Players and Actions

There is a continuum 1 of workers, indexed by i ∈ [0, 1], and a continuum 1 of capitalists, indexed by j ∈ [0, 1]. Each worker is endowed with 1 unit of labor. Each capitalist is endowed with $\overline{K}$ units of capital, ${\underline{K}} \in \left( {0,\overline{K}} \right)$ units of which are immobile and must be invested in domestic market, whereas the remaining ${\overline{K}} - {\underline{K}}$ units can be invested in domestic or foreign markets. The game proceeds in two stages. In stage one, each capitalist decides how to divide his mobile capital between domestic and foreign investments. Let ${k_j} \in \left[ {0,\overline{K} - {\underline{K}}} \right]$ be capitalist j’s domestic investment of his mobile capital, and $K = \int {{k_j}{\rm{<italic>d</italic>}}j \in \left[ {0,\overline{K} - {\underline{K}}} \right]}$ be the aggregate domestic mobile capital. In stage two, each worker observes the total capital investment and decides whether to work or to revolt. If a worker decides to work, he contributes l _i = 1 unit of labor.

Payoffs

Payoffs are realized after the success or failure of the revolution. All players are risk neutral, and maximize their expected payoffs. If the revolution fails, the capitalists receive their returns from domestic and foreign capitals; the workers who worked receive their wages, and those who revolted get 0. If the revolution succeeds, domestic capital is confiscated from the capitalists, and is distributed evenly among all workers, and the workers who worked receive their wages. Moreover, a fraction 1 − L ∈ (0, 1) of workers are potential revolutionaries, and derive warm-glow payoffs s > 0 from participating in a successful revolution.^{Footnote 5} Let $L = \int {{l_i}{\rm{<italic>d</italic>}}i \in \left[ {0,1 - {\underline{L}}} \right]}$ be the aggregate labor input of these potential revolutionary workers. The remaining workers do not gain from participating in a successful revolution, and hence always work in equilibrium. In the rest of the article, when we say workers, we mean potential revolutionary workers who receive warm-glow payoffs s > 0 from participating in a successful revolution.

Markets and Production Technology

Domestic markets are competitive, so that the wage and the return to capital are their marginal revenue products. The production technology is Cobb–Douglas (K + K)^α(L + L)^1−α, with α ∈ (0, 1), and K, L > 0 as described above. Let r _d be the domestic return to capital and w be the wage. Because domestic markets are competitive, ${r_{\rm{<italic>d</italic>}}} = \alpha {\left( {{{{\underline{L}} + L} \over {{\underline{K}} + K}}} \right)^{1 - \alpha }}$ and $w = \left( {1 - \alpha } \right){\left( {{{{\underline{K}} + K} \over {{\underline{L}} + L}}} \right)^\alpha }$, where we normalize the output price to 1. Alternatively, mobile capital can be invested in foreign markets, e.g., treasury bonds or stocks. The rate of return to capital in foreign markets is r _f, which is a random variable with support $\left[ {0,\bar{f}} \right]$.^{Footnote 6}

Revolution Technology and Information Structure

The revolution succeeds whenever the measure of revolters exceeds the uncertain regime strength $\theta \in {\mathbb{R}}$. Capitalists and workers share a prior that θ ∼ G(·), and they receive noisy private signals about θ. Let y _j be a capitalist j’s private signal and x _i a worker i’s private signal. x _i = θ + σ _wε _i, where ε _i ∼ iidF _ε(·), and y _j = θ + σ _c η _j, where η _j ∼ iidF _η(·). We assume that signals and the fundamental θ satisfy the monotone likelihood ratio property.^{Footnote 7} The capitalists observe r _f, but workers receive a noisy public signal ${\tilde{r}_{\rm{<italic>f</italic>}}} = {r_{\rm{<italic>f</italic>}}} + {\varepsilon _{\rm{<italic>f</italic>}}}$ about it, with ε _f ∼ H(·), so that they cannot infer the exact value of θ from aggregate domestic capital investment K. All the noises ε _i, η _j, ε _f and the fundamental θ are independent of each other, and distributed accordingly to twice continuously differentiable distributions with full support on ${\mathbb{R}}$.

