程序代写代做代考 flex Lecture 7: Monetary Policy

Lecture 7: Monetary Policy
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020

Monetary Policy in the New Keynesian Model
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Introduction
• Last week, we assumed a simple monetary policy rule in our 3-equation canonical New Keynesian (NK) model.
• Natural question, however, is how should monetary policy be conducted?
• To analyze this, we’ll consider a variant of our canonical New Keynesian model.
Suppose that the central bank’s goal is to minimize departures of output from its flexible-price level and departures of inflation from zero.
What’s the best monetary policy in this environment?
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Key Equations of Model
• Two key equations of the model, New Keynesian IS curve and New Keynesian Phillips curve:
y =E [y ]−1(i −E[π ])+uIS, t tt+1 t tt+1 t
θ>0 (1) πt = βEt [πt+1] + κ (yt − ytn) 0 < β < 1, κ > 0 (2)
θ􏰏 􏰎􏰍 􏰐 rt
• Assume ytn is the flexible-price level of output (i.e., natural level of output), together with two AR(1) shock processes:
uIS=ρuIS +eIS, −1<ρ <1 tISt−1t IS yn=ρyn +eY,−1<ρ<1 tYt−1t Y where eIS and eY are independent white-noise disturbances. tt 4/33 Objective of Central Bank • The central bank (CB) sets the nominal interest rate it every period, after observing uIS and yn. tt • Suppose the CB would like to minimize: minE 􏰉(y − yn)2􏰊 + λE[π2] where E [·] is the unconditional expectation and λ is the relative weight the CB places on inflation • CB dislikes departures of output from its natural level,1 and dislikes departures of inflation from zero. i.e., CB wants to stabilize output and inflation. 1As an alternative, we could assume the CB would like to minimize departures from the Walrasian level of output, y∗ > yn.
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Optimal Monetary Policy
• It turns out it’s possible for the CB to achieve both objectives, yt = ytn and πt = 0 for all t.
• Suppose that Et [πt+1] = 0. Then, (1) and (2) become: 
y =E [y ]−1i −E[π ]+uIS t tt+1 t tt+1 t
θ 􏰏􏰎􏰍􏰐 =0
π t = β E t [ π t + 1 ] + κ ( y t − y tn ) 􏰏 􏰎􏰍 􏰐
=0
• If the central bank sets it so that yt = ytn, it achieves both its objectives! (Goodfriend & King, 1997)
Blanchard and Galí (2007) call this the divine coincidence. 6/33

Implementing the Optimal Policy
• TheCBwouldliketoachieveyt =ytn andπt =0forallt. • Therefore,imposingy =yn,E[y ]=E[yn ]and
t t t t+1 t t+1 
Et[πt+1] = 0 on (1):
y =E[y ]−1i−E[π ]+uIS
t t t+1 θ t t t+1 t 􏰏􏰎􏰍􏰐 􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐
=yn n =0 t =Et[yt+1]
• Solving for it:
i=θ􏰉E􏰉yn 􏰊−yn+uIS􏰊≡rn,
t t t+1 t t t
where rtn is the economy’s natural rate of interest – i.e., the real interest rate that would prevail with flexible prices.
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Sunspots
• However, the desired equilibrium with yt = ytn and πt = 0 may not prevail if the public does not expect inflation will stay at the target.
• Model is prone to sunspot equilibria – i.e., equilibria with self-fulfilling beliefs.
• Formally, when will this occur?
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Blanchard-Kahn Condition
• Define y ̃t = yt − ytn, and substitute CB’s policy rule
i =θ[E[yn ]−yn+uIS]into(1)and(2),andre-write
t tt+1 t t system of equations as follows:
􏰄y ̃􏰅 􏰄1 1/θ 􏰅􏰄Ey ̃ 􏰅 t= tt+1
πt κ β+κ/θ Etπt+1 􏰏􏰎􏰍􏰐 􏰏 􏰎􏰍 􏰐􏰏 􏰎􏰍 􏰐
xt
xt =AEtxt+1
• Blanchard and Khan (1980): system has a unique solution only if all the eigenvalues of A are inside the unit circle.2,3
2An eigenvalue λ can be a complex number – i.e., λ = a + bi. “Inside the
case, the Blanchard-Kahn condition will be that the eigenvalues of B are all outside the unit circle.
