Risk-Sensitive Real Business Cycles

1. Risk-Sensitive Real Business Cycles Thomas D. Tallarini Jr. Journal of Monetary Economics, 2000 Presented by YaseminSatır Dec 7, 2010

2. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

3. Motivation • Equity Premium Puzzle • Kyland&Prescott (1982), King et al. (1988), Mehra &Prescott (1985) • Hansen-Jagannathan Bound • To bridge the gap between DSGE models and asset markets • Habit formation models by Boldrin et al. (1995), Lettau & Uhlig (1995) • Non-separable preferences across states of the world with increased risk aversion in a “business cycle” model Risk-sensitive real business cycles

4. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

5. Model I – Endowment Economy • Representative agent is non-expected utility maximizer (EZ) • χis the coefficient of the relative risk aversion w.r.t. atemporal wealth gambles • IES=1, so that the different values of χdo not affect the growth tradeoff • Nothing new – Hansen et al. (1999) Risk-sensitive real business cycles

6. Model I – Endowment Economy • Trend stationary vs difference stationary • World 1 : Consumption flow is trend stationary Risk-sensitive real business cycles

7. Model I – Endowment Economy • World II : Consumption flow is random walk • The market price of risk is independent of ϒ in both scenarios whereas β doesn’t affect market price of risk in the random walk world. • As ρ goes to 1, TS model ~ RW model Risk-sensitive real business cycles

8. Model I – Endowment Economy • Those moments will be used to chose the target interest rates and market prices of risk Risk-sensitive real business cycles

9. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

10. Results I (Welfare Costs) • The welfare costs of business cycles : the amount of consumption in all states and dates the representative agent would need to be indifferent between some baseline consumption process and one with a different variance (Lucas, 1987) • Is high RA associated with high interest rates ? (Weil.1989) Risk-sensitive real business cycles

11. Risk-sensitive real business cycles

12. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

13. Model II – Production Economy Risk-sensitive real business cycles • χ is still the coefficient of relative risk aversion w.r.t. atemporal wealth gambles, but now wealth is measured in terms of the composite commodity C1/(1+θ)Lθ/(1+θ) or alternatively, RA can be defined as (χ+θ)/(1+θ) with deterministic leisure.

14. Model II – Production Economy Risk-sensitive real business cycles • The problem needs to be transformed into a stationary problem for ϒ>0 by introducing the effective variables yt=Yt/Xt, ct=Ct/Xt …etc. Labor is already stationary, therefore does not need to be transformed.

15. Model II – Production Economy • Production side: The price of a unit capital is constant at unity Risk-sensitive real business cycles • This model have a closed form solution iffδ=1. This requires taking a second-order Taylor approximation!

16. Model II – Production Economy • The parameters of the model are taken, with the exception of the innovation standard deviation σε, from Christiano & Eichenbaum (1992). σε is chosen to match the variance of consumption. Risk-sensitive real business cycles

17. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

18. Results II (Welfare Costs) Risk-sensitive real business cycles

19. Risk-sensitive real business cycles

20. Results II (Welfare Costs) • The largest EP in the table 0.01% << 2%, why? • ↑χ ↓ r, ↑ m.p.r • Production economy vs endowment economy Risk-sensitive real business cycles

21. Risk-sensitive real business cycles

22. Contents • Motivation • Model I – Endowment Economy • Results I – Welfare Costs • Model II – Production Economy • Results II – Welfare Costs • Final Remarks & Conclusion Risk-sensitive real business cycles

23. Summary & Criticism • The representative agent faces some constant risk. Increasing RA changes her behavior by increasing the volatility of SDF so she can satisfy the Hansen-Jagannathan bound, but does not lead to different dynamics on the business cycle dynamics since risk is nearly constant. • IES=1 • No adjustment cost of capitalthe return on capital is almost riskless • Counterfactual equity premium • Barillas, Hansen, Sargent (2006) : max-min expected utility, detection error probabilities Risk-sensitive real business cycles