Consider a Robinson Crusoe economy. There are two goods x and y. RC is endowed with 100 unit of good x and 0 units of good y. RC's technology allows to generate units of y according to y = ln(x). Consider that RC is endowed with a utility function from problems 4-10

1. Find the optimal use of resources for RC given the technological constraint: quantity of good x that RC devotes to consumption, the quantity of good x that RC devotes to use as an input, the quantity of good y RC produces.

2. Suppose that RC makes decisions as an entrepreneur separately than as a consumer (referred to as decentralized case)

(a) Write down RC's problem as a profit maximizing entrepreneur, solve for factor demand and express the supply function as a function of price, as well as RC's profits.

(b) Write down RC's problem as a consumer that owns the profits to the firm, and solve for the demand of good x for consumption and good y. (

c) Evaluate whether there exists prices that clear the market; i.e. the total demand for good x across its uses equals its supply (endowment), and the demand for good y equals its supply (production).

(d) Evaluate whether the quantities found in the case of optimal resource allocation according to constraints coincide with those found in the decentralized case.