# Chapter 2 Real Business Cycle Theory

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```					Advanced Macroeconomics                               37                          Chapter 2: Real Business Cycle Models

Chapter 2: Real Business Cycle Theory

1.     Stylized Facts on Business Cycles

2.     A Baseline Real Business Cycle (RBC) Model

3.     Empirical Applications

Literature:
Romer, Chapter 4
Felderer / Homburg, Kapitel VIII

___________________________________________________________________________________________
The material in this document is for use in the lecture “Advanced Macroeconomics”, held by Professor Claudia
Buch at the University of Tübingen, only. It should not be duplicated, quoted, or used elsewhere without prior
consent of the author.

1. Stylized Facts on Business Cycles1

Correlations:
o Output movements across sectors are correlated.

Volatility:
o Investment is more volatile than consumption.
o Capital stock is less volatile than output.
o Long-term interest rates are less volatile than short-term rates.
o Employment is as volatile as output.
o Productivity is less volatile than output.
o Consumption is less volatile than investment.

1 Stadler, George W., Real Business Cycles, Journal of Economic Literature, Vol. XXXII, December, pp. 1750-1783

Cyclicality:
o Velocity is countercyclical.
o Prices appear to be counter-cyclical.
o Labor input is pro-cyclical.

Aim of the real business cycle theory is to find a theoretical model which generates responses
of key macroeconomic variables that have these features.

2. A Baseline Real Business Cycle Model

The Basic Model: Prescott (1986), Christiano and Eichenbaum (1992), Baxter and King (1993), Campbell (1994)

Motivation
Walrasian model without imperfections (Ramsey Model) does not generate economic fluctuations.
Real business cycle models extend Walrasian models to encorporate
o source of disturbances (technology shocks, government purchases)
o variations in employment.

Assumptions:
o Large number of identical price-taking firms
o Large number of identical price-taking households

Yt = K tα ( At Lt )
1−α
(1)                            with 0 < α < 1

Y     = output
K     = capital stock
A     = technology parameter
L     = labor

Capital stock in period t+1:

K t +1 = K t + I t − δK t
(2)
= K t + Yt − Ct − Gt − δK t

Government and consumer behavior:
o Government purchases are financed by lump-sum taxes.
o Households are infinitely lived.
o No capital market imperfections.

Ricardian equivalence holds.

Optimization of firms

Real profits are given by:

Π t = Yt − wt Lt − rt K t − δK t

Maximizing profits with respect to L and K yields the first order conditions:

wt = (1 − α )K tα ( At Lt ) At
−α

α
(3)                  ⎛ K ⎞
= (1 − α )⎜ t ⎟ At
⎜       ⎟
⎝ At Lt ⎠
and
1−α
⎛ AL ⎞
(4)    rt = α ⎜ t t ⎟
⎜ K ⎟         −δ
⎝ t ⎠

Optimization of households

∞
Nt
(5)   U = ∑ e − ρt u (ct ,1 − lt )
t =0                    H

u (•) = instantenous utility function
ρ      = discount rate
N      = population
H      = number of households
n      = growth rate of the population
ct     = consumption per household
lt     = work hours per household

Utility of households depends on
o Consumption
o Leisure (number of hours available, normalized to 1, minus number of hours worked)

Population growth:

(6)    ln N t = N + nt          n<ρ

N + nt
Level of N t is given by N t = e               .

Assume log-linear utility:

(7)    ut = ln ct + b ln (1 − lt )           b>0

Technology

No technology shocks:
Technology grows with the rate of technological progress A + gt .

With technology shocks:

~
(8)    ln At = A + gt + At

~
Shock At follows a first-order autoregressive process:

~        ~
(9)     At = ρ A At −1 + ε A,t   −1 < ρ A < 1

with − 1 < ρ A < 1: no explosive process

ρ A shows how persistent shocks are across time!

ε A,t are white noise disturbances (zero mean, no correlation across time).

Government purchases

(10) ln Gt = G + (n + g )t + Gt
~

~           ~
(11) Gt = ρG Gt −1 + ε G ,t   − 1 < ρG < 1

o Government purchases are an additional source of disturbances.
o Government purchases grow with the same rate of growth as the rest of the economy
(technology + labor force growth).
o Otherwise, the government would become arbitrarily large (or small).

Household Behavior under Certainty

One-period model:
o Household lives only one period.
o Household has only one member.

Utility function:         ln c + b ln (1 − l )
Budget constraint:        c = wl

Lagrangian:
(12)   max Λ = ln c + b ln(1 − l ) + λ (wl − c )
c ,l

First-order conditions:

∂Λ 1
(13)      = −λ = 0
∂c c
∂Λ       1
(14)      = −b      + λw = 0
∂l      1− l

1
(13) and the budget constraint    λ=         substitute into (14)
wl
1 1
(15) − b       + =0         Labor supply
1− l l

o Labor supply is independent from wages.
o Reason: Utility is logarithmic in consumption, and household has no initial wealth. Hence, the
income and the substitution effect of an increase in wages offset each other.

