The Lost Mountains of northern Iowa are inhabited by the rare Marshallian deer. Patches of grass are far apart in this rugged land. If a deer finds a fresh patch of grass and spends h hours grazing it, it gets y = √ h units of grass. The deer compete for grass. When there are n deer, it takes a deer n 2 hours to find a fresh patch. A deer can survive if it gets 1 unit of grass every 24 hours.
(a) Find the average cost (measured in time) of a unit of grass if a deer gets y units of grass from each patch.
(b) How much time will an efficient deer spend in each patch when there are n deer? (Hint: Minimize the average cost.)
(c) Since there is free entry into the deer business, the equilibrium population is the maximum number of efficient deer who can survive. How many is this?