Econ 510a (second half)Yale UniversityFall 2004Prof. Tony SmithHOMEWORK #1This homework assignment is due at 5PM on Friday, October 29 in Ted Papageorgiou’s mailbox.1. Consider a growth model with two sectors, one producing consumption goods and oneproducing investment goods. Consumption is given by ct= F (kct, Lct) and investmentby it= G(kit, Lit), where kctis the amount of capital used in time period t in theconsumption-goods sector and kitis the amount of capital used in time period t in theinvestment-goods sector. Similarly, Lctand Litare the amounts of labor used in thetwo sectors. Total capital evolves according to: kt+1= (1 − δ)kt+ it. At each pointin time, capital and labor must be allocated across the two sectors: kt= kct+ kitand L = Lct+ Lit, where the total amount of labor L is fixed. Finally, suppose thatF and G have constant returns to scale. (Recall that this means, for example, thatF (K, L) = Lf(K/L), where f(K/L) ≡ F (K/L, 1).)As in the Solow growth model, we will make ad hoc behavioral assumptions, in thiscase about how to allocate ktand L at a point in time. In particular, assume that,for all t, kct/Lct= kit/Lit, i.e., the capital-to-labor ratios are the same in both sec-tors. In addition, assume that a constant fraction θ of total labor is allocated to theconsumption-goods sector: Lct= θL, where θ ∈ (0, 1).State and prove a “global convergence” result. That is, provide assumptions on theprimitives (F, G, L, δ, θ) and show that, given these assumptions, there is global conver-gence to a steady state (i.e., the dynamic system converges from any initial condition).2. Consider a growth model with capital accumulation equation kt+1= f(kt) if t is evenand kt+1= g(kt) if t is odd. Assume that:(i) f(0) = g(0) = 0.(ii) f0(0)g0(0) > 1.(iii) limk→∞f0(g(k))g0(k) < 1 and limk→∞g0(f(k))f0(k) < 1.(iv) f and g are strictly increasing and strictly concave.Show that there is global convergence to a “two-cycle” in which ktoscillates betweentwo values. How are these values determined?Suppose instead that f(kt) = aktand g(kt) = bkt, where a and b are positive constants.Completely characterize the dynamics of ktas a function of a and b.3. For the neoclassical growth model that we developed in lecture on October 20, wederived the following Euler equation:u0(f(k) − k0) = βu0(f(k0) − k00)f0(k0).Recognizing that k0= g(k) and k00= g(k0) = g(g(k)), where g is the optimal decisionrule, one can state a “functional” version of the Euler equation:u0(f(k) − g(k)) = βu0(f(g(k)) − g(g(k)))f0(g(k)).As described in Section 4.2.3 in Chapter 4 in the lecture notes, the slope of g at thesteady state k∗can be found by differentiating the functional Euler equation, evaluatingit at k∗, solving a quadratic equation in g0(k∗), and then discarding the “unstable” root.Make the following assumptions about functional forms:(i) u has constant elasticity of intertemporal substitution σ−1:u(c) =c1−σ− 11 − σ,where σ > 0 and u(c) = log(c) if σ = 1.(ii) f(k) = Akα+ (1 − δ)k, where A > 0, α ∈ (0, 1), and δ ∈ [0, 1].Discuss how A, α, β, δ, and σ affect the speed of convergence to the steady state. (Asdiscussed in Section 4.2 in Chapter 4 of the lecture notes, the speed of convergencenear the steady state is inversely related to the slope of the decision rule at the steadystate.)4. This problem studies a neoclassical growth model with “time-to-build”, as in the fa-mous article by Kydland and Prescott (Econometrica, 1982) recently awarded theNobel Prize in Economics. Specifically, suppose that it takes two time periods to buildand install new capital (rather than one period as in the standard growth model). Lets2tdenote new investment projects initiated in period t; s2tis a choice variable at allpoints in time. The stock of partially completed investment projects in period t (i.e.,investment projects one period from completion in period t) is denoted s1t. The stocksof partially completed and new investment projects are related by s1,t+1= s2t. Thecapital accumulation equation reads: kt+1= (1 − δ)kt+ s1t, where ktis the stock ofcompleted projects.The resource constraint is: ct+ it= F (kt), where investment it= (1 − φ)s1t+ φs2tandφ ∈ [0, 1]. In other words, starting a new investment proje ct of size s is a commitmentto invest resources φs in the first stage (or period) of its construction and resources(1−φ)s in the second stage of its construction. Finally, assume that F has the standardproperties.(a) Formulate the Bellman equation for the social planning problem in this economy,assuming that a typical consumer’s preferences are given by:P∞t=0βtu(ct), whereu has the standard properties. Be clear about what the state variables are andwhat the choice variables are.(b) Derive the first-order and envelope conditions for this problem.(c) Under the assumptions that F (k) = kαand φ = 1, derive an expression for thesteady-state capital stock in terms of the structural