Homework 5

Now consider the following two models: (i ) Only unexpected money matters, so y t = a ′ zt −1 +be t +v t ; (ii) all money matters, so y t = α ′ zt −1 +βm t +νt . In
each specification, the disturbance is i.i.d. and uncorrelated with zt −1 and e t .
Econ 701 – Survey of Macroeconomics (a ) Is it possible to distinguish between these two theories?
That is,
given
a
Manopimoke, Spring 2015 candidate set of parameter values under, say, model (i ), are there param eter values under model (ii ) that have the same predictions? Explain.
Homework 5 (b ) Suppose that the Federal Reserve
also responds
to some variables that
do not directly affect output; that is, suppose m t = c ′ zt −1 + γ ′ wt −1 + e t
and that models (i ) and (ii ) are as before (with their distubances now
uncorrelated with wt −1 as well as with zt −1 and e t ). In this case, is it posQuestion 1
sible to distinguish between the two theories? Explain.
6.16. Consider an economy consisting of some firms with flexible prices and some
with rigid prices. Let p f denote the price set by a representative flexible-price
firm and p r the price set by a representative rigid-price firm. Flexible-price
firms set their prices after m is known; rigid-price firms set their prices before m is known. Thus flexible-price firms set p f = p i∗ = (1 − φ) p + φm,
and rigid-price firms set p r = E p i∗ = (1 − φ)E p + φE m, where E denotes the
expectation of a variable as of when the rigid-price firms set their prices.
Assume that fraction q of firms have rigid prices, so that p = qp r + (1−q)p f .
(a ) Find p f in terms of p r , m, and the parameters of the model (φ and q).
(b ) Find p r in terms of Em and the parameters of the model.
(c )
(i ) Do anticipated changes in m (that is, changes that are expected as of
when rigid-price firms set their prices) affect y ? Why or why not?
(ii ) Do unanticipated changes in m affect y ? Why or why not?
Question 2 Consider the Lucas Imperfect Information model where producers observe 𝑝! but not p (and hence not 𝑟! ). Consider two economies, one in which the money supply is very stable and one in which the money supply is highly variable. Explain how the aggregate supply curves of these two economies will differ. Question 3 Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what it obtains with 𝑝! = 𝑝∗ are 𝐾 (𝑝! − 𝑝∗ )! , 𝐾 > 0. As usual, we have p*=p+ϕy and y=m-­‐p. Each firm faces a fixed cost Z of changing its nominal price. Initially m is 0 and the economy is at its flexible-price equilibrium, which is y = 0
and p = m = 0. Now suppose m changes to m ′.
(a) Suppose that fraction f of firms change their prices. Since the firms that
change their prices charge p∗ and the firms that do not charge 0, this implies
p=fp∗. Use this fact to find p, y, and p∗ as functions of m′ and f.
The Microeconomic Foundations
of 7Price
Econ 01 – Rigidity
Survey of Macroeconomics Additional Homework Problems Manopimoke, Spring 2015 ECON 3133
(b) Plot a firm’s incentive to adjustDr.
itsKeen
price, K(0 − p∗)2 = Kp∗2, as a function of f.
Be sure to distinguish the cases φ<1 and φ > 1.
1. Suppose that the Lucas supply curve is
(c) A firm adjusts its price if the benefit exceeds Z, does not adjust if the
f
benefit is less than Z, and Y
is=indifferent
if the
n×h×(1 – b)×(P
– Pbenefit
) + Y*, is exactly Z. Given this, can
there be a situation where both adjustment by all firms and adjustment by no
with n×h×(1
– b) =
20,000
Y* =be4,000.
For example,
when
the price
level P is by
1.01
firms
are equilibria?
Canand
there
aFoundations
situation
where
neither
adjustment
alland
The
Microeconomic
of Price
Rigidity
the expected
price Pf is 1.0,
output
Y isHomework
4,200,
or 5%
above
potential
output
Y*
=
4,000.
firms
nor adjustment
by Additional
no firms
is
an equilibrium?
Problems
Suppose that the aggregate demand curve
is 3133
ECON
Question 4 Dr. Keen
Y = 1,101 + 1.288×G + 3.221×MS/P.
a. Suppose
the economy
has been
1. Suppose
that that
the Lucas
supply curve
is at rest for some period with output at potential, and
that no changes in policy are expected for the near
future. The money supply MS is 600
f
YG
= is
n×h×(1
– b)×(P – the
P ) price
+ Y*,level.
and government spending
750. Calculate
with
n×h×(1
– b) =that
20,000
and announces
Y* = 4,000.that
Forit example,
whenthe
themoney
price level
P isfrom
1.01600
andto
b. Now
suppose
the Fed
will increase
supply
f
the expected
price
P
is
1.0,
output
Y
is
4,200,
or
5%
above
potential
output
Y*
=
4,000.
620. What are the new levels of output and the price level?
Suppose that the aggregate demand curve is
c. Now suppose that the Fed announces that it will increase
S the money supply from 600 to
Y = 1,101
+ 1.288×G
+ 3.221×M
/P. of output and the price level?
620 but actually increases
it to 670.
What are
the new levels
a. Suppose that the economy has been at rest for some period with output at potential, and
that nothat
changes
policy are
the near
The money
MS is 600wage
2. Suppose
wage in
contracts
lastexpected
for threefor
years.
Eachfuture.
year, one-third
of supply
the economy’s
and government
spending
G is 750.
Calculate
price level.
contracts
are renegotiated.
Contract
wages
are setthe
according
to
b. NowXsuppose
that the
that it will
increase
the–money
supply
from 600 to
= (1/3)×(W
+ WFed
W+2) – (d/3)×[(U
– U*)
+ (U+1
U*) + (U
+1 +announces
+2 – U*)].
620. What are the new levels of output and the price level?
a. Provide an expression for the average wage rate W.
c. Now suppose that the Fed announces that it will increase the money supply from 600 to
b. 620
Calculate
the wage
set thisit period
a function
X’slevels
and U’s.
Howand
farthe
backwardand
but actually
increases
to 670.as
What
are the of
new
of output
price level?
forward-looking is the wage-determination process? What determines the responsiveness of contract
wages
to current
labor
market
conditions?
2. Suppose
that wage
contracts
last for
three
years.
Each year, one-third of the economy’s wage
Question 5
contracts are renegotiated. Contract wages are set according to
3. “If expectations are rational, monetary policy has no effect on output.” Is this statement true
X Explain
= (1/3)×(W
W+1 + W
– U*) +with
(U+1the
– U*)
+ (U
+2) – (d/3)×[(U
+2 – U*)].
or false?
your+ answer
calling
on both models
Lucas
supply
function and
models
with
wage
contracts
and
sticky
prices.
a. Provide an expression for the average wage rate W.
b. Calculate the wage set this period as a function of X’s and U’s. How far backward- and
forward-looking is the wage-determination process? What determines the responsiveness
of contract wages to current labor market conditions?
3. “If expectations are rational, monetary policy has no effect on output.” Is this statement true
or false? Explain your answer calling on both models with the Lucas supply function and
models with wage contracts and sticky prices.