Topics in Macroeconomics
Lecture 12
Ctirad Slavı́k
email: [email protected]
website (syllabus, lecture notes etc.):
http://www.wiwi.uni-frankfurt.de/professoren/slavik/teach.html
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Summary of Last Lecture
Solve the real intertemporal model.
Temporary vs. permanent changes in G .
Shocks to current and future productivity,
candidate sources of BC fluctuations.
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This Lecture
Dig deeper into the role of productivity shocks.
To do that: introduce an infinite horizon macromodel.
Solow residual, relationship to growth and business cycles.
RBC model: productivity shocks as source of business cycles.
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The Model
Infinite horizon version of the model from previous lecture
(not in Williamson.)
Representative consumer.
Representative firm.
For simplicity omit the government.
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Consumer
Owns capital and can spend time working (or not).
Chooses consumption, labor supply and investment to
maximize lifetime utility:
max
{ct ,it ,nts ,kts }∞
t=0
∞
X
β t u(ct , 1 − nts )
s.t.
t=0
∀t :
ct + it
≤ wt nts + rt kts
s
kt+1
≤ (1 − δ)kts + it
k0s
≤ k0
given
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Firm
Rents capital and labor to maximize profit in each period t:
max
ntd ,ktd
zt F (ktd , ntd ) − wt ntd − rt ktd
Assume zt F (ktd , ntd ) = zt (ktd )α (ntd )1−α .
Recall in data: α = .36.
Note 1: equivalent to dynamic profit maximization.
Note 2: who makes investment is irrelevant.
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Competitive Equilibrium
Definition: competitive equilibrium is:
1
sequence of allocations for the consumer: {ct , it , nts , kts }∞
t=0
2
sequence of allocations for the firm: {ntd , ktd }∞
t=0 , and
3
sequence of prices {wt , rt }∞
t=0 s.t.
1
consumer maximizes utility given prices,
2
firm maximizes profits given prices,
markets clear:
3
goods market in each period t : ct + it = zt F (ktd , ntd ),
labor market in each period t : ntd = nts ,
capital market in each period t : ktd = kts .
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First Welfare Theorem
Competitive equilibrium is Pareto efficient.
Pareto efficient allocation:
a feasible allocation,
there is no other feasible allocation, which gives strictly higher
utility to consumer (here we have only one consumer).
PE allocation is a feasible allocation that maximizes utility.
Use it to find the CE = PE allocation.
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Pareto Efficient Allocation
max
{ct ,it ,kt ,nt }∞
t=0
ct + it
∞
X
β t u(ct , 1 − nt )
s.t.
t=0
≤ zt F (kt , nt )
kt+1 ≤ (1 − δ)kt + it
k0
given
This is called the Neoclassical Growth Model.
What is it good for? To explain the data.
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Macroeconomic Phenomena
The most important data features (Lecture 1):
long run trend,
fluctuations around trend - business cycles.
Next: role of the Solow residual, total factor productivity (TFP).
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Solow Residual, TFP
Y
log Y
= z · F (K , N) = Y = zK .36 N .64
= .36 log K + .64 log N + log z
These series grow over time.
N exogenously (?), K (maybe) because of growth in z.
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Solow Residual
What is the TFP in the data?
log z
= log Y − .36 log K − .64 log N
Can the evolution of TFP help us explain:
business cycle fluctuations (today’s topic),
long run growth in K and Y (next topic).
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Long Run Growth
Log GDP and Solow residual (relative to mean)
0.8
0.6
0.4
0.2
0
TFP
GDP
-0.2
-0.4
-0.6
-0.8
1960
1970
1980
1990
2000
2010
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Business Cycle
HP filtered log GDP and Solow residual
0.05
0.04
0.03
0.02
0.01
0
TFP
GDP
-0.01
-0.02
-0.03
-0.04
-0.05
1960
1970
1980
1990
2000
2010
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Behavior of TFP and GDP
Both series persistent: ρGDP = .86, ρTFP = .73.
Both series volatile:
σGDP
= .015 = 1.5%
σTFP
= .008 = .8%
Positively correlated: corr (TFP, GDP) = .65.
TFP seems to lead GDP.
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TFP leads GDP
corr(GDP, TFP_s)
0.8
0.6
0.4
0.2
0.0
t-5
t-4
t-3
t-2
t-1
t
t+1
t+2
t+3
t+4
t+5
-0.2
-0.4
-0.6
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Conjecture
Maybe fluctuations in TFP cause business cycles.
We saw last time changes in TFP can (qualitatively) explain
some regularities in the data.
Next: Formalize the idea of fluctuations in TFP.
Note: pioneering work of Kydland and Prescott (1982),
Time to build and aggregate fluctuations.
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Stochastic TFP
Assume zt is a stationary (no growth) stochastic process.
Consumer does not know future zt .
Consumer maximizes expected utility.
Can still use the FWT, because CE is PO.
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Real Business Cycle Model
Workhorse model of modern macroeconomics:
max
{ct ,it ,kt ,nt }∞
t=0
ct + it
E0
∞
X
β t u(ct , 1 − nt )
s.t.
t=0
≤ zt F (kt , nt )
kt+1 ≤ (1 − δ)kt + it
k0
given
Abstract from population growth, per capita variables.
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Real Business Cycle Model
Allocations now depend on past zt ’s.
Solving models like this is hard.
Needs to be done on computer (numerically).
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Solving the Real Business Cycle Model
Need to determine the process for zt ’s.
Assume zt = ρzt−1 + εt and estimate the process.
Solve the model for a set of parameters and simulate.
Compare the model generated time series with detrended real
world time series.
Look at cyclicality (correlation with output), standard
deviation, autocorrelation, leads and lags.
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RBC Model Findings
Behavior of model variables consistent with the data
qualitatively (as last time):
C , I , N, w , labor productivity procyclical.
Behavior of model variables consistent with the data also
quantitatively (similar numbers as in the data).
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Drawbacks
Does not capture some regularities
(lot of work over the last 30 years).
Competitive equilibrium is Pareto efficient:
allocations as optimal responses to shocks in TFP,
recessions as optimal responses to decreases in TFP,
no role for the government.
Cannot explain growth (in p.c. variables) if z time stationary.
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Summary and Next
Infinite horizon macromodel.
TFP, Solow residual.
RBC model, productivity shocks as source of business cycles:
Can account for important data features qualitatively.
Can account for important data features quantitatively.
Cannot explain growth.
Next: start studying growth.
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