Answers to Questions on Auctions
1. You have been given a formula for the equilibrium bidding function.
N 1
b(v) =
F (v)N 1
v
xf (x)F N 2 (x)dx
0
When values are distributed uniformly on [0, 1] then f (x) = 1 and
F (x) = x so
b(v) =
v
N 1
v N 1
x1xN 2 dx
Lecture 6, Part II
Game Theory: Normal Form Games
1
C. Nash Equilibrium: How to Calculate
Find the intersection of the players optimal
responses.
Iteratively eliminate all strictly dominated
actions.
Iteratively eliminate all weakly dominated
actions (
EC9011 Economic Analysis: Microeconomics
Problem Set 2, 2010: Model Answers
1. (a) Solve v ( p1 ) ( p2 ) ( p3 ) m , using =1, for m to get
m 1 ( p1 ) ( p 2 ) ( p3 ) v
Then change variables to get
e 1 ( p1 ) ( p 2 ) ( p3 ) u
(b) Using Shepards Lemma, get:
Lecture 7
Auctions
1
A: Introduction
Auctions are a very popular way of selling an object
when the seller does not know the demand function.
A monopolist that knows its demand should just set a
price (Posted-Price Model).
When a monopolist doesnt know
EC9011 Economic Analysis: Microeconomics
Problem Set 2, 2010
1. In question 3 on Problem Set 1, we proved that the indirect utility function for
the consumer took the form v( p1 , p 2 , p3 , m) ( p1 ) ( p 2 ) ( p3 ) m .
(a) Using the inversion method, fin
EC901 Economic Analysis: Microeconomics
Problem Set 3, 2011: Model Answers
1. (i) For exponential Linda, payoffs to not dieting, dieting in period 1, and
dieting in period 2 are, respectively:
Diet in month 1
Diet in month 2
No diet
Period 1 self
-c+ 2b
(
EC901 Economic Analysis: Microeconomics
Problem Set 5, 2011
1. Consider the following two-household, two-good exchange economy.
Household A has an endowment of a units of good 1, and utility function
u A ln x1A (1 ) ln x2A , 0 1 . Household B has an endow
EC901 Economic Analysis: Microeconomics
Problem Set 3, 2011
1. Consider the diet example in the lecture notes with general , and b>c.
(i) For what values of , b,c will an exponential Linda ( <1, =1) diet? When will
she diet, if she does diet? Explain why
EC901 Economic Analysis: Microeconomics
Problem Set 4, 2011
1. Consider the demand for insurance problem in the Lecture handout.
ay
(a) Show that if utility takes the CARA form u ( y ) e , the demand for
insurance, as a function of the price, p, can be e
EC901 Economic Analysis: Microeconomics
Problem Set 4, 2011: Model Answers
1. (a) From the Lecture handout, the FOC for optimal choice of x is:
Eu (Y )
p(1 q)u ' ( y px) q(1 p)u ' ( y l px x) 0
x
ay
if utility takes the CARA form u ( y ) e , u' ( y ) ae
EC901 Economic Analysis: Microeconomics
Problem Set 5, 2011: Model Answers
1. (a) Marshallian demands are x1A a , x 2A
(1 ) p1a
p2 b
B
, x1B x2
p2
p1 p2
(b) Setting p2=1, equilibrium in the market for good 1 requires sum of excess
demands equal zero i.e
EC901 Economic Analysis: Microeconomics
Ben Lockwood and Abhinay Muthoo
1. Recommended Reading
The following textbooks are useful for the course:
G. Jehle and P. Reny, Advanced Microeconomic Theory, Prentice-Hall, 2000 (3rd ed.,
2011)
HG-RR H. Gravelle an
Plan of the Lecture
Introduction
Basics of Behavioural Economics:
Status quo bias
Reference-Dependent Preferences (Endowment effects and loss
aversion)
Framing
Mental Accounting
Optimism Bias
Non-exponential Discounting
Intertemporal Choice and S
LECTURE NOTES ON THE
PRINCIPAL AGENT MODEL
1
SIMPLIFIED MODEL
The Set-Up Two players: a rm and a worker.