Two-period model:

c2          wl
(16)   c1 +              = w1l1 + 2 2
1+ r         1+ r

r     = real interest rate

Lagrangian:
max                Λ = ln c1 + b ln(1 − l1 ) + e − ρ [ln c2 + b ln(1 − l2 )]
c1 , c 2 , l1 , l 2

⎡       wl          c ⎤
(17)                                      + λ ⎢ w1l1 + 2 2 − c1 − 2 ⎥
⎣       1+ r       1+ r ⎦

Four first-order conditions c1, c2 , l1, l2

Only two are needed to show the effect of wages on labor supply:
b
(18)          = λw1
1 − l1
and

e− ρ b   1
(19)          =     λw2
1 − l2 1 + r
1 b
Solve both equations for λ :    (18’) λ =
w1 1 − l1
1 + r e− ρ b
(19’) λ =
w2 1 − l2

1 b        1 + r e− ρ b
=
w1 1 − l1 w2 1 − l2
(18’) = (19’):
1 − l2           w
= (1 + r ) 1 e − ρ
1 − l1           w2

Allocation of labor across time depends on the real wage:
w1
work relatively more today and less tomorrow
w2
(Note: elasticity of substitution between labor supply today and tomorrow is 1 (because of the logarithmic utility
function)

Supply of labor depends on the real interest rate:
r     work relatively more today and less tomorrow (because future income and consumption are
discounted at a higher interest rate)

cycle models (intertemporal substitution of labor supply).

Household optimization under uncertainty

o Shocks to technology and government spending imply that households cannot make
deterministic choices but rather act under uncertainty.
o Optimization is obtained under expectations concerning future realizations of variables.
o Two main issues:

1. The trade-off between consumption today and tomorrow
2. The trade-off between consumption and labor supply

o Main mechanism: In the optimum, gains in expected utility must equal losses in expected utility.

The Trade-Off Between Consumption Today and Tomorrow

Loss in utility in the current period:
Combine utility functions (5) and (7) from above:

∞

∑ e−ρt [ln ct + b ln(1 − lt )]
Nt
(5’) U =
t =0                             H

Marginal utility of consumption per member of the household:

∂U           1 Nt
= e − ρt
∂ct          ct H

Utility cost of the change in consumption:

N t ∆c
e − ρt
H ct

Gain in utility in the following period:
Population growth            change in consumption for the member of the household:
e − n (1 + rt +1 )∆c

Marginal utility of increase period t+1 consumption of the household:

⎡ − ρ (t +1) 1 + rt +1 N t +1 −n ⎤
Et ⎢e                           e ⎥ ∆c
⎣              ct +1 H           ⎦
[Note the expectational operator!]

In the optimum, a small change in c should leave (expected) utility unchanged.

Expected costs of reducing current consumption should equal the expected benefits of
increasing future consumption:

N t ∆c      ⎡             1 + rt +1 N t +1 − n ⎤
(22)   e − ρt          = Et ⎢e − ρ (t +1)                 e ⎥ ∆c
H ct        ⎣               ct +1 H            ⎦

Which can be transformed into:

Nt 1               N             ⎡1 + rt +1 ⎤
e − ρt        = e − ρ (t +1) t +1 e −n Et ⎢
H ct                H            ⎣  ct +1 ⎥ ⎦

1                        ⎡1 + rt +1 ⎤
and, since N t +1 = N t e , into
n           = e − ρ (t +1)+ ρt Et ⎢          ⎥
ct                       ⎣ ct +1 ⎦

1     −ρ  ⎡1 + rt +1 ⎤
= e Et ⎢
ct +1 ⎥
(23)
ct        ⎣          ⎦

Trade-off between current and future consumption depends on the expectations of future
consumption and future rates of returns and on the interaction between the two.

⎡ 1 ⎤
(Note: The term in squared brackets cannot be decomposed into   Et ⎢ ⎥ + Et [1 + rt +1 ]!)
⎣ ct +1 ⎦

−ρ ⎧  ⎡ 1 ⎤                     ⎛ 1            ⎞⎫
= e ⎨ Et ⎢ ⎥ Et [1 + rt +1 ] + Cov ⎜
1
(24)                                         ⎜ c ,1 + rt +1 ⎟⎬
⎟
ct       ⎩ ⎣ ct +1 ⎦                  ⎝ t +1         ⎠⎭

If Cov < 0 : Return to savings is low when the marginal utility of consumption is high.

The Trade-Off Between Consumption and Labor Supply

How should household decide whether to increase its supply of labor by a marginal unit ∆l in
order to increase consumption in the same period?

Nt b
Marginal disutility of working:                 e − ρt
H 1 − lt

Nt b
Utility costs:                                  e − ρt               ∆l
H 1 − lt

Nt 1
Utility benefit:                                e − ρt           wt ∆l
H ct

− ρt   Nt b                N 1
Benefits must equal costs in the optimum:   (25) e                   ∆l = e − ρt t wt ∆l
H 1 − lt            H ct

Transforming (25) gives the following optimality condition:

ct   w
(26)         = t
1 − lt b

[No expectations are involved because only current variables are affected.]

Household behavior is described by equations (23) and (26).

A Special Case of the Model

o Baseline model contains linear and log-linear elements        cannot be solved analytically.
o Simplified version of the model is needed.

