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.
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
Advanced Macroeconomics                                                 87    Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                              88                          Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                              89                       Chapter 2: Real Business Cycle Models



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).
Advanced Macroeconomics                             90                        Chapter 2: Real Business Cycle Models



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).
Advanced Macroeconomics                            91                         Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                                    92                             Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                             93                    Chapter 2: Real Business Cycle Models



                             ~
      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.
Advanced Macroeconomics                                     94                            Chapter 2: Real Business Cycle Models



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
                                                     ~                        ~                ~
Advanced Macroeconomics                              95                     Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                           96               Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                           97                         Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                              98                           Chapter 2: Real Business Cycle Models



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 .
Advanced Macroeconomics                    99   Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                          100                       Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                                   101                      Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                              102   Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                         103                         Chapter 2: Real Business Cycle Models




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.
Advanced Macroeconomics                                        104                        Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                                      105               Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                              106                     Chapter 2: Real Business Cycle Models



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)
Advanced Macroeconomics                            107                        Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                            108                     Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                           109                             Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                                 110                         Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                                       111       Chapter 2: Real Business Cycle Models



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)
Advanced Macroeconomics                        112                         Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                               113                      Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                            114                       Chapter 2: Real Business Cycle Models



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
Advanced Macroeconomics                            115                         Chapter 2: Real Business Cycle Models



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.
Advanced Macroeconomics                          116                      Chapter 2: Real Business Cycle Models




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.