The Dynamic Interactions of Hate, Violence and Economic Well-Being

1 Introduction

Hate and violence have been common features of human history. Secular and religious scholars, as well as leaders and warriors, have discussed their nature throughout history. As far back as the 5th century BC, Thucydides wrote about the nature of violence, observing that wars lead to even worse wars. The Bible, likewise, recognized that^[1] “violence begets violence.” The relationship between hate and violence is complex. What is clear, however, is that the two are very closely intertwined: they affect and are affected by each other. Making matters more complicated is that they are both affected by other factors, which, in turn, also impact their evolution. In the aftermath of the Balkan Wars, the “War on Terror,” and the continued conflicts in the Middle East, hate and violence have become the subject of renewed endless debates and extensive academic research. Indeed, a voluminous literature on the subject encompasses historical, philosophical, social, religious, psychological, political, economic and cultural aspects of hate and violence.^[2]

Naturally, each discipline has its focus, tools, and perspectives in the academic literature, possibly also its pre-conceptions. Thus, for example, many sociologists, psychologists and political scientists explain hate by underlying “root causes.”^[3] Economists, however, tend to explain all phenomena, including hate and violence, as the outcome of underlying optimal decision-making processes (in addition to root causes).^[4] A similar approach can sometimes be found in international relations research. Therefore, individuals’ or governments’ behaviour is explained as the outcome of underlying strategic decisions.^[5] Interestingly, even non-economists often argue that economic considerations such as competition over scarce resources explain hate, extremism and violence.^[6] Hence, government policies, including incitement in the face of such rivalry, are also considered strategic.

Recently, psychologists began looking at the dynamic properties of conflicts as an important element in explaining what is referred to as “intractable conflicts.”^[7] These types of conflicts, typically vicious, persistent, costly and difficult to resolve, seem to defy rationality and are difficult to explain using standard models (in psychology, games theory, bargaining, economics, conflict resolution, etc.). They argue that the apparent lack of rationality suggests that such conflicts are driven by “a psychology-dynamics” that seems to have a “life of its own.” As Vallacher et al. (2010) put it: “It is as though the conflict acts like a gravity well into which the surrounding mental, behavioral, and social-structural landscape begins to slide.”^[8] Thus, in their view, the nature of such conflicts can only be understood within a dynamic framework, or what they call a “dynamical system.”^[9] Specifically, they suggest we can view intractable conflicts as particular equilibria, or “strong attractor states,” of a dynamic interaction process between the groups involved, their emotions, actions and histories. The nonlinear dynamical systems theory was adopted to provide a framework for analyzing chaotic processes in conflicts, psychology, political science and other social sciences (see Guastello (2005)). For example, Guastello (2008) provides conflicts with alternative pathways to chaotic behaviour and empirical techniques for learning the nature of the underlying conflicts.

The role of emotions (not just hate) in conflicts has been examined in theoretical and empirical research in psychology, political science and international relations literature. Indeed, they have been recognized to have an important role. As Bar-Tal, Halperin, and De Rivera (2007) state: “Political conflict arouses strong emotions that shape and are shaped by conflict.” Some argue that the relevance of emotions is, in fact, implicit in many conflict models in international relations (e.g. see Kertzer (2017)). Thus, for example, Schnakenberg and Wayne (2024) provide a game theoretical dynamic model where emotions and strategic considerations interact. In their case, the emotion is anger; in our model, it is hate^[10] but, in principle, the two emotions could play a similar role in a conflict.^[11]’ ^[12] Halperin et al. (2011) provide an empirical study (in the context of the Israeli-Palestinian conflict) that shows that hate plays an important role in a conflict’s nature. Specifically, it affects the willingness to compromise. Similarly, Halperin, Canetti-Nisim, and Hirsch-Hoefler (2009) also find that group-based hatred is the most important variable in explaining political intolerance.

In fact, it is argued and shown that emotions not only play a role, but they often give rise to punitive action (see, for example, García-Ponce, Young, and Zeitzoff (2022) and Wayne (2023)). This punitive action is similar to the hate-induced violence in this paper. Moreover, it has been shown that punitive action, or violence, may bring retributional violence. For example, in a study of the dynamics of the Israeli-Palestinian conflict, Jaeger and Paserman (2008), using fatalities data (for the period 2000–2005), found that there was “evidence of unidirectional Granger causality from Palestinian violence to Israeli violence, but not vice versa.” Another contribution, by Fischer et al. (2018), argues that hate “has the goal to eliminate its target” and is particularly important at the intergroup level. They claim, albeit without evidence, that hate tends to spread quickly in conflict in the face of hate-based violence, which feeds on itself. This type of result is similar to what the model in this paper yields.

