Natalia Lazzati
Mathematics for Economics (Part I)
Note 6: Nonlinear Programming - Unconstrained Optimization
Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and
Blume (1994, Ch. 17).
One objective of using economic mod
THE IMPLICIT FUNCTION THEOREM
1. A
SIMPLE VERSION OF THE IMPLICIT FUNCTION THEOREM
1.1. Statement of the theorem. Theorem 1 (Simple Implicit Function Theorem). Suppose that is a real-valued functions dened on a domain D and continuously differentiable on
The utility maximization program
The expenditure minimization program
The link between the two approaches
Important relations
Complements
http:/selod.ensae.net/m1
Chapter 3 - The classical demand theory
(preference-based approach)
Harris SELOD
Paris Schoo
Sample Solutions to Micro Comprehensive Exam
(Aug 02)
Q1
(a) If a continuously differentiable Walrasian demand function xp, w is generated
by rational preferences, then it must be homogeneous of degree zero, satisfy Walras
law, and have a substitution mat
University of Maryland
Department of Economics
COMPREHENSIVE EXAMINATION
Microeconomic Theory
JANUARY 2003
SAMPLE SOLUTIONS
Question 1.
(a)
[8 points]
(i)
Homogeneous of degree zero.
(ii)
Strictly increasing in w and nonincreasing in each pi.
(iii) Quasic
University of Maryland
Department of Economics
COMPREHENSIVE EXAMINATION
Microeconomic Theory
AUGUST 2003
SAMPLE SOLUTIONS
Question 1.
(a)
[6 points] By homogeneity of degree one with respect to wealth w:
x ! ( p, w) = w x ! ( p,1) .
Differentiating with
University of Maryland
Department of Economics
COMPREHENSIVE EXAMINATION
Microeconomic Theory
JANUARY 2004
Students Social Security Number_
All questions should be answered in examination booklets. Use a separate booklet for each
question. Write the numbe
2.
Implicit function theorems and applications
2.1 Implicit functions
The implicit function theorem is one of the most useful single tools youll meet this
year. After a while, it will be second nature to think of this theorem when you want
to figure out h
Concave and Quasiconcave Functions
Reference: Simon & Blum, chap. 21, Mas-Collels mathematical appendix MC, MD.
1
Convex Sets
A set U Rn is convex if whenever two points x,y U then the line connecting x, y completely lies in U . Some examples:
Formally
Natalia Lazzati
Mathematics for Economics (Part I)
Note 7: Nonlinear Programming - The Lagrange Problem
Note 7 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19).
Introduction to the Lagrange Problem
Let f : Rn ! R, and consi
Natalia Lazzati
Mathematics for Economics (Part I)
Note 8: Nonlinear Programming - The Kuhn-Tucker Problem
Note 8 is based on de la Fuente (2000, Ch. 7) and Simon and Blume (1994, Ch. 18 and 19).
The Kuhn-Tucker Problem
Let f : Rn ! R, and consider the pr
Natalia Lazzati
Mathematics for Economics (Part I)
Note 9: Parametric Optimization - Envelope Theorem and Comparative Statics
Note 9 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 8) and Simon and Blume
(1994, Ch. 19).
Parameterized Optimizatio
Natalia Lazzati
Mathematics for Economics (Part I)
Note 10: Quasiconcave and Pseudoconcave Functions
Note 10 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21).
Monotonic transformations: Cardinal Versus Ordinal Utility
A utility fun
Natalia Lazzati
Mathematics for Economics (Part I)
Note 11: Quasiconcave Programming
Note 11 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 20), Simon and Blume (1994,
Ch. 21) and Dr. Walker Lecture Notes for ECON 501B.
s
Quasiconcave Programmi
Natalia Lazzati
Mathematics for Economics (Part I)
Note 12: Homogeneous and Homothetic Functions
Note 12 is based on Madden (1986, Ch. 9) and Simon and Blume (1994, Ch. 20).
Homogeneous Functions
Many of the functions appearing from solutions of parameter
Preference, Binary Relations, and Utility Functions
Suppose we continue to assume that a particular consumers preference is described by a utility
function, for example u(x, y ) = xy . We dene several sets associated with u(), called binary
relations.
Let
141
5.2 Indirect Utility Function and Expenditure Functions
Each households optimization problem can be written in two forms: (i) as a utility maximization problem for a given budget constraint, or (ii) as an expenditure minimizing problem for a given ut
1
ECONOMICS 581: LECTURE NOTES
CHAPTER 3: CONVEX SETS AND CONCAVE FUNCTIONS
W. Erwin Diewert
March, 2011.
1. Introduction
Many economic problems have the following structure: (i) a linear function is minimized
subject to a nonlinear constraint; (ii) a lin
1.
a).
(i). A preference relation f is homothetic if, for any two bundles x, y X , the
statement that the consumer is indifferent between the two bundles is equivalent
to the statement that the consumer is indifferent between x and y where > 0 .
These pre
University of Maryland
Department of Economics
COMPREHENSIVE EXAMINATION
Microeconomic Theory
AUGUST 2011
Student Identification Number_
All questions should be answered in examination booklets. Use a separate booklet for each question.
Write the number o
University of Maryland
Department of Economics
COMPREHENSIVE EXAMINATION
Microeconomic Theory
AUGUST 2012
Student Identification Number_
All questions should be answered in examination booklets. Use a separate booklet for each
question. Write the number o
In a separate file or with pencil and paper answer the question and then submit the file below.
Draw a graph of a market for a firm in a perfectly competitive industry. Indicate the short run
profit maximizing quantity and the profits for the firm. Explai
In a separate file or with pencil and paper answer the question and then submit the file below.
Explain with the help of graphs how a firm minimizes costs in the long run
The firm can minimize its cost by producing at the level where MC = Average Total Co