ECON 3410/4410: Seminar exercises,
autumn 2007.
28 August, 2007.
The zipped data sets referred to in the exercises can be downloaded from.
http://folk.uio.no/rnymoen/rnyteach.html, follow the Teaching-autumn-2007 link.
Exercise 1
1. Consider the dynamic model made up of equation (1.3) and (1.4) in IDM.
Assume that in the initial situation (in period t = 0), εd,0 = εs,0 = 0 and
P0 = P̄ and X0 = X̄ where P̄ and X̄ are stationary values.
(a) Assume c < −a, and draw a figure similar to figure 1.3 in IDM. Assume
that εd,1 < 0 and that all other values of εd,t and εs,t are zero, ie a
temporary demand shock. Use the graph to illustrate the behaviour of
Pt and Xt in period t = 1, 2 and 3. What happens when t becomes really
large?
(b) Re-do the analysis, but with one change in the set of assumptions: that
c > −a.
2. Using the same model as in question 1, illustrate the behaviour of Pt and Xt
in period t = 1, 2 and 3 in the case of a permanent demand shock, i.e., εd,t < 0
for t = 1, 2, ....... Examine both constellation of slopes of the two curves.
3. Consider the model defined by equation (1.5) and (1.6) in IDM.
(a) What is the expression for the slope of the long-run demand curve?
(b) What happens to the slope of the long-run demand curve if b1 = 1? Can
you think of an interpretation?
(c) Assume that in the initial situation (in period t = 0) εd,0 = εs,0 = 0 and
P0 = P̄ and X0 = X̄ where P̄ and X̄ are stationary values. Assume that
b1 = 1 and c1 = 0. Assume that εs,1 > 0 and that all other values of εd,t
and εs,t are zero, ie a temporary supply shock in period 1.
i. What are the long-run effects of this positive supply shock?
ii. Show, graphically, that in period 1: X1 > X0 and P1 < P0 ; and in
period 2: X2 < X1 and P2 > P1 .
iii. If, instead of a < 0, we set a = 0, what are the effects of the supply
shock on X1 and P1 ? What about period 2 and 3?
4. Classify the following as either stock or flow variables
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(a) The labour force
(b) The rate of unemployment
(c) The trade surplus
(d) The money stock
(e) Private consumption expenditure
(f) Government debt
(g) Government deficit
(h) The price level
(i) Inflation
(j) The supply of foreign currency to the central bank.
5. Stock variables usually change gradually: at each point in time their rate of
change is finite. But some stock variables can jump from one value to another
at any given point in time. In such instances the derivative of the variable with
respect to time is infinite–at least in principle: with actual data (quarterly,
monthly, daily, hourly) the rate of change is finite, but for the observations
when jumps occur, that rate will be huge. Such stock variables are dubbed
jump-variables. Modern economic theory implies that the nominal exchange
rate is a jump variable.
From NORKVAR.zip, which can be downloaded from the course internet page,
obtain the data file NORKVAR.xls.The dataset contains a quarterly time series
CPIVAL, which is an effective exchange rate index for Norway, and two bilateral exchange rates in the dataset: SPEURO and SPUSD, between kroner
and Euro and kroner and USD respectively.
(a) What is the difference between a bilateral exchange rate and an effective
exchange rate.
(b) The base year of the series is 2001. What does that mean?
(c) Are there any examples of jump-behaviour in these series?
(d) Show the correlation between CPIVAL and SPEURO on the one hand,
and CPIVAL and SPUSD on the other.1 Which one is largest–and why?
Hint: calculate the simple correlation coefficient, or use a scatter plot, to
asses the degree of correlation.
(e) Derive a series for the bilateral exchange rate between USD and EURO.
What are the most noteworthy developments over the available period?
6. Calculate the time series of the quarterly rate of change of CPI,the official
Norwegian consumer price index (published by Statistic Norway). What is
the common name of this series? Calculate also the 4-quarter rate of change
in CPI. Plot the two series in a graph. Why is the annual rate of change
“smoother” than the quarterly rate of change?
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Before the introduction of the EURO (and the abolition of national currencies in most EU
countries), the SPEURO series represent the exchange rate between kroner and the European
Currency Unit, ECU.
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7. Calculate also the approximate growth rate, based on the log transform of CPI
(see e.g., the appendix to IDM). How good is the approximation?