The sequential move-structure of the game:
1. Firm proposes a wage contract.
2. Worker decides between accepting or rejecting the wage contract oered.
Lecture 6: Part I
Game Theory: Normal Form Games
1
A. Games in Strategic (or Normal) Form
A game in Strategic (or Normal) form consists of:
A list of players: say, i=1, 2, I.
I can be finite or infinite.
Set of actions or pure strategies for each player
EXERCISE 1 ANSWERS
1. Prisoners' dilemma: The top row is (-1,1) the bottom is (-2,0) and is strictly dominated. A
symmetric argument implies that the right column is strictly dominated.
The left column is (-1,1) the middle and right columns are (-2,0) an
SKETCHED ANSWERS TO EXERCISE ON INFINITELY REPEATED
GAMES, SPENCES JOB MARKET SIGNALLING MODEL AND THE
PRINCIPAL-AGENT MODEL
1. (a) Easy to show that the monopoly quantity is q M = 5/2, and thus one-half of
monopoly output is q M /2 = 5/4. Furthermore, th
Problem Set 1: Strategic Form Games
Here are lots of easy problems to give you some practice at using the concepts. The *
problems are a little harder but not much!
1. Where possible apply the iterated elimination of strictly dominated strategies to
the f
Questions on Auctions:
1) For the case where players values are independently and uniformly distributed on
the interval [0,1] calculate their symmetric equilibrium bids in a first price auction.
2) For the case where players values have a density f(v)=e-v
EXERCISE ON INFINITELY REPEATED GAMES, SPENCES JOB
MARKET SIGNALLING MODEL AND THE PRINCIPAL-AGENT MODEL
1. Consider a market with two rms, 1 and 2. Each rm produces a homogenous
good at a constant marginal cost of 5. The market operates over an innite nu
1
General Equilibrium: Lecture Plan
The Exchange Economy
Definition and Existence of Walrasian Equilibrium
Computing Equilibrium in Examples
General Equilibrium with Uncertainty: Complete Contingent Markets
General Equilibrium with Uncertainty: Spot and A
BL MSc. Micro 2011
Lecture Plan
1. Preferences over Lotteries
2. Utility over Lotteries and the Expected Utility Theorem
3. Risk-Aversion and CRRA, CARA utility functions
4. Applications: Insurance, Portfolio Choice
1
BL MSc. Micro 2011
2
Choice Under Unc
BL MSc. Micro 20
Lecture 1: Consumer Theory
The Axioms of Consumer Preference
The Utility Function
The Utility Maximisation Problem
The Expenditure Minimisation Problem
The Slutsky Equation and the Law of Demand
Utility Maximisation with Endowments
An App
LECTURE NOTES ON SPENCES JOB
MARKET SIGNALLING
1
SIMPLIFIED MODEL
The Set-Up
Two types of workers: HIGH ability ( = 2),
and LOW ability ( = 1), where measures
ability.
Employers dont know the type of any one
worker but have commonly known prior beliefs:
1
Problem Set 7 Model Answers
Question 1
a) Cournot Game
Since firms are identical, it follows that in equilibrium they choose the same output. Hence, substitute for
in the above first-order condition and we obtain that:
, which is the Nash eq/
Substituting
EC9011 Economic Analysis: Microeconomics
Problem Set 1, 2010: Model Answers
1. (a) For the indifference relation: If xIy, yIz, then (i) xRy, yRz and so xRz from
Axiom 2; (ii) yRx, zRy and so z Rx from Axiom 2. So, xRz and z Rx xIz.
For the strict preferen
EC9011 Economic Analysis: Microeconomics
Problem Set 1, 2011
1. (a) Show that the strict preference relation P and the indifference relation I are
both transitive, if Axiom 2 of consumer preference holds.
(b) Does Axiom 1 of consumer preference imply that
BL MSc. Micro 2011
The Theory of the Firm: Lecture Plan
1. Production functions
2. Cost minimization
3. Profit maximisation
1
BL MSc. Micro 2011
1. Production Functions
We study a firm that produces a single product from n inputs
It can produce output y0