Sometimes, hate and violence lay dormant, almost invisible, but simmering under the surface at low-intensity equilibrium levels only to be awakened by external shocks, changes in circumstances, times of transition and uncertainty, new opportunities and old fears. Then, they may erupt again, sometimes in renewed full force. However, while such renewed violence may be triggered during times of change or uncertainty, their existence often predates these triggers. Here are a few examples. Conflicts followed the end of colonialism in Africa, including in Kongo, Nigeria, and Zambia. It was also followed by the 1940s conflict between Ethiopia and Eritrea in the Horn of Africa, the 1947–48 conflict between India and Pakistan and the Indochina War (1946–54). The end of the British mandate in Palestine was followed by the conflict between the Jews and the Arabs and the conflicts in Iraq and Lebanon.

Similarly, the collapse of the Soviet Union led to conflicts in ex-Yugoslavia ^[13] (although this conflict has recently abated) and the Caucasus. The conflict between the Kurds and Turks, as well as Armenians and Turks, followed the collapse of the Ottoman Empire. The collapse of the Austrian Empire led to vicious conflicts between Poles and Ukrainians in what was Austrian Galicia; the same happened soon after Germany invaded the Soviet Union in 1941 (again, this conflict has now been abated). Finally, the internal conflict in Colombia has been going on cyclically since the mid-1800s (see Safford and Palacios (2002)). And there are many more examples. What is interesting is that many of these conflicts were characterized by cycles of hate and violence, and many of them have yet to be fully resolved (see, for example, Horowitz (1985), (2000])). Some of these conflicts (in Eastern Europe) and the role of hate in their evolution are examined by Petersen (2002).

This paper aims to develop a simple model of the dynamic interactions of hate and violence and economic well-being, thus shedding light on the dynamic aspects of intractable conflicts and explaining a wide range of conflicts such as those mentioned above. For example, we show that steady-state equilibria are often characterized by some underlying levels of hate and violence. Moreover, we also show that steady-state equilibria are likely unstable, so small perturbations, in the form of changes in exogenous circumstances, can trigger renewed cycles of hate and violence. Most of the conflicts mentioned above may fall into this category.

Instead of focusing on possible root causes or strategic policy determinants of hate, we focus on its evolution and properties. We do not argue that root causes or strategic considerations do not play a role; we acknowledge that they do. But, although they may affect hate, the level of hate and its evolution are not a matter of choice by individuals; motion (evolution) equations govern them. Even strategic policymakers must take these motion equations into account. We can think of strategic policymakers as using these motion equations to “their advantage.” That is, a strategic policy is a way of “manufacturing” root causes: it can induce hate directly (e.g. by incitement) or indirectly (by affecting the root causes). Paraphrasing Herman and Chomsky (1988), these strategic policies serve to “manufacture dissent.”

Nevertheless, since the evolution of hate is still governed by the motion rules, strategic policies must also consider these rules. Thus, understanding the dynamic properties of hate and violence could help policymakers achieve their objective when facing the evolution of hate and violence as captured by the motion equations derived in this paper.

The paper examines the dynamics of hate and violence in a conflict between two “entities” (which can be viewed as countries, rival groups, communities, etc.) with a history of conflict, hate and violence.^[14] Ethnic, racial or religious conflicts may explain the history of the relationship. It may also be due to geographical, ideological, or economic conflicts. In general, the root causes of the conflict reflect a combination of these underlying factors. But, regardless of the underlying root causes, hate and violence are clearly interdependent within such a relationship: they affect and are affected by each other. Both the literature on the psychology of hate and the political/sociological theory of conflicts recognize that violence and hate may result in a “vicious circle”: violence breeds hate, but hate breeds violence.^[15] Moreover, economic considerations also become part of this vicious circle. Economic well-being directly affects the evolution of hate (presumably, an improvement in economic well-being mitigates the evolution of hate). But, economic well-being itself is affected by costly violence,^[16] which in turn is affected by hate.

This paper captures the interdependent relationships among these factors and their effect on hate evolution within a system of differential equations. We use the model to study the dynamic properties of hate and violence. It is interesting to note that, in focusing on the dynamics of hate and violence rather than on strategic considerations, this paper can be viewed as related, in spirit, to Richardson’s path-breaking two-differential-equation model (two-state variables) to describe the dynamics of the arms race .^[17] However, unlike in Richardson’s model, in this paper, the state variables – hate levels (viewed as “stock variable”) also interact with “flow variables” – violence levels. Moreover, in this paper, the evolution of hate is also affected by (and interacts with) economic conditions.

