Econ 510a (second half)Yale UniversityFall 2004Prof. Tony SmithHOMEWORK #2This homework assignment is due at the beginning of the help session on Thursday, November 4.1. Consider again the two-sector economy that you studied in the first problem on Home-work #1, but replace the ad hoc behavioral assumptions with neoclassical assumptions.Specifically, imagine that a social planner seeks to maximizeP∞t=0βtu(ct), given an ini-tial condition k0> 0, subject to the technological constraints specified in the problemon Homework #1. Assume that u satisfies the usual regularity conditions. Note thatalthough leisure is not valued (i.e., the total amount of labor supply L does not appearin the planner’s objective), the planner must nonetheless decide in each period how toallocate L across the two sectors.(a) Formulate the planner’s optimization problem as a dynamic programming prob-lem. Be sure to distinguish clearly between state variables and control (or choice)variables.(b) Find a set of Euler equations and first-order conditions that an optimal solutionto the planning problem must satisfy.(c) Suppose that F (Kct, Lct) = KαctL1−αct, where 0 < α < 1, and that F = G. Useyour answer from part (b) to find the steady state for this economy as a functionof the structural parameters α, β, and δ.(d) The solution to the planner’s problem is a pair of decision rules describing howto allocate capital and labor across the two sectors given the current state vari-able(s). Use your answer from part (b) to find a pair of functional equationsthat determine the planner’s two decision rules. (Hint: Once you have eliminatedthe derivative(s) of the value function, you will be left with one intertemporalequation—an Euler equation governing the consumption/savings tradeoff—andone intratemporal equation governing the intersectoral allocation of labor at apoint in time.)(e) Show how you can use the two functional equations that you derived in part (d)to determine the derivatives of the two decision rules at the steady state. If thealgebra gets too nasty, you do not need to solve explicitly for these derivatives asa function of primitives (that is, simply display the equations that determine thetwo derivatives but do not solve them for the derivatives).2. Consider a neoclassical growth model similar to the one that we have discussed inlecture, but in which the level of technology oscillates deterministically between twovalues AHand AL, where AH> AL. In particular, period-t output ytequals AHF (kt)if t is even and equals ALF (kt) if t is odd. The planner seeks to maximizeP∞t=0βtu(ct),given k0> 0, subject to ct+ kt+1= yt+ (1 − δ)ktand kt+1≥ 0 for all t.(a) Formulate the planner’s problem recursively. (Hint: Consider two value functions,one for periods in which the level of technology is high and one for periods inwhich the level of technology is low. Find a pair of Bellman equations that thesefunctions must satisfy.)(b) Let the felicity function u be logarithmic, let F (kt) = kαt, and assume that capitaldepreciates fully in one period (i.e., set δ = 1). Use a guess-and-verify method tofind the two value functions in part (a). Describe fully the dynamic behavior ofthe capital stock starting from any initial condition for the capital stock.3. Consider an exchange economy with two (types of) consumers named A and B. Thetwo consumers have identical preferences: they each value consumption streams ac-cording toP∞t=0βtu(ct), where u has a constant elasticity of intertemporal substitutionσ−1. Consumer i’s endowment of consumption goods is {ωit}∞t=0, i = A, B. Consump-tion goods are perishable (i.e., they cannot be stored and used for consumption infuture periods).(a) Carefully define a competitive equilibrium with date-0 trading for this economy.(b) Suppose that ωAt= 2 for all t and ωBt= 1 for all t. Find the competitiveequilibrium allocations and prices.(c) Suppose now that the endowments fluctuate deterministically: consumer A’sendowment stream is {2, 1, 2, 1, 2, 1, . . .} and consumer B’s endowment streamis {1, 2, 1, 2, 1, 2, . . .}. Find the competitive equilibrium allocations and prices.(Hint: Guess that each consumer’s consumption is constant across time and ver-ify that this guess is correct.)(d) In parts (b) and (c) there is no variation in the aggregate endowment across time.Suppose that, as in part (b), consumer A’s endowment is 2 in every period but thatconsumer B’s endowment fluctuates: his endowment stream is {1, 3, 1, 3, 1, 3, . . .}.Find the competitive equilibrium allocations and prices. To simplify the algebra,set σ = 1 (i.e., let the felicity function u be logarithmic).(e) Carefully define a competitive equilibrium with sequential trading for this econ-omy. Use your results from parts (b), (c), and (d) to determine the equilibriuminterest rates for each pair of endowment streams. In addition, for each casedetermine how each consumer’s asset holdings vary over time (assume that eachconsumer starts with zero assets in period