Solution for Problem Set 3

Transcription

1 Solution for Problem Set 3 Q. Heterogeneous Expectations. Consider following dynamic IS-LM economy in Lecture Notes 8: IS curve: y t = ar t + u t (.) where y t is output, r t is the real interest rate, u t is a demand shock. LM curve: m t p t = by t ci t + v t (.2) where m t is the money supply, p t is the price level, i t is the nominal interest rate and v t is the money demand shock. Fisher equation: i t = r t + ~ E t p t+ p t (.3) where ~ E t p t+ denotes the subjective expectation on the price level in period t + : Supply curve (Lucas type): y t = y + p t Et ~ p t (.4) where y is the potential output, for simplicity we set y = 0. Money supply: m t = m t + t (.5) where t is the money supply shock. Finally, we assume there are two types of expectations. fraction of agents form their expectations via adaptive learning, and agents have rational expectations, the subjective expectation on price E ~ t p t is then given by ~E t p t = p t 2 + ( ) E t p t : (.6) Question: Solve the equilibrium price process according to the method we discussed in the notes. (Hint: conjecture that the process of price takes the form of p t = 0 p t + p t m t + 3! t + 4 t ). Answer:

2 2 The dynamic system (.) to (.5) can be simpli ed to m t = p t 2 Et ~ p t c E ~ t p t+! t (.7) where! t = c a u t v t : = b + c a + c + and 2 = c a + b and m t = m t + t : (.8) According to (.6), we conjecture the equilibrium price process is p t = 0 p t + p t m t + 3! t + 4 t : (.9) Plugging (.9) into (.6) and combining with (.8) yields ~E t p t = ( ) 0 p t + [( ) + ] p t 2 + ( ) 2 m t (.0) and ~E t p t+ = ( ) 0 p t + [( ) + ] p t + ( ) 2 m t : (.) Putting (.0) and (.) gives p t = + c ( ) 2 2 ( ) 2 c ( ) 0 m t 2 ( ) 0 + c [( ) + ] c ( ) 0 p t + 2 [( ) + ] c ( ) 0 p t ( ) c ( ) 2 t 0 +! t : c ( ) 0 Comparing last equation with (.9), parameters j can be determined by 0 = 2 ( ) 0 + c [( ) + ] c ( ) 0 = 2 [( ) + ] c ( ) 0 2 = + c ( ) 2 2 ( c ( ) 0 ) 2 3 = c ( ) 0 4 = 2 c ( ( ) 0 ) 2 :

3 3 Q2. RBC model with investment adjustment cost and capacity utilization. Consider following RBC model with investment adjustment cost. The resource constraint is where A t follows AR() process max fc tk t+ I tn tg t=0 Capital accumulates according to where (u t ) = u t > : Questions: X E 0 t [log C t + a n log ( N t )] (2.) t=0 C t + I t = A t (u t K t ) Nt (2.2) log A t log A = (log A t log A) + " t : (2.3) K t+ = ( (u t )) K t + " + 2 # 2 It I t (2.4) I t (a). Denote f t q t g as the Lagrangian multipliers for (2.2) and (2.4), respectively. Write down the full dynamic system including rst order conditions. (b). Suppose in the steady state A = u = : Given deep parameters f N ss g solve the steady states. Note: can be determined by the optimal conditions. (c). Log-linearize the full system you get in (a) around the steady state. Answer: (a). The full dynamic system is given by following rst order conditions where Y t = A t (u t K t ) N t = q t " + 2 t q t = E t =C t = t (2.5) a n N t = t ( ) Y t N t (2.6) q t+ ( t+ ) + t+ Y t+ (2.7) K t+ t Y t = q t 0 (u t ) K t (2.8) u t 2 # # 2 It It It It+ It+ + E t "q t+ (2.9) I t I t I t I t I t

4 4 and additional equations (2.2), (2.3) and (2.4). (b). From (2.9), we have q = : (2.7) implies that (2.2) and (2.4) imply that Y K I=Y = K=Y = = ( ) = : (2.0) ( ) (2.) C=Y = I=Y: (2.2) Given the steady state labor N ss K ss can be solved from the last equation K ss = ( ) N ss : (2.3) And it is straightforward to obtain fy ss I ss C ss ss q ss g. To determine the parameter from (2.8) we have For a n we have (c). Loglinearized system is given by N ss ^C t = ^t = Y K : (2.4) a n = ( ) N ss N ^N ss t = ^ + ^Y t ^Nt ^Y t = ^A t + ^u t + ^K t + ( ) ^N t ^q t = ( ) E t ^q t+ ^ t + ^Y t ^u t = ^q t + ( ) ^u t + ^K t ^ t = ^q t + ^I t E t ^It+ ^u t+ N ss Y C : (2.5) + [ ( )] E t ^t+ + ^Y t+ ^Kt+ ^Y t = C Y ^C t + I Y ^I t ^K t+ = ( ) ^K t ^A t = ^A t + " t : ^u t + ^I t