Exercise 2
1. Answer question 2-5 in the exercises to chapter 2 in IDM.
2. Consider the dynamic model made up of equation (1.3) and (1.4) in IDM. Assume that the models parameters are known with certainty. Give the necessary
and sufficient conditions for
(a) a unique solution, and
(b) an asymptotically stable solution.
3. Answer question 6 of the exercises to chapter 2 in IDM.
4. Consider the model given in question 3 of Exercise 1.
(a) For the case of c1 = 0, find the final equation for Xt .
(b) Is the system stable?
(c) How is stability affected by setting
i. b1 = 1?
ii. a = 0?
iii. b1 = 1 and a = 0?
5. Answer question 8 in the exercises to chapter 2 in IDM
6. In this course we learn how to analyse the dynamics of linear economic models.
Hence it is important to be able recognize a linear model when you see it! The
key property is that the models are linear in parameters, and linear models
may therefore be non-linear in the variables. For example, consider the three
alternative models of the relationship between the two variables Y and X:
(1)
(2)
(3)
Y
ln Y
ln Y
= α + βX
= α + β ln X
= α + βX
(a) A model which is linear (also) in variables has the property that the first
derivative is a constant (independent of X). Which of the three equations
has this property?
(b) A model which is linear (only) in parameters has the property that the
first derivative is itself a function of X. Describe how the first derivatives
of (1)-(3) depend on X.
(c) Which equation implies a constant elasticity of Y with respect to X?
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(d) There are other important linear model specifications that you need to be
comfortable with as well. Let for example Y denote the rate of inflation
in %, and let X denote the rate of unemployment (also in %). What is
the qualitative difference between the ‘linear Phillips curve’ in (1) and the
two ‘convex Phillips curves’ given by the two alternative specifications:
(4)
(5)
Yt = α + β ln Xt , and
1
Y = α+β ?
X
(e) Choose values of the coefficients of equation (4) and (5) in way that makes
both models consistent with the following: At a 4% initial unemployment,
a 1 percentage point increase in unemployment reduces inflation by 1
percentage point.
(f) Equation (5) is called the reciprocal model. Using for example the β
value from question (e), sketch the Phillips curve (i.e., for different values
of U ). Discuss in class: What could become main policy issues if the
reciprocal model was the correct model of inflation in an economy?
7. Using the variables CPIVAL, CPI and CPIKONK in the datafile NORKVAR.xls,
calculate a real exchange rate index for Norway. Draw a graph of the series,
call it REX for example.
Exercise 3
1. Inflation is measured in different ways, using different price indices. Which
operational definition of the consumer price index is used by Norges Bank
(The Central Bank of Norway)? What about Bank of England? Use the
internet for information!
2. A ready-made dataset is available in the file wage price prod.zip on the
course page.
(a) Show inflation and unemployment in a scatter plot, i.e., an empirical
Phillips curve.
(b) Try to draw a line which, intuitively, represents the empirical relationship
between the rates of inflation and unemployment.
(c) Are there signs of a non-linear relationship in your data set? Explain
your findings. (Hint: make a scatter plot with inflation and the log of
the rate unemployment).
(d) Are there periods (“sub samples”) where the Phillips curve “fits better”
than in other periods? If so, do you have you any explanation for this
phenomenon?
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3. Assume that the rate of inflation is given by the augmented price Phillips curve
(1)
∆pt = β 0 + β 1 Ut + α∆pt−1 + εt ,
t = 1, 2, .....
where the subscript t denotes time period (e.g., quarter or year) and ∆ denotes
the difference operator, i.e., ∆pt ≡ pt − pt−1 where pt denotes the (natural)
logarithm of the domestic price level. Ut denotes the unemployment rate (i.e.,
this variable is a rate, it is not log-transformed). εt is the disturbance.
(a) Give (a least) one economic theoretical argument for inclusion of the
lagged inflation rate on the right hand side of the equation.
(b) Show that (1) is an example of an ADL model of the relationship between
inflation and the rate of unemployment.
(c) Assume that the impact multiplier of ∆pt with respect to a change in the
rate of unemployment is −0.01. What does this imply for the value of
β1?
(d) Assume that the long-run multiplier is −0.20. Using the answer to c.,
what is the implied value of α?
(e) In your own words: explain the concept of long-run multiplier in this
application.
(f) A majority of modern economists now routinely sets α = 1. What is
their rationale?
(g) Assume that α = 1, and that the NAIRU rate of unemployment is 0.05.
What is the corresponding value of β 0 ?
