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Theories and Methods of the Business Cycle. Part
1 Dynamic Stochastic General Equilibrium
Models II. The RBC approach Jean-Olivier
HAIRAULT, Professeur à Paris I Panthéon-Sorbonne
et à lEcole dEconomie de Paris (EEP)
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II. The RBC approach 1. Introduction

• F. Kydland and E. Prescott, 1982, Econometrica,
  Nobel Prize in 2005.
• In the line of the Lucas critique to
  Keynesianism Building a model with explicit
  micro-foundations taking part in the general
  equilibrium analysis market clearing, no
  monetary factors, at odds with keynesian
  tradition.
• One-step forward no rationale for macroeconomic
  management the optimal growth model with
  short-run fluctuations induced by productivity
  shocks (stochastic neoclassical growth model in
  the line of Solow (1956), Cass (1965) and
  Brock-Mirman (1972)). Hard-core of the RBC
  approach which has been challenged by a lot of
  works.
• No more methodological opposition between
  business cycle and growth research which was at
  the heart of the neoclassical synthesis.
• Building a successful (relative to data)
  business cycle model imposing a new method based
  on calibration to evaluate the performance of
  business cycle models relative to a new
  definition of the business cycle facts.
  Quantitative Approach.
• The methological innovation has been criticized
  but is now extensively used in macroeconomics
  today, even by proponents of stabilization
  interventions. The methods initiated by Kydland
  and Prescott are now commonly used in monetary
  and international economics, public finance,
  labor economics, asset pricing. In contrast to
  early RBC studies, they involve market failures
  so that government interventions are desirable.

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II. The RBC approach 1. Introduction

• Shock-based approach productivity shocks
• Propagated by intertemporal choices derived from
  dynamic optimization under rational expectations.
• Studying the canonical model first presented by
  King, Plosser and Rebelo (1988), Journal of
  Monetary Economics and reconsidered in King and
  Rebelo (1999), Handbook of macroeconomics.

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II. The RBC approach 2. Measuring cycles

• Any time series can be decomposed as the sum of a
  trend and a cycle.
• Trend and cycle components are not observable.
  This implies to adopt a particular way of
  measuring them.

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II. The RBC approach 2.1 Growth Cycles
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II. The RBC approach 2.2 Trend Cycles
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II. The RBC approach 2.3. Measuring cycles by
using HP filter

• More than identifying the non-stationarity of
  series, we need an economic definition of
  business cycles consistent with the decades of
  works following the seminal approach of Burns and
  Mitchell (NBER tradition).
• The HP filter can make stationary series up
  through four orders of integration.
• It is flexible enough to remove the  undesired 
  long-run frequencies of the stationnary component
  of series.
• See F. Canova 1998 for a detailed analysis of
  the HP filter. Journal of Monetary Economics
• See M. Baxter and R. King 1999, Review of
  Economics and Statistics.

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II. The RBC approach 2.3 Measuring cycles by
using HP filter
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II. The RBC approach 2.3 Measuring cycles by
using HP filter
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II. The RBC approach 2.3 Measuring cycles by
using HP filter

• To understand how HP filter works, it may be
  useful to compare with the measure resulting from
  a band-pass filter procedure the HP filter looks
  like a BP filter which makes the cyclical
  component those parts of output with
  periodicities between 6 and 32 quarters high
  frequencies like seasonnal frequencies and low
  frequencies are removed

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II. The RBC approach 3. Quantifying Business
Cycles

• What are the business cycles features? For Lucas,
  all business cycles would be all alike.
• The stylized facts that any models should aim at
  replicating.
• Amplitude of cycles Variability of macroeconomic
  series, differentials of variability across
  aggregates standard deviation
• Comovements of macroeconomic series correlation
• Persistence of expansions and recessions
  auto-correlation

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II. The RBC approach 3.1 Cyclical dynamics
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II. The RBC approach 3.1 Cyclical dynamics
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II. The RBC approach 3.1 Cyclical dynamics
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II. The RBC approach 3.2 Quantifying Business
Cycles
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The RBC approach 3.2 Quantifying Business Cycles
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The RBC approach 3.2 Quantifying Business Cycles

• High degree of co-movement, except for labor
  productivity.
• Capital governement expenditures are rather
  a-cyclical.
• High serial correlation which makes the evolution
  predictable.

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II. The RBC approach 3. 3 Are business cycles all
alike?

• French Business Cycles (Hairault 1992, Economie
  et Prévision), 1970-1990, quarterly data. See
  also Danthine and Donaldson 1993, European
  Economic Review for an European business cycles
  overview.

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II. The RBC approach 4 Introduction to the
canonical RBC model

• Neoclassical growth model in the line of Cass
  1965
• with stochastic productivity shocks (Brock and
  Mirman 1972) and labor supply (Lucas and
  Rapping 1969).
• See Plosser 1989, Journal of Economic
  Perspectives.

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II. The RBC approach 4. Introduction to the
canonical RBC model

• See Plosser 1989, Journal of Economic
  Perspectives.

