to maximize the aggregate output of the final good, then \(W^{j}=\int _{0}^{N^{j}}v_{h}^{j}\hbox {d}h\).

- 9.
When the number of agents is finite, this conflict externality is harder to justify. Nevertheless, in Appendix, we fully study the case of a finite number of agents (no conflict externality) and find analogous results.

- 10.
Intuitively, all agents in a group agree on how much the group should invest in guns for external conflict. This does not necessarily mean that they do not have a conflict of interest as each agent prefers that others pay for those guns. The government, however, solves this free riding problem using compulsory taxation.

- 11.
See the discussion immediately after Proposition 1. The same ideas apply to groups when there is a continuum of groups. In Appendix A, we also consider a specification with a finite number of groups in which groups do not take the aggregate amount of guns as given. The main results do not change.

- 12.
Note that neither the weak nor the strong form of the paradox of power holds among groups. The strong form of the paradox of power would require that \(V^{nc,j}>V^{nc,k}\) implies \(V^{c,k}>V^{c,j}\) (poor groups under no conflict become rich groups under conflict), which never holds in our model. The weak form of the paradox of power would require that \(V^{nc,j}/V^{nc,k}>1\) implies \(V^{nc,j} /V^{nc,k}>V^{c,j}/V^{c,k}>1\) (poor groups under no conflict reduce their gap with rich groups when conflict is introduced), which does not hold in our model, either.

- 13.
Interestingly, as shown in Appendix A, for groups with a finite number of members, group size provides an advantage for external conflict only if the effectiveness of internal conflict is not too high.

- 14.
Formally, the aggregate production function of the final good has positive and diminishing marginal products and exhibits constant returns to scale. It is also assumed twice continuously differentiable. For a discussion of the links between the characteristics of the production function and the equilibria in contest games see, for example, Cornes and Hartley (2005).

- 15.
When the production function is Cobb-Douglass (5) becomes

$$\begin{aligned} \alpha \left( \frac{\varGamma ^{j}Y}{X^{j}-G^{j}}\right) ^{1-\alpha }=\left( 1-\alpha \right) \left( \frac{X^{j}-G^{j}}{\varGamma ^{j}Y}\right) ^{\alpha }\frac{m\varGamma ^{j}}{G^{j}} \end{aligned}$$It is simple to prove that the left-hand side of this expression is increasing in \(G^{j}\), while the right-hand side is decreasing in \(G^{j}\). Finally, employing the Implicit Function Theorem, we obtain \(\frac{\hbox {d}G^{j}}{\hbox {d}X^{j}}>0\).

- 16.
Note that the individual equilibrium contribution to external conflict is increasing in the effectiveness of external conflict (\(\frac{\partial g_{i}^{j}}{\partial m}=\frac{\left( 1-\alpha \right) \alpha }{\gamma _{i}^{c,j}\left[ \left( 1-\alpha \right) m+\alpha N^{j}\right] ^{2}}>0\)). The reason is that a higher value of

*m*increases the marginal benefit of external conflict. This is closely related to what Münster (2007) calls “the group cohesion effect.” One difference with Münster’s group cohesion effect is that in our specification an increase in*m*does not induce a reduction in internal conflict. The reason is that Münster (2007) assumes that guns for internal and external conflict are produced with the same technology. Similarly, note that a decrease in the effectiveness of internal conflict has an ambiguous effect on individual contributions to external conflict (\(\frac{\partial g_{i}^{j}}{\partial m^{j}}=-N^{j}\int _{\beta _{L}^{j}}^{\beta _{H}^{j}}\left( \beta _{i}^{j}/\beta _{h}^{j}\right) ^{m^{j}}\ln \left( \beta _{i}^{j}/\beta _{h}^{j}\right) \hbox {d}F^{j}\left( \beta _{h}^{j}\right) \)). Thus, in our specification Münster’s (2007) “reversed cohesion effect” might or might not hold. - 17.
Formally, if \(\alpha \ge \left( 1-\alpha \right) m\), then \(g\left( n\right) >1\) for all \(n>1\), while if \(\alpha <\left( 1-\alpha \right) m\), then, there exists \(n^{*}>1\) such that \(g\left( n\right) <1\) for \(n<n^{*}\), \(g\left( n^{*}\right) =1\) and \(g\left( n\right) >1\) for \(n>n^{*}\).

