复旦大学：《宏观经济学》课程教学讲义（PPT）Topics in Macroeconomic Policy：Optimal Monetary and Fiscal Policies

Topics in Macroeconomic Policy: Optimal Monetary and Fiscalpolicies 复9大学经学院

Topics in Macroeconomic Policy: Optimal Monetary and Fiscal Policies

A Standard framework To Study Monetary Policy Production Technology y1=a1+n1Aa1=P2△a1+e o Monopolistic Competition hc.≡W P V=-1 Government Spending Assume GS is a fraction t of output and hence g=-In(l-t) y=c+8,8=P81+ Household behavior maX ∑0 BInc 1+9 st WN1+(1+71)4-PC1+D The supply of labor satisfies P1=c1+m2 复9大学经 By some approximations. we also have s -(-Ex1-p)+ECHp=-hB虎

A Standard Framework To Study Monetary Policy ❖ Monopolistic Competition: a t at nt at a at t y = +  =   + −1 • Production Technology: mct  wt − pt − at −v = − • Government Spending [assume GS is a fraction τ of output and hence g = -ln(1-τ)]: g t t gt gt g gt t y = c + =  + −1 • Household Behavior: ( ) t t t t t t t t t t t t A W N r A PC D st N C = + + − +       + − + −  = +  1 1 0 1 1 . . 1 max ln    • By some approximations, we also have ct = −(rt − Et  t+1 − )+ Et ct+1   −ln  • The supply of labor satisfies: t t t nt w − p = c +

From the labor supply equation, we have +儿v2-a1-g The intertemporal equation vields 90y/=-(-E, m-p)+EJa+( Pg81(2) Finally, assume money demand as Pt=yt-rt Assume flexible price, from (1) and (2) one can see that money policy is neutral 巧=y++g(3) +△4+(-v)-ak(4 wherer=(-1)/(a+o)v=1/(1+) Note: (3)implies first best allocation can be achieved by setting v=u 复9大学经 Moreover,(4)indicates optimal policy rule of real interest rate(mechanism). iF Finally,(3)and(4)show this is a model F with prominent Keynesian properties. F

• From the labor supply equation, we have • The intertemporal equation yields • Finally, assume money demand as mc (1 )(y a ) g v (1) t = + t − t − t − t t t t m − p = y −r (3) t at gt y =  + + ( ) (1 ) (2) t t t t 1 t t 1 g t y = − r − E  −  + E y + −  g + + • Assume flexible price, from (1) and (2) one can see that money policy is neutral: • Note: (3) implies first best allocation can be achieved by setting v = μ. Moreover, (4) indicates optimal policy rule of real interest rate (mechanism). Finally, (3) and (4) show this is a model with prominent Keynesian properties. (1 )(1 ) (4) t a t g t r =  +  a + − −  g   (v −)/( +), 1/(1+) • where

Staggering Price Assume each firm resets price in any period only with probability 1-0. So g othe evolution of aggregate price can be approximated by the following =1+(-0)*(5) Note that this is different from simple sticky price assumption Firm solves the following programming to set p. max ∑,B(HR-C Subject to (5)and p=p With the static price rule p=mc u, we obtain n*=+(1-B2∑()E{mcn}o Here men denotes nominal marginal cost K From (1)and(3), we know mct-u=(+pl, x,=V-y

Staggering Price ❖ Assume each firm resets price in any period only with probability 1-θ. So the evolution of aggregate price can be approximated by the following (1 ) * (5) pt =pt−1 + − pt • Note that this is different from simple sticky price assumption. Firm solves the following programming to set pt *  ( )  = − s t s s s max  IR IC • Subject to (5) and pt = pt*. With the static price rule p = mc + μ, we obtain * (1 ) ( )   (6) 0 n t t s s s pt E mc +   = =  + −     • Here mcn denotes nominal marginal cost. From (1) and (3), we know ( ) t t t t t mc −  = 1+ x x  y − y

Then from(5)and(6), one can obtain a New Phillips curve 1t+1+K gok is a positive parameter(?). From (2) and(4), one can further obtain x=一(-Ex1-)+Ex(8) (7)and ( 8) characterize the dynamics of the model Note that there are 4 distortions in the model First, money-holding cost(Friedman Rule). We ignore this because of the ad hoc money demand assumption. Second static distortion from imperfect competition(solved by the subsidy v). The Third is firms' inability to adjust prices and the Fourth is the relative price distortion( due to the lack of synchronization in pricing ), which induces allocation inefficiency. Intuitively, the optimal 复9大学经济院 monetary policy requires x,n,=0 to eliminate the last two distortions(mechanisms)