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II. The RBC approach 5. The assumptions of the
canonical RBC model
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II. The RBC approach 5. The assumptions of the
canonical RBC model
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II. The RBC approach 6. Stationarization of the
canonical RBC model
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II. The RBC approach 7. Private decisions and
prices in the canonical RBC model
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II. The RBC approach 7.1 Household decisions in
the canonical RBC model

• The value function represents the expected
  life-time utility conditionnal to ks, A and k
  the current flow of utility the expected
  utility that results from starting tomorrow with
  k, K and A and proceeding from then on. ks
  and k are determined today. A will be known
  tomorrow, so we have to compute the expected
  value tomorrow.

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II. The RBC approach 7.1 Household decisions in
the canonical RBC model
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II. The RBC approach 7.1 Household decisions in
the canonical RBC model
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II. The RBC approach 7.1 Household decisions in
the canonical RBC model

• First condition The present marginal utility of
  consumption is equal to the expected and
  discounted marginal value (in terms of utility)
  of capital.
• Second condition The marginal rate of
  substitution between consumption and leisure is
  equal to the real wage.
• Third condition the expected and discounted
  marginal value of capital is given on the
  optimal path by the interest factor evaluated in
  terms of the marginal utility of consumption
  tomorrow.

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II. The RBC approach 7.1 Household decisions in
the canonical RBC model

• The third and the first conditions determine
  together the so-called stochastic Euler (or
  Keynes-Ramsey) condition which relies the
  marginal rate of substitution between current and
  future consumptions to the rental rate

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II. The RBC approach 7.2 Firm decisions in the
canonical RBC model
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II. The RBC approach 8. The competitive
equilibrium in the canonical RBC model
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II. The RBC approach 8. The competitive
equilibrium in the canonical RBC model

• These conditions corresponds to the first best
  allocations of ressources. There is an
  equivalence between the optimal quantities chosen
  by the social planner and those in a competitive
  general equilibrium. Fluctuations are optimal!

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II. The RBC approach 8. Consumption and leisure
smoothing in the canonical RBC model
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II. The RBC approach 9. The steady state in the
canonical RBC model

• The Euler equation can be written at the steady
  state as follows
• Given constant returns to scale, the marginal
  product of capital depends on the capital-labor
  ratio

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II. The RBC approach 9. The steady state in the
canonical RBC model
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II. The RBC approach 10. A closed-form solution
of the canonical RBC model
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II. The RBC approach 10. A closed-form solution
of the canonical RBC model
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II. The RBC approach 11. Transitionnal path in
the canonical RBC model

• Non-linear system of stochastic finite difference
  equations under rational expectations.
• In general no analytical solution, need to rely
  on numerical approximation methods.

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II. The RBC approach 11.1 Expliciting utility and
production function
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II. The RBC approach 11.2 Log-linearizing the
equilibrium conditions
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II. The RBC approach 11.3 Solving linear
difference equations
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II. The RBC approach 11.4 A saddle path
equilibrium
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II. The RBC approach 11.4 A saddle path
equilibrium
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II. The RBC approach 11.5 The saddle path equation
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II. The RBC approach 12. Calibration

• Make explicit use of the model to set the
  parameters
• A lot of discipline
• Let us see it on our baseline model
• Have to set (alpha,gamma,delta,theta,
  beta,eta,,sigma,rho)
• Use data related to growth (k/y, c/y, i/y, h,
  wh/y, r )
• What type of information do we have?
• (k/y, c/y, i/y, h, wh/y, r ) from the model
• (alpha,gamma,delta,theta, beta) the subset of
  parameters can be calibrated

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II. The RBC approach 12.1 Using information on
the growth path
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II. The RBC approach 12.1 Using information from
the growth path
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II. The RBC approach 12.2 Using information from
micro-econometrics
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II. The RBC approach 12.2 Using information from
micro-econometrics
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II. The RBC approach 12.2 Using information from
micro-econometrics
weak
w
high
Nd, Ns
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II. The RBC approach 12.3 Estimating the
productivity stochastic process
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II. The RBC approach 13. Inspecting the
transitional dynamics
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II. The RBC approach 13. Inspecting the
transional dynamics
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II. The RBC approach 13. Inspecting the
transional dynamics
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II. The RBC approach 13. Inspecting the
transional dynamics
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II. The RBC approach 13. Inspecting the
transitional dynamics

• Taking into account a productivity innovation at
  each period.

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II. The RBC approach 14. Responses to
productivity shocks
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II. The RBC approach 14. Responses to
productivity shocks
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II. The RBC approach 14. Responses to
productivity shocks
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Sl II. The RBC approach 15. The Slutsky-Frisch
effect
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II. The RBC approach 16. Stochastic simulations
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II. The RBC approach 16. Stochastic simulations
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II. The RBC approach 16.1 Volatility
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II. The RBC approach 16.2 Persistence and
comovement
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II. The RBC approach 17. Historical simulations