- 18.
The MATLAB code Paradox_of_Power_01.m generates Fig. 1.

- 19.
When

*X*is uniformly distributed, the Gini coefficient is given by \(Gini=1-2\left[ \left( B\right) ^{1+\alpha +m\left( 1-\alpha \right) }-1\right] ^{-1}\left\{ \frac{\left( B\right) ^{2+\alpha +m\left( 1-\alpha \right) }-1}{\left[ 2+\alpha +m\left( 1-\alpha \right) \right] \left( B-1\right) }-1\right\} \). - 20.
In general, when \(v^{nc,j}=\left( \frac{\alpha N^{j}\bar{\beta }^{j}r^{j}}{\alpha +N^{j}m^{j}}\right) ^{\alpha }\left( N^{j}\right) ^{-1}=v^{nc}\) for all

*j*, the Lorenz curve for output per capita is given by \(l\left( p\right) =\left[ \int _{X^{L}}^{X^{H} }\left( X\right) ^{m\left( 1-\alpha \right) }f\left( X\right) \hbox {d}X\right] ^{-1}\int _{X^{L}}^{X\left( p\right) }\left( X\right) ^{m\left( 1-\alpha \right) }f\left( X\right) \hbox {d}X\). - 21.
The MATLAB code Paradox_of_Power_02.m generates Fig. 2.

- 22.
This is the reason we need \(\beta _{L}^{j}>\frac{m^{j}N^{j}\bar{\beta }^{j}}{\alpha +N^{j}m^{j}}\) in Proposition 1.

- 23.
This is not entirely accurate because investments in guns for internal conflict are also employed to determine the distribution of the spoils of external conflict.

- 24.
In Appendix A we also prove a version of Proposition 5 for two groups, each integrated by two agents.

- 25.
For \(j\in \left[ 0,\frac{J}{2}\right] \) we have \(s_{i}^{j}=\frac{Nr\bar{\beta }}{\left( \alpha +N\right) \beta _{1}}\) if \(i\in \left[ 0,\frac{N}{2}\right] \) and \(s_{i}^{j}=\frac{Nr\bar{\beta } }{\left( \alpha +N\right) \beta _{2}}\) if \(i\in \left( \frac{N}{2},N\right] \) and, hence, \(X^{j}=X^{\mathrm{int}}=\frac{\alpha Nr\bar{\beta }}{\alpha +N}\), while for \(j\in \left( \frac{J}{2},J\right] \) we have \(s_{i}^{j}=r\) if \(i\in \left[ 0,\frac{N}{2}\right] \) and \(s_{i}^{j}=\frac{Nr}{2\alpha +Nm}\) if \(i\in \left( \frac{N}{2},N\right] \) and, hence, \(X^{j}=X^{\mathrm{cor}}=\frac{\alpha Nr\left( \beta _{2}+\epsilon \right) }{2\alpha +N}\). Moreover, note that \(X^{\mathrm{cor}} >X^{\mathrm{int}}\).

- 26.
Along this line, Carvalho (2017) argues that more unequal societies tend to exhibit greater cultural fragmentation which, in turn, could boost internal conflict.

- 27.
Morrisson and Snyder (2000) point out that from 1449 until the beginning of the French Revolution, the tax on land (the taille) and on income in general (the taille tariff) expanded to cover most of the provinces. Even more important was the capitation tax, whose implementation in 1695 had, except for the nobility, a universal character.

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We would like to thank Sebastian Galiani, Jorge Streb, the editor and three anonymous reviewers for their insightful comments. We would also like to thank the Ostrom Workshop for its financial support.

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Lopez Cruz, I., Torrens, G. The paradox of power revisited: internal and external conflict.
*Econ Theory* **68, **421–460 (2019). https://doi.org/10.1007/s00199-018-1130-z

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### Keywords

- Paradox of power
- Internal and external conflict

### JEL Classification

- D74
- F50