Question on production economy.

2. [15 points] (This question involves a production economy but you should be able to answer it without having read Chapter 16). Robinson Crusoe is the only inhabitant of an island. He has an endow- ment of 1 unit of leisure and a technology for converting leisure into food. (These are the only commodities on his island). Using commodity 1 to denote leisure and commodity 2 to denote food, a typical commod- ity bundle is denoted (1, T2). Crusoe's endowment is w = (1,0). The technology allows him to convert a units of leisure into VT units of food. Of course, if he consumes , the amount of input he has effect tively devoted to the production activity is 1 - 21. There are no prices or markets on this island. (a) What are the feasible commodity bundles for Crusoe given his en- dowment and technology? This should be described by a function 12 = 9(1), the production possibility curve. [5 points] (b) Crusoe's utility function is u($1, 12) = VT142 and he can choose any commodity bundle (X1, 12) that satisfies the feasibility constraint in (a): 12 = g(x1). Write down his constrained utility maximization problem and solve it by substituting x2 = 9(21) into the utility function. How much of his leisure does he devote to producing food? [10 points]