Theories and Methods of the Business Cycle.Part 1: Dynamic Stochastic General Equilibrium ModelsII. The RBC approach

Jean-Olivier HAIRAULT, Professeur à Paris I Panthéon-Sorbonne et à l’Ecole d’Economie de Paris (EEP)

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.

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.

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.

II. The RBC approach 2.1 Growth Cycles

II. The RBC approach 2.2 Trend Cycles

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.

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

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

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

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

II. The RBC approach 3.1 Cyclical dynamics

II. The RBC approach 3.1 Cyclical dynamics

II. The RBC approach 3.1 Cyclical dynamics

II. The RBC approach 3.2 Quantifying Business Cycles

The RBC approach 3.2 Quantifying Business Cycles

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.

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.

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.

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

See Plosser [1989], Journal of Economic Perspectives.

II. The RBC approach 5. The assumptions of the canonical RBC model

II. The RBC approach 5. The assumptions of the canonical RBC model

II. The RBC approach 6. Stationarization of the canonical RBC model

II. The RBC approach 7. Private decisions and prices in the canonical RBC model

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.

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

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

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.

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:

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

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

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!

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

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:

II. The RBC approach 9. The steady state in the canonical RBC model

II. The RBC approach 10. A closed-form solution of the canonical RBC model

II. The RBC approach 10. A closed-form solution of the canonical RBC model

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.

II. The RBC approach 11.1 Expliciting utility and production function

II. The RBC approach 11.2 Log-linearizing the equilibrium conditions

II. The RBC approach 11.3 Solving linear difference equations

II. The RBC approach 11.4 A saddle path equilibrium

II. The RBC approach 11.4 A saddle path equilibrium

II. The RBC approach 11.5 The saddle path equation

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

II. The RBC approach 12.1 Using information on the growth path

II. The RBC approach 12.1 Using information from the growth path

II. The RBC approach 12.2 Using information from micro-econometrics

II. The RBC approach 12.2 Using information from micro-econometrics

II. The RBC approach 12.2 Using information from micro-econometrics

w

Nd, Ns

weak

high

II. The RBC approach 12.3 Estimating the productivity stochastic process

II. The RBC approach 13. Inspecting the transitional dynamics

II. The RBC approach 13. Inspecting the transional dynamics

II. The RBC approach 13. Inspecting the transional dynamics

II. The RBC approach 13. Inspecting the transional dynamics

II. The RBC approach 13. Inspecting the transitional dynamics

Taking into account a productivity innovation at each

period.

II. The RBC approach 14. Responses to productivity shocks

II. The RBC approach 14. Responses to productivity shocks

II. The RBC approach 14. Responses to productivity shocks

Sl II. The RBC approach 15. The Slutsky-Frisch effect

II. The RBC approach 16. Stochastic simulations

II. The RBC approach 16. Stochastic simulations

II. The RBC approach 16.1 Volatility

II. The RBC approach 16.2 Persistence and comovement

II. The RBC approach 17. Historical simulations