A Etxt+1
√
unit circle” means
3The system can also be written as Etxt+1 = Bxt, where B = A−1. In this
a2 + b2 < 1. 9/33 Intuition for Blanchard-Kahn Condition • For intuition, consider one-dimensional case xt =(1/ρ)Et[xt+1] or Et[xt+1]=ρxt • Suppose x0 = x ̄ ̸= 0. Then: E0[x1] = ρx ̄ E0[x2] = E0[E1x2] = E0[ρx1] = ρ2x ̄ . . E 0 [ x t ] = ρ t x ̄ • Ifρ>1or1/ρ<1,xt explodesforanyx ̸̄=0. Therefore,in this case, only solution is xt = 0. • Ifρ<1or1/ρ>1,manysolutions
• The requirement that the series cannot explode is essential. 10/33

Sunspots in New Keynesian Model
• Recall, the matrix A for the New Keynesian model is 􏰄1 1/θ 􏰅
A= κ β+κ/θ
• To find eigenvalues of A, solve for det(A − λI) = 0, yielding:
(1+β+κ/θ)±􏰟(1+β+κ/θ)2 −4β 2
λ=
The larger eigenvalue is greater than one – hence, there are
multiple equilibria
• Et[πt+1] and Et[y ̃t+1] can increase today, whereby agents to expect inflation and output to gradually return to normal.
This will cause output and inflation to jump today.
Shifts in beliefs are self-fulfilling.
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How to Eliminate Sunspot Equilibria
• Consider an interest rate rule of the form
i t = r t n + φ π E t [ π t + 1 ] + φ y E t [ y ̃ t + 1 ]
• When Et [πt+1] = 0, Et [y ̃t+1] = 0, this becomes it = rtn.
• If φy = 0, system of equations becomes
􏰄 y ̃ 􏰅 􏰄 1 (1 − φ )/θ 􏰅 􏰄 E y ̃ 􏰅 t= π tt+1
πt κ β+κ(1−φπ)/θ Etπt+1
• Multiple equilibria ruled out when φπ > 1 and
φπ < 1 + 2θ(1 + β)/κ (upper bound for φπ is very large for normal parameters). • If CB responds neither “too strongly” or “too weakly” to deviations of expected inflation, indeterminacy is ruled out. 12/33 How to Eliminate Sunspot Equilibria • Alternatively, consider an interest rate rule of the form i t = r t n + φ π π t + φ y y ̃ t • Whenπt =0,y ̃t =0,thisbecomesit =rtn. • If φy = 0, system of equations becomes 􏰄y ̃􏰅 1 􏰄1(1−βφ)/θ􏰅􏰄Ey ̃ 􏰅 t= πtt+1 πt 1 + κφπ/θ κ β + κ/θ Etπt+1 • Multiple equilibria ruled out when φπ > 1.
• The threat to raise the interest rate in response to increases in inflation prevents any increases from occurring, and so never needs to be carried out.
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Practical Shortcomings
• We’ve seen examples of simple policy rules that attain the desired outcome (y ̃t = 0 and πt = 0 for all t).
• However, these rules require the nominal interest rate to be adjusted one-for-one with the rtn, the natural rate of interest.
This is unrealistic – rtn is not observable.
• Similar issue with ytn – this is not observable either (but this
is less of an issue, since the CB can set φy = 0)
• Nevertheless, simpler rules that respond only to inflation and output (deviations from steady state) imply small welfare losses, as long as a high enough weight is placed on inflation.
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Output and Inflation Stabilization
• The canonical New Keynesian model exhibits no long run trade-off in achieving the inflation and output objectives of the policy-maker.
• Most CBs perceive a trade-off between stabilizing inflation and stabilizing the gap between output and desired output.
