1. Traditional Keynesian Theories of Fluctuations (From Romer, chapter 6 part A, section 7.8)
Microfundations
The IS-LM-AD model
The liquidity trap and the post 2008 crisis
The supply curve
The open economy and exchange rate overshooting
The Phillips curve
The natural rate of unemployment
The Canonical New-Keynesian model
Introduction

2. Structure of a Keynesian model
Aggregate demand
IS curve (flow equilibrium: the Keynesian cross)
LM curve (stock equilibrium, financial market, money market)
Aggregate supply
Labour market
Production function P AS AD Y
AS curve not vertical

3. IS-LM model with fixed prices
Nominal rigidities are given (fixed prices)
Time is discrete capital is fixed, closed economy, no government to start with
The production function is:

4. Household optimization problem
Fixed number of infinitely lived households
They get utility from consumption, holding money, and disutility from working
We ignore population growth and normalize the number of household to 1

5. Household objective function is

6. Utility functions (C and M) are CRRA
The assumption that Money enter the utility function is a short cut
Observationally equivalent to transaction purpose (Fenstra 1986)

7. Money and Bonds: Two assets
The two assets are substitute
Bonds bears interest rate but are not liquid (i.e., in the U function)

8. For households, paths of P, W, and I are given
Path of C and M are chosen in order to maximize 6.2 subject to (6.5) and the No-Ponzi game condition
For the moment, L is exogenous
Path of M is set by the central bank

9. New Keynesian IS curve
Following Romer intuitive methodology to derive the Euler equation (last paragraph of page 240) with the real interest rate is:

10. Y(t+1) is added to the traditional IS curve
With uncertainty, we have instead E(Y(t+1)
Inverse relationship between Y(t) and r(t).
Relationship is strengthen with the introduction of capital and investment (chapter 9 of book)

11. LM curve
Again following Romer approach to optimization (bottom of page 241 do it yourself).
First order condition for money holding (an increase in m increase utility of money holding but decrease return on bond holding and consumption (6.9)

12. The LM curve in increasing in output and decreasing in the nominal interest rate (explanation)
Interest rate rule is the modern approach to close the system

13. i
LM(P0)
LM(P1)
The aggregate
demand curve IS Y P P0 P1 AD Y

14. i
LM(P0)
IS(G1)
IS(G0) Y AS P AS
AD1
AD0 Y

15. The liquidity trap
In this case:
LM(P0)
LM(P1) i IS Y Yn

16. -What is the shape of the AD curve in this case?
What happen if the AS curve is vertical (classical)
What about the Pigou effect?
This is related to Summers (1991)

17. Source : Orphanides (2003) Monetary Policy in Deflation: The Liquidity Trap
in History and Practice. Miméo, FED

18. Source : Orphanides (2003) Monetary Policy in Deflation: The Liquidity Trap
in History and Practice. Miméo, FED

21. The remedy should come, I suggest, from a general recognition that the rate
of investment need not be beyond our control, if we are prepared to use our
banking systems to effect a proper adjustment of the market-rate of interest. It
might be sufficient merely to produce a general belief in the long continuance of
a very low rate of short-term interest. The change, once it has begun, will feed
on itself. ...
The Bank of England and the Federal Reserve Board ... should pursue bank-rate
policy and open-market operations a outrance ... [t]hat is to say, they should
combine to maintain a very low level of the short-term rate of interest, and buy
long-dated securities ... until the short-term market is saturated.

22. The aggregate supply curve
Alternative assumptions about wage and price rigidity section 5.3
Four cases studied:
1) Rigid nominal wages (Keynes’s model)
W/P LS
A decrease in the
price level
(AD shock) E A
W/P1
W/P0
W/P counter-cyclical
rejected by facts
W/P moderately
pro-cyclical LD L

23. P AS Y

24. Case 2: Sticky Prices, Flexible Wages, Competitive Labour MarketAD
And Y is determined
by AD, the effective
demand for labour
is determined by AD Y

25. LD LS
W/P
The effective demand
for labour E
Pro-cyclical real wage L
F-1(Y)

26. is the efficiency-wageLD LS
W/P
One reason for this
is the efficiency-wage E
If w flatter than LS, u↑
w(L) L
F-1(Y)
Case 3: Sticky Prices, Flexible wages, Real Labour Market Imperfections

27. Case 4: Sticky Wages, Flexible Prices, Imperfect Competition
Markup function
If μ is fixed, W/P counter cyclical as in case 1
However, if μ is countercyclical (strong evidences), the real wage
can be acyclical or procyclical.