Model Overview

Consider 2 groups of farmers, whose annual income is subject to random rainfall variability and who might fight for control over limited land and labor resources (22). Each year, the decision to attack is taken by weighing the immediate opportunity costs of fighting against future expected returns from the captured resources. The former is given by the current year’s rainfall draw, and the latter is jointly determined by the entire distribution of rainfall, by the probability of victory, and by the endogenous risk of conflict occurring in future years (Materials and Method). Under these conditions, Chassang and Padro-i Miquel (10) show that conflict emerges in bad years, when income falls below a threshold determined by its underlying distribution. Insofar as income is influenced by climate, their model offers a mechanism that can explain the empirical findings that relate climate anomalies to conflicts (16).

We extend the existing model by specifying a rainfall distribution and an income-generating crop function that are analytically tractable and consistent with governing meteorological and hydrological processes (Materials and Methods). Doing so introduces a nonlinear relation between climate and income, whose implications for conflict we discuss in the following section. The parametric distribution of income also allows us to compare predictions of conflict probabilities across distributions by altering parameters to emulate the effect of climate change (Fig. 1). We initially focus on changes in the relative variability of seasonal rainfall, quantified by its coefficient of variation (CVW). The focus on CVW places our study at the intersection of empirical research exploring historic associations between conflict, income, and short-run anomalies of seasonal rainfall (see refs. 7 and 8) and climate modeling research predicting an increase in rainfall variability (e.g., ref. 11). By performing a comparative statics exercise (20), we allow agents to adapt to changed costs and benefits by adjusting their fighting threshold. A changed climate affects both the present opportunity cost and the future returns from conflict, to which agents adapt by shifting the income threshold below which they will decide to fight. For analytical tractability, we favor this rather narrow definition of climate adaptation over a broader interpretation that would allow agents to endogenously optimize income distribution itself, e.g., through crop, policy, and infrastructure selection.

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Fig. 1.

Schematized relation between climate, crop, and conflict models. Moving from left to right, rainfall distributions are related to income distributions via a deterministic model relating seasonal water availability to crop yield, taken as a proxy for income. Income distributions then inform a decision model for conflict. Changes in climate (top to bottom) alter the distribution of water availability, a change that propagates to an altered conflict likelihood.

Mean Climate Can Affect Income Variance

Crop yields do not generally scale linearly with water supply (e.g., ref. 19), and so changes in mean water availability will alter the variance of agricultural income. In particular, a crop chosen to be robust to climate variations will have mean water availability map to a flat region of its yield function (dark blue line in Fig. 2, Top). For such a crop choice, the effect of climate variability on income variability is minimal under existing climate conditions (dark blue line in Fig. 2, Bottom). However, a systematic decrease in water availability will enhance income variability due to the concave nature of the crop yield curve. This effect is particularly pronounced in the low water availability region (low W) of the crop yield curve, where curvature is maximal. There is a broad consensus in climate predictions that points to an increase in rainfall variability and an increase in mean temperatures (see, e.g., ref. 11). The discussion below focuses on changing drought characteristics caused by an increase in rainfall variability. However, the nonlinearity of the climate–income link implies similar conclusions for sustained increases in mean temperature or for excess precipitation (SI Appendix).

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Fig. 2.

Changes in mean climate can cause changes in income variability. A decrease in mean rainfall W (with variance conserved; Top) causes the distribution to map to a more concave region of the crop model (Middle), resulting in distributions of income θ with a larger variance σθ2 (Bottom). This increasing variance compounds the effect of a decreased mean income μθ and increases the coefficient of variation CVθ=σθ/μθ of income. Model parameters (Materials and Methods) are WH=150 mm, θmax= 3 currency units, σW= 69.6 mm, and μW= 270 (dark lines) and 135 mm (light lines).

Nonmonotonic Effect of Climate Variability on Conflict

Despite the stylized nature of the opportunity cost model, changes in the coefficient of variability of water elicit complex nonlinear, and at times nonmonotone, effects on the probability of conflict. Fig. 3 illustrates how conflict probability increases monotonically with climate variability, captured by the coefficient of variation CVW of rainfall, for some parameter combinations (red line), but the relationship becomes nonmonotonic for others (pink line). Indeed, it is possible that conflict prevalence decreases with climate variability for small enough values of CVW and a large enough offensive advantage in the odds of victory (SI Appendix). This behavior suggests that the opportunity cost framework does not consistently predict that a more variable climate will give rise to more prevalent conflicts. This insight is important to consider when using the framework to interpret empirical results. For instance, an empirical study finding an insignificant (Fig. 3, point A) or negative (Fig. 3, point B) relation between climate variability and conflict may not be incompatible with the opportunity cost framework. It also does not dismiss the possibility that a positive relation will emerge as CVW increases under the effect of climate change (as seen in positive slopes at points A′ and B′ in Fig. 3).

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Fig. 3.

