Please answer the following

4. (15 points) Chuck Nolan is on a deserted island off the coast of Brazil (yes, that is the character Tom Hanks plays in Castaway), and is trying to survive by either gathering bacuri (b) or cupuacu (c). The technologies available for these two production processes are given by: b = Lb and c = VL, where Lo is the amount of labour (time) used for gathering bacuri and Le is the amount of labour (time) used for gathering cupuacu. The total time (2) is 8 hours per day, i.e. Lb+ Le= L = 8. Chuck Nolan's preferences are given by the utility function, ((b,c) = bc. Note: both bacuri and cupuacu are fruits indigenous to the Amazon rainforest. a) Construct the production possibility frontier (PPF) for bacuri and cupuacu and illustrate graphically. b) Find the marginal rate of transformation (MRT), and the marginal rate of substitution (MRS) between cupuacu and bacuri. c) Find the optimal combination of bacuri and cupuacu and the optimal allocation of time. Show the competitive equilibrium on the graph