We begin by defining an ideal state of “genuine peace,”^[18] in which there is neither hate nor violence. Given a time-dependent economic growth process (that affects the evolution of hate), we then show that a long-run steady-state is possible, but, generally, such a steady-state will not be characterized by genuine peace. Moreover, we show that a better long-run economic environment does not necessarily result in lower equilibrium levels of hate and violence. Next, we examine the stability properties of the (hate and violence) equilibrium. First, we show that, under reasonable conditions (when both rivals are “congruent” or reciprocating countries who are attuned to each other’s nature), cycles of hate and violence cannot occur. Consequently, the dynamic properties of hate and violence cannot give rise to cyclical patterns of (net) economic well-being; such cyclical patterns can only arise due to cyclical patterns of gross growth rates. We demonstrate that for cycles to occur, we would need to have a case where, for one and only one country, reciprocity (congruency) does not hold. Second, we show that while it is possible to have stable or unstable equilibria under reasonable conditions, it is more likely to have unstable ones. Specifically, the most likely outcome is a saddle point.

We also examine the countries’ underlying characteristics’ role in determining the conflict’s dynamic nature. These characteristics are essentially “psychological/emotional” traits that capture how a country perceives itself and its rival. They include cognitive dissonance aversion, vengefulness, envy and compassion, degree of congruence and reciprocity, etc. Understanding the effects of these underlying traits could give us insight into possible tools that policymakers could use to affect the conflict’s equilibria and its nature (e.g. by trying to modify traits using educational tools) to achieve better outcomes. Better conflict outcomes could also be achieved if policymakers could use underlying root causes as policy instruments. Such instruments could affect the dynamics of the conflict directly or indirectly by changing some of the countries’ traits. But, as we mentioned above, the model that would be needed for this type of problem is beyond the scope of this paper. Specifically, we would need a model with strategic behaviour within a dynamic context. Such a model would require differential games, which combine differential equations and game theory. An example of such a model in the context of Richardson’s arms race is given in Intriligator and Brito (1976).^[19]

Cyclical conflicts have been discussed in the international relations and economics literature. For example, Acemoglu and Wolitzky (2014) develop an overlapping generations model of a conflict where rivals play a sequential coordination game with incomplete information. They show that conflict spirals may be triggered by mistake, then correct themselves, but spiral again. Similarly, while the model in Schnakenberg and Wayne (2024) is dynamic, it is a game-theoretic model with incomplete information, close in spirit to the Acemoglu and Wolitzky (2014) model. It has no direct motion equations describing the evolution of hate and violence and their interaction with economic well-being.

By focusing on the dynamics of hate and violence, this paper provides an example of a formal model that incorporates some of the ideas discussed in psychology’s “dynamical system” literature (see references above). In particular, this paper (i) demonstrates the dynamic interaction of hate and violence and its evolution, (ii) provides conditions required for equilibria of different types, (iii) examines the stability properties of the equilibria and (iv) shows how these are all affected by exogenous variable. Consequently, issues that are otherwise difficult to resolve can be naturally resolved within this model. It should be noted, however, that the main focus of this paper is not on the possibility of or the routes to chaotic patterns.

2 The Model

Consider the dynamic relationship between two countries (rival groups, communities, etc.) with a history of conflict, hate and violence. To model the interdependence of the factors that affect the evolution of hate, we begin by looking at the determinants of violence. Let Country i′s hate toward Country j, (i ≠ j), at each point in time, t, be given by h ⁱ (t). We allow for the possibility that hate may be negative. In such a case, we can think of it as “love.” We view hate as a state (stock) variable that summarizes the state of a country’s antagonism in the conflict. Let Country i′s violence toward Country j, (i ≠ j), at each point in time, t, be given by v _i (t).^[20] We allow for the possibility that violence may be negative, in which case we can think of it as “benevolence.” We view violence as a “flow variable.” To capture the “vicious circle” aspect of violence,^[21] we take a country’s violence toward its rival to depend on three variables: its hate toward the rival, the rival’s level of hate and the rival’s level of violence.

The following two continuously differentiable functions describe these relationships:

We assume that when hate in both countries is zero, the solution to the two equations is v ₁ = v ₂ = 0, namely: V ⁱ [0, 0, 0] = 0, i = 1, 2.