Assumptions:
o No government
o focus on technology shocks        equation (10) and (11) dropped
o 100% depreciation each period (for analytical traceability)     equations (2) and (4) become

(27)    K t +1 = Yt − Ct

1−α
⎛ At Lt ⎞
(28)   1 + rt = α ⎜
⎜ K ⎟   ⎟
⎝ t ⎠

Solving the model:
o Competitive markets
o No externalities
o Individuals live infinitely

Equilibrium must correspond to the Pareto optimum.
solve for the competitive equilibrium

Two variables are of interest:
o Labor supply per person (l)
o Fraction of output that is saved (s)

How do labor supply and savings depend on technology and the capital stock?

Strategy:
o Re-write model in log-linear form.
o Substitute (1 − s )Y / N for C.
o Focus on equations describing optimum of households (23) and (26); remaining conditions
follow from the assumption of competition.

Result:
o s will be independent from technology and the capital stock    s is constant, and model can be
solved analytically.
o Note: The constancy of s is the result of the simplifying assumptions log-utility, Cobb-Douglas
production technology and 100% depreciation, and it is not a general result.

Optimal savings:

Recall the condition for an optimal allocation of consumption across time (= optimal savings):

1     −ρ  ⎡1 + rt +1 ⎤
(23)      = e Et ⎢          ⎥
ct        ⎣ ct +1 ⎦

Substitute (1 − s )Y / N and take logs:

⎡                      ⎤
⎡          Yt ⎤               ⎢ 1+ r                 ⎥
(29) − ln ⎢(1 − st )      = − ρ + ln Et ⎢            t +1
⎥
⎣          Nt ⎥
⎦               ⎢ (1 − st +1 )  Yt +1 ⎥
⎢
⎣               N t +1 ⎥
⎦

Moreover, with 100% depreciation, we have
1−α
⎛ At +1 Lt +1 ⎞
1 + rt +1 = α ⎜
⎜ K             ⎟
⎟
⎝       t +1 ⎠

Yt +1           α −1
1 + rt +1 = α        α
K t +1
K t +1
αYt +1
=
K t +1

and   K t +1 = Yt − Ct = stYt

substituting into (29) gives:

− ln[(1 − st )] − ln Yt + ln N t
⎡                            ⎤
⎢          αYt +1            ⎥
= − ρ + ln Et ⎢                            ⎥
⎢ K t +1 (1 − st +1 ) Yt +1 ⎥
⎢
⎣                     N t +1 ⎥
⎦
(30)
⎡ αN t +1 ⎤
= − ρ + ln Et ⎢
⎣ st (1 − st +1 )Yt ⎥
⎦
⎡ 1 ⎤
= − ρ + ln α + ln N t + n − ln st − ln Yt + ln Et ⎢          ⎥
⎣1 − st +1 ⎦

[Note: In the last step, expectations only of future variables are taken]

⎡ 1 ⎤
This can be simplified into:           (31) ln st − ln[(1 − st )] = − ρ + ln α + n + ln Et ⎢          ⎥
⎣1 − st +1 ⎦

Implications for the optimal savings decision:

A and K do not enter the optimality condition for the allocation of consumption across time (and thus
for optimal savings)
There is a constant value of s that satisfies condition (31).

Assume that st = s . Then, (31) becomes:
ˆ
⎡ 1 ⎤
ln s − ln[(1 − s )] = − ρ + ln α + n + ln ⎢
ˆ           ˆ
⎣1 − s ⎥
ˆ⎦
ln s − ln[(1 − s )] = − ρ + ln α + n + ln 1 − ln[(1 − s )]
ˆ           ˆ                                      ˆ

and          (32)   ln s = ln α + n − ρ
ˆ

or           (33)   s = αe n − ρ
ˆ

Savings rate is constant.

Optimal work decision:

Recall the condition for an optimal increase in labor supply to increase current consumption:

ct   w
(26)         = t
1 − lt b

Ct          Y
With ct =       = (1− s ) t , we can write:
ˆ                             (1 − s ) Yt
ˆ
1    w
= t
Nt          Nt                                    N t 1 − lt b

⎡           Yt ⎤
and, in log-linear form                       (34) ln ⎢(1 − s )
ˆ        ⎥ − ln (1 − lt ) = ln wt − ln b
⎣           Nt ⎦

With a Cobb-Douglas production function, we have
Yt
wt = (1 − α )
Lt N t

Substituting into (34) gives

⎡        Y ⎤                      ⎡          Y ⎤
ln ⎢(1 − s ) t ⎥ − ln (1 − lt ) = ln ⎢(1 − α ) t ⎥ − ln b
ˆ
⎣        Nt ⎦                     ⎣        lt N t ⎦

Written in log-linear form:

ln(1 − s ) + ln Yt − ln N t − ln(1 − lt ) =
ˆ
ln(1 − α ) + ln Yt − ln lt − ln N t − ln b
(35)

Cancelling terms gives:

(36)   ln lt − ln(1 − lt ) = ln(1 − α ) − ln(1 − s ) − ln b
ˆ

Writing (36) in non-logarithmic form and transforming gives:

lt    1−α
=
1 − lt (1 − s )b
ˆ
1−α
lt = (1 − lt )
(1 − s )b
ˆ
⎛      1−α ⎞ 1−α
lt ⎜1 +           ⎟=
⎝    (1 − s )b ⎠ (1 − s )b
ˆ           ˆ
⎛ (1 − s )b + 1 − α ⎞ 1 − α
ˆ
lt ⎜                    ⎟=
⎝      (1 − s )b ⎠ (1 − s )b
ˆ               ˆ

lt =
(1 − α )(1 − s )b
ˆ
(1 − s )b[(1 − s )b + 1 − α ]
ˆ         ˆ

1−α
and thus            (37) lt =                         ≡ lˆ
(1 − s )b + 1 − α
ˆ

If the savings rate is constant, labor supply is constant.

