real business cycles route map stochastic solow-swan growth model the crusoe economy – the simple...

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


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


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)


Model-based Trend Output

• 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.

• Model-based output gap:

where and is the log of the desired mark-

up of price over nominal marginal costs

tt

sx

log t tt

t t

W Ns

PY


UK Full Capacity Output using Model-generated Gap

11.2

11.4

11.6

11.8

12

12.2

12.4

12.6

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

z y

1979-80 Recession

11.84

11.85

11.86

11.87

11.88

11.89

11.9

11.91

11.92

11.93

11.94

11.95

1978.5 1979 1979.5 1980 1980.5 1981 1981.5 1982

z y

1990-91 Recession

12.105

12.11

12.115

12.12

12.125

12.13

12.135

12.14

12.145

12.15

12.155

1989.5 1990 1990.5 1991 1991.5 1992 1992.5 1993

z y


Stochastic Solow-Swan Growth Model

• A is grows at rate each year

• is rate of labour-augmenting technical progress

• N grows at rate n per year

• L grows at rate + n per year

• Aggregate production function:

1

1

0 1

( )

Y K L L AN

Y K L Ky k f k

L L L



• With constant savings rate (s):

( , )K I K s F K L K

( )K

s f k kL

• Divide both sides by L:

k K L Kn

k K L K

K K Kk n k n k

K L L

• Evolution of k:

( ) ( ) ( ) ( )k s f k k n k s f k n k

will be positive if ( ) ( )

will be negative if ( ) ( )

will be zero if ( ) ( )

k s f k n k

k s f k n k

k s f k n k



• With constant savings rate, the evolution of k is given by:

( ) ( )k s f k n k


y*

f(k)

s f(k)

(n++) k

kk*

y

k(0)

0

since ( ) ( )

k

s f k n k

Steady-state equilibrium (k*) is where the addition to the capital stock (through saving) is just enough to keep the capital-labour ratio constant.

In equilibrium, Y is growing at + n since y* is constant.

0

since ( ) ( )

k

s f k n k

Steady-State Equilibrium . ( ) ( )k s f k n k


‘Vibrating’ Production Function

1

1

( , ) where is total factor productivity

( )

t t t

tt t

Y z K L z F K L z

z K Ly z k z f k

L

• Imagine that the production function is stochastic:

'Good times': >1

'Bad times' : <1

'Normal' : =1

t

t

t

z

z

z

Fluctuations in total factor productivity lead to business cycles



• Imagine that z takes on two values - 1.5 and 0.5

s 0.5 f(k)

(n++) k

kk1

y

s 1.5 f(k)

k2

• Start at k = k1 and z = 0.5 (in equilibrium). y is constant so Y is growing at + n

• z rises to 1.5 and k starts to rise to k2 – the new equilibrium



• Y now grows faster than + n to approach higher steady state.

s 0.5 f(k)

(n++) k

kk1

y

s 1.5 f(k)

k2

• Before k2 is reached, let z fall back to 0.5

• k now falls back towards k1 and output growth falls below + n



t0

Transitional paths

log(Y)

Steady State Path for z = 0.5Slope = n +

Steady State Path for z = 1.5Slope = n +

Timet1t2


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


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


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.


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.


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:

2 31 1 2 2 3 3

0( , ) ( , ) ( , ) ( , ) ... ,j

t t t t t t t t t j t jj

U u C L u C L u C L u C L u C L

• Subject to budget and technology constraints:

1

1

( , )

t t t

t t

t t t t t t t

Y C I

N L

Z f N K C K K K

1

1

since

t t t t

t t t t

K K K I

I K K K



• 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)

• To derive analytical solution, McCallum assumes:

1

( ,1 ) log (1 )log(1 )

( , )

1

t t t t

t t t t t t

u C N C N

Z f N K Z N K

1

11

1 (1 )

(1 )

t t t

t t t

t

C N Z K

K N Z K

N N



• Although the shocks are serially independent, their effects on output are serially correlated.

1 11 1 (1 ) (1 ) t t t t tt tC N Z K K N Z K N N

• Taking logs and solving for K and Y:

1 1

2 1

log( ) log( ) (1 )log( ) log( )

which implies

log( ) log( ) (1 )log( ) log( )

t t t

t t t

K N K Z

Y N Y Z


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


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 CyclesStandard Deviations of Percentage Departures from Trend

Variable US Economy Basic Model Kydland-Prescott Hansen ModelModel Model Model

Output 1.76 1.76 1.76 1.76Consumption 1.29 0.55 0.44 0.51Investment 8.6 5.53 5.4 5.71Capital Stock 0.63 0.47 0.46 0.47Hours 1.66 0.91 1.21 1.35Productivity 1.18 0.89 0.7 0.5

Contemporaneous Correlations with Output Departures from Trend US Basic Kydland-Prescott Hansen

Economy Model Model ModelConsumption 0.85 0.89 0.85 0.97Investment 0.92 0.99 0.88 0.99Capital Stock 0.04 0.06 0.02 0.05Hours 0.76 0.98 0.95 0.98Productivity 0.42 0.98 0.86 0.87

Variable

Constructed to be identical

Consumption smoother than investment

But consumption too smooth and investment insufficiently volatile

Hours of work and productivity less correlated with output than models’

predict


Solow-Residual (SR) Method

• Charles Plosser, ‘Understanding real business cycles’, Journal of Economic Perspectives (1989).

• Assume an aggregate production function:

• where X is labour-augmenting technical progress and Z is the technology shock that affects total factor productivity

1t t tt

Y Z XN K

• Taking logs: log log log log (1 )logt t t t tY Z X N K

• SR is the combined contribution of X and Z:

log log log

log log (1 )log

t t t

t t t

SR Z X

Y N K• Calculate SR as residual given knowledge of Y, N, K and



• King and Rebelo assume that log(X) has a deterministic trend and log(Z) is an AR(1) process:

1

1

log log log

log log

log log

so

log log log

t t t

t o

t t t

t o t t

SR Z X

X X t

Z Z

SR X t Z

• From definition of log(SRt-1), we can write log(Zt-1) as:

1 1 1 1 0log log log log log ( 1)t t t tZ SR X SR X t

• Substituting we derive:

1

1

log log (log log ( 1))

log 1 log 1 log

t o t o t

t o t t

SR X t SR X t

SR X t SR



• 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.

1log 1 log 1 logt o t tSR X t SR

log log log (1 )logt t t tSR Y N K


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


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)


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


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


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.

real business cycles route map stochastic solow-swan growth model the crusoe economy – the simple...

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