The cottagers on the shores of Lake Invidious are an unsavory bunch. There are 100 of them, and they live in a circle around the lake. Each cottager has two neighbors, one on his right and one on his left. There is only one commodity, and they all consume it on their front lawns in full view of their two neighbors. Each cottager likes to consume the commodity but is very envious of consumption by the neighbor on his left. Curiously, nobody cares what the neighbor on his right is doing. In fact every consumer has a utility function U(c, l) = c − l2, where c is his own consumption and l is consumption by his neighbor on the left. Suppose that each consumer owns 1 unit of the consumption good and consumes it.
(a) Calculate his utility level.
(b) Suppose that each consumer consumes only 3/4 of a unit. Will all individuals be better off or worse off?
(c) What is the best possible consumption if all are to consume the same amount?
(d) Suppose that everybody around the lake is consuming 1 unit. Can any two people make themselves both better off either by redistributing consumption between them or by throwing something away?
(e) How about a group of three people?
(e) How about a group of three people? No.
(f) How large is the smallest group that could cooperate to benefit all its members?