Political Rents, Promotion Incentives, and Support for a Non-Democratic Regime

The isorent (11) is a downward-sloping and convex curve. It behaves approximately as a
hyperbolic curve Tb^-α with α > 1 n the vicinity of Tb = 0 (since f(Na) is a concave function)
and approaches a horizontal asymptote at Na = f ^-1(rC) as rTb approaches infinity.

The derivative of feasible supply (9) with respect to boss tenure is:

= ¹A κ^ ∣^- Tb ’^x (1 - e - ^rT^b Ÿ + Tb ^^x (1 - e " ^rT^b )¹² re " ^rT^b

Rearranging terms and substituting (9) into the expression above obtains:

(12)          ^dNa- = N- (- T_b- + r (e^rT^b -1)^-1 )< OforanyTb >0.¹8

dTb   2

Therefore, feasible supply is a downward-sloping curve in the (Tb, Na) plane. Its maximum
value is reached at Tb = 0 and equals:

(13)        Nr=_xn' ■ δr

which sets the upper boundary on the number of activists under given regime parameters.¹⁹

The feasible supply curve is also convex, but its curvature is systematically lower than
that of an isorent. ²⁰ This guarantees the existence of a unique interior solution to problem
(10). A typical configuration of an isorent and the feasible supply constraint is presented in
Figure 1. The first-order condition to the problem (10) is:

¹⁸ The term in brackets, - T_b^-1 + r(e^rTb - 1)^-1 < 0 , can be rearranged to obtain rTb + 1 < e ^rTb, which
holds for any Tb >0 by the properties of the exponential function.

¹⁹ By the L’Hopital rule, lim (1 - e^-rTb) / T_b = re^-rTb = r. In fact, the maximal sustainable number of
Tb→0

activists falls short of that given by (13) and is determined by the bosses’ participation constraint
discussed later in this in section.

²⁰ Proof of this statement can be obtained from the author.

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