Country i′s violence toward Country j (≠ i) is assumed to be an increasing function of its hate, the other country’s hate and the other country’s violence:

Let us now examine the role of economic considerations. Consider some measure of economic performance at time t in Country i. For example, this may be Country i′s GDP, denoted as x _i (t). We assume that any type of violence in the conflict (regardless of its source) has an economic cost. Suppose a fraction of gross GDP is lost due to violence in the conflict. Let this fraction, denoted as c ⁱ , be captured by the continuously differentiable cost function,

We assume that the cost function is increasing in violence (in either country), but when there is no violence, the costs are zero. Namely,

Thus, net (of the cost of violence) GDP, denoted as y _i (t), is given by:

Now, define Country i′s net economic growth rate as:

where g _i (t) is the gross rate of economic growth, and r ⁱ [v ₁(t), v ₂(t), t] is the rate of growth in the fraction of GDP that is not lost due to violence (1 − C ⁱ ). Note that since ∂C ⁱ [v ₁(t), v ₂(t)]/∂v _j  > 0, we have:

Thus, since in the absence of violence, the net and gross rates of economic growth are the same, we have:

Let us assume that the gross rate of economic growth converges to some “long-run” value of,

We can now turn to the evolution of hate. We assume that the evolution of hate depends on the countries’ levels of hate, violence, and net economic growth.^[22] The evolution of hate can, therefore, be described by the following two differential equations:^[23]

where H ¹ and H ² are continuously differentiable functions and where, for notational simplicity, the time variable, t, is dropped for the rest of the paper whenever it is not required. The dynamics of hate are complex because they depend on hate, violence and economic conditions; however, violence levels are determined simultaneously and depend on hate levels and affect net economic conditions.

Before considering the likely properties of the evolution equations, we should note that these properties may not be global for general and possibly nonlinear systems. Namely, properties may be different for different values (and signs) of the right-hand side variable in (8). Thus, in the following, when we refer to properties of the functional forms, they should be understood as local rather than global properties.

In principle, the response of a country’s evolution of hate to its rival’s hate falls into one of two categories: it may or may not be reciprocating (∂H ⁱ /∂h _j≠i  < 0, or ∂H ⁱ /∂h _j≠i  > 0). Essentially, this captures the country’s degree of congruence (attuned to its rival’s nature). Similarly, the response of a country’s evolution of hate to its hate falls into one of two categories: it may have a short or a long “memory,” that is, high or low “depreciation” (∂H ⁱ /∂h _i  < 0, or ∂H ⁱ /∂h _i  > 0). There are, therefore, four possible combinations to consider. These describe what we refer to as a country’s (local) intrinsic type. They are shown in Table 1 below.

Table 1:

The Nature of Country i – Effects of Hate.

As we will show later, however, a country may have a (local) “perceived” type which is different from its (local) intrinsic one. This difference may occur for the following reasons. Violence and economic conditions depend on hate; consequently, a change in hate has both direct and indirect effects. The direct effect is what we referred to above as the intrinsic type (captured by ∂H ⁱ /∂h _j ). The indirect effect captures the effect of a change in hate on the evolution of hate through its effects on violence and economic conditions (or other so-called “root causes”). The overall, or “perceived” effect, is the sum of the two, so there is no reason the overall effect should have the properties (say, sign) as the intrinsic type. We show this below.

As for the local effects of violence on a country’s evolution of hate, here, too, we may have four cases. A country may be locally “masochistic” or vengeful (∂H ⁱ /∂v _j≠i  < 0, or ∂H ⁱ /∂v _j≠i  > 0) with respect to violence by its rival. Moreover, a country’s violence may give rise to a “need to justify” its actions by minimizing cognitive dissonance, hence (locally) affecting the evolution of hate positively (∂H ⁱ /∂v _i  > 0). On the other hand, it may exhibit (local) dissonance “affinity” or incongruence so that its own violence and the evolution of its hate are negatively related (∂H ⁱ /∂v _i  < 0). These cases are shown in Table 2:

Table 2:

The Nature of Country i – Effects of Violence.

