Instructor : Tony Smith November 6, 2007T.A. : Evrim Aydin SaherCourse : Econ 510a (Macroeconomics)Term : Fall 2007 (second half)Problem Set 1 : Suggested Solutions1. (a) In the Solow growth model we have:kt+1= (1  )kt sf(kt)wheref(kt) = LtF (KtLt; 1)i. We know that in the steady state kt+1= kt= k , and sok = (1  )k  sf(k )k = sf(k )If  = 0 and if we assume there is a non-zero savings rate, i.e., s 6= 0, thenk = 0:And so there is no non-trivial steady state.ii. The growth rate of capital stock is de…ned as:kt+1 ktkt=sf(kt)kt limt!1kt+1 ktkt= limt!1sf(kt)kt limt!1= limt!1sf(kt)kt(since  = 0)= limt!1sf0(kt) (by l’Hôpital’s rule)= 0 (by Inada condition).(b) In the neoclassical growth model we have the Euler equation:u0(F (K)  K0) = u0(F (K0)  K00) F0(K0)F (K) = f(K)  (1  )K1At the steady state kt+1= kt= k , and so:u0(F (K )  K ) = u0(F (K )  K ) F0(K )F0(K ) = 1f0(K )  (1  ) = 1f0(K ) = 1+ (1  )If  = 0 we havef0(K ) = 1+ 1Since all of our assumptions on f still hold, a steady state exists.2. Recall that near the steady state, the speed of convergence is inversely related to the slope of the optimaldecision rule (policy function) at the steady state:k0= g(k) g(k ) + g0(k )(k  k )And so we have:k0 k = g0(k )(k  k )To …nd an expression for the slope of the decision rule we start with our functional Euler equation:u0(f (k)  g (k)) + u0(f (g (k))  g (g (k))) f0(g (k)) = 0Di¤erentiate both sides with respect to k; we have0 = u00(f (k)  g (k)) (f0(k)  g0(k))+u00(f (g (k))  g (g (k)))hf0(g (k)) g0(k)  (g0(k))2if0(g (k))+u0(f (g (k))  g (g (k))) f00(g (k)) g0(k)Evaluate this equation at k = k , and notice that g (k ) = k and f (k ) = 1, we have0 = u00(f (k )  k ) (f0(k )  g0(k ))+u00(f (k )  k )hf0(k ) g0(k )  (g0(k ))2i+u0(f (k )  k ) f00(k ) g0(k )Plug in c = f (k )  k , after some manipulation, we get(g0(k ))21 +1+u0(c )u00(c )f00(k )f0(k )g0(k ) +1= 02Or:21 +1+u0(c )u00(c )f00(k )f0(k ) +1= 0where  is equal to g0(k ) :Looking at this quadratic equation we can see that curvature of the production f unc tion will speed upconvergence, while curvature of the utility function will slow it down.The economic intuition is as follows:(a) the greater the curvature of the production function in the steady state, the greater th e change in themarginal return of capital when we are perturbed from steady state. And so, p eople want to invest moreto make use of this opportunity, which speeds up convergence(b) the greater the curvature of the utility function in the steady state, the greater the change in themarginal utility when perturbed from the steady state. Since people want to smooth cons ump tion, theywill consume more today to o¤set the change, which slows down the capital accumulation.To analyze the parameters we plug in the utility and production functions:21 +1+c  (1  ) (k )2 +1= 0in whichk = (f0)11=1   (1  )11c = f (k )  k = k k 1 De…neB = 1 +1+c  (1  ) (k )2= 1 +1+k k 1  (1  ) (k )2= 1 +1+ (1  )(k )1k 1 = 1 +1+(1  ) (1   + ) (1   +   )We know that the solutions of characteristic equation are =B pB2 412We are interested in the smaller root, i.e.1=B pB2 4123First, the e¤ect of : Di¤erentiating w.r.t. , we get@1@=12 1 BpB2 41!@B@Obviously the …rst term is negative. The second term is@B@=(1   + )12(1   +   ) +1 1()= (1   + )1   +1  22< 0Therefore, we have@1@> 0. Since the total e¤ect of increasing  reduces the curvature (@B@< 0), itmakes the speed of convergence slower (higher ).Second, the e¤ect of :Di¤erentiating w.r.t. , we get@1@=12 1 BpB2 41!@B@Obviously the …rst term is negative. The second term is@B@= (1  ) (1   + ) (1   +   )2< 0Therefore, we have@1@> 0. The economic intuition goes as follows. Since in the steady state thereare no intertemporal issues,  won’t in‡uence steady-state capital level. Therefore, the only e¤ect of on convergence wil be on the direct e¤ect of intertemporal substitution. If  increases, the curvature ofthe utility function increases and convergence slows down. Consequently, an increase in  causes slowerconvergence (higher ).3. Our …rst step is to derive the Euler equation for this problem, and then we can solve the questionnumerically.The consumer’s problem is:vt(kt) = maxkt+1ln(f(kt)  kt+1) + vt+1(kt+1)Note that here value function depends on the time subscript t. F.O.C. for t  (T  1) is1ct+ v0t+1(kt+1) = 04From the Envelope Theorem, we have:v0t(kt) =1ctf0(kt)If we update and then substitute for v0t+1(kt+1) we have our Euler equation:1ct+1f0(kt+1) =1ctWe havect= f(kt)  kt+1f(kt) = Akt+ (1  )ktand sokt+2= f(kt+1)  (f (kt)  kt+1)f0(kt+1)kt+2=Akt+1+ (1  )kt+1 (Akt+ (1  )kt kt+1)Ak1t+1+ (1  )kt+2= A(1 + )kt+1+ (1  )(1 + )kt+1 (Akt+ (1  )kt kt+1)Ak1t+1+ (1  )We can see that it is a second-order nonlinear di¤erence equation, with the boundary condition k0= 10and kT +1= 0.Before we go on to the numerical calculation, we solve for A. The in…nite horizon steady state k solvesf0(k ) = 1A =1 (1  )(k )1=(0:95)1 (1  0:1)0:4(100)0:41A = 6:0476(a) Savings: 0:48; 0:45; 0:42; and so on(b) For an error bound jkT +1j < 0:01, when time horizon T exceeds 37, the highest value of capitalbetween periods 0 and T exceeds 90. The approximate period-one capital stock for T = 37 isk1= 16:3833123.4. We have the functional equationv(k) = u (F (k)  g(k)) + v(g(k))which we di¤erentiate twice5v0(k) = u0(F (k)  g(k)) (F0(k)  g0(k)) + v0(g(k))g0(k)v00(k) =u00(F (k)  g(k)) (F0(k)  g0(k))2+ u0(F (k)  g(k)) (F00(k)  g00(k)) +hv00(g(k)) (g0(k))2+