Does this imply that technology does not matter?

o No. Due to the special assumptions of the model, the effects of an improvement in technology
merely cancel out.
o Improvement in technology:
o Current wages increase relative to future wages      current labor supply
o Savings       expected interest rate        labor supply

Due to the construction of the model, the two effects simply cancel out.

Is the solution to the model unique?

o The remaining parameters of the model can be found without additional optimization.
o Hence, we have found one solution to the model with s and l being constant.
o This solution describes the optimization problem of the representative household.
o Standard results about optimization imply that this problem has a unique solution.

Discussion of the results:

o Real shocks drive output movements.
o Because there are no externalities or market failures, movements are the optimal responses to
shocks.

No role for the government to mitigate fluctuations.

How large are the implied fluctuations?

Re-write the production function

Yt = K tα ( At Lt )
1−α

as
(38) ln Yt = α ln K t + (1 − α )(ln At + ln Lt )

With K t = sYt −1 and Lt = l N t , we get
ˆ               ˆ

ˆ           (
ln Yt = α ln Yt −1 + α ln s + (1 − α ) ln At + ln lˆ + ln N t   )
= α ln Yt −1 + α ln s + (1 − α )(A + gt ) + (1 − α )At
~
ˆ
(               )
(39)
+ (1 − α ) ln lˆ + N + nt

~
since (6) ln N t = N + nt and (8) ln At = A + gt + At .

On the right-hand-side (RHS) of equation (39), there are two variables that do not follow a
deterministic path, i.e. that are stochastic: α ln Yt −1 and (1 − α ) At .
~

(40) Yt = αYt −1 + (1 − α ) At
~      ~              ~
Hence, we can write (39) in the following form:

which gives the difference between the output that would be achieved in the absence of any shocks
to A (i.e. if At = A + gt ) and the output in the presence of shocks.

If (40) holds for each period, we can also write:

Yt −1 = αYt − 2 + (1 − α ) At −1
~        ~                 ~

~
or, solving for At −1

(1 − α ) At −1 = −αYt −2 + Yt −1
~         ~       ~
~        ~
(41) ~         Yt −1 − αYt −2
At −1 =
1−α

Recalling that A follows an autoregressive process of the following form

~        ~
At = ρ A At −1 + ε A,t

and substituting this and (41) into (40) gives

~    ~
⎛ ⎛ Yt −1 − αYt − 2 ⎞ ⎞
Yt = αYt −1 + (1 − α )⎜ ρ A ⎜
~     ~
⎜ ⎜ 1 − α ⎟ + ε A ,t ⎟
⎟ ⎟
⎝ ⎝                 ⎠ ⎠

This can be simplified into

Yt = αYt −1 + ρ A (Yt −1 − αYt − 2 ) + (1 − α )ε A,t
~     ~            ~         ~

= (α + ρ A )Yt −1 − ρ AαYt − 2 + (1 − α )ε A,t
(42)              ~             ~

Or, using lag operators:

= (α + ρ A )LYt − ρ AαL2Yt + (1 − α )ε A,t
~                ~          ~
(42’) Yt

o Departures of output from its normal path follow a second order autoregressive process, i.e.
output movements are persistent.
o Output can be written as a linear combination of its two previous values plus a white-noise
disturbance.

Implications of the simplified version of the model:

~                                     ~
o Because of the positive coefficient on Yt −1 and the negative coefficient on Yt − 2 , the adjustment is
hump-shaped.
o Because all of the adjustment takes place in two periods, the simplified version of the model
does not allow for temporary technology shocks to have long-lasting effects on output.
o Constant savings rate implies that consumption and investment are equally volatile.
o In reality, however, investment is much more volatile than consumption.
o Labor input does not vary.
o In reality, however, labor input is strongly pro-cyclical.
o Wage rate rises one-to-one with output.
o In reality, however, wages are moderately pro-cyclical.

Modifications of the simplified model:

(1) Less-than-full depreciation:
Positive technology shock
Marginal productivity of capital increases.
Households save more.
Consumption growth is positive.
Interest rate increases.
Current labor supply increases.
With incomplete depreciation, investment and employment respond more to shocks (because
change in capital stock is more persistent).

(2) Shocks to government purchases:
Household’s tax liability increases.
Life-time wealth decreases.
Labor supply increases (to maintain the level of consumption).
Output increases and real wages fall.
Wages are not procyclical anymore.

Solving the model in the general case

Analytical solution of the model is not available.

Two alternatives to analytical solutions:

1) Calibration techniques
o Solve the model numerically by choosing some parameter values.
o Campbell (1994): Calibration provides little information on sources of model’s implications
How general are the results? Do they depend on the parameter values chosen?