Timing

Capitalists observe the return to foreign investment r _f and their signals y _js about the regime’s strength θ, and decide how to divide their capital between domestic and foreign markets. Workers observe aggregate domestic capital, a public signal of foreign returns ${\tilde{r}_{\rm{<italic>f</italic>}}}$, and their signals x _is about the regime’s strength θ, and then decide whether or not to revolt. The success or failure of revolution is determined, payoffs are realized, and the game ends.

We maintain the following assumptions throughout the article.

Assumption 1. If a worker is sure that the revolution will succeed, then he has a dominant strategy to revolt: $s > \left( {1 - \alpha } \right){\left( {{{\overline{K}} \over {\underline{L}}}} \right)^\alpha }$.

Assumption 2. If a capitalist is sure that no one will revolt, he has a dominant strategy to invest domestically: $\bar{f} \lt \alpha {\left( {{1 / {\overline{K}}}} \right)^{1 - \alpha }}$.

Assumption 1 ensures that when the regime is very weak, the workers have a dominant strategy to revolt. That is, the payoff from participating in a successful revolution s is larger than the upper bound on wages, which is obtained if the supply of capital is at its maximum $\overline{K}$ and the supply of labor is at its minimum L. Assumption 2 ensures that when the regime is very strong, then even if all the mobile capital is invested domestically (thereby reducing domestic capital returns), a capitalist wants to invest domestically. It also implies that the reason for capitalists to invest in foreign markets is to avoid the political risk of regime change. Assumption 1 implies that workers prefer regime change over regime stability, whereas Assumption 2 implies that capitalists prefer regime stability to regime change. These Assumptions generate a lower dominance region in θ < 0 for the interactions among the workers, and a higher dominance region in θ > 1 for the interactions among the capitalists.

A pure strategy for a capitalist j ∈ [0, 1] is a mapping ${\rho _j}{\rm{:}}\;{\mathbb{R}} \times \left[ {0,\bar{f}} \right] \to \left[ {0,\overline{K} - {\underline{K}}} \right]$ from his private signal y _j and the foreign rate of return r _f to a decision of how much capital ${k_j} \in \left[ {0,\overline{K} - {\underline{K}}} \right]$ to invest domestically. A pure strategy for a worker i ∈ [0, 1] is a mapping ${\sigma _i}{\rm{:}}\;{{\mathbb{R}}^2} \times {{\mathbb{R}}_ + } \to \left\{ {0,1} \right\}$ from his signals x _i and ${\tilde{r}_{\rm{<italic>f</italic>}}}$ and the aggregate domestic capital K + K to a decision whether to work or revolt, where ${\sigma _i}\left( {{x_i},{{\tilde{r}}_{\rm{<italic>f</italic>}}},K} \right) = 0$ indicates that he works and ${\sigma _i}\left( {{x_i},{{\tilde{r}}_{\rm{<italic>f</italic>}}},K} \right) = 1$ indicates that he revolts. We focus on symmetric strategies, so that ρ _j(·,·) = ρ(·,·) for all j and σ _i(·,·,·) = σ(·,·,·) for all i. The equilibrium concept is Perfect Bayesian. Adapting a global games approach to equilibrium selection, we characterize the equilibrium in the limit when first the noise in the workers’ private signals becomes vanishingly small and then the noise in the capitalists’ private signals becomes vanishingly small.

Before we proceed, we highlight two limitations of the model. One concerns the order of limits: we characterize the equilibrium when first the noise in the workers’ signals goes to zero, and then the noise in the capitalists’ signals goes to zero. A more general solution would characterize the equilibrium for all manners in which the noise goes to zero. The question of how to characterize the equilibria in this more general case remains open. However, three observations are worth highlighting. (1) If the workers move first, before the capitalists, then one can mirror our approach and obtain a solution with the reverse order of limits. (2) That the workers’ information about the regime’s strength is far more precise than the capitalists’ can reflect, for example, that workers have better information about the attitudes and sentiments of security forces as well as rank-and-file government employees, which play a key role in determining the regime’s strength. This is more so if the capitalists are foreign investors. (3) Given this order of limits, the model makes reasonable predictions. The second issue concerns the workers’ information about their wages. An implication of the model is that, when workers decide whether to revolt or work, they estimate their future wages, reflecting the strategic uncertainty about other workers’ behavior, which will determine labor supply. That workers do not observe their wage before deciding to work reflects the simplifying choice of collapsing repeated decisions over a time period into a single decision of whether to work or revolt.^{Footnote 8} Abstracting from these more dynamic considerations, the model focuses on how strategic uncertainty affects the workers’ beliefs about their economic opportunities—a worker recognizes that other workers’ behaviors influence his economic opportunity.