• Then from (5) and (6), one can obtain a New Phillips curve (7) t t t 1 t  = E +x + • κ is a positive parameter (?). From (2) and (4), one can further obtain ( ) (8) t = − t − t t+1 − t + t t+1 x r E r E x • (7) and (8) characterize the dynamics of the model. • Note that there are 4 distortions in the model. First, money-holding cost (Friedman Rule). We ignore this because of the ad hoc money demand assumption. Second, static distortion from imperfect competition (solved by the subsidy v). The Third is firms’ inability to adjust prices and the Fourth is the relative price distortion (due to the lack of synchronization in pricing), which induces allocation inefficiency. Intuitively, the optimal monetary policy requires xt = πt = 0 to eliminate the last two distortions (mechanisms)

The Taylor Rule gAylor(1993)finds U.S. monetary policy follows a simple rule(why not M?) r1=con+1.5x,+0.5x Generally the Taylor Rule is not optimal compared with (4). So why we need it? The answer is positive since xt -T=0 is not implementable by the optimal rule(4). To see this, substituting(4) into(8), together with (7), one can obtain Ex K B+KE Clearly x, =T=0 is a solution. Uniqueness requires both of the eigenvalues of the coefficient matrix lie in the unit cycle, to eliminate bubble solutions(see Yuan and Song, pp 67-70 for a non-technical explanation). However, one can verify that only the smaller one lies in [O, 1]. Hence, the F optimal rule(4) cant implement x=兀=0.晚

The Taylor Rule • Taylor (1993) finds U.S. monetary policy follows a simple rule (why not M?): t t t r = con +1.5 + 0.5x • Generally the Taylor Rule is not optimal, compared with (4). So why we need it? The answer is positive since xt = πt = 0 is not implementable by the optimal rule (4). To see this, substituting (4) into (8), together with (7), one can obtain:             + =      + + 1 1 1 1 t t t t t t E x E x      • Clearly xt = πt = 0 is a solution. Uniqueness requires both of the eigenvalues of the coefficient matrix lie in the unit cycle, to eliminate bubble solutions (see Yuan and Song, pp. 67-70 for a non-technical explanation). However, one can verify that only the smaller one lies in [0,1]. Hence, the optimal rule (4) can’t implement xt = πt = 0

Use a specific Taylor Rule can avoid the indeterminacy r1=71+中n1+ Then one can verify (?)that uniqueness requires the following condition k(4-1)+(0-)>0(1 However, many economists argue that (9) itself is not practicable since central bank doesnt have sufficient knowledge about equilibrium real interest rate(why?). One can instead assume a simple Taylor rule =+nm1+2x1(11) It is straightforward to see(1O)ensures 3 uniqueness. But there are many other possible rules to avoid indeterminacy. So again we have to answer why we need s5 Taylor Rule? Gali(2003)compares Taylor Rule with two other candidates Money growth Peg and Interest Rate Peg (with implementation rule)

• Use a specific Taylor Rule can avoid the indeterminacy (9) t t t x t r r    x = +  + • Then one can verify (?) that uniqueness requires the following condition ( −1)+ (1− )  0 (10)    x • However, many economists argue that (9) itself is not practicable since central bank doesn’t have sufficient knowledge about equilibrium real interest rate (why?). One can instead assume a simple Taylor Rule: (11) t t x t r     x = +  + • It is straightforward to see (10) ensures uniqueness. But there are many other possible rules to avoid indeterminacy. So again we have to answer why we need Taylor Rule? Gali (2003) compares Taylor Rule with two other candidates: Money Growth Peg and Interest Rate Peg (with implementation rule)

