Eve (consumer A) and June (consumer B) growing old together on a remote island on which only fish (good 1) and coconut (good 2) are available for consumption. Eve's preferences are described by the utility function u(xA1 , xA2 ) = (xA1 )(xA2 ), where xAj is Eve's good j consumption with j = 1, 2. Similarly, June's utility function is u(xB1 , xB2 ) = (xB1 )3(xB2 ), where xBj is June's good j consumption with j = 1, 2. Also, Eve's initial endowment is ?A = (?1A, ?2A) = (2, 1) while June's initial endowment is ?B = (?1B , ?2B ) = (3, 3).
a) Draw the Edgeworth box for this economy. Mark the point indicating the initial endowment of each consumer.
b) Explain if it is Pareto efficient for Eve and June to consume their endowments.
c) What is the set of allocations that could be the outcome under barter in this economy?
d) Let the price of fish be p1 while the price of coconut be normalized to 1 without loss
of generality. For each consumer, solve the utility maximization problem, i.e., derive
xi? forj=1,2andi=A,Basafunctionofthepricep. j1
e) Calculate the equilibrium prices and demands for the economy.
f) Explain if the equilibrium you find in part e) is Pareto Optimal.
g) Suppose now that Eve discovered a new territory of coconut trees. As a result, Eve's endowment of coconuts (i.e., ?2A) has increased slightly. Explain how this alters the equilibrium price of fish.
Consider the following Diamond economy (Le., the Overlapping Generations Model: OLG). In every period f with t = 0, 1,2, .., there always exist two types of individu- als, young and old who are continuously born or are continuously dying, respectively. L two-period-lived Individuals are born in period & and the population grows at a rate n (Lt = (1 + n)Lt-1). Each young individual supplies one unit of labor when he or she is young and divides the resulting labor income between first-period consumption and saving In the second period, the individual simply consumes the saving and any interest he or she earns. Let's assume that instantaneous (period) utility is given by the following constant-relative- risk-aversion (CRRA) utility function: 8 >0. (1) Show that the period utility function is increasing and concave in Ci. (2) Show that the coefficient of relative risk aversion (defined as - 90,162) is actually constant. Let Che and Car denote the consumption in period : of young and old individuals. Thus the (lifetime) utility of an individual born at t, denoted U, depends on Cu and Car+ and is given by U = [Cu) + 1+p where p is the rate of time preference. A production function Y, = F(K,, AL,) , where Ke represents capital, A, the technology or the effectiveness of labor, and &, labor itself, is assumed. Suppose that F( ) follows the usual properties, i.e., it has constant returns to scale and satisfies the Inada conditions. Also assumed is At grows at exogenous rate g (so At = (1 + g)A-1). Markets are competitive; thus labor and capital earn their marginal products, and firms earn zero profits. (3) Define be = 2,t, the amount of capital per unit of effective labor, a = 7,5, output per unit of effective labor, and me = f(k) the production function in intensive form. Assuming no depreciation, show that the real interest rate is given by n = f'(k) and the wage per unit of effective labor is given by un = /(1) - bef(kat).II. (50 points) Suppose that a firm has a Leontief production function given by y = min {Z/2, A/4), and that the rental rates for labor and capital are given by w=1 and v=4. Suppose also that K is fixed at 200 in the short run. (1) Write down the short-run cost minimization problem of the firm. (2) Find the conditional factor demand function for labor and calculate the firm's short-run cost function. (3) Calculate the firm's average fixed, average variable, average and marginal cost functions. (4) Draw the graph of all short-run cost curves. (5) Now suppose the firm can choose the level of capital stock K in the long run. Write down the firm's long-run cost minimization problem. (6) Find conditional factor demand functions for both inputs and calculate the firm's long-run cost function. (7) Calculate the firm's long-run average and marginal cost functions. (8) Draw the graph of all long-run cost curves. (9) Explain graphically the relationships between long run and short-run cost curves for the production function given above. (10) Explain the relationships between production functions and cost functions for this problem. III. (50 points) Answer all the questions in Part II again when the firm has a Leontief production function given by y = [min {L/2, K/4) 1 1/2 or y = [min {L/2, K/4}]2