Benchmark 1: Complete Information

Suppose the regime’s strength θ is known. In this complete information setting, if θ ≥ 1 − L, then even if all potential revolutionaries revolt, the regime survives. Anticipating this, all capitalists invest all their capital domestically, and no worker revolts. By contrast, if θ < 0, the regime collapses for exogenous reasons, even absent any significant active revolters.^{Footnote 9} Anticipating this, all capitalists move all their mobile capital abroad, leaving only the immobile capital K in the country. However, due to market congestion externalities, workers’ decisions depend on each other. A worker’s decision to revolt depends on the difference between his forgone wages and the rewards s that he gets from participating in a successful revolution. But the wage varies depending on how many workers revolt, going from (1 − α)(K/1)^α if almost no one revolts to (1 − α)(K/L)^α if almost all potential revolutionaries revolt. Assumption 1 implies that when θ < 0, so that a worker is sure that the regime collapses, then he has a dominant strategy to revolt. For intermediate values of regime strength, both these equilibria exist. In sum:

Proposition 1. Consider the complete information setting where the regime’s strength θ is known. There are multiple equilibria:

• • If θ ≥ 1 − L, there is a unique equilibrium in which capitalists invest all their capital domestically, no worker revolts, and there is no regime change.

• • If θ < 0, there is a unique equilibrium in which capitalists move all their mobile capital abroad, all potential revolutionaries revolt, and there is a regime change.

• • If θ ∈ [0, 1 − L), then both equilibria coexist.

The model with complete information has three shortcomings: the multiplicity of equilibria hinders empirical predictions, it is unreasonable to assume that the regime’s strength is known, and there are no useful comparative statics. In fact, the capitalists play a passive role. That is, the political risk of regime change affects the economy by influencing the capitalists’ investment decisions, but the capitalists’ decisions do not influence the workers’ decisions or the political risk.

Benchmark 2: Incomplete Information with an Exogenous Economy

Now, suppose domestic capital returns r _d and wages w are exogenous, but the regime’s strength is uncertain as described in the model. We assume s > w and r _d > r _f to avoid trivial cases where no worker ever revolts, or no capitalist ever invests domestically. To simplify exposition, we focus on symmetric cutoff strategies. Suppose a potential revolutionary worker revolts whenever his signal about the regime’s strength is below a threshold, x _i < x _e. Then, for any given regime strength θ, the measure of revolters is Pr(x _i < x _e|θ)(1 − L). This measure is decreasing in θ, crossing the 45° line at a unique point. Calling that point θ _e, the measure of revolters exceeds the regime’s strength for all θ < θ _e, causing a regime change. Otherwise, the regime survives. That is, the equilibrium regime change threshold θ _e is exactly the measure of revolters at θ = θ _e, which we will show to be (1 − L)(1 − w/s). Let p(x _i) ∈ [0, 1] be citizen i’s belief that the regime collapses, so that p(x _i) = Pr(θ < θ _e|x _i). Different regime strengths θ induce different signal distributions among the workers, and hence difference distributions of beliefs p(x _i). If we knew the distribution of these beliefs in an equilibrium, because those with p(x _i) > w/s will revolt, we could calculate the equilibrium regime change thresholds.

