PS 5052
Mathematical Modeling in Political Science
Fall 2007
Monday & Wednesday
10:00-11:30
Eliot 316 |
page last revised 12/5/2007
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The Course evaluation website is now open. Please visit it near the end of the semester. Your responses will help guide the future design of the course.
Jump directly to current topics & assignments
Professor
Randall Calvert home
page
E-mail: calvert at wustl.edu
Office
hours: TBA
|
Office: Eliot
328
Phone: 5-5846 |
Teaching Assistant
Ugur Ozdemir
Help Sessions: Friday 1:30-3:00 in Eliot 316
This course is an introduction to mathematical techniques used to
model phenomena studied in political science, with special attention
to the analysis of individual action. Mathematical topics covered
include: sets, functions, and graphs; matrix algebra; differential
calculus and optimization; probability and risk; integral calculus;
and sequences, series, and limits. All these topics are useful in
many settings in political science, including game theory, dynamic
modeling, and statistics.
This course website will be updated to reflect any changes in
schedule, topics covered, or assignments, as well as to provide
relevant links to materials associated with the course.
Textbooks
The course will not closely follow a textbook, because there seems
to be none covering the appropriate material. Silverman is an
elementary calculus text with several virtues (it's clean,
well-written, lightweight, and cheap) and we will appeal to it for
outside help when possible. Gill is a more comprehensive text, some
of whose coverage is beyond the scope of this course, but it will
prove a useful reference both now and in future work. In addition it
will be a source of some assignments.
- Richard A. Silverman, Essential Calculus with
Applications, Dover Books 1989.
- Jeff Gill, Essential Mathematics for Political and Social
Research, Cambridge Univ. Press 2006.
The Silverman book is available at the bookstore; the Gill text is
a late entry, and I am asking you to obtain it on-line from Amazon,
Barnes and Noble, Cambridge, or some other such source. I will make
other readings available from time to time if I identify useful
ones.
Course requirements
Exams 75%. Grades for the course will be compiled primarily from your
performance on three examinations, given at the points indicated in
the course outline. I will give you at least two weeks warning of
any changes in exam dates.
Homework and occasional short quizzes 25%. I will provide frequent problem sets along the way, whose solutions we can discuss in class. I encourage you to consult with one another in working on exercises, although exams should represent each individual's work alone. Grading of these will be based primarily on (1) completion and (2) effort. From time to time, I will also administer a short quiz on the easier aspects of the problems and other material being covered; these will be graded more for accuracy than are the homeworks.
Course Outline and Approximate
Schedule
Summer Review Sessions
To meet in Eliot 314 at 2:00-4:00 pm on the dates shown below:
- Click here to download exercises for
summer review sessions (PDF file)
- Thursday 8/23 Algebra
- see also Silverman 1.3, 1.4, 1.5
- see also Gill 1.2, 1.3, 1.4, and 1.7 (pages 1-18, and 34-40)
- Friday 8/24 Sets
- see also Silverman 1.2
- see also Gill 7.3 (pages 291-306)
- Monday 8/27 Statements and Proof
- see also Silverman 1.34, 1.37 on some methods of proof,
and also 1.4 and 1.5 for some simple proofs of theorems
- Tuesday 8/28 Functions and Graphs
- see also Silverman 1.6-1.9, 2.1-2.3
- see also Gill 1.5 (pages 18-34) and 2.1 and 2.2 (pages 51-54)
--- semester begins ---
Sets, relations, functions
- Wednesday 8/29 and Wednesday 9/5
- Monday 9/10 and Wednesday 9/12
- for your review: the real numbers (Silverman 1.3)
- aside: the method of proof by mathematical induction (Silverman 1.37)
- sequences and limits (Silverman 2.23. 2.94. 2.95)
- limits (Silverman 2.41, 2.44, 2.45)
- algebraic operations on limits; continuous functions; one-sided limits (2.6)
- limits involving infinity; asymptotes; (Silverman 2.91, 2.92, 2.93)
- sums and series (Silverman 2.96; see also Gill, pp. 259-266)
- application: choices over time (discounted present value)
- Exercises due Monday 9/17 (in Silverman):
- p. 10 #12, 19 (on proof by mathematical induction)
- p. 45 #7(a-d), 9
- p. 58 #6(a-e), 7, 12
- p. 73 #4(a-d), 5, 6, 7
- pp. 98-99 # 1(a-c), 2(a-c), 7, 8, 9, 14(a-d), 16
- AND:
- In the Powerball Lottery for September 12, 2007, the jackpot is $20 million. As the web page shows, if you (alone) win, you can take the winnings in 30 yearly payments (the first one immediately) that actually add up to $20 million, or as a much smaller lump sum. After withholding of federal and Missouri taxes, the yearly payment is $473,333, and the lump-sum payment is $6,368,500. Use this information to answer the following:
- What discount factor would make you exactly indifferent between the lump-sum payment and the 30 annual payments? What would you choose if your personal discount factor were higher than that?
- Suppose that your personal discount factor were based solely on the average interest rate at which you expect to be able to invest the money every year. What interest rate would make you exactly indifferent? What would you choose if the interest rate were higher than that?
Differential calculus
Generally, the material in this section is covered in Silverman,
chapters 2 and 3.
- Monday 9/17
- slope of a secant line
- the Newton quotient
- Wednesday 9/19
- Monday 9/24 and Wednesday 9/26
- techniques of differentiation: sums, products, and quotients
- techniques of differentiation: inverse and composite functions
- Monday 10/1
- application: optimization
- implicit differentiation
- Exercises due Monday 9/24 (in Silverman):
- p. 58 #1, 2
- p. 66 #2, 4, 6, 9, 13
- Exercises due Monday 10/1:
- Silverman: p. 80-81 #1-3 (all parts), 8-11
- Silverman: p. 87-88 #3 (a-c), 7, 8
- Gill: p. 230 #5.5, first eight of the ten exercises given
- [We'll have a short quiz on elementary differentiation, as illustrated in these exercises, in class Monday 10/1.]
