Economics 113
University of California, San Diego
Winter 2010
Prof. R. Starr, Mr. Troy Kravitz
Lecture Notes, February 16, 2010
Fundamental Theorems of Welfare Economics
Pareto Efficiency
Definition: An allocation x i , i H , is attainable if there is y j
Chapter 5
Existence of equilibrium
So far we have studied the welfare properties of the Arrow-Debreu equilibrium,
assuming its existence. This is a dangerous avenue, for it makes no sense to study
something unless it exists. Once a mathematics professor t
Chapter 2
Convex analysis and convex
programming
2.1
Convex sets
A set C RN is said to be convex if the line segment generated by any two
points in C is entirely contained in C. Formally, C is convex if x, y C implies
(1 )x + y C for all [0, 1] (Figure 2.
Chapter 1
Arrow-Debreu model
1.1
What is general equilibrium?
The term general equilibrium (GE for short) is the antonym of partial equilibrium (PE for short). Partial equilibrium is what we learn in introductory
microeconomics, for example the demand cur
Chapter 6
Uniqueness of equilibrium
6.1
Sonnenschein-Mantel-Debreu theory
It would be nice if an economy has a unique equilibrium. To study this question,
let E = cfw_I, (ei ), (ui ) be an economy with strongly monotonic, strictly quasiconcave, and smooth
Mathematical Economics
Midterm Exam 2
Prof. Alexis Akira Toda
May 20, 2014
Name:
Instruction:
Dont start the exam until instructed.
Turn off any electronic devices and put them in your bag.
Dont put anything on your desk except the exam sheet, pens, pe
Chapter 9
Finance
Section 9.1 presents the classical portfolio selection problem. Section 9.2 presents
the Capital Asset Pricing Model. Section 9.3 explains the no-arbitrage pricing
theory.
9.1
Portfolio selection
9.1.1
The problem
Suppose you live in a w
Chapter 7
Computation of equilibrium
Computing the equilibrium is usually difficult (at least by hand) because for
a given price we need to compute the demand of each agent (solving as many
constrained optimization problems as the number of agents) and th
Chapter 8
International trade
8.1
Numerical example of Ricardos model
Consider an economy with two countries, A, B. Think of country A as developed
and B as developing. Country A and B have labor endowment eA = 1 and
eB = 2, respectively. Each country pro
Chapter 4
Welfare properties of
equilibrium
In this chapter we ask whether the market mechanism is good in some sense.
To answer the question, we must first define what is good. The minimal
requirement is Pareto efficiency.
4.1
Pareto efficiency
Let E = c
Chapter 3
Quasi-linear model
Having introduced the mathematics for studying convex sets and convex optimization problems, we can proceed to economics.
Let E = cfw_I, (ei ), (ui ) be an Arrow-Debreu economy. We say that E is a
quasi-linear economy if
the
RN
Limits in RN
Closed sets in RN
Compact Sets in RN
Continuous Functions in RN
Image of a compact set under a continuous mapping is compact:
A continuous real-valued function on a compact set achieves its maximum
Convex Sets in RN
Brouwer FPT
Firms: Comp
UCSD Economics 113
Mr. Troy Kravitz
Winter 2010
Prof. R. Starr
Lecture Notes for January 14, 2010
What this course is about: Classic Arrow-Debreu general equilibrium model of the
economy.
Economic General Equilibrium
General Equilibrium Theory: Who was Pr
UCSD Economics 113
Mr. T. Kravitz
Winter 2010
Prof. R. Starr
Lecture Notes, January 21, 2010, and following
Convexity
A set of points S in R N is said to be convex if the line segment between any two points
of the set is completely included in the set.
S
CB046/Starr
LN1
December 9, 2009
11:46
Lecture 1: Existence of general equilibrium in an
economy with an excess demand function
General equilibrium theory focuses on nding market clearing (equilibrium)
prices for all goods simultanously. Since there are d
CB046/Starr
LN012810
January 21, 2010
10:52
Lecture Notes for January 28, 2010, and following:
Households
12.1 The structure of household consumption sets and preferences
Households are elements of the nite set H numbered 1, 2, . . ., #H . A household i H
CB046/Starr
LN020910
February 5, 2010
17:17
1
Economics 113 Prof. R. Starr UCSD Winter 2010
Lecture Notes for February 9, 2010
A market economy
Firms, prots, and household income
H , F , ij R+ , iH ij = 1 ,
r i.
r
iH
j (p) supcfw_p y |y Y j p S j (p)
Th
Economics 113
Winter 2010
UCSD
Prof. Ross Starr, Mr. Troy Kravitz
Lecture Notes, February 23, 25, 2010
Social Choice Theory, Arrow Possibility Theorem
Bergson-Samuelson social welfare function W(u1(x1), u2(x2), , u#H(x#H)
W
with
> 0 all i.
u i
Let the all
Economics 113
UCSD
Prof. R. Starr, Mr. Troy Kravitz
Winter 2010
Lecture Notes, March 9, 2010
Salvaging Majority Rule: Single Peaked Preferences and the Median
Voter Theorem
Arrow Possibility Theorem implies that majority rule or any similar
decision-makin
Economics 113
Mr. Troy Kravitz, UCSD
Prof. R. Starr
Winter 2010
Lecture Notes, January 7 & 12 , 2010
The Edgeworth Box
2 person, 2 good, pure exchange economy
Fixed positive quantities of X and Y, and two households, 1 and 2.
Household 1 is endowed with X