2) First-order Taylor approximation
o Solve model using log-linear version of the model around balanced growth path.
o Investigate properties of approximated models.
o Analyze response of models to shocks.

Log-linearizing the model around the balanced growth path

State variables:
o Capital stock inherited from the previous period
o Technology
o Government purchases

Endogenous variables:
o Consumption
o Employment

Log-linearization around non-stochastic balanced growth path:

~           ~   ~        ~
(43) Ct ≅ aCK K t + aCA At + aCG Gt   consumption
~           ~   ~        ~
(44) Lt ≅ aLK K t + aLA At + aLG Gt   labor

a     = functions of the parameters of the model
= difference between actual and balanced-growth-path value, e.g. At = ln At − ( A + gt )
~
∼

o Consumption and labor supply are linear functions of the logs of K, A, and G (the state
variables).
o Consumption and labor supply are on their balanced growth path if K, A, and G are on their
balanced growth path.

Method of undetermined coefficients:

o a’s are unknown and must be obtained from the model.
o For a’s to be a solution to the model, they must meet the first order conditions for the
household’s optimum.
o Aim: Find general functional form that solves the model.

Application to the intratemporal and the intertemporal first order conditions from above.

Intratemporal First Order Condition:

Household’s trade off between consumption and labor supply:

ct   w
(26)         = t
1 − lt b

and optimal wages:

α
⎛ K ⎞
(3) wt = (1 − α )⎜ t ⎟ At
⎜ AL ⎟
⎝ t t⎠

imply the following log-linear form:

⎛1 − α ⎞
(45) ln ct − ln (1 − lt ) = ln⎜     ⎟ + (1 − α )ln At + α ln K t − α ln Lt
⎝  b ⎠

First-order Taylor-series approximation

What is the difference between the variables in equation (45) and their balanced growth
paths?

Right-hand-side (RHS):

(1 − α ) A t
~         ~       ~
+ α K t − α Lt

Left-hand side (LHS):
o Population growth is exogenous and is thus not affected by the shock.
o Deviation of total consumption is determined by deviation of consumption per worker
~ ~         ~ ~
Ct = ct and lt = Lt

Derivatives of LHS:

∂[ln ct − ln (1 − lt )]      ∂[ln ct − ln(1 − lt )]           l*
= 1,                               =
∂ ln ct                      ∂ ln lt         l =l *
1− l *
t

log-linearizing (45) around the balanced growth path yields:

l* ~
Lt = (1 − α ) At + αK t − αLt
~                         ~     ~      ~
(46) Ct +
1 − l*

*
where l = value of l on the balanced growth path

Exkurs: Taylor Series of a Polynomial Form
(Chiang, p. 256)

Aim: Expand quadratic form f ( x ) = 2 + 4 x + 3 x around any point x0 .
2

Interpret any value of x as a deviation from x0 :

f ( x ) = 2 + 4( x0 + δ ) + 3( x0 + δ )
2

f ' ( x ) = 4 + 6( x0 + δ )
f ' ' (x) = 6

Since x0 is fixed, only δ can be treated as a variable, and we can write:

g (δ ) = 2 + 4( x0 + δ ) + 3( x0 + δ )
2
[≡ f ( x )]
g ' (δ ) = 4 + 6( x0 + δ )
g ' ' (δ ) = 6

Expansion of g around zero δ = 0 yields the following Maclaurin series (= expansion around 0):

g (0 ) g ' (0 )    g ' ' (0 ) 2
g (δ ) =         +         δ+           δ
0!      1!           2!

Since x = x0 + δ , δ = 0 implies x = x0 , hence:

g (0 ) = f ( x0 ) g ' (0 ) = f ' ( x0 ) g ' ' (0 ) = f ' ' ( x0 )
And thus

f ( x0 ) f ' ( x0 )
f ( x )[= g (δ )] =           +           ( x − x0 ) + f ' ' ( x0 ) ( x − x0 )2
0!        1!                       2!

For f ( x ) = 2 + 4 x + 3 x , we obtain
2

f ( x0 ) = 2 + 4 x0 + 3 x0           f ' ( x0 ) = 4 + 6 x0   f ' ' ( x0 ) = 6
2

Which gives the Taylor series formula as

f ( x ) = 2 + 4 x0 + 3x0 + (4 + 6 x0 )( x − x0 ) +
6
2
( x − x0 )2
2
= 2 + 4 x + 3x 2

Hence, the Taylor series correctly represents the original function.

Taylor’s theorem:
Given an arbitrary function φ ( x ) and knowing the value of the function at x = x0 as well as the values
of its derivatives at x0 , the function can be expanded around x0 by using the following formula ( Rn
= ‘reminder’):
f ( x0 ) f ' ( x0 )
φ (x) =         +               ( x − x0 ) + f ' ' ( x0 ) ( x − x0 )2
0!            1!                       2!
f ' ' ( x0 )
+ ... +               ( x − x0 )n + R n
n!
= Pn + Rn

Using (43) and (44):
~ ⎛ l*           ⎞
+ α ⎟(aLK K t + aLA At + aLG Gt )
~         ~                               ~         ~        ~
aCK K t + aCA At + aCG Gt + ⎜
⎜1 − l*     ⎟
(47)                             ⎝           ⎠
= (1 − α ) At + αK t
~      ~
~ ~ ~
Equation (47) must hold for all values of K t , At , Gt . Otherwise, households could increase their utility
by changing current consumption and labor supply.

