Download Real Business Cycles

Real Business Cycles
Route Map
•
Stochastic Solow-Swan Growth Model
•
The Crusoe economy – the simple intuition of RBC
•
The formal model
•
Assessment
New Classical Principles
•
Real variables affected by other real variables
•
No profitable opportunities unexploited - market equilibrium
•
Rational Expectations
Real Business Cycles
Price Misperceptions Model
• Follows these principles
• Empirically weak:
• Anticipated money matters (Mishkin, Gordon)
• King & Plosser AER 1984 introduced money to
RBC model in its transactions function
• Profit-seeking banks raise money supply in periods
of expansion.
• High correlation of output with inside money rather
than outside money
• ‘Reverse’ causation
Real Business Cycles
Price Misperceptions Model
• Empirically weak:
• RBC theorists challenged the informational basis
for misperceptions model
• Barro (Modern Business Cycle Theory 1989)
‘people could expend relatively few resources to
find out quickly about money and prices’ (p.2)
Real Business Cycles
Model-based Trend Output
st  m
xt 
 
W N 
where st  log  t t  and m is the log of the desired mark-
• Model-based output gap:
t t 
 PY
up of price over nominal marginal costs
• This is based on a simple production function Y = AN
• Interesting to compute z = y – x using the model-based measure
of x and compare the time series behaviour of z and y.
Real Business Cycles
1979-80 Recession
11.95
11.94
11.93
y
UK Fullz Capacity
Output using Model-generated Gap
11.92
11.91
11.9
12.6
11.89
11.88
11.87
11.86
12.4
11.85
z
11.84
1978.5
1979
12.2
1979.5
1980
1980.5
y
1981
1981.5
1982
12
1990-91 Recession
11.8
12.155
12.15
12.145
11.6
z
y
12.14
12.135
12.13
11.4
12.125
12.12
12.115
11.2
1960
12.11
1965
1970
1975
1980
1985
12.105
1989.5
1990
1990
1995
1990.5
1991
2000
1991.5
1992
2005
1992.5
1993
Real Business Cycles
Stochastic Solow-Swan Growth Model
• Aggregate production function:
Y  K  L1
0    1 L  AN

Y K  L1  K 
y  
    k   f (k )
L
L
L
• A is grows at rate l each year
•
l is rate of labour-augmenting technical progress
• N grows at rate n per year
• L grows at rate l + n per year
Real Business Cycles
Stochastic Solow-Swan Growth Model
• With constant savings rate (s):
K  I   K  s  F (K, L)   K
• Divide both sides by L:
K
 s  f (k )   k
L
• Evolution of k:
k K L K



 n  l 
k
K
L
K
k 
K K
K
 n  l  k 
 n  l  k
K L
L
k  s f (k )   k  (n  l ) k  s f (k )  (n  l   ) k
Real Business Cycles
Stochastic Solow-Swan Growth Model
• With constant savings rate, the evolution of k is given by:
k  s f (k )  (n  l   ) k
k will be positive if s f (k )  (n  l   )k
k will be negative if s f ( k )  (n  l   )k
k will be zero if s f (k )  (n  l   )k
Real Business Cycles
Steady-State Equilibrium k  s f (k )  (n  l   ) k .
(n+l+) k
k  0
y
since s f (k )  (n  l   )k
f(k)
y*
s f(k)
k  0
since s f (k )  (n  l   )k
k(0)
k*
k
Steady-state
equilibrium (k*) is
where the
addition to the
capital stock
(through saving)
is just enough to
keep the capitallabour ratio
constant.
In equilibrium, Y
is growing at l + n
since y* is
constant.
Real Business Cycles
‘Vibrating’ Production Function
• Imagine that the production function is stochastic:
Y  zt K  L1  zt F (K , L)
where zt is total factor productivity
zt K  L1
y 
 zt k   zt f (k )
L
'Good times': zt >1
'Bad times' : zt <1
'Normal'
: zt =1
Fluctuations in total factor productivity
lead to business cycles
Real Business Cycles
‘Vibrating’ Production Function
(n+l+) k
y
•
Imagine that z
takes on two
values - 1.5 and
0.5
•
Start at k = k1 and
z = 0.5 (in
equilibrium). y is
constant so Y is
growing at l + n
•
z rises to 1.5 and
k starts to rise to
k2 – the new
equilibrium
s 1.5 f(k)
s 0.5 f(k)
k1
k2
k
Real Business Cycles
‘Vibrating’ Production Function
(n+l+) k
y
•
Y now grows
faster than l + n
to approach
higher steady
state.
•
Before k2 is
reached, let z fall
back to 0.5
•
k now falls back
towards k1 and
output growth falls
below l + n
s 1.5 f(k)
s 0.5 f(k)
k1
k2
k
Real Business Cycles
‘Vibrating’ Production Function
log(Y)
Steady State
Path for z = 1.5
Slope = n +l
Transitional
paths
Steady State
Path for z = 0.5
Slope = n +l
t0
t1
t2
Time
Real Business Cycles
Stochastic Solow-Swan v RBC model
• The full RBC models differ from the stochastic Solow-Swan
model in a number of ways:
• the intertemporal substitution effect
• labour supply does not grow simply at rate n
• more effort supplied when productivity (wages) high
• output rises both because productivity is high and
because workers offer more effort
• endogenous savings rather than a fixed savings rate
• start with utility function and solve for consumption
and savings
Real Business Cycles
Crusoe Economy
• RBC models usually use representative agent assumption
• Use Robinson Crusoe economy for intuition
• Decision: how many coconuts to (a) eat and (b) invest (plant)
• Given preferences and technology (yield on each tree) and
without shocks, Crusoe will choose an optimal balance:
• enough planted to maintain stock of trees
• enough consumed to satisfy inter-temporal optimality
• Shocks of two types: transitory and permanent
Real Business Cycles
Crusoe Economy: Transitory Shocks
• Effects of unusually good weather
• Crusoe may work harder – amplifies the effect of good weather
• Smooth out the benefits of his good luck by planting more trees
– investing much of the extra harvest
• The island economy will have increased output most of which
will be invested but some of it will be consumed.
• Implications:
• work & output correlated given intertemporal substitution
• consumption smoother than investment
• although shocks (weather) are serially uncorrelated, they
lead to serially correlated changes in output.
Real Business Cycles
Crusoe Economy: Permanent Shocks
• Effects of ‘anti-monkey’ technology
• No intertemporal substitution
• Less need to smooth out consumption through investment
• The island economy will have increased output most of
which will be consumed but some of it will be invested.
• Combination of transitory and permanent shocks could be
used to explain movements in work, output, consumption
and investment.
Real Business Cycles
Formal RBC Model
[Taken from McCallum 1989]
• Economy consists of large number of identical agents – explain
aggregate behaviour using the ‘representative-agent’ model
• The agent maximizes the following utility function:


U  u(Ct , Lt )   u(Ct 1, Lt 1)   u(Ct  2 , Lt  2 )   u(Ct  3 , Lt  3 )  ...    j u Ct  j , Lt  j
2
3
j 0
• Subject to budget and technology constraints:
Yt  Ct  It
Nt  Lt  1
Zt f (Nt , K t )  Ct  K t 1  K t   K t
since Kt 1  Kt   Kt  It
It  Kt 1  Kt   Kt

Real Business Cycles
Formal RBC Model
[Taken from McCallum 1989]
• To derive analytical solution, McCallum assumes:
u(Ct ,1  Nt )   logCt  (1   )log(1  Nt )
Zt f (Nt , Kt ) 
Zt Nt Kt1
 1

Ct  1  (1   )   N Zt Kt1
Kt 1  (1   )  N Zt K t1
Nt  N
• Because of utility function, income and substitution effects cancel
so there is no intertemporal substitution effect
• Given these assumptions analytical solutions for consumption and
investment can be derived (handout has details)
Real Business Cycles
Formal RBC Model
[Taken from McCallum 1989]
Ct  1  (1   )   N Zt Kt1
Kt 1  (1   )  N Zt Kt1
Nt  N
• Taking logs and solving for K and Y:
log(Kt 1)  1   log(N )  (1   )log(Kt )  log( Zt )
which implies
log(Yt )  2   log(N )  (1   )log(Yt 1)  log( Zt )
• Although the shocks are serially independent, their effects on
output are serially correlated.
Real Business Cycles
Formal RBC Models
• More general assumptions (e.g. about preferences and
depreciation rates) means models:
• cannot be solved analytically
• require numerical solution methods.
• Two approaches:
• set the variance of the technology shocks (Z) to replicate
exactly the variance of output
• use the ‘Solow Residual’ as the measure of the technology
shocks
Real Business Cycles
Testing Methodology
• Compare variables’ variance and covariance in the model with
those in the data – ‘eye-ball’ testing.
• Data for actual US Economy: detrended using the Hodrick-Prescott
trend.
• The Basic Model: the McCallum model model described earlier in the
previous section, but allowing depreciation to be incomplete.
• Kydland-Prescott Model:
• ‘Time to build’ model: it takes four quarters to build productive capital;
• ‘Fatigue effect’: the harder workers have worked in the past, the more
they value leisure today;
• the technology shock has both permanent and transitory components,
which agents cannot distinguish.
• Hansen Model: Similar to Basic Model, but workers cannot choose the
hours they work - they either work fixed hours or do not work any.
Real Business Cycles
Standard Deviations of Percentage Departures from Trend
Variable
Output
Consumption
Investment
Capital Stock
Hours
Productivity
US Economy
1.76
1.29
8.6
0.63
1.66
1.18
Basic Model
Model
1.76
0.55
5.53
0.47
0.91
0.89
Kydland-Prescott
Model
1.76
0.44
5.4
0.46
1.21
0.7
Hansen Model
Model
1.76
0.51
5.71
0.47
1.35
0.5
Contemporaneous Correlations with Output Departures from Trend
Variable
Consumption
Investment
Capital Stock
Hours
Productivity
US
Economy
0.85
0.92
0.04
0.76
0.42
Hours of work and productivity less
correlated with output than models’
predict
Basic
Model
0.89
0.99
0.06
0.98
0.98
Kydland-Prescott
Model
0.85
0.88
0.02
0.95
0.86
Hansen
Model
0.97
0.99
0.05
0.98
0.87
Consumption smoother than investment
But consumption too smooth and investment
insufficiently volatile
Constructed to
be identical
Real Business Cycles
Solow-Residual (SR) Method
•
•
Charles Plosser, ‘Understanding real business cycles’, Journal of
Economic Perspectives (1989).

