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:
dNa
dTb
= 1A κ^ ∣^- Tb ’x (1 - e - rTb Ÿ + Tb ^x (1 - e " rTb )12 re " rTb
Rearranging terms and substituting (9) into the expression above obtains:
(12) dNa- = N- (- Tb- + r (erTb -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, - Tb-1 + r(erTb - 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) / Tb = 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.
18