Coefficients on these variables on both side of the equation must be equal:

⎛ l*        ⎞
(48) aCK     +⎜
⎜1 − l* + α ⎟aLK = α
⎟
⎝           ⎠
⎛ l*        ⎞
(49) aCA + ⎜
⎜1 − l* + α ⎟aLA = 1 − α
⎟
⎝           ⎠
⎛ l*        ⎞
(50) aCG + ⎜
⎜1 − l* + α ⎟aLG = 0
⎟
⎝           ⎠
Equations (48)-(50) inform us on response of households to changes in technology, capital
stock, and government spending.

How do C and l respond to changes in the state variables?

Note: Utility must stay constant (in the steady state)!

Changes in government consumption (50):
Government consumption does not affect wages for a given level of labor supply (eq. (45)).
But: Increase in labor supply due to an increase in government spending has an impact:
Wages fall, marginal disutility of working rises.
Households will increase labor supply only if marginal utility of consumption is higher (which is
the case if consumption has declined).
Labor supply and consumption move into opposite directions (and, according to eq. (50), these
movements are just offsetting each other).

Changes in technology (49):
o Improvement in technology raises wage for a given level of labor supply (higher productivity).
o Households could raise their utility by not changing labor input and consumption.
o In order to keep marginal utility equal, households must increase either labor supply (marginal
productivity of labor and thus wages fall) or consumption (marginal utility of consumption falls).

Changes in capital stock (48):
o Increase in capital stock raises wage for a given level of labor supply (higher productivity).
o Households could raise their utility by not changing labor input and consumption.
o In order to keep utility constant, households must increase either labor supply (marginal
productivity of labor and thus wages fall) or consumption (marginal utility of consumption falls).

The Intertemporal First Order Condition

First-order condition for household’s trade off between current and future consumption:

1             ⎡1 + rt +1 ⎤
(23)      = e − ρ Et ⎢          ⎥
ct            ⎣ ct +1 ⎦

o Due to the expectational terms on the RHS, retrieving the coefficients for the intertemporal first
order conditions is less straight forward than for the case of the intratemporal first order
conditions.
o Eventually, solving the model requires numerical (calibration) techniques.

General solution strategy:

~         ⎛ 1 + rt +1 ⎞                               1 + rt +1
Define Z t +1 = ln⎜
⎜             ⎟ – ln (balanced growth path of
⎟                                         )
⎝ ct +1 ⎠                                     ct +1

From (43), we have

~           ~          ~           ~
(51) Ct +1 ≅ aCK K t +1 + aCA At +1 + aCG Gt +1

~                                                         ~ ~ ~
Z t +1 can be expressed as a function of future values of K t +1 , At +1 , Gt +1

~
K t +1 is endogenous, and thus given by

~           ~        ~         ~
(52) K t +1 ≅ bKK K t + bKA At + bKG Gt

~
Substitute this into Z t +1 .
~
Form expected value of Z t +1 .
3 additional restrictions on ‘undetermined’ coefficients a

Use equations (43), (44) and (52) to obtain the optimal responses of consumption, employment,
and the capital stock with respect to shocks to technology and government spending.

Overview of solution strategy for the model with expectational terms

Shocks to technology
Shocks to government spending

(43) consumption
(44) labor
and (52) capital

Responses of consumption, employment, capital stock

remaining equations of the model

Responses of output, investment, wage, interest rate (e.g. from (44), it follows)

Yt = αK t + (1 − α )(At + Lt )
~      ~             ~ ~

(53) = αK t + (1 − α )(At + aLK K t + aLA At + aLG Gt )
~             ~          ~        ~          ~

= [α + (1 − α )aLK ]K t + (1 − α )(1 + aLA ) At + (1 − α )aLG Gt
~                        ~                ~

What are the implications of this model? Calibration results

Numerical assumptions (quarterly data!):

1            share of capital
α=
3
g = 0.5%         growth rate of technology
n = 0.25% growth rate of the labor force
δ = 2.5% depreciation
ρ A = 0.95 autoregressive parameter of technology shock (= degree of persistence)
ρG = 0.95        autoregressive parameter of government spending shock (= degree of persistence)
*     Steady-state level of government expenditure to output
⎛G⎞
⎜ ⎟ = 0.2
⎝Y ⎠
r * = 1.5%       Steady-state real interest rate
1          Steady-state level of labor input
l* =
3

The Effects of Technology Shocks
(see Romer, Graphs 4.2-4.4)

Positive technology shock (1%) in t = 1:

o Current capital stock (= inherited from past) is unchanged.
o Labor supply and consumption rise on impact.
o Output increases.
o Wages and interest rates increase on impact.

o Technology is above normal (by 0.95%).
o Capital stock has increased.
o Labor supply, consumption, and output have increased.

o Capital stock reaches its peak after 20 quarters and declines slowly.
o Labor supply declines slowly and temporarily falls below normal.
o Output declines slowly to normal levels.
o Consumption responds less and more slowly to shock than output.
o Investment is more volatile than consumption and output. (cf. the sharp increase in the capital
stock in t = 1) Results of the model match the stylized facts!
o Wage adjustments are relatively moderate: interest rates are the main driving force of changes
in labor supply.