Gali finds that Taylor rule is superior to Money growth Peg and Interest Rate Peg specifically, under some reasonable parameters, the volatility of inflation and g output gap is close to zero(0. 21 and 0.16 and the welfare loss is impressively close to zero(0.002%), substantially lower than MFG(2.30,0.98and0.22%) and IRP(2.31, 0.99and0.22% 1) Taylor Rule overwhelming MGP and IRP is not a robust result. One can see if the dynamics is characterized by more than two variables. 1. e x, and I it is possible that Taylor rule performs worse than somed simpler policy(e.g. capital -CKM?) 2) There is no inflation-output trade-off in K this model(since v=u). Consequently there is no time inconsistency problem 5 (why?), which seems unpleasant. So one ip can assume(why?) 丌1=BE,m1+1+xx1+l

• Gali finds that Taylor Rule is superior to Money Growth Peg and Interest Rate Peg. Specifically, under some reasonable parameters, the volatility of inflation and output gap is close to zero (0.21 and 0.16) and the welfare loss is impressively close to zero (0.002%), substantially lower than MPG (2.30, 0.98 and 0.22%) and IRP (2.31, 0.99 and 0.22%). • 1) Taylor Rule overwhelming MGP and IRP is not a robust result. One can see if the dynamics is characterized by more than two variables, i.e. xt and πt , it is possible that Taylor Rule performs worse than some simpler policy (e.g. capital - CKM?). • 2) There is no inflation-output trade-off in this model (since v = μ). Consequently there is no time inconsistency problem (why?), which seems unpleasant. So one can assume (why?) t t t t ut  = E +1 +x +

Some Related Researches Benhabib, Schmitt-Grohe and Uribe(2001, AER 02002, JET) find that Taylor Rules are possible to generate multiple equilibria(indeterminacy and bifurcation)and hence lead to high inflation. The mechanism lies in the active response of interest rate to inflation, i. e. p>l in (9) SU(2003, JET)assume distortion from monopolistic competition cant be removed by subsidy(with sticky price we have inflation incentive to increase employment) and government has no access to lump-sum tax (hence inflation plays as a non-distortionary tax to finance government spending). They also assume sticky price and Cia (welfare loss of inflation to firms and agents ). Then the optimal (with commitment) volatility of inflation is close to zero(compared with Gali, 2003) ACC(2002, pp.30)find that lack of commitment together with CIa and sticky price can generate multiple equilibria(non-monotone cost function of inflation- first decrease and then increase 复9大学经学院 compared with ACC, 2003. which assumes different money demand and technology

Some Related Researches • Benhabib, Schmitt-Grohe and Uribe (2001, AER, 2002, JET) find that Taylor Rules are possible to generate multiple equilibria (indeterminacy and bifurcation) and hence lead to high inflation. The mechanism lies in the active response of interest rate to inflation, i.e. фπ > 1 in (9). • SU (2003, JET) assume distortion from monopolistic competition can’t be removed by subsidy (with sticky price we have inflation incentive to increase employment) and government has no access to lump-sum tax (hence inflation plays as a non-distortionary tax to finance government spending). They also assume sticky price and CIA (welfare loss of inflation to firms and agents). Then the optimal (with commitment) volatility of inflation is close to zero (compared with Gali, 2003). • ACC (2002, pp.30) find that lack of commitment, together with CIA and sticky price, can generate multiple equilibria (non-monotone cost function of inflation – first decrease and then increase – compared with ACC, 2003, which assumes different money demand and technology)

Some Indirect Evidences Jordi Gali and his coauthors hope that the above model could be a standard g framework to study monetary policy (from both of the normative and positive aspects ). They provide some supportive evidences to convince their colleagues First, small variance of money supply is able to generate large volatility of output This wins support from many monetary economists Second, it is possible to generate liquidity trap (low 1/0). This wins support from many Keynesian economists Finally, it is consistent recent empirical finding: positive technological shock leads to decrease in employment. There 复9大学经 are many explanation to this interesting henomenon(example). Next we discuss 2 pl the explanation from Gali (1999, AER). Ba

Some Indirect Evidences • Jordi Gali and his coauthors hope that the above model could be a standard framework to study monetary policy (from both of the normative and positive aspects). They provide some supportive evidences to convince their colleagues. • First, small variance of money supply is able to generate large volatility of output. This wins support from many monetary economists. • Second, it is possible to generate liquidity trap (low 1/σ). This wins support from many Keynesian economists. • Finally, it is consistent recent empirical finding: positive technological shock leads to decrease in employment. There are many explanation to this interesting phenomenon (example). Next we discuss the explanation from Gali (1999, AER)