Subtle changes in economic parameters can substantially alter the qualitative relationship between conflict probability and climate variability. A slight (2%) increase in the probability of first strike victory π (Materials and Methods) introduces a nonmonotone relationship between the coefficient of variation of seasonal rainfall (CVW) and the predicted probability of conflict (Pwar): a higher climate variability successively decreases and then increases the probability of conflict for a higher value of π (pink line), whereas the relationship remains monotonically increasing for a lower first strike advantage (red line). The shaded area shows the mean (vertical line) and 99% confidence interval of CVW observed for seasonal (3-monthly) rainfall, constructed from daily observations at 671 locations within the United States (23). Model parameters (Materials and Methods) are π1=0.5148 (red), π2=0.5252 (pink), c=0.9, δ=0.9, μW=WH=150 mm, and θmax= 3.

Governing Statistics and Strategic Adaptation

Changes in rainfall variability (CVW) might cause farmers to alter the income threshold below which they will engage in conflict. This adaptation response can strongly influence the probability of conflict (Pwar) as farmers weigh the current opportunity costs of attack against expected future profits. Opportunity costs are lower during a negative climate shock due to decreased crop productivity. Attacking then increases potential future profits for 2 reasons. First, the victor will capture her opponent’s resources and permanently increase her own agricultural profits. Second, triggering a conflict during an income shock hedges against the possibility of a conflict ever occurring in future periods, when opportunity costs are higher on average. In the stark language of our our simple model, both incentives rely on the assumption that the defeated opponent exits the game forever. However, qualitatively similar incentives emerge if the defeated agent temporarily loses his land and ability to fight back, particularly when agents place a high emphasis on short-term profits.

Consequently, the opportunity cost model points to 3 fundamental statistics of the income distribution that govern the relationship between climate change, adaptation response, and conflict: 1) The frequency of income shocks, defined as the probability F(θ∼) that income falls below agents’ conflict threshold θ∼, has a direct effect on the probability of conflict. Any change to the income distribution that increases shock frequency will directly increase Pwar. However, this direct effect also causes war to occur sooner in expectation, thereby reducing the future profits from (current) peace. This gives rise to a second, indirect, effect of an increase in income shock frequency: because peace becomes less advantageous, farmers respond by increasing the income thresholds below which they attack, an adaptation that further exacerbates Pwar. 2) A decrease in mean income E[θ], for instance, associated with a permanent decrease in mean rainfall, makes victory less profitable. Agents adapt to this by adjusting their income threshold for conflict. This causes a decrease in agents’ threshold and, all other statistics being held constant, a decrease in Pwar. 3) Expected income during conflict-inducing shocks E[θ∣θ<θ∼], as a measure of the intensity of income shocks, is proportional to the destructivity of conflicts and affects agents’ incentive to fight through the hedging motive described above. More intense income shocks cause smaller expected losses in income during conflict years. Future conflicts are then less costly on average, which incentivizes agents to postpone fighting. Consequently (and perhaps surprisingly), the anticipation of more intense income shocks has a negative effect on Pwar.

Changes in the 3 income statistics discussed above have independent and consistent (either positive or negative) effects on conflict prevalence, as summarized in Table 1. However, the influence of changes in climate on all 3 income statistics is complex and driven by the specific shape of the distribution that governs interseasonal climate variability. For a realistic distribution of water availability (Materials and Methods), these relations are displayed in Fig. 4, Top, and show that changes in water variability (CVW) elicit different changes in each of the 3 income statistics, in terms of sign and magnitude. In particular, the income shock frequency response can be either positive or negative, depending on the value of CVW and the equilibrium threshold for fighting (Fig. 4, Top, dark blue). In Fig. 4, Bottom, we decompose the overall changes in conflict probability caused by increased climate variability (∂Pwar∂CVW, in red) into its previously described fundamental components (Table 1). Depending on the relative magnitude of the responses, the overall relation between climate variability and conflict may itself be nonmonotone (Figs. 4 and 3, pink). In particular, Fig. 4 shows that the relation can be dominated by agents’ response to changes (dashed) in both mean income (gray dashed) and the intensity of income shocks (i.e., conflict destructivity; light blue dashed). This insight is relevant in the context of recent literature focusing almost exclusively on the effect of changes in the frequency of income shocks on conflicts (e.g., ref. 7). Our theoretical results suggest that farmer adaptation to other climate-driven income statistics, such as the intensity of income shocks, may be equally important to consider.

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Fig. 4.

Components of the climate–income–conflict relationship. (Top) A marginal increase in rainfall variability affects each of the 3 governing statistics S of the income distribution: shock frequency F(θ∼), mean income E[θ], and conflict destructivity cE[θ∣θ<θ∼] (as a measure of shock intensity). The magnitude of each effect is expressed as a partial derivative with respect to CVW. Variables θ, θ∼, c, and CVW indicate annual income (a random variable with cumulative density function F), the income threshold for conflict, the opportunity cost parameter, and the coefficient of variation of rainfall, respectively (Materials and Method). (Bottom) Income shock frequency has a direct and axiomatic impact on the probability of conflict Pwar=F(θ∼) (solid blue). However, changes in all 3 income statistics affect Pwar because agents adapt by changing their income threshold θ∼ (dashed lines; Table 1). The total contribution of these effects determines the nonmonotonic response of Pwar (red), which is also expressed as a partial derivative with respect to CVW. Parameters (Materials and Methods) are π=0.524, c=0.9, δ=0.9, μW=WH= 150 mm, and θmax=3.

