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1. The RBC approach (Real-Business-Cycle Model) (Romer chapter 5, based on Prescott, 1986, Christiano and Eichenbaum, 1992, Baxter and King, 1993, Campbell, 1994. RBC Objectives: explain business cycles with walrasiens equilibrium (market clearing conditions), no frictions Solution: technology shocks (supply shocks) No room for stabilisation policies (monetary and fiscal policy) Point of departure: Ramsey model, adding uncertainty and work/leisure decision in consumer optimization problem.
2. Discret time, G financed by lump-sum taxation, balanced budget; labour and capital earn their marginal product:
3. The representative household maximises the expected value of:
4. Technology and government purchases are the two key variables: Where A tild reflects the effects of the shocks. We suppose that A tild follows a first-order autoregressive process: Where the ε are white-noise disturbances (zero-mean shocks, uncorrelated with one another). For A, ρ is generally > 0, discussion, effect of technology shocks gradually disappear through time.
5. Where the εG are not correlated with the εA and are white-noises disturbances that can be viewed as demand (real) shocks.
6. Household behaviour: intertemporal substitution of labour supply2 differences with Ramsey: 1) intertemporal substitution of LS and 2) uncertainty Consider first the case of one household that live for two periods, without initial wealth, no uncertainty. The household’s budget constraint is: The Lagrangian is: The control variables are:
7. The first order conditions for After manipulation (to demonstrate) we get: Lucas and Rapping, 1969 Intertemporal substitution of labour
8. 2 - Optimisation under uncertainty, consider at time t, a marginal reduction of c that is used to increase wealth and increase consumption next period in such a way that the household is let on the optimal path. This term is on the left-hand side of 4.22, for the right-hand side:
9. ↑c per member in t+1 is: and: Demonstrate that this simplifies to 4.23
10. Without uncertainty, this is the Euler equation:
11. With uncertainty however, important to note the expectation E: If r(t+1) is high when c(t+1) is also high, then Cov < 0 and this makes saving less attractive since the return to saving is high when the marginal utility of consumption is low. And the intratemporal trade-off condition between consumption and labour supply is (to demonstrate) :
12. Analysis of a special case of the model (Long and Plosser, 1983) Eliminate government Assume 100 percent depreciation It can be show that in this case, both the saving rate s and labour supply ℓ are constant (Romer section 4.5, first part). We concentrate in the analysis that follows. The model resumes to:
13. Which implies: And given that:
14. We then get the following expression for the departure of the log of output from its normal path: It follows a second order autoregressive process. Consider α = 1/3 and ρ = 0.9, we get: Consider the effect of a one time shock of 1/(1-α) to ε(t). The time path of the effect of the shock on output is: 1, 1.23, 1.22, 1.14. 1.03, 0.94,… (see analysis on top of page 185) What happen if ρ(A)=0 ??? However, for ρ close to 0.9, the model yields interesting output dynamics (hump-shaped) (Blanchard, 1981).
15. Calibrating (Romer 2012 section 5.8)
16. Objections to RBC models Omission of monetary disturbances (empirically monetary disturbances affect output) solution : Lucas assymetric information or imcomplete nominak adjustments à la new-keynesian Technology shocks and the Solow residual : the large swings in the Solow residual from between quarters does not appear to be related with technology shocks (for the US, Hall 1988 shows that the SR is determined by the political party of the president, change in military expenditures, and oil price) Intertemporal substitution in labour supply Dynamics of the basic RBC does not look like a business cycle (unless a precise dynamics is assumed for the shocks).
17. Extensions of RBC models ‘Real extension’ : indivisible labor (Rogertson and Hansen), labor supply is either 0 or 1. Make the model labor supply more sensible to wage. Increase the SD of output in 5.4 from 1.3 to 1.73. Distortionary taxes and many others (see footnote 33 page 231) Introducing nominal rigidity read page 231 to 233.