1
Assume an aggregate production function: Yt  Zt  XN t Kt
•
where X is labour-augmenting technical progress and Z is the
technology shock that affects total factor productivity
•
Taking logs: logYt  log Zt   log X t   log Nt  (1   )log Kt
•
SR is the combined contribution
of X and Z:
•
Calculate SR as residual given
knowledge of Y, N, K and 
log SRt  log Zt   log X t
 logYt   log Nt  (1   )log Kt
Real Business Cycles
Solow-Residual (SR) Method
•
•
•
King and Rebelo
assume that
log(X) has a
deterministic
trend and log(Z)
is an AR(1)
process:
From definition of
log(SRt-1), we can
write log(Zt-1) as:
Substituting
we derive:
log SRt  log Zt   log X t
log X t  log X o   t
log Zt   log Zt 1   t
so
log SRt   log X o   t   log Zt 1   t
log Zt 1  log SRt 1   log X t 1  log SRt 1    log X 0   (t  1)
log SRt   log X o   t   (log SRt 1   log X o   (t  1))   t
log SRt     1     log X o   1     t   log SRt 1   t
Real Business Cycles
Solow-Residual (SR) Method
log SRt  logYt   log Nt  (1   )log Kt
log SRt     1     log Xo   1     t   log SRt 1   t
•
King and Rebelo use US data to calculate logSR
•
and then regress logSR on a time trend and lagged logSR
•
They estimate  to be 0.979 – a very high degree of persistence.
•
They combine the Solow Residual with an RBC model to derive
predicted variances and co-variances of the key variables.
Real Business Cycles
King & Rebelo Model
Actual and Simulated Standard Deviations
Variable
US Economy
King & Rebelo Model
Output
1.81
1.39
Consumption
1.35
0.61
Investment
5.3
4.09
Hours
1.79
0.67
Productivity
1.02
0.75
•
RBC models ‘explain’ about 78% (1.39/1.81) of the business cycle –
perhaps not surprising since SR is based on actual output data.
•
Simulated consumption still too smooth
•
Simulated Hours & Productivity also too smooth
Real Business Cycles
Assessment
•
To explain the actual business cycle we required unreasonably large and
persistent technology shocks.
•
Insufficiently strong internal propagation mechanisms to generate the
persistence in output - models only succeed if the shocks themselves are
serially correlated
•
Simulated hours more highly correlated with output than in the data - the
models over-emphasize the importance of intertemporal substitution.
• Microeconomic evidence suggest that the intertemporal substitution
effect is in fact very weak (an example is Joseph Altonji’s (Journal of
Political Economy, 1986) study of US household panel data).
• Actual correlation of hours and output due to a mechanism not included
in the RBC models.
• Keynesians suggest it is due to disequilibrium in the labour market
(unemployment falls when output rises)
Real Business Cycles
Assessment
•
The ‘productivity puzzle’
• RBC models predict high correlation between productivity and
employment (typically 0.9)
• Data suggests it is far lower – possibly negative
•
Galí (AER 1999):
•
price stickiness – so output is demand-determined
•
favourable technology shock requires less labour to produce required output
•
hence employment (hours) and productivity negatively correlated.
•
RBC models are promising once embedded in a new-Keynesian (sticky-price)
framework
Real Business Cycles
Assessment
•
Weak testing methodology – comparing simulated with actual ‘second
moments’.
• Hartley, Salyer and Sheffrin (1997) simulated a Keynesian economy
(using Ray Fair’s (1990) model), and then compared the simulated
economy with the RBC model.
• The RBC model did quite well in mimicking the variances and
correlations of the simulated data.
•
RBC models have difficulty in explaining recessions – falling output.
•
Alan Kirman (1992) questioned the representative agent assumption. Adding
a small measure of heterogeneity can have destructive consequences for
what we observe in the aggregate
Real Business Cycles
McCallum’s Conclusion:
•
It would seem to be virtually indisputable that the RBC literature has
provided a substantial number of innovative and constructive technical
developments that will be of lasting benefit in macroeconomic analysis. ...
the RBC studies have provided a healthy reminder that a sizeable portion of
the output and employment variability that is observed in actual economies
is probably the consequence of unavoidable shock, that is, disturbances not
generated by erratic monetary or fiscal policy makers.