(Solved by Humans)-We have so far assumed that the interest rate adjusts immediately

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?We have so far assumed that the interest rate adjusts immediately to the target level given by the Taylor rule.

However, empirical research has found that central banks tend to adjust their interest rates only gradually

towards the target level because they do not like to change the interest rate too abruptly. We may model

such "interest rate smoothing" by assuming that:

rt = (1 ? #) r?t + #rt?1

rt ? rt?1 = (1 ? #) (r?t ? rt?1) , 0 ? # < 1,? (61)

where #? is a parameter indicating the sluggishness of interest rate adjustment, and where r?t? is the target

real interest rate given by the Taylor rule:

r?t = ?r + h (?t ? ??) + b (yt ? ?y) .? (62)

In the main text we have analyzed the special case where # = 0,? but here we will focus on the general case

where # > 0 . The goods market equilibrium condition may still be written as:

yt ? ?y = ?1 (g ? ?g) ? ?2 (rt ? ?r) + vt,? (63)

and the economy?s supply side is still given by:

?t = ?e

t + " (yt ? ?y) + st.? (64)

As a new element, we will assume that economic agents are suffi ciently sophisticated to realize that on

average over the long run, the inflation rate must equal the central bank?s inflation target. Hence we assume

that the expected inflation rate is:

?e

t = ??, 8t.? (65)

This completes the description of our revised AD-AS model.

1. Discuss whether the assumption made in (65) is reasonable.

2. Define ?yt ? yt ? ?y? and ??t ? ?t ? ??,? and show that the AD curve is given by the equation:

?yt = ??yt?1 ? ???t + zt ? #zt?1,

?where ? ? !

1+?2b(1?!), ? ? ?2h(1?!)

1+?2b(1?!) ,? and zt ? ?1(gt??g)+vt

1+?2b(1?!) .

?3. Show that the model can be reduced to the diff erence equation:

?yt = ???yt?1 + ? (zt ? #zt?1) ? ??st, ? ?

1

1 + "?

,? (67)

4. Prove that, in the absence of shocks, the economy will converge on a long-run equilibrium where

?yt = ??t = 0.? How does the parameter #? aff ect the economy?s speed of adjustments? Explain.

ECONOMICS 602: GRADUATE MACROECONOMICS
HOMEWORK ASSIGNMENT 4

CHAPTER 18: Inflation, Unemployment and Aggregate Supply
Exercise 1: Intersectoral Labor Mobility and the Expectations-augmented
Phillips Curve
In Section 2 we abstracted from intersectoral labor mobility by assuming that individual workers are educated
and trained to work in a particular sector. In that case a worker?s outside option (the income he could expect
to earn if he were not employed by his current employer) is simply equal to the rate of unemployment benefit.
In this exercise we ask you to show that one can still derive the expectations-augmented Phillips curve from
a trade union model of wage setting even if unemployed workers can move between sectors in their search
for jobs.
To simplify matters a bit, we now set our productivity parameter B = 1 and work with a linear production
function with constant marginal productivity of labor, corresponding to ? = 0 in Eq. (3) in the main text.
For ? = 0 and B = 1, it follows from (7) in Section 2 in the main text (or Lecture Notes 09, Equation (3) )
that the representative firm will set its product price as:
Pi = m p W i ,

mp &gt; 1.

(43)

As in Section 2, an optimizing monopoly union for workers in firm (sector) i will set the nominal wage rate
Wi to attain an expected real wage which is a mark-up over the expected real value of its members? outside
option, denoted by v e . The union expects the current price level to be P e . Hence it will set the wage rate:
Wi
= mw ? v e ,
Pe

mw &gt; 1

(44)

This equation is a parallel to Eq. (11) in Section 2 in the main text (or Lecture Notes 09, Equation (14)
) where we assumed that v e is simply equal to the real rate of unemployment benefit, b. Here we assume
instead that workers who are initially members of the trade union for section i are qualified to apply for a job
in other sectors if they fail to find one in section i. For an average job seeker, the probability of finding work
in equal to the rate of employment, 1 u, where u is the unemployment rate which gives the probability
that an average job seeker will remain unemployed. If the ratio of unemployment benefits to the average
wage level is equal to the constant c, we thus have:
v e = (1

u) we + ube = (1

u + uc) we ,

0 &lt; c &lt; 1,

(45)

where we is the expected average level of real wages, and be = cwe is the expected real rate of unemployment
benefit.
Since the outside option v e is the same across all sectors and all unions are assumed to hold the same
price expectations, it follows from (44) and (43) that all unions will charge the same nominal wage rate, W ,
and that all firms will charge the same output price, P :
(46a)

Wi = W
and

(46b)

Pi = P

According to (43) the average real wage will then be W/P = 1/m . Let us suppose that union wage setters
have a realistic estimate of the average real wage so that:
p

we = 1/mp .

(47)

1. Show that the relationship between the actual and the expected price level may be written as:
P = mw (1

u) P e ,
1

?1

c &gt; 0.

(48)

2. Use (48) to derive an expectations-augmented Phillips curve of the same form as equation (17) in
Section 2 in the main text (or Lecture Notes 09, Equation (23) ). (Hint: use the approximation
ln (1
u) ?
? and define u
? ? ln mw / .)

CHAPTER 19: Explaining Business Cycles
Exercise 2: Interest Rate Smoothing in the AD-AS Model
We have so far assumed that the interest rate adjusts immediately to the target level given by the Taylor rule.
However, empirical research has found that central banks tend to adjust their interest rates only gradually
towards the target level because they do not like to change the interest rate too abruptly. We may model
such &quot;interest rate smoothing&quot; by assuming that:
rt = (1
rt

rt

1

= (1

) rt? + rt

1

) (rt?

1) ,

rt

0

&lt; 1,

(61)

where is a parameter indicating the sluggishness of interest rate adjustment, and where rt? is the target
real interest rate given by the Taylor rule:
rt? = r? + h (?t

? ? ) + b (yt

(62)

y?) .

In the main text we have analyzed the special case where = 0, but here we will focus on the general case
where &gt; 0. The goods market equilibrium condition may still be written as:
yt

y? = ?1 (g

g?)

?2 (rt

(63)

r?) + vt ,

and the economy?s supply side is still given by:
?t = ?te + (yt

(64)

y?) + st .

As a new element, we will assume that economic agents are su?ciently sophisticated to realize that on
average over the long run, the inflation rate must equal the central bank?s inflation target. Hence we assume
that the expected inflation rate is:
?te = ? ? ,
8t.
(65)
This completes the description of our revised AD-AS model.

1. Discuss whether the assumption made in (65) is reasonable.
2. Define y?t ? yt

y? and ?
? t ? ?t

? ? , and show that the AD curve is given by the equation:
y?t = ??
yt

where ? ?

1+?2 b(1

),

??

?2 h(1
)
1+?2 b(1
),

??
?t + zt

1

and zt ?

zt

1,

?1 (gt g
?)+vt
1+?2 b(1
).

3. Show that the model can be reduced to the di?erence equation:
y?t = ??
yt

1

+

(zt

zt

1)

? st ,

?

1
,
1+ ?

(67)

4. Prove that, in the absence of shocks, the economy will converge on a long-run equilibrium where
y?t = ?
?t = 0. How does the parameter a?ect the economy?s speed of adjustments? Explain.

2



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