5 5 Q3. RBC model with general form of preference. Consider following RBC model with general form of preference. where max fc tk t+ X tn tg t=0 E 0 X t=0 t C t N t X t (3.) X t = C t X t : (3.2) We assume that 0 < <, >, > 0, > 0, and 2 [0 ] : The presence of X t means that preferences is non-time-separable in consumption and labor. The resource constraint is where A t follows AR() process C t + K t+ ( ) K t = A t K t N t (3.3) log A t log A = (log A t log A) + " t : (3.4) Questions: (a). Write down the full dynamic system including rst order conditions. (b). Suppose = 0: This preference is called Greenwood-Hercowitz-Hu man (GHH) preference. Derive the labor supply curve through the rst order conditions you obtain in (a). Note: wage rate W t = Y t =N t : Show that under GHH preference, there is no income e ect on the labor supply, i.e., an increasing in income due to the positive technology shock does not shift the labor supply curve. is N ss : (c). Solve the steady state for the system in (a), where 2 [0 ], A = and steady-state labor (d). Log-linearize the system in (a) around the steady state you obtain in (c). Answer: (a). Denote Lagrangian multipliers for (3.2) and (3.3) as t and t respectively. The full dynamic system is given by following FOCs C t N t X t + t C t X t = t (3.5) C t Nt X t Nt X t = t ( ) Y t N t (3.6) C t Nt X t Nt + t = ( ) E t t+ C t+ X t (3.7) t = E t t+ Y t+ + ( ) K t+ (3.8)

6 6 and additional equations (3.2), (3.3) and (3.4). (b). If = 0 X t is a constant (for simplicity, we set X t = ). (3.5) and (3.6) imply that N t = ( ) Y t N t : (3.9) Obviously, the increasing in output does not shift the labor supply curve (3.9) since the t does not appear in (3.9). (c). In the steady state, (3.8) and (3.3) imply K Y = ( ) C Y = K Y (3.0) Given N ss we can solve fk ss C ss Y ss g : (3.2) implies X ss = C ss : From (3.6) and (3.7), we have (N ss ) = Y C ss = (N ss ) : From (3.5), we have ss = (C ss ) h (N ss ) i + (N ss ) : (d). Loglinearized system is given by C t N t X t + t C t X t = t (3.5) " ^C (N ss ) (N ss ) t (N ss ) ^N t + ^X # t + ( ) ^N t + ^X t = ^ t + ^Y t ^Nt (3.6) 8 >< ^C 9 (N ss ) t (N + ss ) ( ) ^N (N >: ss ) t + ^X >= t + + ^N ( ) > ^ t = E t ^ t+ + ^C t+ ^X t (3.7) t Y=K ^ t = E t^t+ + E t ^Yt+ ^Kt+ (3.8) Y=K + ^X t = ^C t + ( ) ^X t (3.9) C Y ^C t + K Y ^K ( ) K t+ ^K t = Y ^Y t : (3.0) ^Y t = ^A t + ^K t + ( ) ^N t (3.) ^A t = ^A t + " t : (3.2)

7 7 Q4. RBC model with habit formation and indivisible labor. Consider the following RBC model with habit formation preference. max fc tk t+ N tg t=0 where 2 (0 ) : The resource constraint is where A t follows AR() process X E 0 t [log (C t C t ) a n N t ] (4.) t=0 C t + K t+ ( ) K t = A t K t N t (4.2) log A t log A = (log A t log A) + " t : (4.3) Questions: (a). Write down the full system including rst order conditions. (b). Solve the steady state given steady-state labor is N ss and A = : (c). Log-linearize the system around the steady state. Answer: (a). The full dynamic system is given by C t C t E t C t+ C t = t (4.4) and (4.2), (4.3). a n = t ( ) Y t (4.5) N t t = E t t+ Y t+ + ( ) (4.6) K t+ Y t = A t Kt Nt (4.7) (b). (4.6) and (4.2) imply that K Y = ( ) C Y = K Y : (4.8) Given the steady state N ss we can solve fc ss K ss Y ss g : From (4.4), we have And (4.5) gives the parameter value of a n : ss = : (4.9) Css

8 8 (c). Loglinearized system is given by ^ t = ^C t ^C t 0 = ^ t + ^Y t ^Nt : Y=K ^ t = E t^t+ + E t ^Yt+ ^Kt+ Y=K + ^Y t = C Y ^C t + K Y ^K ( ) K t+ ^K t Y ^Y t = ^A t + ^K t + ( ) ^N t ^A t = ^A t + " t : + E t ^C t+ ^C t