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ECON6020/8020 – Macroeconomic Theory A

Course Information

Course profile

Webpage – Blackboard

- Lecture notes and tutorials

- Announcements

- http://blackboard.elearning.uq.edu.au

Assessment

- One assignment – total 10%

- Midterm exam – total 40%

- Final exam – total 50%

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Lecture 1 - Real Business Cycle Theory

Relevant reading:

Chapter 4 of Romer (2001)

Kydland, F. and E. Prescott (1982), “Time to build and aggregate fluctuations”, Econometrica, Vol.

50, 1345-1370.

Hansen, G. (1985), “Indivisible labor and the business cycle”, Journal of Monetary Economics, Vol.

16, 309-327.

Some facts about economic fluctuations

Modern economies undergo significant variations in aggregate output and employment. At some times,

output and employment fall while at others, output and employment rise rapidly.

Some important features of fluctuations:

- Fluctuations do not exhibit any simple regular or cyclical pattern. Comment [l1]: economists think that

the economy is affected by disturbances

of various types and sizes at more or less

random intervals.

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- Fluctuations are distributed very unevenly over the components of output (consumption, investment, net

export, etc.).

- There are asymmetries in output movements.

A baseline real business cycle (RBC) model:

The economy consists of a large number of identical competitive firms and price taking households.

The production is:

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Y K A Lt t t t

, 0 1

Output is divided between consumption, investment, and government expenditure which is financed by

lump-sum taxes.

Capital evolves according to:

1K K I K K Y C G Kt t t t t t t tt

Comment [l2]: Output seems to be characterized by relatively long periods

when it is slightly above its usual path,

interrupted by brief periods when it is relatively far below.

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Factor markets are competitive so each input receives its marginal product:

1Ktw At t A Lt t

1A Lt trt Kt

The representative household’s utility maximization:

Max ,10

Nt tU e u c lt t Ht

Here, (.) ln ln 1 ttu c b l , 0b is utility each period

Nt is population (growing at rate n so that N ntN et )

H is total households

Comment [l3]: These can be derived from profit maximization conditions.

Comment [l4]: Utility is derived from consumption and leisure

Comment [l5]: Nt

His average number

of people in the household. The total

number of household is fixed for simplicity.

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Ctct Nt is consumption per each member of the household

Ltlt Nt is amount of time devoted to work for each member.

The zero profit condition implies:

y w l r kt t t t t

The budget constraint is:

(1 )1

C K Y G Kt t t t t

And in per worker term:

(1 )1

nc k y g k et t t t t

(1 )1

nr k w l g k et t t t t t

The Bellman’s equation is:

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max ( ,1 )1,

1

nV k u c l e E V kt t t t tk ltt

, n

The optimal conditions are:

1kt

: 1. 0

1 1

Vu c nt t te Etc k kt t t

Or

1 11

1

uut te E rt tc ct t

(1)

The household also chooses optimal labor supply. The trade-off between consumption and leisure is:

lt

: . 0u ut twtc lt t

(2)

Here, a reduction in leisure leads to an increase in consumption which is equal to the wage rate wt .

Comment [l6]: Note that

.V u ct t t

k c kt t t

,

1

c nt ekt

,

and 1 1

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ct r

tkt

Comment [l7]: This equation says that the marginal utility of forgone consumption

is equal to the present value of expected gain in marginal utility of consumption in

the future.

Comment [l8]: The condition is

. 0u c ut t t

c l lt t t

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Given that (.) ln ln 1u c b ltt then 1utcc tt

and

1

u btl lt t

. Equation (1) becomes:

1 1 11

1e E rtc c tt t

For simplicity, assume no government and 100 percent depreciation each period ( 1 ).

Hence 1c s yt t t , 1

s yt tk nt e

, 11

11

ytr

t kt

, and 1 ytwt lt

Substituting these in gives:

1 1 11. .

1 1 111 1 1

y nete E e Et t s yks y s y s t ttt t t t t

Or

1 1.1 1

1

ne Etss stt t

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Let 1

s st t

then ns e .

And equation (2) becomes:

1 1

1 1 1

w w l yb t t t tcl c l s y l s ltt t t t t t t t

Solving for lt yields:

1

1 1l

b s

Thus labor supply is also constant.

Note: When the utility function is more general such as a CES form

1

1 1u c l

the saving

rate and labor supply would respond to the state of the economy. However, there may not be explicit

solution to this model as in Cobb-Douglas case.

Comment [l9]: The saving rate is constant and unresponsive to the realization

and expectation of A .

Comment [l10]: The reason is that movements in either technology or capital

have offsetting impacts on the relative wage

and interest rate effects on labor supply. An improvement in technology raises current

wages and raises labor supply. However, by

raising the amount saved, it also lowers the expected interest rate and hence reduces

labor supply.

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With regard to technology, assume that ln A A gt At t where A reflects the effects of the shocks and is

assumed to follow a first order autoregressive process:

1A At A t At

, 1 1A

(3)

where At

’s are white-noise disturbances – a series of mean-zero shocks that are uncorrelated with one

another.

With a note that 1

K sYt t

and L lNt t , from the production function:

ln ln (1 )(ln ln )Y K A Lt t t t

ln ln ln ln ln1

s Y A l Nt tt

ln ln (1 )( ) (1 ) (1 )(ln )1

s Y A g t A l N nttt A

(4)

The two components on the right that do not follow deterministic paths are 1

Yt

and At . The result in (4)

can be rewritten as:

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(1 )1

Y Y At tt

(5)

where Yt is the difference between lnYt and the value it would take if 0At .

