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. The control variables are:
Household behaviour: intertemporal substitution of labour supply 2 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
Relative labour supply respond
to relative wages and to r

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. After manipulation (see Romer (2012) p. 205, 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, , 0.94,…
(see analysis on top of page 206)
However, for ρ close to 0.9, the model yields interesting output dynamics (hump-shaped) (Blanchard, 1981).

15. Method: Least Squares
Date: 06/04/13 Time: 09:34
Sample: 1955Q1 2013Q1
Included observations: 233
Convergence achieved after 5 iterations
Variable
Coefficient
Std. Error
t-Statistic
Prob.   C
0.0000
AR(1)
AR(2)
R-squared
    Mean dependent var
Adjusted R-squared
    S.D. dependent var
S.E. of regression
    Akaike info criterion
Sum squared resid
    Schwarz criterion
Log likelihood
    Hannan-Quinn criter.
F-statistic
    Durbin-Watson stat
Prob(F-statistic)
Inverted AR Roots
      1.00
          .33

16. Calibrating (Romer 2012 section 5.8)

18. Objections to RBC models
Omission of monetary disturbances (empirically monetary disturbances affect output) solution : Lucas asymmetric information or incomplete nominal 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).

19. 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.