12
the utility maximizing contribution is implicitly characterized and define
(6) ∏λ) = P1 (Ah, Am) xhτ - u'(λ (w - r)).
λsh
This is the marginal utility of type z when incomes and reservation prices for mobilization have
both been multiplied by λ (for given Ah and Am). Clearly, ^>(1) = 0 for all individuals for whom
the non-negativity constraint on r is not binding. The sign of the change of φ as λ increases will
tell us if λr — the amount needed to exactly pay for the higher compensation rate — exceeds
or falls below the utility maximizing best reply. Differentiating with respect to λ, evaluatingthe
result at λ = 1 and using (3) we see that for all contributing individuals
√(1) = -p1 (Ah,Am) ɪ - (w - r) u''(w - r) =
sh
= -u'(w - r) - (w - r) u"(w - r) =
,, ( Γ (w - r) u"(w - r)
= -u'(w - r) 1 + ʌ-----j---Ц---= =
u'(w - r)
= -u,(w - r)(1 - η),
where η is the elasticity of the marginal utility of consumption. Therefore, the sign of ^,(1)
depends on the elasticity of the marginal utility being smaller or greater than unity.
In particular, if η lies between 0 and 1,15 φ'(1) < 0, which implies that the contributions do not
rise enough to compensate for the higher payment that must now be made to activists. We can
deduce that a scaling of wealth by a common factor λ to each member of one group will reduce
the group’s aggressiveness by shifting their aggregate response function down if η < 1. We
reach the opposite conclusion when the elasticity of the marginal utility η is larger than unity.
We bring these results together in the following Proposition.
PROPOSITION 3. Suppose that a change occurs in the parameters relevant to one single side. Then
(i) an increase in population or a uniform increase in radicalism results in an increase in activism; while
(ii) a uniform increase in wealth has ambiguous results on activism, depending on the elasticity of the
marginal utility of consumption. When η < 1 (η > 1) a scaling of w will unambiguously reduce (in-
crease) the group’s activism.
4.3. Within-Group Heterogeneity. The discussion above suggests that the distribution of eco-
nomic resources within a group might have a more powerful effect on activism than a uniform
change in the relevant parameters for one group. If this were the case, this would imply that
intra-group dispersion might play a more important role in conflict than average levels.
We start by performing the two exercises separately: the effects of a change in inequality in
the distribution of x within a group and of the change in the distribution of w. This allows us
to compare the role of within-group heterogeneity in radicalism versus higher heterogeneity in
wealth. We end this subsection by examining the effect of changes that increase the correlation
between religious extremism and wealth and hence make the two characteristics more aligned
with each other.
15Consider the family of utility functions
u(y) = Z' y1 η.
1-η
This family of utility functions is increasing and concave for all η ≥ 0 and its marginal utility has a contant elasticity
of η. Notice that lim u(y) = ln y. Hence, η < ' means that u is assumed to be “less concave” than lny.
η↑1