Although there are several possible configurations, not all are equally likely. Specifically, it does not seem likely that, in any region, an increase in the rival’s levels of hate or violence would negatively affect a country’s evolution of hate. Thus, we can expect that (globally),^[24]

Furthermore, the literature on the psychology of hate has found that (due to the need to minimize cognitive dissonance) violence against a rival can lead to increased hate toward the rival.^[25] Thus, locally, we may have:^[26]

The local effect of a country’s hate on the evolution of its hate, however, is not that clear. On the one hand, as is common in macroeconomic models, persistence or inertia may exist. On the other hand, there may be “depreciation” or willingness to forget.^[27] Hence, it is not clear how the current level of a country’s own hate affects its evolution of hate; ∂H ⁱ /∂h _i may be locally positive, negative, or zero.^[28]

Finally, it is well recognized in the literature that economic well-being affects conflicts’ nature, prevalence and severity.^[29] Thus, we should generally expect a country’s net economic growth to negatively affect its hate evolution. In other words, locally, we would have:^[30]

Since w _i  = g _i  + r ⁱ , this implies that we would have:

Our model is, therefore, described by equations (1), (5) and (8). To study its dynamic properties, we first solve for the flow (non-state) variables v ₁, v ₂, w ₁, w ₂ in terms of the state variables h ₁ and h ₂. Let us begin with the violence equations in (1). Write these equations as:

The corresponding Jacobian, denoted by F, is given by:

For a solution to exist, we must have the following:^[31]

Assuming that this condition holds, let the solution to equations (1) be given by:

Now, consider the effects of an increase in hate on violence. Using equations (1) we have:

Thus,

Or,

But, since it is unreasonable that the overall (total) effect of an increase in hate on violence should be negative (i.e. that violence should decrease with hate), we assume that,^[32]

Furthermore, we also have,

In other words, there is also no violence if there is no hate.

Now, plugging the solution for v i * , i = 1,2 into equations (4), (5) we get the solution for net growth as:

Then, from equations (5), (6) and (16) we have:

Furthermore, when h ₁ = 0, h ₂ = 0, we, w i * ( 0,0 , g i ) = g i .

We can now plug the solutions for the variables v ₁, v ₂, w ₁ and w ₂ in terms of the state variables h ₁ and h ₂ and growth variables g ₁ and g ₂ into the two differential equations to obtain:

Equations (18) capture the dynamic properties of the system. To understand the nature of the system, we need to examine the partial derivatives of the G ⁱ functions. Define h ≡ (h ₁, h ₂) and G j i ≡ ∂ G i ( h ; g i ) / ∂ h j , i , j = 1,2 . Assuming that G ⁱ (h; g _i ), i = 1, 2, is continuously differentiable, define the Jacobian matrix corresponding to these two equations, at (h; g _i ), as:

where,

The determinant of J is given by:

What can we say about the signs of the elements of the matrix J? Since ∂ H i ∂ V * j > 0 , ∂ r i ∂ V * j < 0 for all i, j and ∂ H i ∂ w i < 0 , for all i, we have,^[33]

But, since ∂ H i ∂ h j > 0 , ∂ V * i d h j > 0 for all i ≠ j, we have,

Unfortunately, since the effect of a country’s own hate on the evolution of its hate ∂ H i / ∂ h i is not that clear, we cannot determine the sign of G i i . Since the signs of the diagonal terms are unknown, this also means that so is the sign of the determinant of J. How likely is it for ∂ H i ∂ h i to be positive? We know that a sufficient condition for  G i i > 0 is that ∂ H i ∂ h i > 0 and a necessary and sufficient condition is that ∂ H i ∂ h i > − ∂ H i ∂ V * 1 ∂ V * 1 ∂ h i + ∂ H i ∂ V * 2 ∂ V * 2 ∂ h i + ∂ H i ∂ w i * ∂ r * i ∂ h i . But the effects of depreciation or forgetfulness, even when they exist, may not be sufficient to outweigh the effects of violence (which consist of its direct and indirect effects through its impact on the economic cost of the conflict). The case of when G i i > 0 , therefore, seems quite likely.

Finally, note that while the properties of the Jacobian matrix J are related to the intrinsic styles (properties) shown in Tables 1 and 2 above, there is no a priori reason why G j i and H j i should always have the same signs (although here we have that H j ≠ i i and G j ≠ i i are both positive). A country’s intrinsic type is captured by the properties of H ⁱ , whereas its perceived type is captured by (the properties of) G ⁱ . The properties of G ⁱ reflect not only the intrinsic nature of a country but also all other factors that depend on hate and play a role in its evolution. In our example, these other factors include violence and economic conditions. But, in a more general model, they may also include other root causes. For example, factors that affect and are affected by hate. Some of these other factors may reflect underlying strategic (game) considerations (e.g. by political actors). Confusion over what is intrinsic and what root causes are may explain why there is so much debate and disagreement over the causes of hate, violence, extremism and terrorism.

