1. Suppose the City of Berkeley decides to grant allocations of water from Strawberry Creek according to the appropriative doctrine. Next, due to significant administrative burdens, suppose only two individuals decide to apply for water allocations:
Tom [Bates] is the more senior user, and Nicholas [Dirks] is the more junior user. Further, suppose the annual water flow in Strawberry Creek follows a uniform distribution from 0 to 100 acre-feet per year. Tom's profits as a function of water use A are: ΠT (A) = 5000 ln(A), and Nicholas' profits as a function of water use are: ΠN(A) = 12000 ln(A). Both Tom and Nicholas share the same cost function of water diversion capacity: C(A) = 5A2 + 1000. Finally, both Tom and Nicholas are risk neutral.
Hint: recall the quadratic formula for a quadratic equation of the form ax2 + bx + c:
x = -b±√b2-4ac/2a
(a) Draw graphs of both the PDF (g(•)) and CDF (G(.)) of Strawberry Creek's annual flow. Be precise with your axes.
(b) How much water diversion capacity does Tom build?
(c) How much water diversion capacity does Nicholas build?