Econ 525a (first half)Fall 2007Yale UniversityProf. Tony SmithHOMEWORK #2This homework assignment is due on October 3.1. Consider a two-period growth model with endogenous leisure but no uncertainty. Thereare two types of consumers, A and B. Type-A consumers comprise fraction θ ofthe population and type-B consumers comprise fraction 1 − θ of the population. Allconsumers have time-separable preferences with a felicity function given by:log(c) + D log(`),where c is consumption and ` is hours of leisure in any given time period. A typicaltype-A consumer has initial holdings of capital equal to kA, while a typical type-Bconsumer has initial holdings of capital equal to kB< kA.Find the equilibrium prices and quantities in this economy. Show that changes in kAand kBthat leave the initial aggregate capital stock unchanged have no effect eitheron prices or on aggregate quantities (i.e., total capital accumulation and total laborsupply in either period).2. Consider an infinite-horizon, complete-markets exchange economy with two types ofconsumers, A and B. Both types of consumers have time-separable preferences with afelicity function that exhibits constant elasticity of intertemporal substitution (EIS).The EIS of a type-i consumers is γi, i = A, B. At any point in time there are fourpossible states of the economy. In states 1 and 2, the aggregate endowment is high; instates 3 and 4, the aggregate endowment is low. In states 1 and 3, type-A consumersreceive fraction θ of the aggregate endowment; in states 2 and 4, type-A consumersreceive fraction λ of the aggregate endowment. The state of the economy follows aMarkov chain.Find an equilibrium relationship between the (log) consumption of a typical type-Aconsumer and the (log) consumption of a typical type-B consumer that holds at everydate and state. Do you think this relationship would be likely to hold in observeddata?3. Imagine a consumer who either has a low wage (w = w1) or a high wage (w = w2) andwhose asset holdings a are restricted to lie in the set {a1, a2, a3}, where a1< a2< a3.1The wage w follows a discrete-state Markov chain with transition probabilities πij=P (w0= wj|w = wi). Let the consumer’s savings decision rule a0= g(a, w) take thefollowing form:(i) a1= g(a1, w1), a1= g(a2, w1), a2= g(a3, w1)(ii) a2= g(a1, w2), a3= g(a2, w2), a3= g(a3, w2)(a) Suppose that π22= 0.95 and π11= 0.5. Find the invariant distribution overthe two wage levels w1and w2. That is, find two numbers p1and p2such that iffraction piof consumers have wage equal to witoday, then these fractions replicatethemselves in the next period.(b) Now find the invariant distribution pijover the discrete state space {a1, a2, a3} ×{w1, w2}.(c) Suppose that consumers are distributed initially according to the invariant distri-bution that you computed in part (b). What is the probability that a (randomlychosen) consumer with the lowest level of asset holdings today has the highestlevel of asset holdings three periods from now?(d) Suppose that initially consumers are spread uniformly over the state space. Com-pute the dynamics of the distribution of consumers as time evolves and verifynumerically that this distribution converges to the one that you computed in part(b). (You may want to write a program, say, in Matlab, to automate the