W3213 - Intermediate Macroeconomics
Recitation 1 - Eciency, GDP and Price indices
February 2, 2010
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Economic Activity over time
We saw in class that we have to be careful in comparing quantities across time.
We have focused on GDP and price indices.
We s
The rollback/backward induction solution is that if Warden chooses to guard the wall, Prisoner
chooses Dig Tunnel and if Warden chooses regular inspections, Prisoner chooses Climb Wall.
The solution will be different, as the Prisoner can perfectly conditi
Split the dollar(Lecture 3 and 5)
Set-up Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name
shares they would like to have, s1 [0, 1] and s2 [0, 1], respectively. If s1 + s2 1, then the players
receive the shares
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Getting Started
1.1 Reading these notes
These notes will be given out in parts to accompany the rst seven weeks
of class. The notes do not replace the readings but should help with the
lectures and should summarize so
2.1 No Predictions Possible?: Arrows Theorem
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2.1.3 Examples of SWFs and Consequences of Axiom
Violation
We have then that these four properties can not all be simultaneously
satised for a SWF. However it is also the case that all of these properties
ar
3.3 Proof by Contradiction
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can be large, it can often be divided into simple groups that share a common
property. Consider the following direct proof.
Example 22 To prove (directly) that the number 3 is an odd integer.3
This is equivalent to proving th
3. a) [1/2 point] Assigning a payoff of 4 to best, 3 to second best, 2 to third best and
1 to worst, the normal form of this game would be written as:
Pats
Ball Game
Choice Concert
Sams Choice
Ball Game
Concert
4,3
3,2
1,1
2,4
b) [1/2 point] The unique Na
(b)
The subgame perfect equilibrium is the same as the Nash equilibrium of the simultaneous
move game, Child chooses Bad and Parent chooses Not Punish.
(c)
In this case, the subgame perfect equilibrium now differs from the Nash equilibrium of
the simultan
Player 1
Dont Invest
Invest
Nature
0
Fail w.p. 4/5
Succeed w.p. 1/5
5 million
-1 million
Payoffs are in net terms: Earnings investment, if any.
(b) Expected payoff from the investment in net terms is: (1/5)*$5million +(4/5)*(-$1million)= $200,000.
Since $
probability 1-p she gets a payoff of 0. Therefore, player 1s expected payoff from D is
2p+0(1-p)=2p. On the other hand, if she played C against player 2s who always played D
(D,D), she would earn an expected payoff of 0p+2(1-p)=2(1-p). So playing D is pla
Second Price Auction(Lecture 2)
An art object is auctioned. You have n players who participate in the auction. The players can bid
for the object. The winner is the one with the highest bid, and the winner pays the second highest bid.
Everybody bids simul
W3213 - Intermediate Macroeconomics
Recitation 5 - Problem Set 2, Romer model,
Natural experiment and growth stu
February 25, 2010
1
Problem Set 2, Exercise 4
See solutions posted on Courseworks
2
Romer model
We will solve exercise 6, page 164 in Jones. I
W3213 - Intermediate Macroeconomics
Recitation 9 and 10 - Moral Hazard, Principal Agent
Problems, Leverage and Risk, Game Theory, Problem set 5
April 19, 2010
Contents
1 Asymmetric Information
1.1 Moral Hazard (Problems in the Incentive System) . . . . .
Game Theory
Sbastien Turban
e
Columbia University
June 7, 2011
Sbastien Turban (Columbia University)
e
Lecture 10
June 7, 2011
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Outline
1
Announcements
2
From Yesterday
3
Zero-Sum games
Playing in a zero-sum game
Theorems
For the midterm
Sbastien Tu
Game Theory
Sbastien Turban
e
Columbia University
June 7, 2011
Sbastien Turban (Columbia University)
e
Lecture 9
June 7, 2011
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Outline
1
Announcements
2
From Friday
3
Mixed Strategies
Mixed Best Responses
Nash equilibrium in Mixed Strategies
Discuss
Game Theory
Sbastien Turban
e
Columbia University
June 3, 2011
Sbastien Turban (Columbia University)
e
Lecture 8
June 3, 2011
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Outline
1
Announcements
2
From yesterday
3
Mixed strategy
Introduction
Theory
Mixed Best Responses
Solving a game with mix
Game Theory
Sbastien Turban
e
Columbia University
June 2, 2011
Sbastien Turban (Columbia University)
e
Lecture 7
June 2, 2011
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Outline
1
Announcements
2
From yesterday
3
The Commons Problem
Introduction
A simple model
Social Optimum
Extensions
Discu
Game Theory
Sbastien Turban
e
Columbia University
June 2, 2011
Sbastien Turban (Columbia University)
e
Lecture 6
June 2, 2011
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Outline
1
Announcements
2
From yesterday
3
Models of rm competition
4
Variants of Cournot
Sbastien Turban (Columbia Univer
Game Theory
Sbastien Turban
e
Columbia University
June 1, 2011
Sbastien Turban (Columbia University)
e
Lecture 5
June 1, 2011
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Outline
1
Announcements
2
From Thursday
3
Models of rm competition
Sbastien Turban (Columbia University)
e
Lecture 5
June
Game Theory
Sbastien Turban
e
Columbia University
May 27, 2011
Sbastien Turban (Columbia University)
e
Lecture 4
May 27, 2011
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Outline
1
Announcements
2
From yesterday
3
Nash Equilibrium
Sbastien Turban (Columbia University)
e
Lecture 4
May 27, 2011
Game Theory
Sbastien Turban
e
Columbia University
May 26, 2011
Sbastien Turban (Columbia University)
e
Lecture 3
May 26, 2011
1 / 29
Outline
1
Announcements
2
From yesterday
3
Solving a game with dominated strategies
Sbastien Turban (Columbia University)
Game Theory
Sbastien Turban
e
Columbia University
May 25, 2011
Sbastien Turban (Columbia University)
e
Lecture 2
May 25, 2011
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Outline
1
Announcements
2
Reminders
3
Solving a game, Phase I: dominance solvability
Sbastien Turban (Columbia University)
Game Theory
Sbastien Turban
e
Columbia University
May 23, 2011
Sbastien Turban (Columbia University)
e
Lecture 1
May 23, 2011
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Outline
1
Administrative things
2
Introduction
3
Examples
4
Game forms
5
Strategic form
6
Solving a game, Phase I: dominan
Cournot Model(Lecture 6)
1
Simple 2-rm Cournot Model
Set-up Two rms are competing to produce the same product. They can choose any quantity
Demand is given by Q(P ) = max(0, P ). We can write down
P = max(a bQ, 0)
where b =
1
and a =
Note that is the elas
Bertrand Price Competition - Nash Equilibrium(Lecture 4)
Set-up Two rms are competing to produce the same product. They can choose any price they want.
Demand is given by Q(P ) > 0: demand is positive for all prices. We assume that Q (P ) < 0, i.e. the
de