Econ 510a (second half)Yale UniversityFall 2006Prof. Tony SmithHOMEWORK #3This homework assignment is due at NOON on Friday, November 17 in Marnix Amand’s mailbox.1. This problem introduces wealth inequality into a two-period economy with productionlike the one discussed in lecture on Friday, November 10.(a) Suppose that there are two types of consumers distinguished by their initial en-dowments of capital. In particular, type-1 consumers (who comprise fraction θ ofthe population) are richer than type-2 consumers (who comprise fraction 1 − θof the population): type-1 consumers are endowed with k10units of capital andtype-2 consumers are endowed with k20units of capital, where k10> k20. The twotypes of consumers are identical in all other respects. Each consumer takes pricesas given (in particular, each consumer takes the aggregate, or total, capital stockin period 1 as given) when making savings decisions in period 0. The equilibrium(or consistency) condition is that the total savings of the two types of consumersin period 0 must equal the aggregate capital stock that consumers take as givenwhen deciding how much to save.Assume that each consumer’s utility function takes the form u(c0) + βu(c1), withu(c) = log(c). The production technology available to firms is: y = kαn1−α, with0 < α < 1, where y is the firm’s output and k and n are the services of capitaland labor, respectively. (Consumers do not value leisure; if each consumer’sendowment of time is normalized to one, then n = 1 in each time period.)Derive the equilibrium aggregate capital stock in period 1 as a function of primi-tives (i.e., the parameters α, β, and θ and the initial capital stocks k10and k20).(b) Use your answer to part (a) to show that changes in k10and k20that keep aggre-gate capital in period 0 (i.e., θk10+ (1 − θ)k20) constant have no effect either onequilibrium aggregate savings or on equilibrium prices. This is a version of an ag-gregation theorem for this economy: holding the total amount of capital in period0 constant, the behavior of the aggregates in this economy does not depend onthe distribution of capital in period 0.(c) Suppose that the felicity function takes the form: u(c) = (1 − σ)−1(c1−σ− 1),where σ > 0. Does an aggregation theorem like the one described in part (b) holdfor this economy? Explain why or why not.2. 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)?3. This problem studies a neoclassical growth model with an externality in produc tion.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 aggregatecapital stock in competitive equilibrium. (Hint: Find a typical consumer’s Eulerequation and then impose equilibrium conditions.) Use this equation to find anexpression for the steady-state aggregate capital stock in competitive equilibrium.(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 subsidiz es savings at a proportional rate τand finances these subsidies by means of a lump-sum tax on consumers. Theinvestment subsidy is c onstant 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