Consider the following economy with production. There is a worker endowed with 1 unit of time, which can be allocated either to leisure or labor. There is a firm with a technology that turns a worker's labor (the complement of his leisure) into bread. That technology satisfies:

b(l) = l^1/2

where l[0, 1] denotes the worker's labor and b(l) is the amount of bread produced as a function of labor input. The worker's utility function from labor supplied and bread consumed is:

u(l, b) = ln(b) l, 0 < < 2

1. What is the set of Pareto optimal allocations in this economy?

2. Provide prices and an allocation that, together, constitute a competitive equilibrium in this economy.