Monetary policy and business fluctuations
Seminar problems, Spring 2007
Seminar 1
i)
What are business cycles? Compare the definition of Burns and Mitchell (1946)
with that of Lucas (1977).
ii)
What is meant by “business cycle facts”? Describe some important stylized facts
of business cycles that are common in most countries.
iii)
What is meant by leading and lagging indicators? What kind of data would you
look at if you want to make forecasts?
iv)
What is a spurious relationship?
v)
Describe briefly the structure of the basic Real Business Cycle model. What are
the key differences with a traditional Keynesian empirical model? What was the
rationale for developing RBC models?
vi)
Describe the key assumptions that are necessary for an RBC model to have a
steady state path. Explain why one ensures that the models have a steady state
path.
vii)
Show that the utility function U(C, L) = log (C) + ν log (L), where C is
consumption and L is leisure, is consistent with steady state growth
Seminar 2
i)
Consider the “high substitution economy” described by King and Rebelo. Explain
briefly the main features that lie behind this characteristic, and the motivation for
choosing them.
ii)
Derive the optimal capacity utilization of a representative firm, and interpret the
result.
iii)
Discuss the effect of the size of ξ, the elasticity of the derivative of the
depreciation rate with respect to the rate of capital utilization, Dδ(zt).
iv)
Describe briefly Gali’s empirical evidence suggesting that a positive technology
shock has a negative impact on firm’s use of labour input, and discuss the
relevance of this finding for the basic RBC model with shocks to total factor
productivity.
v)
What are the effects of a temporary increase in government consumption on output
and private consumption in a standard RBC model? Compare with the effect
within a traditional Keynesian model.
Seminar 3
Consider the paper by Gali, Gertler and Lopez-Salido, and the paper by Gali.
i)
Explain briefly the following concepts
a. wage markup
b. price markup
c. inefficiency gap
What is the basis for GGL-S’s argument that some form of wage rigidity is central
to business fluctuations?
Based on GGL-S and G, discuss the merits of using fiscal policy to stabilize the
economy. Try also to include additional arguments, in both directions, that are not
included in their analysis.
What is the basis for Gali’s argument that “there appears to be a global trend
towards more countercyclical fiscal policies”. How would Norway fit into this
picture? (In Norwegian, consider e.g. the National budget 2007, 3.2.3. In English,
consider e.g. the Inflation report 3/2006, figure 3.17).
ii)
iii)
iv)
Seminar 4
A
A simplified RBC model with additive technology shocks
Consider an economy consisting of a constant population of infinitely lived individuals. The
representative individual maximizes the expected value of


t 0
u (Ct )
1   
t
,
 0
The instantaneous utility function u(Ct), is u(Ct) = Ct – θCt2, θ > 0. Assume that Ct is always
in the range where u’(Ct) is positive.
Output is linear in capital, plus an additive disturbance: Yt = AKt + et. There is no
depreciation, so Kt+1 = Kt + Yt – Ct, and the interest rate is A. Assume that A = ρ.
Furthermore, assume that the disturbance follows a first-order autoregressive process:
et = φet-1 + εt, where -1 < φ < 1 and where the εt’s are mean-zero, i.i.d shocks.
(i)
(ii)
(iii)
(iv)
Find the first-order condition (Euler equation) relating Ct and expectations of Ct+1.
(Tip: use the informal perturbation method where you suppose the individual
reduces period-t consumption by ΔC. He then uses the greater wealth in period t+1
to increase consumption above what it otherwise have been.)
Guess that consumption takes the form Ct = α + βKt + γet. Given this guess, what
is Kt+1 as a function of Kt and et?
What values must the parameters α, β and γ have for the first-order condition in
part (i) to be satisfied for all values of Kt and et?
What are the effects of a one-time shock to ε on the paths of Y, K and C?
B
Consider a basic RBC (e.g. as in King and Rebelo)
i)
ii)
How does a productivity shock affect the economy?
Explain the difference between temporary, persistent and permanent productivity
shocks, and the difference in their effects on the economy. Why do King and
Rebelo argue that productivity shocks must be persistent?
Why cannot the productivity shocks be replaced by other types of shock?
iii)
Seminar 5
A
A simplified RBC model with additive taste shocks
This follows the setup in problem 4A, but the technology shocks are replaced by taste shocks.
Consider an economy consisting of a constant population of infinitely lived individuals. The
representative individual maximizes the expected value of


