Econ 511b (Part I)Yale UniversitySpring 2004Prof. Tony SmithHOMEWORK #2This homework assignment should be handed in by 5PM on Friday, January 23 to JinhuiBai’s mailbox in the basement of 28 Hillhouse.1. Consider the planning problem for a simple finite-horizon neoclassical growth model:max{ct, kt+1}Tt=0TXt=0βtlog(ct),given k0= 10 and subject to the constraint that ct+ kt+1= Akαt+ (1 − δ)kt. Setβ = 0.95, δ = 0.1, and α = 0.4. Choose A so that the steady-state value of capital inthe corresponding infinite-horizon model is 100.Solve the model numerically (say, in Matlab) using the “shooting” method describedin lecture on January 14: start by guessing a value for k1, solve for k2from the Eulerequation at time 0, then solve for k3from the Euler equation at time 1, and so on,until kT +1is found. Then vary k1and repeat until the appropriate value of kT +1(i.e.,0) is found. Find the lowest value for T such that the highest value of capital betweenperiods 0 and T exceeds 90.2. Consider a neoclassical growth model with two sectors, one producing consumptiongoods and one producing investment goods. Consumption is given by Ct= F (KCt, LCt)and investment is given by It= G(KIt, LIt), where Kjtis the amount of capital in sectorj at the beginning of period t and Ljtis the amount of labor used in sector j in periodt. The total amount of lab or in each period is equal to L (leisure is not valued). Laborcan be freely allocated in each period between the two sectors: L = LCt+LIt. Capital,on the other hand, is sector-specific: once it is installed in a given sector, it cannot bemoved to the other sector. Investment goods, however, can be used to augment thecapital stock in either sector. In particular, the capital stocks in the two sectors evolveaccording to:Kj,t+1= (1 − δ )Kjt+ Ijt, j = C, I,where It= ICt+ IIt.The social planner seeks to maximizeP∞t=0βtu(Ct), given KC0and KI0, subject tothe constraints on technology. Note that although leisure is not valued (i.e., the total1amount of labor supply L does not appear in the planner’s objective), the planner mustnonetheless decide in each period how to allocate 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−αCtand G(KIt, LIt) = KγItL1−γIt. Use youranswer from part (b) to find the steady state for this economy as a function ofthe structural parameters.3. Consider an exchange economy with two consumers named A and B. The two con-sumers have identical preferences: they each value consumption streams according toP∞t=0βtu(ct), where u has a constant elasticity of intertemporal substitution σ−1. Con-sumer i’s endowment of consumption goods is {ωit}∞t=0, i = A, B. Consumption goodsare perishable (i.e., they cannot be stored and used for consumption in future periods).(a) Carefully define a competitive equilibrium with date-0 trading for this economy.(b) Suppose that ωAt= 3 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 {3, 1, 3, 1, 3, 1, . . .} and consumer B’s endowment streamis {1, 3, 1, 3, 1, 3, . . .}. 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 acrosstime. Suppose that, as in part (b), consumer A’s endowment is 3 in every pe-riod but that consumer B’s endowment fluctuates: his endowment stream is{1/2, 3/2, 1/2, 3/2, 1/2, 3/2, . . .}. Find the competitive equilibrium allocationsand prices. To simplify the algebra, set σ = 1 (i.e., let the felicity function ube 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