How are interest rates and consumption linked?

Case 1: Inelastic labor supply
1−α
⎛ AL ⎞
First order condition for a profit maximum implies        rt = α ⎜ t t ⎟
⎜ K ⎟       −δ
⎝ t ⎠
Hence, an increase in A raises r.
o Effect of A dies out slowly: r must remain high (unless K increases rapidly).
o But: Depreciation is low (ca. 1% p.a.). Hence, there cannot be a rapid change in the capital
stock.
o Households raise savings but not by enough to lower r to its normal value.
o Rapid adjustment of savings (and thus K) would violate the households’ first order condition.

Case 2: Elastic labor supply
Part of the adjustment would fall on L.

Less persistent shocks:
o wealth effects are smaller
o intertemporal substitution effect is larger
o sharper, shorter output fluctuations
Key parameter of the model is the persistence of the technology shock ρ A .

The Effects of a Government Spending Shock
(see Romer, Graphs 4.5-4.7)

Negative wealth effect:
o Consumption falls.
o Labor supply increases.
o Capital stock declines temporarily.

3. Empirical Applications: The Persistence of Output Fluctuations

Real business cycle models (RBC models):
o Shifts in technology are the main mechanism generating output fluctuations.
o But: Shifts in technology are assumed to be temporary.
o In reality, changes in technology also have a permanent or persistent component.
o These can be incorporated into RBC-models.

Fluctuations can be permanent.

Keynesian models:
o Fluctuations of output are due to monetary and fiscal policies, coupled with slow adjustment of
nominal prices.
o Output fluctuates around a deterministic trend path.

Fluctuations are generally not permanent.

Actual persistence of output fluctuations can be used to discriminate between the two
models empirically.

The Test by Nelson and Plosser (1982)

Output fluctuations around a deterministic trend imply

Output above trend         Output growth less than normal
Output below trend         Output growth above normal

Regress changes in output on past levels of output:

(54) ∆ ln yt = a + b{ln yt −1 − [α + β (t − 1)]} + ε t

ln y                           = log GDP
ln yt −1 − [α + β (t − 1)]     = deviation from trend output growth
a                              = constant term
εt                             = error term
α + βt                         = trend in GDP
b                              = coefficient to be estimated

b < 0:        output reverts to the trend
b = 0:        no trend reversion

Re-write (54) as:

(55) ∆ ln yt = α '+ β ' t + b ln yt −1 + ε t

α ' = a − bα − bβ
β ' = −bβ

Estimate (55) and test whether b = 0.

Null-hypothesis ( H 0 ):
o Output does not revert to the trend.
o Output is non-stationary.
o Output has a unit root.

Alternative hypothesis ( H1 ):
o Output reverts to the trend.
o Output is stationary.
o Output does not have a unit root.

Empirical result:
o Nelson and Plosser analyze real GNP, real GNP per capita, industrial production, and
employment.
o OLS estimates of b are between –0.1 and –0.2 and are insignificant.
o H 0 that fluctuations have a permanent component cannot be rejected.
o Results would thus be more in line with RBC than with Keynesian models of economic
fluctuations.

Econometric problem: Non-stationarity of the data

o Under H 0 , ordinary least squares (OLS) estimates of b are biased towards negative values.
o If b = 0: ln yt −1 = ln y0 + (t − 1)α '+ε 1 + ε 2 + ... + ε t −1   correlation of dependent variable with past
values of the error term
o Even if the true output series is not trend reverting, OLS estimate of b might indicate trend
reversion.

Use Dickey-Fuller unit-root test:
o Dickey and Fuller have used Monte-Carlo simulations to obtain critical values for b under the
H 0 that the dependent variable is non-stationary.
o Critical values for rejection of H 0 are higher than standard critical values.
o Using standard critical values, H 0 would thus be rejected too often.

The Test by Campbell and Mankiw (1987)

Test by Nelson and Plosser does not provide information on the magnitude of the permanent
component of output fluctuations.

Campbell and Mankiw measure persistence by estimating a third-order autoregressive process for y:

(57) ∆ ln yt = a + b1∆ ln yt −1 + b2 ∆ ln yt − 2 + b3∆ ln yt −3 + ε t

Forecast the implied response of the level of y to a one unit shock of ε t .

Measure of persistence:
Level of output to which forecasted output converges.

How should forecast of output be adjusted if current output is 1% higher than
expected?

Output is trend-stationary                       Do not adjust forecast.
Output follows a random walk ( ∆yt = a + ε t )   Permanent output increases by 1%.

Empirical results:
Measure of persistence exceeds 1!
(which would imply that output permanently diverges from its current value and follows an explosive
path)

Discussion of Time Series Empirical Tests of RBC Models

Statistical problem:
o Data for limited time spans provide insufficient information on long-term relationships in the
data.
o Persistence within a given sample might be consistent with very slow mean reversion in a
longer-term data set.
o Output might follow a longer-term AR process and not just an AR(3) process.