From (5):

11 1 21

A Y Yt t t

Substituting this result into (3) and then (5):

(1 )1 1

Y Y At t A t At

(1 )1 1 2

Y Y Yt A t t At

(1 )1 2

Y YA t A t At

Hence, the departures of log output from its normal path follow a second-order autoregressive process. Y

can be written as a linear combination of its two previous values plus a white noise disturbance.

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The combination of a positive coefficient on the first lag of Y and a negative coefficient on the second lag

can cause output to have a “hump-shaped” response to disturbances.

When is not large, the dynamics of output are determined largely by the persistence of the technology

shocks A

.

Although the model yields interesting output dynamics, it does not have any mechanism that translates

transitory technology disturbances into significant long-lasting output movements.

The saving rate is constant (so that consumption and investment are equally volatile) and labor input does

not vary.

In practice, investment varies much more than consumption.

In actual fluctuations, the real wage appears to be only moderately procyclical (move in the same direction

with output) while the model predicts that the real wage is highly procyclical: the real wage rises one-for-

one with output since L does not respond to technology shocks.

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In terms of improving the model to better capture many of the major features of observed output

movements, a number of things can be considered such as introducing depreciation of less than 100

percent and shocks to government purchases.

Almost all RBC models cannot be solved analytically.

Papers in this area generally address this difficulty by solving the models numerically.

That is, once a model is presented, parameter values are chosen, and the model’s quantitative implications

for the variances and correlations of various macroeconomic variables are discussed.

Discussion

There are two problems with the investigation of the persistence of fluctuations:

Statistically, it is difficult to learn about long-term characteristics of output movements from data covering

limited time spans. The existence of a permanent component to fluctuations and the asymptotic response

of output to an innovation concern the characteristics of the data at infinite horizons.

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The fact that output movements are highly persistent in the sample may be consistent with the presence of

permanent component to fluctuation but may also reflect that output reverts extremely slowly to a

deterministic trend.

Alternatively, if output returns rapidly to some trend in a sample is consistent not only with trend

stationary but also with a view that a small portion of output movements is not just permanent but

explosive – so that the reaction is to revise the forecast of output in the future.

Samples of plausible length contain few independent, long subsamples. As a result, no procedure is likely

to provide decisive evidence about the long-term effects of shocks.

Various approaches to studying persistence have been employed. The point estimates generally suggest

considerable persistence. However, at horizons of more than 5 years, the estimates are not very precise.

The theoretical difficulty is that there is only a weak case that the persistence of output movements, even if

it could be measured precisely, provides much information about driving forces of economic fluctuations.

Besides technology which may have an important trend-reverting component RBC models take into

account shocks from other sources as well.

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Extensions of RBC model

Since the work by Kydland and Prescott (1982, Econometrica), research in this area has considered many

variations and extensions. Here, we focus on the indivisible-labor version by Hansen (1985, Journal of

Monetary Economics) and Rogerson (1988, JME).

l for each individual is either employed, 0l , or unemployed, 0. There are fixed costs of working.

Once the number of workers employed is determined, individuals are divided between employment and

unemployment randomly. The number of workers employed is

0

LtEt l . The probability of being employed

is / /0

L l Nt t

.

The expected utility from leisure is:

0 0ln 1 ln10

L Lt tNtl lb l b

N Nt t

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This expression is linear in Lt : individuals are not averse to employment fluctuations.

In contrast, when all individuals work the same amount, utility from leisure is ln 1LtbNt

which is

decreasing in Lt at an increasing rate (the second derivative with respect to Lt is negative), i.e. there is

increasing marginal disutility of working.

Thus, in the divisible labor case, the labor input is less responsive to labor productivity and wages than in

the indivisible labor case.

Objections to RBC models

Technology shocks

The models posit technology shocks with a standard deviation of 1% each quarter. It is difficult to have

such a large quarter-to-quarter swing in the real world.

Intertemporal in labor supply

Variations in incentives to work in different periods drive employment fluctuations in the model.

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Microeconomic studies, however, have had little success in detecting significant intertemporal elasticities

of substitution in labor supply.

The omission of monetary disturbances

Fluctuations are due to real rather than monetary shocks.

However, there is strong evidence that monetary shocks have important real effects (incomplete

adjustment of nominal prices or wages).

Lack of propagation mechanisms

The dynamics of output follow the dynamics of the shocks quite closely. The model produces realistic

output dynamics only to the extent that it assumes them in the driving processes.

However, there are other predictable movements in output, consumption, and hours in actual economy but

not in the baseline RBC model.

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RBC theoretical modeling

Papers on RBC tradition have considered a wide range of non-Walrasian features: rigid nominal

prices/wages and monetary disturbances, externalities, efficiency wages, job search and matching, etc.

These models have the following characteristics:

- The “default” model setting choices are Walrasian which imposes discipline: it is not free to make any

non-Walrasian assumptions to generate the results one likes.

- The RBC modeling focuses on general equilibrium which, in general, makes the analysis more

complicated.

As a result, the analysis must take a simpler approach to modeling the central issue of interest.

- Models are evaluated by calibration so their success is judged by the capacity to match major variances

and covariances in the data. Because the models are not tested against alternatives, there may be

completely different models that can match the moments as well.