(1)$${\theta _{\rm{<italic>e</italic>}}} = \left( {1 - {\underline{L}}} \right)\Pr \left( {p\left( {{x_i}} \right) > {w / s}{\rm{|}}{\theta _{\rm{<italic>e</italic>}}}} \right).$$

A key statistical property simplifies the analysis (Guimaraes and Morris Reference Guimaraes and Morris2007; Loeper, Steiner, and Stewart Reference Loeper, Steiner and Stewart2014; Morris and Shin Reference Morris, Shin, Dewatripont, Hansen and Turnovsky2003):

Lemma 1. Recall that x _i = θ + σ _wε _i. Fix $\hat{\theta }$, and for all $\hat{x}$, define $p\left( {\hat{x}} \right) = \Pr \left( {\theta \lt \hat{\theta }{\rm{|}}{x_i} = \hat{x}} \right)$. Let H(p|θ) be the cdf of p(x _i) conditional on θ. Then, when the noise is vanishingly small (σ _w → 0), $H\left( {p{\rm{|}}\theta = \hat{\theta }} \right) = p$. That is, p is distributed uniformly at $\theta = \hat{\theta }$.

Applying Lemma 1 to the equilibrium conditions (1) yields a unique equilibrium regime change threshold.

Proposition 2. Consider the setting with exogenous wage and capital returns. When the noise in private signals becomes vanishingly small, there is a unique symmetric monotone equilibrium in which the regime collapses if and only if θ < θ _e, where

$${\theta _{\rm{<italic>e</italic>}}} = \left( {1 - {\underline{L}}} \right)\left( {1 - {w / s}} \right).$$

Critically, neither foreign returns r _f nor the magnitude of capital mobility $\overline{K} - {\underline{K}}$ affect the likelihood of regime change. The political risk of revolt affects the capitalists’ behavior: when θ _e is higher, more capital moves abroad. However, with no model of economy to determine wages and capital returns endogenously, capital allocations have no influence on political risk, and there is no coordination problem among the capitalists. In fact, the capitalists’ problem is barely strategic: given θ _e that comes from the anticipated behavior of the workers, each capitalist simply estimates the likelihood of regime change and his expected returns, and decides how to allocate his capital.

Capitalists’ Problem

The capitalists’ equilibrium behavior determines the aggregate domestic capital. A capitalist i with signal y _i invests ρ(y _i, r _f) units of his mobile capital abroad. We focus on monotone strategies, so that ρ(y _i, r _f) is increasing in y _i for a given r _f. Given a level of regime strength θ, the aggregate domestic mobile capital is $K\left( \theta \right) = \int {\rho \left( {{y_i},{r_{\it{f}}}} \right)} f\left( {{y_i}{\rm{|}}\theta } \right){\rm{<italic>d</italic>}}{y_i}$. When the regime is stronger, aggregate capital is higher:^{Footnote 11} K(θ) is increasing in θ, rising from $\mathop {\lim }\limits_{\theta \to - \infty } K\left( \theta \right) = 0$ to $\mathop {\lim }\limits_{\theta \to \infty } K\left( \theta \right) = \overline{K} - {\underline{K}} > 0$. Thus, as θ traverses the real line, θ**(K) from Proposition 3 falls from θ**(0) to ${\theta ^{**}}\left( {\overline{K} - {\underline{K}}} \right)$. This implies that there exists a unique θ* ∈ (0, 1) such that the regime collapses if and only if θ < θ*, where

(6)$${\theta ^*} = \left( {1 - {\underline{L}}} \right)\left( {{{1 - {w^{**}}\left( {K\left( {{\theta ^*}} \right)} \right)} / s}} \right).$$

The capitalists’ strategic interactions feature forces for both strategic complements and substitutes. When other capitalists are more likely to move their capital abroad, the workers’ productivity and hence their wages fall, increasing the likelihood of revolution, and raising a capitalist’s incentives to move his capital abroad. However, the smaller supply of domestic capital raises its returns, increasing a capitalist’s incentives to invest domestically.

Given the strategy of other capitalists and the workers, a capitalist i with signal y _i maximizes his expected payoff:

$$\eqalign{\mathop {\max }\limits_{{k_i} \in \left[ {0,\overline{K} - {\underline{K}}} \right]} {r_{\rm{<italic>f</italic>}}}\left( {\overline{K} - {\underline{K}} - {k_i}} \right) + \Pr \left( {\theta \,\ge\, {\theta ^*}{\rm{|}}{y_i}} \right) \times E\left[ {{r_{\rm{<italic>d</italic>}}}\left( \theta \right){\rm{|}}\theta \ge {\theta ^*},{y_i}} \right] \times \left( {{\underline{K}} + {k_i}} \right),$$

