复旦大学：《宏观经济学》博士生课程教学资源（讲义）Topics in Macroeconomic Policy：Time Inconsistency in Monetary and Fiscal Policies

Topics in Macroeconomic Policy Time Inconsistency in Monetary and Fiscal Policies 复9大学经学院

Topics in Macroeconomic Policy: Time Inconsistency in Monetary and Fiscal Policies

Time Inconsistency in Monetary Policy 90 Central Bank's Object u 2 min ∑ BE t+s +6-L stu,= - 丌)+E, which yields LL+.+ 1+6 Note that this solution deviates from 复9大 agents' expectation, or equivalentI central bank's ex-ante commitment. This is the time inconsistent problem found by 4 Kydland-Prescott(1977, JPE)and Barro Gordon(1983, JPE)

( ) ( ) t e t t t T s t s t s t s st u u u E       = − − +       + − = + + . . 2 2 min 0 2 2 Central Bank’s Object: which yields     + − + + = 1 t e t t u Note that this solution deviates from agents’ expectation, or equivalently, central bank’s ex-ante commitment. This is the time inconsistent problem found by Kydland-Prescott (1977, JPE) and Barro￾Gordon (1983, JPE). Time Inconsistency in Monetary Policy

Agents Response Update Expectation s Find Rational Expectation (Fixed Point Problem) E d +Tt +Er 1+b u The Consequent equilibrium Inflation becomes u t O 1+6 Note that this time consistent solution is inferior to the solution 复9大学经学院 with zero inflation commitment. 3

Agents’ Response: Update Expectation ❖ Find Rational Expectation (Fixed Point Problem): u e t   1  = −         + − + + −  = −      1 1 1 t e t t e t t t u E E • The Consequent Equilibrium Inflation becomes t u  t    + = − + 1 1 1 • Note that this time consistent solution is inferior to the solution with zero inflation commitment

Four Solutions . Eirst-Best Allocation: government is allowed to manipulate agents g expectation to achieve zero unemployment and inflation . Cheating (Time-Inconsistent) Solution: government would like to deviate from her precommitment because she always has incentive to reduce unemployment by unexpected inflation &Precommitment Solution government is assumed to commit to her ex-ante announced policy. &s Time Consistent Solution government is unable to commit to her policy 复9大学经学院 wElfare Comparison

❖ First-Best Allocation: government is allowed to manipulate agents’ expectation to achieve zero unemployment and inflation. ❖ Cheating (Time-Inconsistent) Solution: government would like to deviate from her precommitment, because she always has incentive to reduce unemployment by unexpected inflation. ❖ Precommitment Solution: government is assumed to commit to her ex-ante announced policy. ❖ Time Consistent Solution: government is unable to commit to her policy. ❖ Welfare Comparison: 1 > 2 > 3 > 4 Four Solutions

Time Inconsistency in Fiscal Policy s The following model, developed g 0 by Fischer(1981, JEDC), is a benchmark to start with ☆ Household's behavior maxUC.c, L 2 g Subject to +k=2=(+了k+(=r1 This yields households response function Ig) k 56 A benevolent government would like 3 to maximize households'welfare,45 subject to g=ckk+rzl

Time Inconsistency in Fiscal Policy ❖ The following model, developed by Fischer (1981, JEDC), is a benchmark to start with. ❖ Household’s Behavior maxU(c ,c ,l, g) 1 2 Subject to c k y c ( )Rk ( )zl k l + = = 1− + 1− 1 2 This yields households’ response function c ( g) c ( g) k( g) l( g) k l k l k l k l , , , , , , , , 1 2         • A benevolent government would like to maximize households’ welfare, subject to g k zl k l = +

Four Solutions First-Best Allocation: maximize g households welfare subject to resource constraint . o Cheating (Time-Inconsistent) Solution: government would like to deviate from her previous policy at the second period when k has been accumulated, because she always has incentive to reduce distortion by only taxing k &s Precommitment Solution: Open Loop Solution in the Stackelburgh Game 复9大学 ☆ Time Consistent solution: Subgame经 Perfect Equilibrium o Welfare Comparison >2>3

Four Solutions ❖ First-Best Allocation: maximize household’s welfare subject to resource constraint. ❖ Cheating (Time-Inconsistent) Solution: government would like to deviate from her previous policy at the second period when k has been accumulated, because she always has incentive to reduce distortion by only taxing k. ❖ Precommitment Solution: Open￾Loop Solution in the Stackelburgh Game. ❖ Time Consistent Solution: Subgame Perfect Equilibrium. ❖ Welfare Comparison: 1 > 2 > 3 > 4

