Econ 510a (second half)Yale UniversityFall 2004Prof. Tony SmithFINAL EXAMINATIONThis is a closed-book and closed-notes exam. You have three (3) hours to complete the exam.There are four (4) questions on the exam for a total of 100 points. The points allocatedto each question are indicated below. Please put the answer to each question in a differentblue book. To receive full credit, you must provide convincing explanations to support youranswers. Please write as neatly as possible.1. (25 points) Consider an exchange economy with two (types of) consumers (labelledA and B), three time periods (labelled 0, 1, and 2), and a single good in each perio d.Consumer A is endowed with 4 units of the good in each of the three time periods,while consumer B is endowed with 0 units of the good in period 0, 12 units of the goodin period 1, and 0 units of the good in period 2. The consumer’s endowments are notstorable. The two consumers value consumption streams according to:P2t=0βtu(cit),where citis the consumption of consumer i in period t and β ∈ (0, 1). The felicityfunction u is strictly increasing and strictly concave.(a) Carefully define a competitive equilibrium with date-0 trading (i.e., an Arrow-Debreu equilibrium) for this economy.(b) Let u(c) = log(c) and β = 1/2. Solve for the competitive equilibrium allocationsand prices.(c) Carefully define a competitive equilibrium with sequential trading for this econ-omy, assuming that each consumer is endowed with zero assets in period 0. Un-der the assumptions in part (b) determine the equilibrium rates of return onone-period loans issued in periods 0 and 1, respectively.2. (30 points) Consider a neoclassical growth model in which the production of outputcreates pollution as a byproduct. The aggregate amount of pollution ¯stcreated inperiod t is proportional to aggregate output in period t:¯st= θf(¯kt, ¯nt),1where¯ktis the aggregate stock of capital, ¯ntis aggregate labor supply, and the ag-gregate pro duc tion f exhibits constant returns to scale. Capital depreciates at rate δ.There is a continuum of identical consumers whose preferences are given by:∞Xt=0βtu(ct, ¯st),where ctis a typical consumer’s level of consumption in period t. The felicity functionu is strictly increasing and strictly concave in its first argument and strictly decreasingin its second argument. Consumers do not value leisure, so without loss of generalityset ¯nt= 1 for all t.(a) Carefully define a recursive competitive equilibrium for this economy (assumingthat consumers own the factors of production).(b) Does the steady-state aggregate capital s tock depend on θ, and if so, how ?(c) Explain how to calculate the speed of convergence of the aggregate capital stockto its steady-state value (in a neighborhood of the steady state). You do nothave to compute the speed of adjustment explicitly, but you should describe theprocedure in sufficient detail so that someone who can do algebra—but who knowsnothing about economics—could implement it.(d) State the social planning problem for this economy. Do the social planning andcompetitive equilibrium allocations coincide in this economy? Explain as fully asyou can.3. (25 points) Consider the following model of growth: there is one consumption goodper period, one (type of) consumer who lives forever, and there are two sectors, aconsumption-goods sector and an investment-goods sector. Production in the consump-tion-goods sector satisfies:c = Akαcn1−α,where kcis the amount of capital used in this sector, n is the amount of labor used inthis sector, and 0 < α < 1. Production in the investment-goods sector satisfies:i = Bki,where B > 0 and kiis the amount of capital used in this sector. The total stock ofcapital accumulates according to:k0= (1 − δ)k + i,2and the stock of capital in any perio d can be freely allocated across the two sectorssubject to:k = kc+ ki.The consumer has one unit of labor in every period but does not value leisure; hispreferences are given by:∞Xt=0βtlog(ct).(a) State (in recursive form) the social planning problem for this economy. Identifyclearly the state variables and the control (choice) variables.(b) Show that the solution to the planning problem features exact balanced growth:that is, the economy is always on its balanced growth path. Express the growthrate in terms of primitives.4. (20 points) Consider an economy with overlapping generations of consumers who livefor two periods. Young consumers are born with zero assets and are endowed with oneunit of labor. Old consumers are endowed with λ units of labor, where 0 ≤ λ ≤ 1.Output y is produced by a competitive firm according to: y = kαn1−α, where k is theaggregate capital stock and n is the aggregate supply of labor. Capital depreciates fullyin one perio d. Consumers do not value leisure; the preferences of a young consumerborn in period t are given by:log(c1t) + β log(c2,t+1),where c1tis his consumption when young and c2,t+1is his consumption when old.(a) Find the equilibrium law of motion of the aggregate capital stock.(b) Show that the steady-state of this economy is dynamically efficient (for all α andβ) provided that λ is greater than some “cutoff” value¯λ ∈ (0, 1). (You do notneed to find an explicit expression for¯λ.) Give an intuitive explanation for