Download ECON 7800-001 Quantitative Methods in Economics

Fall 1999
Department of Econoinics
ECON 7800
(Ph.D. Math Preparatory Course)
Schedule of course offering for Fall 1999
August 9- 13
9:00 - 12:00 p.m., Econ 119
August 16-17
9:00 - 12:00 p.m., Econ 119
August 18 - REVIEW SESSION
9:00 - 12:00 p.m., Econ 119
August 20 - FINAL EXAMINATION
9:00 - 12:00 p.m., Econ 119
DEPARTMENT OF ECONOMICS
ECON 7800
(PhD Math Preparatory Course)
August 1999.
Instructor
Office/tel
Email
Office Hours
Ron Smith
To be Announced
To be Announced
By appointment
Schedule
All meetings 9:00-12:00, held in Econ 119
August 9-13 Lectures
August 16-17 Lectures
August 18 Review Session
August 19 Final Exam.
Econ 7800 is a two week refresher course in matrix algebra, calculus, probability and
statistics. It is designed to prepare students for studying the core courses in economics at the
PhD level: microeconomic and macroeconomic theory, statistics and econometrics.
Tentative Course Outline.
1. Numbers, functions, sets, systems of linear equations.
2. Matrices and vectors, operations, types of matrices, inverses, quadratic forms,
determinants.
3. More matrices including idempotent matrices, eigenvalues and eigenvectors (characteristic
roots and vectors).
4 . Calculus derivatives and integrals, rules, limits continuity. Multivariate calculus, partial
derivatives and differentials.
5. Optimisation, unconstrained and constrained, lagrange multipliers, difference and
differential equations.
6. Probability, random variables, probability distributions.
7. Expectations, variances, moments; joint conditional and marginal distributions.
References. There is no set text, this material is covered in many books. You will need a
maths for economists book as background for your main courses. Examples are: Hoy,
Livernois, McKenna, Rees and Stengos, Mathematics for Economists, Addison Wesley;
Chiang, Fundamental Methods of Mathematical Economics, McGraw Hill; Simon & Blume,
Mathematics for Economists, Norton. For matrix algebra and statistics W.H Greene,
Econometric Analysis (used on the econometrics courses), and many other econometrics
texts, provides good surveys in their early chapters or in appendices, Amemiya Introduction
to Statistics and Econometrics, Harvard University Press.
r
Econ 7800. Exam. 20 August 1999.
Ans,;,.·er all questions. make sure you do all the subsections. Show all work.
You haYe tluee hours to complete the exam. Please write neatly. Good luck.
l. (a) Simplify the formula
(b) Suppose quantity demanded and supplied are determined by:
qd
01 -
5
q
0:2
31p
+ J2P-
Derive equilibrium price and quantity. What restrictions are required to
give a sensible answer?
2. (a) Let
X' = [ 1
1
1
Xn ]
X1
1
; Y = [Y1,Y2, .. ,,yn]·
What are X'X, I X'X \, (X'X)- 1 , X'y, and (X'X)- 1 X'y? (b) Suppose
X is an n x k matrix of rank k. Show that X' X is positive definite.
( c) Suppose Yt is a k x 1 vector determined by the first order difference
equation: Yi = AYt-i- Explain how you would transform the system to
diagonalise A. What are the stability conditions for the system?
3. Find the derivatives of y with respect to x for the following functions
and simplify your answers: (a) y = (ln2x3 )(x- 2 ) (b) y = x 3 /3x- 1 (c)
f = 3x2 + x 2 y + iy 2 - .jy = 6.
4. (a) Give the first tluee terms of the Taylor Series expansion for f(x) == fi
around x 0 = 4. ( b) The profits of a monopolist facing constant marginal
costs are IT = pq - cq. What is its profit maximising price if it faces
a demand curve: q = 0:p-;J? What restrictions are required to give a
sensible answer ?
.5. Find the integrals of t he following functions and check your answer by
differentiation. (a) f = :z: 1 + 2.r3 +-lx+ 10. (b) f = x 2 13 . (c) f = exp(bx).
6. Suppose a project pays D(t) = De1; 1 from t
constant. Its present value is
=0
to t
= y,
where D is a
\vhere r > g is a constant interest rate. \Vh at is the value of the project?
l
i. Find any stationary points of the following fu nctions and indicate whether
they are maxima minima or saddle points: (a) y = 4x 2 - 2x - 2; (b)
y = (2/ 3)x 3 - (3/2)x 2 - 2x + 4: (c) y = (-3 / 2)x 2 - 2z 2 -'- 4xz - 2x.
8. A consumer has a utility function and budget constraint in terms of income
y, quantities of goods q,, and prices p; for i = 1. 2 :
Derive the utility maximising expenditure on each good. p; q;.
9. (a) For random variables x and y with joint pdf f(x. y) show that
·) =
!( y I x
J(x I y)f(y) .
f(x )
(b) Show that the variance of a random variable x, with pdf f (x) and
expected value µ can be written:
( c) Find the expected value and variance of the uniform distribution over
[a, b] : f(x) = (b - a)- 1 ; a~ x ~ b; f (x) = 0 otherwise.
10. (a) For observed dependent variable y and independent variables X , and
unobserved parameters (3 and disturbances u: let y = X f3 + u , where y
and u are n x 1 vectors; Xis an x k matrbc of rank k . Find the estimator
of (3 , say
which minimises u'u. Check the second order conditions. (b)
Suppose the probability of finishing a PhD in 3+t years is given by f (t)
= Aexp(-At), t=0,1,2, .... The university has data on ti for past students
i = 1. 2 . ... , n. What is the maximum likelihood estimator of A? Check the
second order conditions.
/3,
2