Econ 510a (second half)Yale UniversityFall 2004Prof. Tony SmithHOMEWORK #4This homework assignment is due at the beginning of the help session on Thursday, November 18.1. Consider an infinite-horizon one-sector growth model with an externality in production.Leisure is not valued and the (representative) consumer has time-separable preferenceswith discount factor β ∈ (0, 1). Consumers own the factors of production. Capitaldepreciates at rate δ. There is a large number of identical firms each of which has thefollowing production technology:F (k, `,¯k) = Akα`1−α¯kγ,where k is the capital rented by the firm,¯k is the aggregate capital stock, and theparameters α and γ satisfy 0 < γ < 1 − α and α ∈ (0, 1). Thus there is a productiveexternality from the rest of the economy: a higher aggregate capital stock increasesthe productivity of each firm. A typical (small) firm takes the aggregate capital stockas given when choosing its inputs.(a) Define a recursive competitive equilibrium for this economy. Be clear about whichvariables consumers and firms take as given when they solve their optimizationproblems. Find a second-order difference equation that governs the evolution ofthe economy’s aggregates. (Hint: Find a typical consumer’s Euler equation andthen impose equilibrium conditions.)(b) Write the planning problem for this economy in recursive form. The plannerinternalizes the externality in production: his production technology isG(¯k, `) ≡ F (¯k, `,¯k) = A¯kα+γ`1−α.Is the competitive equilibrium allocation Pareto optimal? (Hint: Compare theplanner’s Euler equation to the second-order difference equation that you foundin part (a).)(c) Now introduce a government that subsidizes savings at a proportional rate τand finances these subsidies by means of a lump-sum tax on consumers. Theinvestment subsidy is constant across time but the lump-sum tax varies over timeso as to balance the government’s budget in every period. Define a recursivecompetitive equilibrium for this economy.(d) For what subsidy rate τ is the competitive equilibrium steady-state aggregatecapital s tock equal to the steady-state aggregate capital stock in the planningproblem?2. Consider a neoclassical growth model with logarithmic felicity function, Cobb-Douglasproduction function F (¯k, `) = A¯kα`1−α, full depreciation of the capital stock in oneperiod (the rate of depreciation is equal to 1), and inelastic labor supply (leisure isnot valued). In this problem, you will solve explicitly for the recursive competitiveequilibrium of this economy (assuming that the economy is decentralized in the mannerthat we have discussed in class).(a) Suppose that aggregate capital evolves according to k0= G(k) = sF (k, 1). (Youwill verify the validity of this conjecture below.) Find explicit formulas for thevalue function v(k, k) and the decision rule k0= g(k, k) of a “small” (or typical)consumer who takes the law of motion for aggregate capital as given. The func-tions v and g depend on s as well as on primitives of technology and preferences.(Hint: Guess that v(k,¯k) = a + b log(k + d¯k) + e log(¯k) and then find expressionsfor the unknown coefficients a, b, d, and e in terms of the structural parametersα and β and the be havioral parameter s.)(b) Find the competitive equilibrium value of s by imposing the consistency conditionG(k) = g(k, k). Verify that the resulting law of motion for aggregate capital solvesthe planning problem for this economy. Display v and g for the equilibrium valueof s.(c) How does an increase in aggregate capital affect the savings behavior and the(indirect) utility of a typical consumer (holding fixed the consumer’s ow n holdingsof capital)?(d) How does the equilibrium utility of a typical consumer vary with aggregate capital(taking into account that the consumer’s own holdings of capital equal aggregatecapital in equilibrium)?3. Consider the planning problem for a neoclassical growth model with logarithmic utility,full depreciation of the capital stock in one period, and a production function of theform y = zkα, where z is a random shock to productivity. The shock z is observedbefore making the current-period savings decision. Assume that the capital stock cantake on only two values: i.e., k is restricted to the set {¯k1,¯k2}. In addition, assume thatz takes on values in the set {¯z1, ¯z2} and that z follows a Markov chain w ith transitionprobabilities pij= P (z0= ¯zj|z = ¯zi).(a) Let ¯z1= 0.9, ¯z2= 1.1, p11= 0.95, and p22= 0.9. Find the invariant distributionassociated with the Markov chain for z. Use the invariant distribution to computethe long-run (or unconditional) expected value of z; that is, compute E(z) =π1¯z1+ π2¯z2, where π1and π2determine the invariant distribution.(b) Let β = 0.9, α = 0.36,¯k1= 0.95kss, and¯k2= 1.05kss, where kssis the steady-statecapital stock in a version of this model without shocks and with no restrictions oncapital (i.e., kss= (αβ)1(1−α)). Let g(k, z) de note the planner’s optimal decisionrule. Prove that g(ki, zj) = kj.(c) The decision rule from part (b) and the law of motion for z jointly determinean invariant distribution over (k, z)-pairs. Find this distribution. (That is, findprobabilities πij= P (k = ki, z = zj) that “repro duce” themselves: if πijis theunconditional probability that the economy is in state (ki, zj) today, then it is alsothe unconditional probability that the economy is in this state tomorrow. For amore complete discussion of this concept, see pp. 78 and 79 in the lecture notesby Per Krusell.) Use your answer to compute the long-run (or unconditional)expected values of the capital stock and of output.(d) In Matlab, use the optimal decision rule, the law of motion for z, and a randomnumber generator to create a simulated time series {kt, yt}Tt=0, given an initial con-dition (k0, z0). Compute T−1PTt=1ktand T−1PTt=1ytfor a suitably large value ofT and confirm that these sample means are close to the corresponding populationmeans that you computed in part (c). (You may find useful the Matlab code byLjunqvist and Sargent for simulating a Markov chain that I have posted on thecourse web