where k _i = ρ(y _i, r _f) is i’s mobile capital that he invests domestically. The capitalist’s problem can be written as

$$\mathop {\max }\limits_{{k_i} \in \left[ {0,\overline{K} - {\underline{K}}} \right]} \left\{ {\Pr \left( {\theta \,\ge \,{\theta ^*}{\rm{|}}{y_i}} \right)E\left[ {{r_{\rm{<italic>d</italic>}}}\left( \theta \right){\rm{|}}\theta \,\ge \,{\theta ^*},{y_i}} \right] - {r_{\rm{<italic>f</italic>}}}} \right\} \times \,{k_i}$$

$${\rm{with}}\;{r_{\rm{<italic>d</italic>}}}\left( \theta \right) = \alpha {\left( {{{{\underline{L}} + L\left( \theta \right)} \over {{\underline{K}} + K\left( \theta \right)}}} \right)^{1 - \alpha }}.$$

When a capitalist’s signal increases, he believes that the regime is stronger, raising his incentives to invest domestically: Pr(θ ≥ θ*|y _i) increases. But he also believes that both workers and other capitalists also receive higher signals, and hence workers become more inclined to work and the capitalists become more inclined to invest domestically. However, this increase in capital supply has another effect: it suppresses capital returns, thereby reducing incentives to invest domestically. Of course, in calculating domestic returns, capitalists must also condition on the fact that they only receive domestic returns if the regime survives, i.e., θ ≥ θ*. In sum, the net expected payoff from moving a unit of capital abroad need not be monotone. However, we show that Assumption 2 delivers that this net expected payoff has the single-crossing property, and the best response to a monotone strategy is a finite-cutoff strategy.

Lemma 4. Suppose all capitalists j ≠ i use a cutoff strategy in which they invest their mobile capital domestically whenever their private signals are above a finite threshold y*. There exists a threshold $\bar{\sigma } > 0$ such that if ${\sigma _{\rm{<italic>w</italic>}}} \lt \bar{\sigma }$, then capitalist i’s best response also takes a cutoff form, in which he invests all his mobile capital domestically if and only if his signal is above a finite threshold.

The idea of the proof is similar to that of Lemma 2, but it is complicated by the facts that capitalists have to anticipate the worker’s behavior and that they receive their domestic returns only if the regime survives. Let Γ(y _i) be the net expected payoff from investing one unit of capital in domestic versus foreign markets for capitalist i with signal y _i. If a capitalist knew θ, his net expected payoff would be

(7)$${\Pi} \left( \theta \right) = {{\bf{1}}_{\left\{ {\theta \ge {\theta ^*}} \right\}}}\alpha {\left( {{{{\underline{L}} + \Pr \left( {{x_k} \ge {x^*}{\rm{|}}\theta } \right)\left( {1 - {\underline{L}}} \right)} \over {{\underline{K}} + \Pr \left( {{y_j} \ge {y^*}{\rm{|}}\theta } \right)\left( {\overline{K} - {\underline{K}}} \right)}}} \right)^{1 - \alpha }} - {r_{\rm{<italic>f</italic>}}}.$$

But capitalist i does not know θ, and has to use his information to estimate his net expected payoff:

$$\Gamma \left( {{y_i}} \right) = \int_{ - \infty }^\infty {\Pi \left( \theta \right)} p{\rm{<italic>d</italic>}}f\left( {\theta {\rm{|}}{y_i}} \right){\rm{<italic>d</italic>}}\theta .$$

Now, when the regime is very weak, Π(θ) = −r _f < 0. Assumption 2 implies that when the regime is very strong,

$$\mathop {\lim }\limits_{\theta \to \infty } \Pi \left( \theta \right) \,\ge \,\alpha {\left( {{{1 / {\overline{K}}}} } \right)^{1 - \alpha }} - {r_{\rm{<italic>f</italic>}}} > 0.$$

