econ 245
solution to problem set 2
Max
January 29, 2012
Exercise 1
Simon and Blume: Problems 15.1 and 15.2 on page 342.
15.1
Solution: a)
Let F (x, y ) := x2 xy 3 + y 5 17. Note that F (5, 2) = 0 and
that the rst partial derivatives:
F1 (x, y ) = 2x y 3 ,
Lecture 21:
Optimization and
Curvature II
Cecilia Fieler
Economics 245
Lecture 21: Optimization and Curvature II
Cecilia Fieler
April 12, 2012
Concave and
quasi-concave
programs
Readings
Lecture 21:
Optimization and
Curvature II
Cecilia Fieler
Simon and B
Lecture 20:
Optimization and
Curvature I
Cecilia Fieler
Taylor expansion
Economics 245
Lecture 20: Optimization and Curvature
Cecilia Fieler
April 10, 2012
Sucient
conditions
Convexity
Readings
Lecture 20:
Optimization and
Curvature I
Cecilia Fieler
Simon
Lecture 18:
Production
Economy
Cecilia Fieler
An economy with
production
Economics 245
Lecture 18: An economy with production
Cecilia Fieler
April 3, 2012
Second Welfare
Theorem
First welfare
theorem
Readings
Lecture 18:
Production
Economy
Cecilia Fieler
Lecture 19:
Existence and
Denite Matrices
Cecilia Fieler
Existence
Economics 245
Lecture 19: Existence and Denite Matrices
Cecilia Fieler
April 5, 2012
Denite Matrices
Readings
Lecture 19:
Existence and
Denite Matrices
Cecilia Fieler
Existence: Simon and
Lecture 16: The
Envelope Theorem
Cecilia Fieler
Value function
Envelope Theorem
Economics 245
Lecture 16: The Envelope Theorem
Cecilia Fieler
March 22, 2012
Why envelope?
Short run and long
run
Readings
Lecture 16: The
Envelope Theorem
Cecilia Fieler
Chap
Lecture 15: Value
Functions
Cecilia Fieler
Value function
Envelope Theorem
Economics 245
Lecture 15: Value Functions
Cecilia Fieler
March 13, 2012
Shephards lemma
Readings
Lecture 15: Value
Functions
Cecilia Fieler
chapter 5 in Dixit, 19.2 in Simon and Bl
Lecture 13: The
Second
Fundamental
Welfare Theorem
Cecilia Fieler
Economics 245
Lecture 13: The Second Fundamental
Welfare Theorem
Welfare optimum
benchmark
Competitive
equilibrium
Non-uniqueness of
( p, I 1 , . . . , I C )
Second Welfare
Theorem
Cecilia
Lecture 14: The
First Fundamental
Welfare Theorem
Cecilia Fieler
Economics 245
Lecture 14: The First Fundamental Welfare
Theorem
Cecilia Fieler
March 1, 2012
Competitive
equilibrium
First Welfare
Theorem
Readings
Lecture 14: The
First Fundamental
Welfare
Lecture 17:
Optimization and
Curvature III
Cecilia Fieler
Economics 245
Lecture 22: Optimization and Curvature III
Cecilia Fieler
April 19, 2012
Separating
Hyperplane
Theorem
An economy with
production
Readings
Lecture 17:
Optimization and
Curvature III
C
ECON 245
ps1 solution
Max
January 23, 2012
Exercise 1
f : R R is dierentiable, and let A = [a, b] be a closed interval
< a < b < . State the necessary rst-order conditions (i.e., ones
involving just rst derivatives) for x A to maximize f on A. (Hint: the
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Problem Set 2
Due Tuesday, January 31
Simon and Blume: Problems 15.1 and 15.2 on page 342.
1. Demand for a good is given by a function D(p, I ), where p is the price of the good
and I is
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Problem Set 1
Due Jan. 24
1. Suppose f : R R is dierentiable, and let A = [a, b] be a closed interval with
< a < b < . State the necessary rst-order conditions (i.e., ones involving
just
Find the competitive equilibrium in an economy with two consumers, 1 and 2 and two
goods x and y . The endowment of consumers is (x1 , y1 , x2 , y2 ). Utility is ui = ax + yi
i
for i = 1, 2.
Solution Let pY = 1. The problem is to nd a price p = px and an
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Mock Midterm 1
80 points total. Closed book and notes. 1.25 hours max.
1. (30 pts) When a gas station sells a gallon of gas, it receives the price
p t, where p is the price the consumer p
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Mock Midterm 2 - solutions
100 points total. Closed book and notes. 1.15 hours max.
1. (30 pts) Consider this problem:
(P)
max xy such that
x,y 0
x + 2y 20 and 2x + y 50.
(a) (5 pts) Writ
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Midterm 2
100 points total. Closed book and notes. 1.15 hours max.
1. (30 pts) Consider this problem:
(P)
max xy such that
x,y 0
x + y 10 and x + 2y 20.
(a) (5 pts) Write the Lagrangian f
Economics 245
Spring 2012
Ana Cecilia Fieler
University of Pennsylvania
Solutions to Midterm 2
Thursday, March 29
110 points total. Closed book and notes. 1 1 hours max.
3
1. (20 pts) Consider this problem:
(P)
max x + y such that
x,y
3x + y 7 and x + 1 y
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Midterm 1 Solutions
80 points total. Closed book and notes. 1.25 hours max.
1. (30 pts) There is a market in a good with one supplier and one consumer. When the
supplier sells a unit of g
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Midterm 2
Thursday, March 29
110 points total. Closed book and notes. 1.15 hours max.
1. (20 pts) Consider this problem:
(P)
max x + y such that
x,y
3x + y 7 and 1 x + y 3.
2
(a) (5 pts)
Lecture 5:
Constraint
Qualication and
the Lagrangian
Approach
Cecilia Fieler
Economics 245
Lecture 5: Constraint Qualication and the
Lagrangian Approach
Substitution
method using the
IFT
Constraint
qualication
Method of
Lagrange
Cecilia Fieler
January 26,
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Problem Set 7
Due Tuesday, April 17
1. Consider the denition of a competitive equilibrium with production from the lecture
notes 17:
A competitive equilibrium (CE) for the incomes distrib
Econ245-2012
ps7_sol
April 6, 2012
max
Exercise 1
The sum of the rms revenues is gG p g g . The sum of the consumers
incomes is cC I c . Consumers budget constraints say that I c = gG p g x c .
g
But supply equals demand in equilibrium: cC x c = g , g. So
ECON245
problem set 6 sol
max
March 13, 2012
The problems below are about a private ownership exchange economy in which each
consumer
c
has a given endowment vector,
endowment of good
g
is
Xg :=
xc ,
of goods, but no dollar income. The total
xcg .
c
Quest
Economics 245
Spring 2012
Cecilia Fieler
University of Pennsylvania
Problem Set 6
Due Tuesday, March 20
The problems below are about a private ownership exchange economy in which each
consumer c has a given endowment vector, xc , of goods, but no dollar i