Relation to Empirical Regularities

Chassang and Padro-i Miquel (10) point to 2 stylized facts that persistently emerge from the empirical literature on income and conflict: 1) conflicts tend to happen during bad income shocks and 2) conflicts are more prevalent in low-income countries. At first sight, these regularities may appear at odds with our theoretical predictions suggesting that more intense income shocks and lower average income both decrease the propensity for conflict (Table 1).

At a closer look, however, stylized fact 1 is a statement about low individual draws from a given distribution (horizontal direction in Fig. 1), whereas Table 1 concerns sustained shifts in the distribution of income (vertical direction in Fig. 1). In line with ref. 10, conflict occurs in our model when income falls below a certain threshold. Table 1 is saying that a sustained shift in the distribution toward more extreme droughts causes agents to lower that threshold. In other words, agents fight in anomalously dry years for a given distribution, but they think twice about fighting for a given draw if dry years become the new normal.

Regarding the second stylized fact, it is important to point out that the theoretical results in Table 1 concern marginal changes in each income statistic, with the 2 other statistics held constant. Any nonmarginal change in distribution will also change the other 2 statistics because they are themselves determined by the threshold θ∼. For instance, scaling annual income by a constant factor affects all 3 statistics in a way that they exactly cancel out (SI Appendix). This gives rise to the invariance of Pwar to income scaling noted by Chassang and Padro-i Miquel (10). Similarly, a constant upward shift in income results in a decrease in Pwar under reasonable assumptions, as shown in SI Appendix. The reality may be best captured by a combination of the 2: rich countries have more income and also less volatile income. The model would then indeed predict a lower probability of war.

Practical Implications

The theoretical arguments in this paper are a strong simplification of reality. The economic incentives we discuss represent a small subset of the social, political, and historical processes that together give rise to violent conflicts. Nonetheless, they capture important dynamics through which climate-related income shocks may cause rational agents to be amenable to conflict. Theoretical insights from the model have 3 important implications that can guide policy and empirical research.

First, it is important to distinguish climate from income variability when examining their implications for conflicts. The nonlinear and highly local effect of climate on agricultural income has been highlighted in several studies (e.g., refs. 24 and 25) and has a strong qualitative impact on conflict incentives. It emerges from a combination of natural (timing of rain events (26)), technical (crop choice (27)), economic (agricultural prices (28)), and institutional (insurance and regulation (29)) processes that were often put in place precisely to decouple income from climate variability (30). However, as climate variability begins to exceed historical ranges, these hedging mechanisms may become less effective. For instance, a crop that is adapted to a certain precipitation range will be more susceptible to variation at lower rainfall levels due to the increased curvature of the crop function (Fig. 2). This curvature causes a change in mean climate to affect the variability of income, which propagates to conflict incentives. This stylized example highlights the necessity of a careful empirical characterization of the climate–income relationship to understand implications for conflicts.

Second, theoretical results may inform empirical research that seeks to disentangle opportunity cost motives from other mechanisms that predict conflict during bad years. Alternative hypotheses (see ref. 16) include weakened government structures (caused by a drop in tax revenue), increased (perceived) inequality, and climate-induced migration, as well as cognitive and physiological factors that contribute to aggression. All of these competing mechanisms also predict that current conflict is negatively correlated with current income. However, since none of the alternative explanations are forward-looking, they would predict either no, or perhaps a negative, correlation between current conflict and the income in prior years (see discussion in SI Appendix). Opportunity costs are different: if agents update their belief about future incomes in a Bayesian way (some evidence of it is given in ref. 31), a sequence of good years leads agents to expect greater gains from attack and thus renders them more, not less, aggressive in subsequent years. This is a testable implication that is unique to the opportunity cost argument and can thus serve to empirically assess its explanatory power.

Finally, caution must be exercised in using microeconomic income shock arguments to interpret empirical analyses of historic data and draw extrapolations for climate change. While the model does suggest a positive correlation between weather anomalies and conflict, it does not support the argument that conflicts will always be more prevalent if these anomalies occur more frequently due to climate change. Rather, the theory suggests a complex, and potentially nonmonotonic, relation between climate variability and conflict. This complexity emerges both from nonlinear climate to income relationships and from strategic adaptation by agents to a changing income distribution. By affecting the entire distribution of climate, climate change will effectively define a new normal. Agents strategically adapt to multiple facets of climate change by adjusting their response to income variability. In doing so, they redefine the very notion of climate anomalies and associated negative income shocks as they pertain to climate-related conflicts.