Wednesday 10/3: Discussion and review
First exam out; due back in Monday 10/8
Vector and Matrix Algebra
recommended readings from Gill:
- Chapter 3: 3.1-3.5 (omit examples 3.21, 3.22)
- Chapter 4: 4.3, 4.4, 4.6, 4.7, 4.9 (except example)
- Monday 10/8
- Why matrix notation?
- Vectors and vector operations: addition, scalar multiplication, dot product; geometric interpretations
- Wednesday 10/10
- matrices: basic definitions
- basic arithmetic operations: addition, scalar multiplication, matrix multiplication
- Exercises due Mon. 10/15
- For each of the following expressions, perform the indicated calculation and illustrate it with a graph.
- find the scalar product: 3 times (1, 2)
- find the vector sum: (1,2) + (3,1)
- find the vector difference: (1,3) - (1,-1)
- find the dot product: (1,2) times (2, 1)
- find the dot product: (2,1) times (-2,4)
- find the dot product: (2,1) times (-2,2)
- show whether each of the following sets of vectors is linearly independent:
- {(3,4,5), (1,1,2), (2,2,4)}
- {(1,3,2), (6,7,5), (4,1,1)}
- {(2,0,0), (0,3,0), (0,0,1)}
- From Gill, p. 127, problems 3.10 & 3.11
- Monday 10/15 & Wednesday 10/17
- operations on matrices: determinant; rank
- matrix inverse
- application: solving systems of linear equations
- Exercises due Mon. 10/22
- Monday 10/22 & Wednesday 10/24 Further applications and discussion
- Regression models
- Quadratic forms [Gill 4.9]; quadratic utility in spatial voting models
- Eigenvalues and eigenvectors; stable states in Markov processes [Gill 4.8]
- Exercises due Mon. 10/29
Multivariate calculus
Some of the material in this section is covered in Silverman, chapter 6.
- Monday 10/29 and Wednesday 10/31
- functions of real vectors; multivariate calculus
- Exercises due Monday 11/5:
- Silverman: p. 222 #11 & 12 (all parts). Note that
- d/dx ex = ex
- d/dx ln x = 1/x.
- Silverman: p. 227, # 1 and 2 (all parts); in problem 2 just use the Chain Rule and not substitution.
- Monday 11/5
- optimization: first-order conditions
- comparative statics
- the Hessian matrix and second-order conditions
- concave and convex functions
- Wednesday 11/7
- optimization with equality constraints
- the method of Lagrange
- Exercises due Monday 11/12:
- Silverman: p. 233-234, # 1 (all parts), 3, 12, 13.
- For problems 12 and 13: perform a comparative statics analysis to determine the effects of changes in the parameters P1, P2, and q on the profit-maximizing choices of Q1 and Q2.
- Monday 11/12
Second exam out Wednesday 11/14; due Monday 11/19
Probability
- Wednesday 11/14
- Monday 11/19
- conditional probability and Bayes's rule
- --- Thanksgiving break ---
- Monday 11/26
- application: expected utility and choice under risk
- lotteries and risk aversion
- Wednesday 11/28
- probability with continuous outcomes: PDF, CDF
- Exercises due Mon. 12/3
The Course evaluation website is now open. Please visit it near the end of the semester. Your responses will help guide the future design of the course.
Integral calculus
The basic material in this section is covered in Silverman, chapter 4.
- Monday 12/3
- Wednesday 12/5
- the antiderivative; fundamental theorem of calculus
- definite integrals
- Exercises due Monday 12/10
- Exercise 1. U s e a l i m i t o f s u m s o f r e c t a n g u l a r a r e a s t o c a l c u l a t e t h e a r e a u n d e r t h e f u n c t i o n f ( x ) = 3 x f r o m x = 1 t o x = 3 . H i n t : y o u m a y a s s u m e t h a t t h e f o l l o w i n g f o r m u l a h o l d s : f o r a n y p o s i t i v e i n t e g e r n ,
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1 + 2 + . . . + n = n ( n + 1 ) / 2 .
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- Silverman p. 121 #2, 4, 5
- Silverman p. 151 #2, 4
- Exercise 6. U s e i n t e g r a t i o n t o f i n d t h e C D F , m e a n , a n d v a r i a n c e o f a r a n d o m v a r i a b l e X w h o s e P D F i s f ( x ) = 1 f o r x i n [ 0 , 1 ] a n d f ( x ) = 0 e v e r y w h e r e e l s e .
- Exercise 7. F i n d t h e m e a n a n d v a r i a n c e o f a " t r i a n g u l a r d i s t r i b u t i o n" o n [ 0 , 2 ] d e f i n e d a s f o l l o w s :
- f ( x ) = x f o r x i n [ 0 , 1 ] ;
- f ( x ) = 2 - x f o r x i n [ 1 , 2 ] ; a n d
- f ( x ) = 0 e v e r y w h e r e e l s e .
- Exercise 8. F i n d t h e C D F , m e a n a n d v a r i a n c e f o r a r a n d o m v a r i a b l e X h a v i n g t h e u n i f o r m d i s t r i b u t i o n o n [ a , b ] , w h e r e a < b .
- Review and discussion
Final exam out Tuesday 12/12; due Monday 12/17
This page written by Randall Calvert © 2007
Email comments and questions to calvert at wustl.edu