Theoretical problem:
o Empirical models are insufficiently specified to truly be able to discriminate between different
theoretical models.
o Keynesian models are not necessarily incompatible with long-run (i.e. persistent) adjustment
processes.

How should we evaluate whether an RBC model fits the data?

Calibration Approach (Kydland and Prescott 1982)
o Choose parameter values based on microeconomic evidence.
o Compare model’s predictions concerning variances and covariances of the data with empirical
variances and covariances.

Comparison of calibration methods over time series methods:
o Better microeconomic foundation
o Tests of statistical significance more difficult or even impossible

Results by Hansen and Wright (1992)

US            Baseline RBC
data          model
Volatility of output                                          σY            1.92                     1.30
Relative volatility of consumption                         σ C /σY          0.45                     0.31
Relative volatility of investment                           σ I /σY         2.78                     3.15
Correlation between labor input and output per
Corr (L, Y / L )    –0.14                    0.93
worker

Model correctly replicates some features of the data:
o Consumption is less volatile than output.
o Investment is more volatile than output.
o Model’s output fluctuations are similar to those observed in practice.

But: Model does not do a good job in predicting response of labor supply.

Note: Productivity movements are measured through the Solow residual.

Exkurs: Finding the Solow Residual

Y = AF (K , L ) = AK α L1−α               A = technology = total factor productivity

Decomposing the growth in productivity:

∆Y                           ∆K
=       α             +     (1 − α ) ∆L     +
∆A
Y                             K                      L                               A
Changes in                 Contribution of       Contribution of         Change in factor productivity /
=                     +                     +
production                    capital                labor                  technological progress

Empirical observations for:
o Growth in production
o Growth in capital stocks
o Growth in employment
o Shares of capital and labour

derive technological progress as the residual (Solow-Residual)
∆A   ∆Y     ∆K            ∆L
=    – α    – (1 − α )
A   Y       K             L

Empirical results for the US:

Contributions of economic growth by factors

5                                                                 5
4                                                                 4
3                                                                 3
2                                                                 2
1                                                                 1
0                                                                 0
1950-      1960-   1970-       1980-   1990-   1950-
1960       1970    1980        1990    1999    1999

K       L    A      Y

Source: Mankiw (2003, p. 273)

Productivity Movements and the Great Depression

Two potential sources for changes in the Solow residual:
o Changes in technology
o Other sources of changes in the measured Solow residual
o For example: increasing output under increasing returns to scale      Solow residual
increases even in the absence of changes in technology

Can we identify ‘other’ sources of movements in output?

Bernanke and Parkinson (1991)
o Use the Great Depression episode as a ‘natural’ experiment.
o Decline in output was too large to be explained solely by a change in technology.
o How does the measured Solow residual move with output during the Great Depression and
during the post-war period?
o Solow residual and output are expected to move together only in the post-war period (when
technology was the main source of output fluctuations

Regress change in output on change in number of person-hours:
(58) ∆ ln yit = a + bi ∆ ln Lit + ε it

Assumption:
capital stock exhibits little short-run fluctuation
Solow residual can be proxied through:
Change in output – labor share × hours worked

Hypothesis:

Depression sample: technology shifts are                    b roughly equals the economy’s labor share
unimportant                                                 (0.5)
Postwar sample: technology shifts matter                    Higher b

Results (see Romer, Table 4.5):
o Estimates for b in the depression sample are around 1.
o Estimates for b tend to be lower in the postwar than in the depression sample.
o This would be consistent with
(i) Depression being caused mainly by technology shocks (unrealistic) or
(ii) Solow residual being a poor proxy for technological change.

Extensions of RBC Models

Indivisible labor decision:
o Individuals move in and out of employment: l = 0 or l = 1
o Responsiveness of labor input to shocks increases.

Distortionary taxes:
o e.g. proportional tax on output, which corresponds to equal tax rates on capital and labor
o similar effect as changes in technology
o typically analyzed in combination with government spending
o aggregated output tends to fall since tax-induced incentives for intertemporal substitution tend
to outweigh interest-rate effects

Multiple sectors and sector-specific shocks:
o Can be used to analyze transmission of shocks across sectors
o Relocation of labor is time consuming: output in sector affected by shock is affected more than
output of other sectors
o Difficult to obtain results for aggregated output

Objections to RBC Models

1. Technology shocks
o What explains relatively large technological changes of 1% in technological innovations per
quarter?
o Short-run variations in Solow residual are likely to reflect factors other than technological
change.
o Other possible factors: political changes, oil price shocks

2. Propagation mechanism
o Key propagation mechanism (= transmission channels of shocks) is the intertemporal
substitution in labor supply.
o Empirical results, however, suggest that labor supply adjusts less frequently and, moreover,
responds to different factors than those stressed by the model.

3. Omission of monetary disturbances
o All fluctuations are due to real shocks.
o There is no role for incomplete price adjustment (sticky nominal prices).
o There is no role for market imperfections to generate fluctuations.

4. Empirical tests
o Parameters used for the calibration exercises are not always measured with precision.
o How can one test for significance of results?

Integrate RBC models with models featuring nominal rigidities.

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