t 0
u (Ct )
1   
t
,
 0
The instantaneous utility function u(Ct), is u(Ct) = Ct – θ(Ct + νt)2, θ > 0. The νt’s are taste
shocks, assumed to be mean-zero, i.i.d shocks. Furthermore, we assume that Ct is always in
the range where u’(Ct) is positive.
Output is linear in capital: Yt = AKt, there is no depreciation, so Kt+1 = Kt + Yt – Ct, and the
interest rate is A. Assume that A = ρ.
(i)
(ii)
(iii)
(iv)
B
Find the first-order condition (Euler equation) relating Ct and expectations
of Ct+1.
Guess that consumption takes the form Ct = α + βKt + γνt. Given this guess,
what is Kt+1 as a function of Kt and νt?
What values must the parameters α, β and γ have for the first-order
condition in part (i) to be satisfied for all values of Kt and νt?
What are the effects of a one-time shock to ν on the paths of Y, K and C?
Aggregate demand and long-run unemployment
(i)
(ii)
(iii)
Describe the empirical basis for Ball’s conclusion that passive monetary policy
during a recession can lead to permanently higher unemployment
Explain briefly the economic mechanisms that Ball argues may explain his result.
Discuss alternative explanations for Ball’s results, and the possible policy
conclusions we can draw from this.
Seminar 6
A.
Wages set one period in advance
Assume that nominal wages are set for one period but that they can be indexed to the price
level:
wtc  wt0  b( pt  Et 1 pt )
where w0 is a base wage and b is the indexation parameter ( 0  b  1 ).
a. How does this change modify the aggregate supply equation (5.17) in Walsh?
b. Suppose the demand side of the economy is represented by a simple quantity
equation, mt  pt  yt and assume mt  vt , where vt is a mean zero shock. Assume the
indexation parameter is set to minimize Et 1 (nt  Et 1nt* )2 and show that the optimal
degree of wage indexation is increasing in the variance of v and decreasing in the
variance of e.
B
The New Keynesian Phillips curve
Consider the following New Keynesian Phillips curve:
 t   Et t 1   yt
a. Find the ‘forward solution’ of the Phillips curve (where only current and future output
gaps enter on the right-hand side).
b. Assume that the process for the output gap is given by
yt   yt 1   t , 0    1 , where  t is a white noise shock.
Find the solution for inflation as a function of yt and  t only.
Seminar 7
1. Assume that output is given by
(0.1)
yt   ( t  Et 1 t )  ut
where Et 1ut  0 .
Consider two alternative specifications of the preferences of the monetary authorities:
(0.2)
1
Lt  [( t   * )   ( yt  y * ) 2 ]
2
(0.3)
Lt  12 ( t   * ) 2   yt
A. Give an economic interpretation of the difference between the two specifications.
B. Derive the solution for inflation and output under a discretionary policy for the two
loss functions. Compare and discuss the results.
2. Consider two monetary policy regimes; (i) discretionary policy, and (ii) strict inflation
targeting, i.e.,  t   * . Let var(ut )   2 . Compute the expected loss, i.e., ELt, under the two
regimes based on the loss function (0.2). Is one regime always superior to the other? How
does the relative performance of the two regimes depend on the output target y*?
Seminar problem 8
Suppose that the economy can be represented by the following New Keynesian model:
 t   Et t 1   yt  e t
(0.4)
(0.5)
yt  Et yt 1 
1

(it  Et t 1 )  ut
where et and ut are white noise processes.
Suppose that the central bank sets the interest rate according to the following “Taylor rule”:
it    t   y yt
(0.6)
A. Discuss advantages and disadvantages with simple interest rate rules like (1.3)
B. Solve the model, i.e., write the endogenous variables as functions of the exogenous
shocks. (Hint: Et t 1  Et yt 1  0 because of no autocorrelation)
C. Suppose that the central bank’s loss function is
1
L  [ t2   yt2 ]
2
Can the central bank achieve optimal policy by using a rule like (1.3)?
Seminar problem 9
Consider the following model:
(0.7)
yt  Et yt 1 
1

(it  Et t 1 )  ut
(0.8)
 t   Et t 1   yt  et
(0.9)
L  Et   k [( t2k   yt2k )

k 0
A. Explain in words the difference between discretion and commitment.
B. Derive the first-order conditions under discretion and commitment respectively. What
characterizes the difference?
C. It can be shown that the commitment solution implies a stationary price-level. Vestin
(2001) shows that the optimal monetary policy can be replicated under a discretionary
policy if the inflation target is replaced by a price-level target. Write a loss function
that can be interpreted as price-level targeting.
D. Discuss similarities and differences between inflation targeting and price-level
targeting.
Seminar problem 10
Consider the following model:
(0.10)
yt  Et yt 1  t (it  Et t 1 )  ut
(0.11)
 t   Et t 1   yt  et
(1.3)
1
Lt  [ t 2   yt2 ]
2
The shocks ut and et are white-noise processes.
1. Suppose that the demand shock ut cannot be observed perfectly by the central bank,
but the central bank can observe everything else. Assume further that the central bank
receives a noise signal of the shock, such that the bank’s estimate is given by
ut  ut  zt , where zt is white noise with variance E ( zt2 )  var( zt )   z2 .
Write the first-order condition for optimal policy under discretion. Solve for
the
interest rate. (Hint: All expected future variables are zero, due to
discretion and no
auto-correlation).
How does uncertainty about ut affect monetary policy?
2. Suppose that αt is given by t     t , where εt is a white-noise shock with
variance E ( t2 )  var( t )   2 . Suppose further that the central bank sets the
interest rate before εt is realised. Write the first-order-condition for optimal
policy under discretion. Solve for the interest rate.
How does uncertainty about αt affect monetary policy? Explain the difference
between additive uncertainty and multiplicative uncertainty.