ABSTRACT
Proposition 3 The social value of a public project of uncertain value increases as the degree of rivalry increases.
4. Changes in population
With a larger population, the coefficient ω is smaller, but the value of W is greater, resulting in potentially opposing effects on the private cost of risk bearing. Differentiating the individual risk premium with respect to n yields
dπ=dn ¼ ½�απω þ cπW �=n2 (17)
since ωn ¼ �α=n2 and Wn ¼ c=n2. Differentiating Equation (4), which defines the risk premium, with respect to W yields Eθ½V 0ðW þ ωθÞ � θ� ¼ V 0ðW þ ωθ � πÞ � ð1� πW Þ, which implies
πW ¼ 1� Eθ½V 0ðW þ ωθÞ � θ�=V 0ðW þ ωθ � πÞ: (18)
Substituting from Equations (6) and (18) into Equation (17) yields
n2dπ=dn ¼ �α θ � Eθ½V 0θ�
V 0
þ c 1� Eθ½V
0� V 0
; (19)
where V 0 ¼ V 0ðW þ ωθ � πÞ. It follows that
c 1� Eθ½V 0�
V 0
<α θ � Eθ½V
0θ� V 0
¼ αθ 1� Eθ½V
0θ� V 0θ
(20)
is necessary and sufficient for the private cost of risk bearing to decline as the population increases.
