ECON 3141/4141: International macro and finance. Term
paper, spring 2005.
Annotated version
Ragnar Nymoen
1. See individual comments to your answers.
2. In chapter 3 of Introductory Dynamic Macroeconomics, you find the following
expression, as well as an explanation of symbols and of variables:1
Ut = 0.005 + 0.005 · S1989t + 0.8Ut−1 + εU,t
Show that according to this equation, the steady-state rate of unemployment
is 0.025 before 1989, and 0.05 after 1989.
ANSWER SUGGESTIONS:
(note: originally there was a typo as the coefficient of S1989t was 0.005, both
were corrected though).
The equation is an ADL model of the rate of unemployment, Ut . S1989t is
the only exogenous variable (corresponding to ”xt ” in the lectures). Since
0.8 < 1 the model has a stable solution for any given initial value of U, and
the actual/chosen values of S1989t and εU,t . (In the case of S1989t the values
are known form the specification of the dummy, while the εU,t are set to zero
by us). From IDM we know the expression for the stable steady-state solution
of Ut , which we can denote U ∗ .
0.005
= 0.025 if S1989t = 0
0.2
0.01
= 0.05 if S1989t = 1
=
0.2
U∗ =
U∗
This means that, if we for example take 1985 as the initial period, and assume
that U1985 = 0.025 then the solution will be U1986 = U1987 = U1988 = 0.025 and
U1989 = 0.03, U1900 = 0.034, and so on: a gradual rise of actual unemployment
to the new equilibrium of 5%.
3. Assume that, in a closed economy, the rate of unemployment in year t, denoted
Ut is given by:
(1)
Ut = γ 0 + γ 1 (it − π t ) + γ 2 (it−1 − π t−1 ) + γ 3 Ut−1 ,
γ 1 + γ 2 > 0, 0 ≤ γ 3 < 1
where it denotes a nominal interest rate and πt denotes the rate of inflation.
Hence it − π t denotes the real interest rate in year t. In order to simplify the
notation, the random disturbance term is omitted from (1)–and also from (2)
and (3) below.
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Note that the typo in IDM has been corrected (i.e., Ut−1 ).
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(a) Explain intuitively (on the basis of your earlier studies for example) the
economic rationale for the distributed lag of the real interest rate in equation (1).
ANSWER SUGGESTIONS:
Some points that might be mentioned: U is a variable which is affected
both by factors that influence labour demand and labour supply. It is
most reasonable that the effect of real interest rate is via demand, which
in turn depends on (the demand of) GDP output: A rise in the real
interest rate affects both private consumption (both a substitution effect
and an effect though disposable income) and private investments in real
capital. Moreover, adjustment lags both in consumption and investment
decisions make the distributed lag specification in (1) reasonable. Due
to such decisions lags, and if we think of the time period as annual for
example, then it might be realistic to assume that γ 2 > γ 1 , i.e., the
second year effect of a rise in the interest rate is larger than the first year
(impact) effect.
Note 1: You can of course also establish the link between U and GDP
demand by invoking Okun’s law, as we do in the lectures.
Note 2: You are not expected to comment on the autoregressive part of
(2), but a reasonable remark is that both (further) adjustment lags in
labour demand and separate lagged adjustments of labour supply when
labour market conditions change, motivate the presence of Ut−1 in the
equation, with a positive coefficient.
(b) Assume that both the nominal interest rate and the inflation rate are exogenous variables (this assumption also applies to c) and d) below). What
is the expression for the long-run multiplier of the rate of unemployment
with respect to the real interest rate?
ANSWER:
Since this is an ADL model, we have from the general properties of ADLs
that the long-run multiplier in this case is given by
∂U ∗
γ + γ2
.
= 1
∂(i − π)
1 − γ3
(Note that the parameter restriction γ 3 < 1 is important, as γ 3 = 1
corrupts the multiplier.)
(c) Draw a typical graph of the dynamic multipliers of Ut with respect to a
permanent change in the real interest rate. Explain.
ANSWER:
If both γ 1 and γ 2 are positive: a monotonously increasing multiplier (of
course maintaining 0 ≤ γ 3 < 1 as given in (1)). Many of you draw a
graph that approaches 1 asymptotically. Thus the long run multiplier
is restricted to 1. However, take care to note that there is nothing in
the question (or altogether in the concept of the long run multiplier)
that restricts its value to 1 (In many applications, such a restriction
might however be theoretically appealing. For example: the consumption
function or a PPP based model of the domestic price level. Why?).
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(d) Calibrate the parameters of (1) so that the steady-state rate of unemployment is 0.03 for a constant real interest rate of 0.01. For a constant
real interest rate of 0.06, what is the corresponding steady-state rate of
unemployment?
ANSWER:
Again, since this is an ADL model, the expression for the steady-state
rate of unemployment is
U∗ =
γo
γ + γ2
+ 1
0.01
1 − γ3
1 − γ3
Calibration means “choosing parameter values so that the model matches
some criterion” (theoretical or empirical: in this case the criterion is a
long run value of Ut equal to 0.03). Several constellations of the three
parameters are consistent with U ∗ = 0.03. For example, and for simplicity
set γ 0 = 0 and set γ 3 = 0.8. Then the calibrated value of γ 1 + γ 2 becomes
0.6. (Note that we don’t need do identify values of γ 1 and γ 2 separately,
only of the sum).
For the same parameter values, and if the constant real interest rate is
0.06, we get U ∗ = 0.18, so a high real interest rate has a huge impact on
long run unemployment in this calibrated model.
4. Simplify equation (1) by setting γ 1 = 0 and γ 3 = 0, to obtain
(2)
Ut = γ 0 + γ 2 (it−1 − πt−1 ).