• In practice, why might cause the divine coincidence to fail?
• This can happen if the CB would like to keep output close to y∗ (Walrasian output) instead of yn.
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What Might Break Divine Coincidence?
• Through New Keynesian Phillips curve, there is a long-run tradeoff between inflation and output.
πt = βEtπt+1 +κ(yt −yn) ⇒ π(1−β) = κ(y −yn) > 0 If inflation steady at π > 0, output steady at y > yn.
If y∗ > yn, CB willing to trade-off higher inflation to get
output closer to y∗.
• Suppose instead y∗ − yn = εn, where εn is a mean-zero
white-noise disturbance. Suppose CB conducts policy so that E[π ]=0,E[y ]=E[yn ]=E[y∗ ]=0:
t t+1 t t+1 t t+1 t t+1 y=E􏰉yn 􏰊−it+uIS
t t t+1 θ t π t = κ ( y t − y tn )
To achieve output objective, set it so that yt = yt∗. To achieve inflation objective, set it so that yt = ytn.
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Time Inconsistency of Monetary Policy
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Motivation
• We’ve seen the optimal policy leads to πt = 0.
• Actual policy often appears to be different from this.
• High inflation in many industrialized countries in the 1970s. • Countries have a tendency to pursue “activist,
countercyclical monetary policies.”
• Could there be an inflationary bias in monetary policy?
• Underlying source of inflationary bias is likely existence of an
output-inflation tradeoff.
⇒ According to New Keynesian Phillips curve:
Increase in πt ⇒ increase in yt, given Et[πt+1].
Yet, no significant tradeoff in long-run if Et[πt+1] increases.
• Kydland and Prescott (1977) and Barro and Gordon (1983): inability of policy-makers to commit to low-inflation policy can give rise to excessive inflation.
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Overview
• Suppose policy-makers have discretion to pick an appropriate policy each period.
• Given expectations, policy-maker would like to increase π to increase y to y∗ > yn.
• Public anticipates this, and expects high inflation.
• End result is high inflation, but output y = yn.
• Suppose instead policy-makers can commit to follow a low-inflation policy rule.
• Public expects low inflation.
• Policy-maker does not deviate from rule, as it’s binding. • End result is low inflation and output y = yn.
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Main Assumptions of Model
• Log output (y) determined by the Lucas supply curve: y=yn+b(π−πe), b>0 (3)
yn is log flexible-price output, πe is expected inflation.
• y > yn when π > πe (“surprise inflation”)
• (3) differs slightly from New Keynesian Phillips curve (which would have similar implications)
• Policy-maker directly controls inflation π.
⇒ given πe, policy-maker can control y by choosing π.
• Assume yn < y∗ 20/33 The Objective of the Policy-Maker • Target output of policy-maker is y∗ > yn.
• Target inflation of policy-maker is π∗.
• Policy-maker minimizes quadratic in inflation and output:
L = 1 (y − y∗)2 + 1a (π − π∗)2 (4) 22
• The parameter a reflects the relative importance of output and inflation in social welfare.
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General Problem of Policy-Maker
• The general problem faced by the policy-maker is min L = 1 (y − y∗)2 + 1a (π − π∗)2
π22 subject to
y =yn +b(π−πe)
• Re-write problem, substituting for Lucas supply curve:
minL= 1(yn +b(π−πe)−y∗)2 +1a(π−π∗)2. π22
• Two ways π and πe can be determined:
(1) Rules: Policy-maker makes binding commitment about π
before πe is determined
(2) Discretion: Policy-maker chooses π taking πe as given.
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The Rule/Commitment Solution
• In the rule/commitment solution, we assume the policy-maker makes a binding commitment about what π will be before πe is determined.
• Since commitment is binding, π = πe. From Lucas supply curve (3) ⇒ y = yn
• Policy-maker’s problem simplifies to
min L = 1 (yn − y∗)2 + 1a (π − π∗)2
π22
• FOC is
Hence, the solution is π = π∗. Output is y = yn.
a(π−π∗)=0 ⇒ π=π∗
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The Discretion Solution
• In the solution under discretion, the policy-maker chooses π, taking πe as given.