3 Steady State

Let us first examine the special case when the two growth rates are constant. We assume that for any given (finite column) vector h = (h ₁, h ₂), when g _i is “sufficiently” low, we have d h 1 d t > 0 and when g _i is “sufficiently” high we have d h 1 d t < 0 . Given continuous G ⁱ (h ₁, h ₂; g _i ) functions and since ∂H ⁱ /∂g _i  < 0, for all h, g _i , this implies that for any given (finite) h, there exists, at least, one fixed rate of economic growth in Country i, given by g i h , such that,^[34]

Specifically, this implies that, in the absence of hate or violence in either country, there exists some fixed rate of economic growth in Country i, given by g i 0 , such that,

Namely, the fixed growth rates g 1 0 , g 2 0 are consistent with zero hate in both countries. We refer to them as “normal” growth rates, but only in the sense that they are required to keep hate levels in both countries at zero. Clearly, in general, there is no reason why g i 0 should be the same as the (fixed) long-run growth rate (LRGR) g i * .^[35]

Let us now examine the existence and nature of a steady-state (SS) solution. First, time-dependent growth rates cannot yield an SS. Second, for fixed growth rates g ₁, g ₂, an SS (if it exists) is defined by the two conditions:

Hence, if g i = g i 0 = c o n s t a n t , i = 1, 2, then h ₁ = h ₂ = 0 satisfies equations (20). In other words, if growth rates in both countries are constant and equal to the normal rates, then we have an SS with zero hate. But can we have an SS with zero hate when we do not have constant normal growth rates in both countries? Since G i 0,0 ; g i 0 = 0 and ∂G ⁱ (0, 0; g _i )/∂g _i  < 0 for all g _i (and regardless of what g _j≠i is), we know that there is no other fixed value of g _i that satisfies G ⁱ (0, 0; g _i ) = 0. Hence, we can have an SS with zero hate if and only if both countries have (fixed) normal growth. Since there is no reason why growth in both countries should be fixed and equal to the normal growth rates, it is clear that, in general, we will not have an SS with zero hate. In other words, genuine peace is not likely as an SS.

But does an SS with any level of hate exist? Since the only constant growth rates are the two long-run growth rates g 1 * and g 2 * , the question is whether there exists a vector h * ≡ h 1 * , h 2 * such that:

Define the corresponding Jacobian matrix evaluated at h*; g* as

where a j i ≡ G j i h * ; g i * . Assuming that the determinant of the Jacobian A does not vanish at h*; g*, there exists a vector h * ≡ [ h 1 * ( g * ) , h 2 * ( g * ) ] , where h*(g*) is continuously differentiable and satisfies the two conditions in equations (21). Although it is obvious that

in general, we expect to have g i * ≠ g i 0 , so we conclude that it is not likely that we will have: h i * = 0 , for i = 1, 2.

To be able to understand the nature of the SS, we examine the properties of the two conditions (isoclines, or demarcation curves) G 1 h ; g 1 * = 0 (denotes as IC ₁) and G 2 h ; g 2 * = 0 (denoted as IC ₂) around the SS h* (and given the long-run growth rates g 1 * , g 2 * ) . Total differentiation yields the slopes of IC ₁ and IC ₂, at h*, g*, denoted as S ₁ and S ₂, respectively, as:

While we know that a j i > 0 , f o r i ≠ j , we do not know the sign of a i i . Thus, we do not know the signs of S ₁ and S ₂ at h*. Moreover, comparing the two slopes, we get that:

Again, since we do not know the signs of a 2 2 and A , we do not know which isocline is steeper (we do not know the sign of S). It may be possible, however, to infer the likelihood of the two possible cases (S > 0 and S < 0) by examining the effects of a change in long-run growth rates on the SS. A Change in g i * shifts IC _i , thus affecting the SS values of both h 1 * and h 2 * . From equations (21) we get:

where the 2 × 2 matrices d h * d g * and A _g are given by:

where a g i i ≡ ∂ G i h * ; g i * / ∂ g i . Thus, the effect of a change in long-term growth rates is given by:

where A ⁻¹ is the inverse of A. Since ∂G ⁱ (h; g _i )/∂g _i  = ∂H ⁱ /∂w _i ∂w _i /∂g _i  < 0, we also have: a g i i ≡ ∂ G i h * ; g i * / ∂ g i < 0 .