Thus, Π(θ) and Γ(y _i) both have at least one sign change. A stronger version of Assumption 2 that $\alpha {\left( {{{\underline{L}} / {\overline{K}}}} \right)^{1 - \alpha }} > \bar{f}$ would immediately imply that Π(θ) has exactly one sign change: if θ < θ*, then Π(θ) = −r _f < 0; if θ ≥ θ*, then $\Pi \left( \theta \right) > \alpha {\left( {{{\underline{L}} / {\overline{K}}}} \right)^{1 - \alpha }} > \bar{f}$. With the current Assumption 2, however, domestic return is generally nonmonotone in θ [see the ratio in equation (7)]. Still, the proof shows that Π(θ) switches sign only once when the noise in the workers’ signals is sufficiently small. Here, the idea is to use $\mathop {\lim }\limits_{{\sigma _{\rm{<italic>w</italic>}}} \to 0} {x^*}\left( {{\sigma _{\rm{<italic>w</italic>}}}} \right) = {\theta ^*}$. Then, one can show that for any θ > θ*, labor supply [the numerator in equation (7)] approaches 1, allowing one to invoke Assumption 2. But this approach needs to be modified to prove uniform convergence, so that one can find a threshold $\bar{\sigma } > 0$ such that if ${\sigma _{\rm{<italic>w</italic>}}} \lt \bar{\sigma }$, then Π(θ) has one sign change—as opposed to the easier task of proving a separate threshold $\left( {{{\bar{\sigma }}_\theta }} \right)$ for each θ.

Lemma 4 implies that the marginal capitalist whose signal is at the equilibrium threshold y* must be indifferent between investing in the country or abroad. Thus, symmetric monotone equilibria are characterized by cutoffs (x*, y*, θ*):

(8)$$\Pr \left( {\theta \,\ge \,{\theta ^*}{\rm{|}}{y_j} = {y^*}} \right)E\left[ {{r_{\rm{<italic>d</italic>}}}\left( \theta \right){\rm{|}}\theta \,\ge\, {\theta ^*},{y_j} = {y^*}} \right] = {r_{\rm{f}}}.$$

(9)$${r_{\rm{<italic>d</italic>}}}\left( \theta \right) = \alpha {\left( {{{{\underline{L}} + L\left( \theta \right)} \over {{\underline{K}} + K\left( \theta \right)}}} \right)^{1 - \alpha }}.$$

(10)$$\eqalign{{\theta ^*} = \left( {1 - {\underline{L}}} \right)\left( {1 - {{{w^{**}}\left( {K\left( {{\theta ^*}} \right)} \right)} \over s}} \right),\;{\rm{with}}\,{w^{**}}\left( {K\left( \theta \right)} \right) = {{{{\left( {{\underline{K}} + K\left( \theta \right)} \right)}^\alpha }\left( {1 - {{\underline{L}}^{1 - \alpha }}} \right)} \over {\left( {1 - {\underline{L}}} \right)}},$$

where aggregate supply of capital and labor (conditional on θ) are:

$$\eqalign{K\left( \theta \right) = \Pr \left( {{y_j} \,\ge\, {y^*}{\rm{|}}\theta } \right)\left( {\overline{K} - {\underline{K}}} \right)\;{\rm}\;L\left( \theta \right) = \Pr \left( {{x_i} \,\ge \,{x^*}\left( {K\left( \theta \right)} \right){\rm{|}}\theta } \right)\left( {1 - {\underline{L}}} \right),$$

and x*(K) is the workers’ equilibrium strategy from the second stage, in which the workers observe the aggregate domestic capital.

These conditions reflect both within-group and between-group interactions among capitalists and workers. For example, the very shape of the equation (10) reflects the interactions among the workers, which capitalists anticipate, and the appearance of K(θ) in it reflects that each capitalist recognizes the effect of other capitalists on the likelihood of regime change. Using an approach similar to what we discussed in our characterization of the workers’ behavior, we show that when the noise is vanishingly small, in equilibrium, the marginal capitalist believes that domestic supply of capital is uniformly distributed: ${\underline{K}} + K\left( \theta \right){\rm{|}}{y_i} = {y^*} \sim U\left[ {{\underline{K}},\overline{K}} \right]$. Although the capitalists’ problem is more complex than the workers’ (e.g., the workers’ behavior must be taken into account, and domestic capital returns must be conditioned on the regime’s survival), we show that the equilibrium is unique and the equilibrium regime change threshold θ* takes a simple closed form.

