Econ 510a (second half)Yale UniversityFall 2005Prof. Tony SmithHOMEWORK #4This homework assignment is due at the beginning of class on Wednesday, November 16.1. Consider a neoclassical growth model in which consumers have time-separable prefer-ences given by:P∞t=0βtu(ct). Let the aggregate production (or resource) function takethe form:f(¯k, n) = A¯kαn1−α+ (1 − δ)¯k,where δ is the rate of depreciation of capital. The parameters satisfy: 0 < β < 1, A > 0,0 < α < 1, and 0 < δ ≤ 1. Consumers are endowed with one unit of time in eachperiod but do not value leisure (so that n = 1). In this problem, you will solve explicitlyfor the recursive competitive equilibrium of this economy under the assumptions thatu(c) = log(c) and δ = 1. (Assume too 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 behavioral 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 own holdingsof capital)?2. This problem studies a neoclassical 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, who own the factors of production, areendowed with k0units of capital in period 0 and with one unit of time in each period.There is a large number of identical profit-maximizing firms each of which has thefollowing production technology:f(k, n,¯k) = Akαn1−α¯kγ+ (1 − δ)k,where k is the amount of capital rented by the firm, n is the amount of labor hiredby the firm,¯k is the aggregate capital stock, δ is the rate of depreciation of capital.The parameters satisfy: 0 < γ < 1 − α, 0 < α < 1, and 0 < δ ≤ 1. Thus there is aproductive externality from the rest of the economy: a higher aggregate capital stockincreases the productivity of each firm. A typical (small) firm takes the aggregatecapital stock as given when choosing its inputs.(a) Carefully define a sequential competitive equilibrium for this economy.(b) Carefully define a recursive competitive equilibrium for this economy.(c) Find a second-order difference equation that governs the evolution of the econ-omy’s aggregates in competitive equilibrium. (Hint: Find a typical consumer’sEuler equation and then impose equilibrium conditions.)(d) Display the Bellman equation for the social planning problem in this economy.The planner internalizes the externality in production: his production technologyish(¯k, n) ≡ f (¯k, n,¯k) = A¯kα+γn1−α+ (1 − δ)¯k.Is the competitive equilibrium allocation Pareto optimal? (Hint: Compare theplanner’s Euler equation to the second-order difference equation that you foundin part (c).)(e) Now introduce a government that subsidizes savings at a prop ortional 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.(f) For what subsidy rate τ is the competitive equilibrium steady-state aggregatecapital stock equal to the steady-state aggregate capital stock in the planningproblem?3. Consider a neoclassical growth model in which consumers value leisure: a typicalconsumer’s preferences over sequences of consumption and leisure are described byP∞t=0βtu(ct, `t), where ctis consumption in period t and `tis leisure in period t. Thefelicity function u is strictly increasing in both arguments, strictly concave, twice con-tinuously differentiable, and satisfies limc→0uc(c, `) = ∞. In all other respects, thiseconomy is identical to the one that we have discussed in class.(a) Carefully define a sequential competitive equilibrium for this economy.(b) Carefully define a recursive competitive equilibrium for this economy. (Hint: You needtwo functions to describe the behavior of the aggregate economy.)(c) Show that the competitive equilibrium allocation for this economy solves a social plan-ning problem. (Hint: Compare first-order conditions.)(d) Let the aggregate production function take the form f(¯k, ¯n) =¯kα¯n1−α+(1−δ)¯k, where¯n is aggregate labor supply, and let u(c, `) = λ log(c) + (1 − λ) log(`). The parameterssatisfy: 0 < α < 1, 0 < δ ≤ 1, and 0 < λ < 1. Solve explicitly (in terms of parameters)for the steady-state values of aggregate capital and aggregate leisure in