1. 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)

2. 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.

3. 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.

4. 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.

5. II. The RBC approach 2.1 Growth Cycles

6. II. The RBC approach 2.2 Trend Cycles

7. 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.

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

10. 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

11. 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

12. II. The RBC approach 3.1 Cyclical dynamics

15. II. The RBC approach 3.2 Quantifying Business Cycles

16. The RBC approach 3.2 Quantifying Business Cycles

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

18. 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.

19. 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.

20. II. The RBC approach 4. Introduction to the canonical RBC model See Plosser [1989], Journal of Economic Perspectives.

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

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

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

25. 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.

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

28. 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.

29. 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:

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

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

32. 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!

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

34. 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:

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

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

38. 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.

39. II. The RBC approach 11.1 Expliciting utility and production function

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

41. II. The RBC approach 11.3 Solving linear difference equations

42. II. The RBC approach 11.4 A saddle path equilibrium

44. II. The RBC approach 11.5 The saddle path equation

45. 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

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

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

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

50. w Nd, Ns weak high

51. II. The RBC approach 12.3 Estimating the productivity stochastic process

52. II. The RBC approach 13. Inspecting the transitional dynamics

53. II. The RBC approach 13. Inspecting the transional dynamics

56. II. The RBC approach 13. Inspecting the transitional dynamics Taking into account a productivity innovation at each period.

57. II. The RBC approach 14. Responses to productivity shocks

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

61. II. The RBC approach 16. Stochastic simulations

63. II. The RBC approach 16.1 Volatility

64. II. The RBC approach 16.2 Persistence and comovement

65. II. The RBC approach 17. Historical simulations