We know that an increase in g i * shifts IC _i . In fact, since the off-diagonal elements in A are positive and a g i i < 0 , we know that when g 1 * increases, IC ₁ shifts up, and when g 2 * increases, IC ₂ shifts to the right. But, unfortunately, since (as was shown above) we do not know the sign of the determinant of A or the signs of its diagonal elements, we do not know if the two isoclines are upward or downward sloping, and we do not know which is steeper. Consequently, even though we know in which direction isoclines shifts, we cannot tell what happens to the SS (the intersection of IC ₁ and IC ₂). Thus, the effects of a change in long-run growth rates are generally ambiguous. Nevertheless, let us examine these effects further.

From equation (22) we get:

which is ambiguous. From equation (22) we also get:

Since a g j j a j i < 0 , we know that sign d h i * / d g i * = − sign ( | A | , i ≠ j . Thus, for both countries, the effects in equation (24) must have the same sign: sign d h 1 * / d g 2 * = sign d h 2 * / d g 1 * . There are, therefore, two possible cases: (1) both d h 1 * / d g 2 * and d h 2 * / d g 1 * are negative, (2) both d h 1 * / d g 2 * and d h 2 * / d g 1 * are positive.

Although it may be obvious, it is still useful to note that (assuming that a g i i and a j i , i, j = 1, 2, are all non-zero) a change in the LRGR in one country affects the SS hate levels in both countries. This “cross effect” simply reflects the interaction of the effects on the two isoclines. However, it introduces two important elements to the analysis. First, even though g j * does not appear in country i′s motion equation, a higher LRGR in country j will decrease or increase the SS level of hate in country i ≠ j. In other words, “indirect” elements of envy/compassion are introduced into the model by the interaction of the two isoclines. Second, even though, by assumption, g i * negatively affects a country’s hate evolution a g i i < 0 , an increase in g i * may either decrease or increase its own SS level of hate. In other words, the SS equilibrium response to an increase in its growth rate may be inconsistent with its “inherent” response (as captured by a g i i ; the response of its evolution of hate).

Altogether, there are six possible cases, depending on the properties of the A matrix (which determine the slopes, S _i , and the difference in slopes, S, of the isoclines). These cases are summarized in Table 3 below.^[36]

Although there are six possible cases, not all seem equally “reasonable.” Specifically, it seems reasonable that a country’s SS level of hate should decrease when its own LRGR increases. This property should be true at least for one of the two countries (namely, we should have d h i * / d g i * < 0 for at least one country). It is, of course, possible that this may not be true, but it would indeed seem unreasonable if, for both countries, h i * should increase with g i * . If this requirement is true for both countries, all cases except for cases (i) and (iia) would be eliminated. If true for at least one country, cases (iii) and (iv) will be eliminated. What about the “cross effect” of a change in g i * on h j * ? Do we expect d h i * / d g j * , i ≠ j to be positive or negative? If d h i * / d g j * > 0 , we have what we referred to above as indirect envy/compassion (when g j * increases/decreases h i * increases/decreases). Although the envy part does not sound appealing, it does not seem unreasonable. What about the other case where d h i * / d g j * < 0 ? In this case, a country’s SS level of hate will be low when its rival’s growth rate is high and high when its rival’s growth rate is low. Regardless of ethical considerations, this seems less “reasonable.” If, in addition to eliminating cases without d h i * / d g i * < 0 , for at least one country, we were also to eliminate the cases where d h i * / d g j * < 0 , we would end up with three cases only: cases (iia), (iib) and (iic).

Let us consider the implications of cases (i)-(iic) above regarding the nature of the SS. Specifically, we are interested in finding out the likely SS solutions, particularly whether they are likely to involve no hate. First, as was pointed out above, in general, there is no reason why we should have g i * ≠ g i 0 , so it is not likely that we will have zero hate in both countries. Second, whether the SS levels of hate are positive or negative depends on (i) whether the long-run growth rates are higher or lower than the normal rates and (ii) which of the six cases above occurs. The possible cases are summarized in Table 4 below (an entry/pair +− means that h 1 * > 0 and h 2 * < 0 ; more than one entry means that, at least for one country, the sign is ambiguous – it depends on g 1 * − g 1 0 , relative to g 2 * − g 2 0 ).

As Table 4 shows, a given configuration g 1 * , g 2 * and g 1 0 , g 2 0 does not necessarily tell us if h 1 * and h 2 * are positive or negative. Specifically, just because g 1 * > g 1 0 , it does not mean that h 1 * < 0 (e.g. in case (iib), we have h 1 * > 0 ) . Furthermore, an SS generally involves non-zero levels of hate (love). Table 4 also shows that, in general, an SS involves non-zero levels of hate (love).