Assume next that the nominal interest rate is exogenous, but that the rate of
inflation is endogenized by the Phillips curve:
(3)
π t = β 0 + β 1 Ut + β 2 πt−1 ,
β 1 < 0, 0 ≤ β 2 ≤ 1.
(a) Assume that β 1 = −0.5. Calibrate the other parameters of the system
made up of equation (2) and (3) in such a way that the equilibrium rate
of unemployment (identical to the natural rate in this case) is equal to
0.06. Also give an expression which defines the corresponding equilibrium
rate of inflation.
ANSWER SUGGESTION:
Since this is a Phillips curve system, the steady state rate of unemployment is given by
−β 0 + (1 − β 2 )π ∗
U∗ =
β1
However, unless β 2 = 1, U ∗ depends on the steady-state rate of inflation
(corresponding to a downward sloping rather than a vertical long-run
Phillips curve). Hence, the easiest way to answer this question is to set
β 2 = 1, which is admissible since equation (3) specifies 0 ≤ β 2 ≤ 1. In
the case of β 2 = 1, for example β 0 = 0.03 and β 1 = −0.5 gives U ∗ = 0.06.
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There is an important “tactical lesson” here: (nearly) all of you attempt
to answer the question for the more general case of 0 ≤ β 2 < 1 which
is of course very commendable, but also much more difficult! In the
school exam, an “A” would typically (I expect) be based on the explicit
assumption of β 2 = 1!
However, returning to the case of 0 ≤ β 2 ≤ 1,we have the following long
run system, defining the equilibrium values π ∗ and U ∗ in a situation where
it = it−1 = i:
U ∗ + γ 2 π∗ = γ 0 + γ 2 i
−β 1 U ∗ + (1 − β 1 )π∗ = β 0
Solving this long run model for U ∗ gives:
U∗ =
(γ 0 + γ 2 i)(1 − β 2 ) − β 0 γ 2
(1 − β 2 ) + γ 2 β 1
Unless β 2 = 1, U ∗ is seen to depend on i, the nominal interest rate, so
a relevant critique of the question (always a good thing to include in an
exam answer (if you are right that is!!)) is that it is wrong to say that U ∗
is “identical to the natural rate in this case”. Presumably, a natural rate
should not depend on nominal variables (see for example your essays on
the natural rate!).
Having said that, one calibration may be β 2 = 0.95, β 1 = −0.5, β 0 = 0.03,
γ 2 = 0.6, γ 0 = 0.01 and i = 0.09:
(0.01 + 0.6 × 0.09) × (1 − 0.95) − 0.03 × 0.6
≈ 0.06.
(1 − 0.95) + 0.6 × (−0.5)
(b) Explain what we mean by a solution to the system defined by (2) and
(3).
ANSWER:
See for example chapter 1.9 of IDM: Assume all parameters are known.
Assume also that π 0 is known and that we have a known series of values i0 ,
i1 , i2 , ...iT . Then (2) and (3) delivers two unique times series U1 , U2 , ...UT
and π1 , π 2 , ...π T which represent the solution. It is found recursively:
First U1 and π 1 , then U2 and π 2 and so on.
(c) Show that a sufficient condition for a stable solution of the system is:
β 2 − γ 2 β 1 < 1.
ANSWER:
Substitute the expression for Ut in (3) and apply the condition for a stable
solution of an ADL. See for example chapter 1.9 in IDM. Specifically, the
so called final equation for π t is:
π t = β 0 + β 1 γ 0 + β 1 γ 2 it−1 + (β 2 − γ 2 β 1 )πt−1
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(d) Is a vertical long-run Phillips curve consistent with a stable solution of
the system? Explain.
ANSWER:
In this model a vertical Phillips curve is synonymous to setting β 2 = 1.
If γ 2 ≥ 0 (as in 3 d), the system is then dynamically either unstable or
explosive: From known initial conditions, the solution does not converge
to U ∗ , unless U = U ∗ to start with.
Intuition: assume that, for some reason or another, that we get a higher
initial value (π 00 ) than the value π 0 which is consistent with U1 = U ∗ :
then π 1 increases with the same amount from the lagged term alone. In
addition, U1 is reduced (below U ∗ ) which increases π 1 further, this leads
to instability.
Most believers in the Phillips curve would not like this implication, since
a vertical long run Phillips curve is seen as theoretically sound it will
be seen as problematic (an internal inconsistency almost) that the model
is unstable in the case of β 2 = 1. When we reconsider equation (2)
is is clear that the “problem” is that the rate of inflation only enters
through the real interest rate (in fact the AD schedule in this case has
the “wrong slope” compared to the text-book case). Hence one would
have to propose other effects of inflation on Ut than the effect through
the real interest rate, and include them in the model (so that the slope
of the AS schedule is turned around). In the closed economy case this
may not be easy, although so called effects of real balances on aggregated
demand (real money in the consumption function for example) represent
one possibility. In the case of open economy models, the real exchange
rate offers a way out of this puzzle, which in fact is what we have in the
AD-AS model in B&W.
(e) Derive the dynamic multipliers of inflation with respect to the nominal
interest rate as fully as you can–or at least: the impact and long-run
multipliers. Set β 1 = −0.5, γ 2 = 0.1, β 2 = 0.75.
ANSWER NOTE:
The first multiplier (impact) is 0. The second multiplier is δ1 = β 1 γ 2
which becomes −0.05. The third multiplier: δ 2 = β 1 γ 2 × 1 + (β 2 −
γ 2 β 1 )δ 1 = −0.09. The long-run multiplier:
β1γ2
= −0.25.
1 − β2 + γ2β1
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