• In this case, the policy-maker’s problem is:
minL= 1(yn +b(π−πe)−y∗)2 +1a(π−π∗)2.
π22 • The FOC:
b(yn +b(π−πe)−y∗)+a(π−π∗)=0 • Re-write FOC:
b(y∗ −yn −b(π−πe))=a(π−π∗) 􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐
MB MC
Policy-maker equates marginal benefit of additional inflation (i.e., y closer to y∗) with marginal cost (π above π∗).
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Determination of Optimal π if πe = π∗
MB = b (y∗ − yn − b (π − πe)), MC = a (π − π∗) Given πe = π∗, policy-maker chooses π > π∗
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Marginal Cost/Benefit

Determination of Optimal π if πe > π∗
Marginal Cost/Benefit
MB = b (y∗ − yn − b (π − πe)), MC = a (π − π∗) Given higher πe, policy-maker chooses a higher π
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Optimal Inflation Under Discretion
• The FOC:
b(y∗ −yn −b(π−πe))=a(π−π∗) 􏰏 􏰎􏰍 􏰐 􏰏 􏰎􏰍 􏰐
MB
MC
• Solving for π:
π = π∗ + a + b2 (y∗ − yn) + a + b2 (πe − π∗) (5)
b b2
Optimal π is an increase function of πe (slope < 1) • The public, being rational, anticipates that π will be set using (5). Hence, they use (5) to form their expectations: b b2 πe = π∗ + a + b2 (y∗ − yn) + a + b2 (πe − π∗) 27/33 Determination of Equilibrium Inflation 28/33 Equilibrium Under Discretion • To summarize, in equilibrium, we must have: b b2 πe = π∗ + a + b2 (y∗ − yn) + a + b2 (πe − π∗) • Solving for πe yields the equilibrium inflation rate π=πe =πEQ. πe =π∗+b(y∗−yn) a 􏰏 􏰎􏰍 􏰐 πEQ • According to the Lucas supply curve, Equation (3), y=yn becauseπ=πe. ⇒ Compared to the commitment solution, inflation is higher but output is still y = yn. 29/33 Summary • Policy-maker is worse off under discretion. Why? • This is because the optimal policy is time inconsistent (or dynamically inconsistent) • Before expectations have been formed, the policy-maker would like to choose π = π∗. • If the public were to form their expectations so that πe = π∗, it’s optimal for the policy-maker to deviate from this policy (i.e., generate surprise inflation) • The public anticipates this. So, the end result is higher inflation but with output still equal to yn. • This is a possible explanation for why inflation was too high in the real world where discretionary policy was widespread. 30/33 Resolving the Time-Inconsistency Issue (1) Rules • Must be binding on the policy-maker • Very inflexible, limits the use of monetary policy to address shocks that hit the economy. • Yet, there may be ways of resolving time-inconsistency without resorting to fixed rules. (2) Reputation • The policy-maker will have an incentive to carry through optimal monetary policy if she cares enough about her reputation. (3) Delegation to an independent authority with different preferences/incentives • Example: an independent central bank which focuses only on inflation 31/33 Delegation • Rogoff (1985): appointment of a central banker with a greater distaste for inflation than society • Assume society’s loss function is L, but “conservative” central banker’s loss function is L′: L=1(y−y∗)2+1a(π−π∗)2, a>0 22
L′=1(y−y∗)2+1a′(π−π∗)2, a′>a 22
• Equilibrium π closer to the outcome under commitment:
π′ = π∗ + b (y∗ − yn) < π∗ + b(y∗ − yn) = π EQ a′ a EQ • Equilibrium converges to commitment outcome as a′ → ∞: lim π∗+b(y∗−yn)=π∗ a′→∞ a′ 32/33 Effect of Delegation Central banker with a′ > a ⇒ πEQ moves closer to π∗
b b2
π = π∗ + a′ + b2 (y∗ − yn) + a′ + b2 (πe − π∗)
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