4. Using wage price prod.zip: is there a wage Phillips curve in this data set?
Note that there are three unemployment series, you have to choose one of
them for your analysis. Are there sub-periods where the relationship is more
pronounced? Does it matter whether the rate of unemployment is in log or
not?
Exercise 4
1. Use the data in Norw wage shares.zip, and formulate a view on the following
issues
(a) The degree of correlation between the exposed sector wage rate and the
components of the “main-course” (labour productivity and the product
price) in the long-run (use a scatter plot).
(b) What about the short-run correlation?
(c) Comment on the degree of constancy over time in the two wage shares
(you may want to use the smoothed version of the series)
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(d) Are there any evidence that the rate of unemployment is correlated with
the e-sector wage-share, and that it can explain some of the shifts in the
wage-share?
Note that the file Norw wage shares.txt explains the variable definitions.
2. What would you say is the Norwegian model’s counterpart to the modern
concept of “core inflation”?
3. In the ECM version of the Norwegian model of inflation, with a direct link
from (lagged) profitability to wage increases, there is no implied natural rate of
unemployment. This is different from the Phillips curve version of the model,
and also from any other Phillips curve variant. Does this mean that if the
ECM main-course model is correct, no long-run rate of unemployment exists?
Discuss.
4. Answer question 1-6 in the exercises to chapter 3 in IDM.
5. Exercise 5 to chapter 18 in IAM.
Exercise 5
1. Exercise 2 in chapter 15 in IAM.
2. Exercise 3 and 4 in chapter 16 in IAM.
Exercise 6
Assume that the rate of inflation and the output-gap of an economy can be represented by the following two equations:
(1)
(2)
∆pt = as0 + asy yt + asz zs,t
yt = ad0 + adp ∆pt−1 + adz zdt
where the subscript t denotes time period (e.g., quarter or year) and ∆ denotes the
difference operator, i.e., ∆pt ≡ pt − pt−1 where pt denotes the (natural) logarithm of
the domestic price level. yt denotes the output-gap in period t (deviation from full
employment output). zs,t and zd,t are catch-all indicators of important exogenous
supply-side and demand-side shocks. We could have included disturbances εs,t and
εd,t , but omit them for simplicity.
1. Explain, intuitively, how you would sign the two slope coefficients asy and adp .
2. In (1), substitute yt by the right hand side of (2) to derive the so called final
form equation for the rate of inflation, and show that it takes the form of an
ADL model with two exogenous variables, zs,t and zd,t .
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3. Once you have found the final form equation for ∆pt , and have used that
equation to calculate the inflation multipliers (for example ∂∆pt+j /∂zs ), it is
possible to also find the multipliers for yt by taking the derivative of (2) with
respect to zs,t or zd,t . Use this method to answer the following:
(a) Assume a permanent increase in zs,t . Calculate the impact multiplier,
the first four cumulated dynamic multipliers and the long-run multiplier,
for both the rate of inflation and for the output gap. Use the following
coefficient values for the calculations: asy = 0.1, asz = 0.5 and adp =
−0.01.
(b) Are the multipliers of y with respect to zd very different from the multipliers in a.?
4. Try to illustrate the dynamics in a diagram with AD/AS curves (i.e., after a
shift in the AS curve).
5. Using the data set in Dp.zip, investigate whether the inflation dynamics that
you found in your answer to question 4 is realistic for that data set. (note:
read the text file carefully, it contains explanations and essential hints!).
6. Discuss in seminar: How can the model (1)-(2) be modified so that it can
(logically) accommodate more realistic inflation response to a change in zs ?
7. Exercise 3 in chapter 19 in IAM.
Exercise 7
1. Exercise 3 in chapter 20 in IAM
2. Exercise 1 in chapter 20 in IAM.
Exercise 8
1. Exercise 2 in chapter 21 in IAM
2. Answer the first three questions in exercise 1 in chapter 23 in IAM.
3. Using the variables CPIVAL, CPI and CPIKONK in the datafile NORKVAR.xls,calculate a real exchange rate index for Norway. Draw a graph of
the series, call it REX.
(a) According to an economic hypothesis called purchasing power parity
(PPP), real exchange rates typically have constant means. Is there any
evidence in support of PPP to be hauled from the real exchange rate
REX that you have calculated?
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(b) In macroeconomics we often make the assumption that REX is constant
in each period (a stronger version of the PPP hypothesis). Comment on
the realism of such an assumption.
(c) Take the (natural) log of REX, and obtain its (approximate) growth rate.
