TremblingHand Perfect Nash Equilibrium
intuition : a good property for a solution concept is that players
still want to play their strategies even if theres a small
probability that the other player makes a mistake
L
R
t (8, 4) (3, 3)
b (5, 0) (3, 2)
ther
Public Goods
(1) nonrivalrous (Samuelson, MasColell, Whinston and
Green) : increase in consumption by person i does not reduce
consumption by person h = i
(2) non-excludable (Samuelson) : person i cannot be excluded
from consuming the good, even if she do
SinglePeaked Preferences
suppose that the alternatives in X can be ordered in some way
left to right, smallest to biggest, etcetera (denition 21.D.2 in
MasColell, Whinston and Green provides a formal denition of
what it means for the alternatives to be o
First Fundamental Theorem of Welfare Economics
Theorem : Assume that all peoples preferences are locally
nonsatiated. If (x , y , p) is a Walrasian equilibrium, for some
distribution of endowments i (i = 1, 2, . . . , I) and distribution of
rm ownership i
GrovesLoeb Mechanism
rules :
1. each person reports a marginal benet function ri (Q)
2. the planner then provides the level Q of the public good
obeying
I
ri (Q ) = c
i=1
3. each person pays an equal share
public good
cQ
I
of the cost of the
4. each pers
Social Welfare Functionals
Start with
a nite set X of (mutually exclusive) alternatives
a nite set of I people
a (transitive) preference relation
i
for each person i
then a social welfare functional F is a rule, which maps
proles of individuals orderings
Example 19.C.2
(page 693 of MasColell, Whinston and Green)
L = 1, I = 2, S = 2
= (2, 1)
i
i
U i (xi ) = u(x1 ) + u(x2 )
u < 0 i = 1, 2
so riskaverse people, and state of the world #1 is better than
state of the world #2
added assumption : u is homothetic
Incomplete Markets : NonExistence of Equilibrium
(example 19.H.3 from MasColell, Whinston and Green)
2 goods, 2 people, 2 states of the world
each person has the same endowment (3, 3) in each state of
the world
but person 1 observes the state of the world
GS/ECON 6100
Answers to Assignment 2
February 2012
Q1. Find all the subgame perfect Nash equilibria to the 3person game for
which the extensive form is shown in this link.
A1. This game is a game of perfect information, so that it can be solved by
backwar
GS/ECON 6100
Answers to Assignment 1
January 2012
Q1. Suppose that the prisoners dilemma game (8.B.1 in Mas-Colell, Whinston and Green)
were modied, so that player #2 got to observe player #1s action before he (player #2) made his
own move.
Write down the
GS/ECON 6100
Assignment 1
due : Wednesday Jaunary 25
W201112
before class
Do all 5 questions. Each counts 20%.
1. Suppose that the prisoners dilemma game (8.B.1 in Mas-Colell, Whinston and Green) were
modied, so that player #2 got to observe player #1s ac
Assets
Let good 1 be the numeraire in each state of the world.
(demands in the spot market in each state of the world s are
homogeneous of degree 0 in the L spot prices for that state, so
that we can pick one numeraire for each state)
An asset k is now de
TwoPerson Model
two types of people with different productivities
w i : output per hour of person of type i (i = 1, 2)
convention : w2 > w1
productivities w i are exogenous, and private information
hours worked H i are endogenous, and private information
YORK UNIVERSITY
Faculty of Graduate Studies
Midterm Examination
February 13 1995
Economics 6100.03AW : Topics in Microeconomic Theory
S. Bucovetsky
time=80 minutes
Do any 3 of the following 4 questions.
All questions count equally.
1. Suppose that a rms l
Taxes
let T (Y ) be the total amount of income taxes payable on a total
income of Y
so that T (Y ) is the marginal tax rate
and a person will have X = Y T (Y ) to spend on consumption
if she has a gross income of Y
so a tax schedule T () denes an opportun
The Index of an Equilibrium
new normalization of prices : let good L be the numeraire, so
that pL 1
Walrass Law ensures that ZL (p) = 0 if
Z1 (p) = Z2 (p) = = ZL1 (p) = 0
so a Walrasian equilibrium is now a vector of prices
(p1 , p2 , . . . , pL1 ), with
Iterated Deletion of Dominated Strategies
left
centre right
A (2, 0) (10, 2) (8, 5)
B (3, 12) (6, 3) (5, 6)
C (8, 2) (4, 8) (6, 1)
here player #2 has no dominated strategies, but for player #1,
strategy B is strictly dominated by the mixed strategy : A wi
Partial Equilibrium
I people, 1 private good, 1 public good
cost of the public good (in terms of the private good) is c
U i (x i , Q) = x i + i (Q)
with i > 0, i < 0
so i (Q) is person is MRS between the public good and the
private good (a.k.a. her Lindah
denitions
I people
J rms
L different commodities
so no distinction here between inputs and outputs ; commodity
3 might be a produced nal good (haircuts, episodes of some
television show), and commodity 7 might be an input (hours of
unskilled labour, kilos
Excess Demand Functions in an Exchange Economy
x a pattern of endowment ownership ( 1 , 2 , . . . , I );
then
zi (p) xi (p, p i ) i
is the excess demand function of person i
note that the second argument of zi is person is income
mi p i
(since there are n
Existence of Walrasian Equilibrium
let be the unit simplex in L dimensions
Existence of Walrasian Equilibrium
let be the unit simplex in L dimensions
L
cfw_p 0|
pk = 1
k=1
Existence of Walrasian Equilibrium
let be the unit simplex in L dimensions
L
cfw_
YORK UNIVERSITY
Faculty of Graduate Studies
Final Examination
April 21 1997
Economics 6100.03Y : Topics in Microeconomic Theory
S. Bucovetsky
time 2.5 hours
Do any 5 of the following 8 questions. All count equally.
1. Suppose that a monopoly serves two di
The Second Fundamental Theorem of Welfare
Economics
Proposition : Suppose that all peoples preferences are convex
and nonsatiated, and all production sets Yj are convex. Let
(x , y ) be any Pareto optimum. Then there exists a price
vector p, and a vector
Social Welfare Functionals : Examples
notation : #cfw_i|x i y is the number of people who prefer
(strictly) alternative x to alternative y
Plurality Rule
let Pi (x) equal 1 if person i prefers strictly alternative x to each
other alternative in X i.e. if
private ownership
in a private ownership economy, endowments and rms are
owned by people
so take an economy with I people, L goods, J rms, with
aggregate endowments and preferences i for people,
and production sets Yj for rms
and now assign person i her
Pareto Principle
seems a minimal condition we should require in a social welfare
functional
that the social ranking must somehow reect the preferences of
the people
so we require : if x
social ranking
i
y for every person i, then x
y for the
Independence
GS/ECON 6100
Answers to Assignment 3
March 2012
Q1. Find all the Pareto optima in the following economy :
There are two people, three goods, and two rms. Person 1s preferences
can be represented by the utility function
U 1 (x1 , x1 , x1 ) = 2 ln (x1 ) + l
A General Equilibrium Mechanism
(cf. Walker, M. A Simple Incentive-Compatible Scheme for
Attaining Lindahl Allocations, Econometrica, 1981)
an indirect mechanism, which implements Pareto optima as
Nash equilibria
person i has some utility function U i (x
PART A
Define fully the following terms (5 x 2 = 10 marks)
You must use diagrams and/or formulas whenever necessary.
1
2
3
4
5
Inflation and disinflation
Law of one price
Foreign exchange market (fixed exchange rates)
Fiscal Policy
Deflation
PART B
Short-