Econ 510a (second half)Yale UniversityFall 2007Prof. Tony SmithHOMEWORK #1This homework assignment is due at noon on Friday, November 2. Please put your assign-ment in Evrim Aydin’s mailbox.1. (a) In the Solow growth model that we developed in lecture, suppose that the rateof depreciation is zero. Show that, in this case, there is no steady state but,nonetheless, the growth rate of the capital stock converges to zero.(b) Does the neoclassical growth model have a steady state if the rate of depreciationis zero? Explain.2. In the neoclassical growth model that we developed in lecture, assume that u hasconstant elasticity of intertemporal substitution σ−1(i.e., let u(c) = (1 − σ)−1c1−σ)and that F (k) = kα+ (1 − δ)k. Discuss how changes in σ and α affect the speedof convergence to the steady state. (Recall that, near the steady state, the speed ofconvergence is inversely related to the slope of the optimal decision rule at the steadystate.) Try to give economic intution for your findings.3. Consider the planning problem for a simple finite-horizon neoclassical growth model:max{ct, kt+1}Tt=0TXt=0βtlog(ct),given k0= 10 and subject to the constraint that ct+ kt+1= Akαt+ (1 − δ)kt. Setβ = 0.95, δ = 0.1, and α = 0.4. Cho ose A so that the steady-state value of capital inthe corresponding infinite-horizon model is 100.(a) Set T = 20 and solve the model numerically (say, in Matlab) using the “shooting”method described in lecture: start by guessing a value for k1, solve for k2from theEuler equation at time 0, then solve for k3from the Euler equation at time 1, andso on, until kT +1is found. Then vary k1and repeat until the appropriate valueof kT +1(what is it?) is found. Calculate the savings rate at each point along theoptimal path.(b) Find the lowest value for T such that the highest value of capital between periods0 and T exceeds 90.14. By differentiating the functional Euler equation in the neoclassical growth model, weshowed in lecture that the derivative of the optimal decision rule, evaluated at thesteady state, satisfies a quadratic equation with two roots, one between 0 and 1 andone larger than β−1. (Aside: Be sure that you know how to show this!) One wayto rule out the explosive root (the one larger than β−1) is to argue that the optimalpath for capital converges to a (unique) steady state for any (positive) initial capitalstock. This problem explores another way to rule out the explosive root, under theassumption that both the value function and the decision rule are twice differentiable(this does not hold in general, but does hold for many specific choices for u and F , andwe will assume it here).Given the optimal decision rule k0= g(k), the value function solves the followingfunctional equation:v(k) = u(F (k) − g(k)) + βv(g(k)).Differentiate this equation twice, evaluate it at the steady state k∗, and then solve forv00(k∗). (Hint: Use the first-order condition for capital to eliminate some terms.) Thenuse the fact that v is strictly concave to argue that (g0(k∗))2< β−1, implying thatg0(k∗) cannot be larger than
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