4 Stability

Rather than examining the details of the solution for the two differential equations system, we now focus on the system’s stability (the solution is given in Appendix 6.1).

We begin the stability analysis by examining the solution to the autonomous system (AS), given by d h d t = A h (see Appendix 6.1). The characteristic equation corresponding to the AS is given by,

and its characteristic roots are:

where Δ ≡ a 1 1 + a 2 2 2 − 4 A is the corresponding discriminant. But, since a i j ≡ ∂ G i h * , g i * / ∂ h j > 0 , i ≠ j , i, j = 1, 2, we have:

Thus, we conclude that the solution involves two distinct real roots. Consequently, a cyclical pattern of hate in the AS is not possible. Moreover, given that cyclical patterns of hate cannot occur in the AS and given the monotonic relationship between violence and hate (in equation (16), we have dV* ⁱ /dh _j  > 0, all i, j = 1, 2), it also follows that the AS (itself) cannot give rise to cyclical patterns of violence. Similarly, this also implies that, for fixed values of gross rates of economic growth, we cannot have cyclical patterns of net rates of economic growth ( w 1 * ( h 1 , h 2 ; g 1 ) . Cyclical patterns of hate, violence and net economic growth can, therefore, occur only due to possible cyclical patterns in the non-autonomous component, η i ( t ) = q i + a g i i g i ( t ) , which in turn reflect cyclical patterns in g _i (t).

Before continuing the stability analysis, we should note that other possible non-autonomous components may play a similar role as the non-autonomous economic considerations. For example, such considerations may capture political, social or global changes (and the uncertainty associated with such changes) can also be considered. These exogenous non-autonomous effects could represent wars, the collapse of empires, the end of colonialism, severe climate change, etc. Thus, they could explain the triggers of the cyclical patterns of some conflicts mentioned in the introduction.

Continuing with the stability analysis, it is clear from what we have shown thus far that the only question is whether the SS equilibrium in the AS is stable or unstable. In principle, the following cases are possible in the AS: (I) if both roots are negative, we have a stable node; (II) if both roots are positive, we have an unstable node; and (III) if the roots have opposite signs, we have a saddle point.

Let us look at the six possible cases above. In the three “most reasonable” cases ((iia), (iib) and (iic)), we have A < 0 . Since

λ ₁ and λ ₂ must have opposite signs. Hence, we conclude that the solution to the AS is a saddle point in all three “most reasonable” cases.

In cases (i), (iii), and (iv) that were deemed to be “less reasonable,” we have the following stability properties. In case (i), we have A > 0 , so λ ₁ and λ ₂ must have the same sign. But, since a j j < 0 , we have λ 1 + λ 2 = a 1 1 + a 2 2 < 0 . Consequently, λ ₁ and λ ₂ must both be negative; hence, the solution to the AS is a stable node. In case (iii), we have a i i < 0 , and A < 0 , so we have a saddle-point. In case (iv), we have A > 0 and a i i > 0 , so λ ₁ and λ ₂ must be both positive, so we have an unstable node.

Since all three reasonable cases have a saddle point, we conclude that the most likely outcome is, in fact, a saddle point. Namely, the AS is most likely to be unstable. But, since the AS is unstable in the most likely cases, so is the corresponding NAS. At the same time, given growth rates that converge to fixed (finite) long-run rates, it follows that, in the less likely case when the AS is stable (case(i)), so is the NAS.

Can we refine this even further? Remember that to have a i i < 0 , we require that (at h * ; g i * ): ∂ H i ∂ h i < − ∂ H i ∂ V * 1 ∂ V * 1 ∂ h i + ∂ H i ∂ V * 2 ∂ V * 2 ∂ h i + ∂ H i ∂ w i * ∂ r * i ∂ h i ; namely, the effects of depreciation, or forgetfulness, if they exist, must outweigh the effects of persistence plus the direct and indirect effects of violence. Since this is likely not to occur, cases (iib) and (iic) are not likely to occur. Therefore, the most likely case is (iia), which yields a saddle point. We should remember, however, that hate neither conforms to nor is based on reason. Asking whether a particular case is reasonable may, in itself, not be desirable or even reasonable. Hence, it is perhaps best not to exclude cases that seem unreasonable. Hence, it is perhaps best not to exclude cases that seem unreasonable.