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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.
Advanced Macroeconomics 38 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 39 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 40 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 41 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 42 Chapter 2: Real Business Cycle Models
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 ⎠
Advanced Macroeconomics 43 Chapter 2: Real Business Cycle Models
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)
Advanced Macroeconomics 44 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 45 Chapter 2: Real Business Cycle Models
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).
Advanced Macroeconomics 46 Chapter 2: Real Business Cycle Models
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).
Advanced Macroeconomics 47 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 48 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 49 Chapter 2: Real Business Cycle Models
Two-period model:
Lifetime budget constraint:
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
Advanced Macroeconomics 50 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 51 Chapter 2: Real Business Cycle Models
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)
Adjustment of labor supply is crucial to business cycle fluctuations in the real business
cycle models (intertemporal substitution of labor supply).
Advanced Macroeconomics 52 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 53 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 54 Chapter 2: Real Business Cycle Models
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:
Advanced Macroeconomics 55 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 56 Chapter 2: Real Business Cycle Models
⎡ 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.
Advanced Macroeconomics 57 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 58 Chapter 2: Real Business Cycle Models
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).
Advanced Macroeconomics 59 Chapter 2: Real Business Cycle Models
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 ⎠
Advanced Macroeconomics 60 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 61 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 62 Chapter 2: Real Business Cycle Models
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 ⎥
⎦
Advanced Macroeconomics 63 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 64 Chapter 2: Real Business Cycle Models
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 ⎦
Advanced Macroeconomics 65 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 66 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 67 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 68 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 69 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 70 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 71 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 72 Chapter 2: Real Business Cycle Models
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 .
Advanced Macroeconomics 73 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 74 Chapter 2: Real Business Cycle Models
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) ~ ~
Advanced Macroeconomics 75 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 76 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 77 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 78 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 79 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 80 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 81 Chapter 2: Real Business Cycle Models
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.
Advanced Macroeconomics 82 Chapter 2: Real Business Cycle Models
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 ⎠
Advanced Macroeconomics 83 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 84 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 85 Chapter 2: Real Business Cycle Models
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
Advanced Macroeconomics 86 Chapter 2: Real Business Cycle Models
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
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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
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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.
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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).
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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).
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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.
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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
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~
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.
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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
~ ~ ~
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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
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The Effects of Technology Shocks
(see Romer, Graphs 4.2-4.4)
Positive technology shock (1%) in t = 1:
Adjustment 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.
Adjustment in t = 2:
o Technology is above normal (by 0.95%).
o Capital stock has increased.
o Labor supply, consumption, and output have increased.
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Adjustment in t > 2:
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.
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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 .
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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)
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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.
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Additional Empirical Applications: Calibration
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
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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.
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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
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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)
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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
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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.
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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
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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.
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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.