Comment on how much the three components have contributed to the
rate of change of REX.
Exercise 9 (exam, May 2004)
1. Consider the ADL model
(1)
yt = β 0 + β 1 xt + β 2 xt−1 + αyt−1 + εt ,
where εt denotes the disturbance term in period t. The other Greek letters denote coefficients. Assume that xt is an exogenous variable (i.e., not influenced
by yt or yt−1 ).
(a) Consider a permanent shock to the exogenous variable. Give the expression for the impact multiplier, the 2nd multiplier, and the long run
multiplier.
(b) What is the condition for stability of (1)?
(c) Figure 1 contains graphs of two solutions of an ADL equation of the type
given in (1). The solutions are based on identical coefficient values, and
the same numbers for the exogenous variable have been used in both
cases. The thicker line shows a solution where the first solution period is
1990. The thinner line has 2000 as the first solution period.
Consider the following statement:
The ADL model is unstable, or even explosive, since both graphs show
persistent growth in yt .
Do you agree? Give a brief justification for your answer.
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2.2
y t solution, start 1990
y t solution, start 2000
2.1
yt
2.0
1.9
1.8
1990
1995
2000
time
2005
2010
Figure 1: Two solutions of an ADL equation for yt .
2. Consider the case where yt in (1) denotes the hourly wage rate in the exposed
sector of an open economy, and where xt is an exogenous main-course variable.
Assume that the main-course model’s hypothesis about a constant long run
wage share holds.
(a) Re-write the ADL model (1) as an error-correction model for the hourly
wage rate.
(b) Give a brief economic interpretation of wage setting in the exposed sector.
3. Make use of the AD-AS model given below. Analyse the short and long run
effects of a permanent reduction in the exogenous tax level. Consider both a
fixed and a floating exchange rate regime.
Y = C(Y − T ) + I(ρ) + G + P CA(Y, Y ∗ , σ)
0 < CY < 1, Iρ < 0, P CAY < 0, P CAY ∗ > 0, P CAσ < 0
SP
σ= ∗
P
ρ = i − π̄,
M
= L(Y, i), LY > 0, Li < 0
P
i = i∗ − se (S), s0e (S) S 0 (i.e., unsigned)
π = π̄ + a(Y − Ȳ ) + z, a > 0
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Volumes
Y
GDP (output)
Ȳ
full employment trend output
T
net taxes
G
government consumption and investments
Prices, interest rates
i
nominal interest rate
ρ
real interest rate
σ
real exchange rate
S
nominal exchange rate, foreign currency per unit of domestic currency
P
price level
π
inflation
π̄
“core inflation”
z
variable representing a supply side shock.
Exercise 10 (exam May/Aug 2005)
1. Assume that the rate of unemployment (Ut ) and inflation (π t ) is endogenous
in the dynamic system made up of equation
(1)
π t = β 0 − β 1 Ut + β 2 π t−1 ,
β 1 ≥ 0, 0 ≤ β 2 ≤ 1.
and
Ut = γ 0 + γ 1 (it−1 − πt−1 ) + γ 2 (π t−1 − π ∗t−1 ),
γ 2 ≥ 0,
γ 1 ≥ 0,
(2)
where it denotes the domestic interest rate and π ∗t is the foreign rate of inflation, both taken as exogenous. It is also assumed that the nominal rate of
foreign exchange is exogenous and constant (for simplicity, it is not specified
as a separate variable in equation (2)).
(a) Suggest an economic interpretation of equation (2).
(b) Explain what is meant by a solution to the system defined by (1) and (2).
(c) Show that the condition for a stable solution of the system is:
(3)
β 2 + β 1 γ 1 − β 1 γ 2 < 1.
(d) Is a vertical long-run Phillips curve consistent with a stable solution of
the system? Explain.
(e) Assume that condition (3) holds. Denote the steady-state (long-run)
solution of the two endogenous variables by π̄ and Ū. Draw a figure that
illustrates how π̄ and Ū are determined.
(f) Give the expression for Ū. How does this expression compare to your
answers to question 1 (c)?
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(g) Can inflation be controlled by monetary policy in this model?
2. Try to answer the following questions concisely but without the use of mathematics.
(a) Consider an open economy of the type studied in the course. What will
be the effect of an increase in government spending? How does the answer
depend on the exchange rate regime in this economy?
(b) Explain briefly what are: i) Rational expectations, ii) The Lucas critique,
and iii) Ricardian equivalence
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