Proposition 4. When the noise in private signals becomes vanishingly small (where we first let σ _w → 0 and then let σ _c → 0), there is a unique symmetric monotone equilibrium in which the regime collapses if and only if θ < θ*, where

(11)$$\eqalign{{\theta ^*} = \left( {1 - {\underline{L}}} \right)\left( {1 - {{{w^*}} / s}} \right),\;with\;{w^*} = {{\left( {{{\overline{K}}^\alpha } - \left( {\overline{K} - {\underline{K}}} \right){r_{\rm{<italic>f</italic>}}}} \right)\left( {1 - {{\underline{L}}^{1 - \alpha }}} \right)} / {\left( {1 - {\underline{L}}} \right)}}.$$

By analogy with Proposition 2, Proposition 4 shows that we can treat our model as if we are in our second benchmark, with an exogenous wage given in equation (11). Calling w* the “effective wage,” we can define effective supply of labor and capital that would generate this wage in a competitive market with Cobb–Douglas production technology. To do so, we can use Proposition 3 to derive an effective labor supply, and then use it together with w* in Proposition 4 to calculate an effective capital supply. From Proposition 3

(12)$$\left( {1 - \alpha } \right){\left( {{{{\underline{K}} + K} \over {{\rm{effective}}\;{\rm{labor}}\;{\rm{supply}}}}} \right)^\alpha } = {\left( {{\underline{K}} + K} \right)^\alpha }{{1 - {{\underline{L}}^{1 - \alpha }}} \over {1 - {\underline{L}}}},$$

and hence

$${\rm{Effective}}\;{\rm{labor}}\;{\rm{supply}} = {\left( {1 - \alpha } \right)^{1/\alpha }}{\left( {{{1 - {\underline{L}}} \over {1 - {{\underline{L}}^{1 - \alpha }}}}} \right)^{1/\alpha }}.$$

Now, using w* from Proposition 4, we have

$$\eqalign{\left( {1 - \alpha } \right){\left( {{{{\rm{effective}}\;{\rm{capital}}\;{\rm{supply}}} \over {{\underline{L}} + L}}} \right)^\alpha } = \left( {{{\overline{K}}^\alpha } - \left( {\overline{K} - {\underline{K}}} \right){r_{\rm{<italic>f</italic>}}}} \right){{1 - {{\underline{L}}^{1 - \alpha }}} \over {1 - {\underline{L}}}}.$$

Finally, substituting the effective labor supply from equation (12) for L + L yields

$${\rm{Effective}}\;{\rm{capital}}\;{\rm{supply}} = {\left( {{{\overline{K}}^\alpha } - \left( {\overline{K} - {\underline{K}}} \right){r_{\rm{<italic>f</italic>}}}} \right)^{1/\alpha }}.$$

If changes in L did not have a strategic effect due to the workers’ within-group interactions, then the effective labor supply would be linear: ${\underline{L}} + q_{\rm{<italic>w</italic>}}^*\left( {1 - {\underline{L}}} \right)$ for some equilibrium value $q_{\rm{<italic>w</italic>}}^*$ that would not depend on L. Similarly, if changes in K did not have a strategic effect, then the effective capital supply would be linear: ${\underline{K}} + q_{\rm{<italic>c</italic>}}^*\left( {\overline{K} - {\underline{K}}} \right)$ for some equilibrium value $q_{\rm{<italic>c</italic>}}^*$ that would not depend on K. These nonlinearities in effective supply of labor and capital stem from the players’ strategic interactions.

Corollary 1. Increases in immobile capital K or total capital $\overline{K}$ both decrease the likelihood of regime change. In contrast, increases in the foreign returns to capital r _f, the warm-glow from participating in a successful revolution s, or the fraction of potential revolutionary workers 1 − L all raise the likelihood of regime change.