What causes Time Inconsistency sufficient Policy Instruments: One can easily find that there is no time go inconsistency in the second model if the government is allowed to levy lump-sum tax (or proportional income tax? ) However, this explanation seems not satisfactory since it can't explain why time inconsistency rises in the first model Persson and Tabellini(1994)argue that ti is due to the sequential nature of policy-making. This, again, is only partially correct Now most economists believe that TI is induced by the heterogeneity of interests, or equivalently some 复9大学经学院 externality This is first pointed out by F Chari, Kehoe and Prescott (1989)

What Causes Time Inconsistency • Insufficient Policy Instruments: One can easily find that there is no time inconsistency in the second model if the government is allowed to levy lump-sum tax (or proportional income tax?). However, this explanation seems not satisfactory since it can’t explain why time inconsistency rises in the first model. • Persson and Tabellini (1994) argue that TI is due to the sequential nature of policy-making. This, again, is only partially correct. • Now most economists believe that TI is induced by the heterogeneity of interests, or equivalently, some externality. This is first pointed out by Chari, Kehoe and Prescott (1989)

Atomistic Assumption wIthout loss of generality, we can write down a"semi-indirect utility function?" for the ith household K.T Precommitment Solution(the first equation is the households behavior and the second is the government's choice) Oak a 0 =0 Time Consistent Solution(the first equation is the households behavior and 旦 the second is the government's choice) 0 0 Clearly the discrepancy lies in atomistic i agents, or similarly, the externality of F aggregate capital K

Atomistic Assumption ❖ Without loss of generality, we can write down a “semi-indirect utility function” for the ith household: v (k ,K, ) i i • Precommitment Solution (the first equation is the households’ behavior and the second is the government’s choice): 0 = 0           +     =   i i i i i K v K v k v   • Time Consistent Solution (the first equation is the households’ behavior and the second is the government’s choice): 0 = 0   =   i i i i v k v  • Clearly the discrepancy lies in atomistic agents, or similarly, the externality of aggregate capital K

Application for Monetary Policy . One can easily find that time inconsistency in our first model lies in the Phillips Curve s However, the microfoundation of the phillips curve again lies in the heterogeneity of interests everyone hopes others to underforecast inflation &So, the phillips curve and consequently the time 复9大学 inconsistency problem no longer 9 exist in communist world o conclude, time inconsistency i seems to be a general outcome from the world with heterogeneous interests

Application for Monetary Policy ❖ One can easily find that time inconsistency in our first model lies in the Phillips Curve. ❖ However, the microfoundation of the Phillips Curve again lies in the heterogeneity of interests: everyone hopes others to underforecast inflation. ❖ So, the Phillips Curve and consequently the time inconsistency problem no longer exist in a “communist world”. ❖ To conclude, time inconsistency seems to be a general outcome from the world with heterogeneous interests

A Formal Model to stud Time Inconsistent Monetary Product Market Policy Monopolistic Competition: Price W/M s One cash good and one credit good. p part of goods are determined at the begin of each period, i.e. the price, Psi, is sticky during one period. The flexible price is denoted by Pi Aggregate price is P .dP dPf=1-H 今 QusehglAlsen+hnC2+yh( st GA=wn+B+M-p PC +T+D M1≥B1CnA1=M1+B 复9大学经 Given prices, households choose M, B and ng RP,Cl=P W P (2) P Clearly the Precommitment Solution follows Friedman Rule. ie.R=1. For Time Consistent Solution, Government chooses G ex-ante, or equivalently pt, subject to(1)and(2)

A Formal Model to Study Time Inconsistent Monetary Policy ❖ Product Market: ❖ Monopolistic Competition: Price = W/λ ❖ One cash good and one credit good. μ part of goods are determined at the begin of each period, i.e. the price, Ps i , is sticky during one period. The flexible price is denoted by Pf i . ❖ Aggregate price is Pi . dPi / dPf i = 1-μ. ❖ Households  : ( ) t t t t t t t t t t t t t t t t t t t t t t t M P C A M B GA W n R B M P C P C T D C C n st  = + = + + − − + + + + − +  = 1 1 1 1 1 2 2 0 1 2 max  ln ln  ln 1 . . • Given prices, households choose M, B and n. (1) Rt P1t C1t = P2t C2t (2) 1 1 1 2 2 2 2 2 t f t t t t t t t P P C C C P W n C =  − − =  − • Clearly the Precommitment Solution follows Friedman Rule, i.e., R = 1. For Time Consistent Solution, Government chooses G ex-ante, or equivalently Pf , subject to (1) and (2)