Consider a model of a negative consumption externality. Both households have Cobb-Douglas preferences over composite (x^A, x^B respectively) and a hours of activity per day chosen by household A measured by 0 <= t <= 24. The household's utility functions are:

u^A(x^A,t) = x^(1/2)*t^(1/2) and u^B(x^B,t) = x^(1/2)*(24-t)^(1/2)

Suppose the initial endowment of composite is e^A = 5 and e^B = 5. Law limit Household A's daily activity to be at most 12 hours, ie/ the legal maximum is = 12.

1. In an Edgeworth Box, identify the set of allocations that would make both households better off than they are at the initial endowment (it is called the set of mutually beneficial trades relative to the initial allocation).
2. Explain why the set of Pareto Efficient allocations (contract curve) is represented by the 45 line connecting the bottom left to the top right of the Edgeworth box, ie = .
3. Introduce a market for units of t traded relative to the initial value, = 12 . Find household demands, market equilibrium allocation and price.