(13)$${{\partial {\theta ^*}} \over {\partial {\underline{K}}}},\;{{\partial {\theta ^*}} \over {\partial \overline{K}}},\;{{\partial {\theta ^*}} \over {\partial {\underline{L}}}} \lt \,0\, \lt {{\partial {\theta ^*}} \over {\partial {r_{\rm{<italic>f</italic>}}}}},\;{{\partial {\theta ^*}} \over {\partial s}}.$$

The effects of global economy r _f and capital mobility K are of particular interest. Improvements in global markets that increase foreign returns r _f raise the capitalists’ incentives to move their capital abroad, and increases in capital mobility (smaller K) enable them to do so. Both these changes raise the likelihood of regime change domestically. Thus, globalization and market integration, as well as improvements in transportation technology that reduce the costs of moving capital, or economic modernization that changes the focus of the economy from relatively immobile sectors to more mobile ones (e.g., from the agricultural sector to service/finance sectors), can amplify the likelihood of social conflict and revolution.^{Footnote 12} When are these destabilizing effects stronger? We focus on the marginal effect of increases in foreign return—the effect of capital mobility is similar.

Corollary 2. The marginal effect of foreign returns to capital r _f is higher when either the warm-glow from participating in a successful revolution s or the capital share α is lower.

$${{{\partial ^2}{\theta ^*}} \over {\partial \alpha \partial {r_{\rm{<italic>f</italic>}}}}},\;{{{\partial ^2}{\theta ^*}} \over {\partial s\partial {r_{\rm{<italic>f</italic>}}}}} \lt \,0.$$

Corollary 2 implies that the destabilizing effect of globalization (marginal increases in r _f) is higher where ideological convictions for regime change (captured by s) or the capitalists’ share of income (captured by α) are lower. Labor share 1 − α is considered as a measure of inequality (Acemoglu et al. Reference Acemoglu, Johnson, Robinson and Yared2008; Piketty Reference Piketty2014), and there is a theme in the literature that associates inequality with conflict. We can capture this effect of inequality by positing that s is an increasing function of α. Then, increases in capital share α increase the likelihood of regime change by raising the workers’ motivation to revolt (Corollary 1). But increases in α also affect the likelihood of regime change by changing the technology, (K + K)^α(L + L)^(1−α). Thus, for example, when total capital $\bar{K}$ is sufficiently high, increases in capital share can raise the effective wage, thereby reducing the likelihood of regime change. These conflicting effects are consistent with inconclusive empirical findings on the relationship between inequality and conflict (Blattman and Miguel Reference Blattman and Miguel2010, 26–7). Corollary 2 adds to this debate by highlighting the mediating effect of capital share on the destabilizing effect of globalization in conflict-prone societies. Now, the effect is unambiguous, as both effects move in the same direction:

$${{\rm{<italic>d</italic>}} \over {{\rm{<italic>d</italic>}}\alpha }}{{\partial {\theta ^*}} \over {\partial {r_{\rm{<italic>f</italic>}}}}} = {{{\partial ^2}{\theta ^*}} \over {\partial \alpha \partial {r_{\rm{<italic>f</italic>}}}}} + {{{\partial ^2}{\theta ^*}} \over {\partial s\partial {r_{\rm{<italic>f</italic>}}}}}{{\partial s\left( \alpha \right)} \over {\partial \alpha }} \lt \,0.$$

Higher labor share always exacerbates the destabilizing effect of increases in foreign returns (or capital mobility), which can stem from globalization, modernization of the economy, or improvements in technologies that reduce international transportation costs.

We end this section by highlighting two additional observations. First, one may wonder what would happen if, instead of a continuum of capitalists, there was only one capitalist. Then, in the limit when the capitalist’s noise goes to zero, he prevents revolution whenever possible. That is, if the regime survives when all the capital is invested domestically, then the capitalist invests all his capital domestically; otherwise, he invests all his mobile capital abroad. In particular, the equilibrium regime change threshold does not depend on foreign returns or capital mobility. The reason is that the workers’ decisions, which determine the regime change threshold, depend on foreign returns and capital mobility only through the capitalist’s decisions. When the capitalist has a very precise estimate of the regime’s strength, absent strategic risk, he knows whether or not he can stop the regime change, and does so if he can, independent of foreign returns and capital mobility.

Second, our analysis so far applies as much to decisions of foreign investors to invest in a country as to decisions of domestic capitalists to send their capital abroad. In the following section, we focus on the latter interpretation (capital flight), and investigate the capitalists’ decisions to give the state authority over their decisions to remedy their collective action problem.