Economics 425
Problem Set #1
Due: Wednesday, Sept. 19, 2012
1. Consider the matrix
[
] and the vector
[ ]. Show that the product Ax is a
linear combination of the columns of A.
a b c
2. Find the dete
Economics 425
Midterm Exam
Wednesday, Oct. 26, 2011
Part A. Answer three questions. Each question is worth 10 points.
1. Consider the function f(x,y) = (x2)2 + (y x)2 + 2.
a) Find the stationary point
Economics 425
Problem Set #2
Due: Wednesday, Oct. 10, 2012
1. Find the maxima and/or minima of the function f(x,y) = x2 2xy + 2y2 2x. Use the Hessian
to determine whether the points are maxima or mini
Matrix algebra
An mn matrix has m rows and n columns. For example,
a11 a12
a
a22
A 21
.
.
am1 am 2
. a1n
. a2 n
.
. .
. amn
A linear transformation, T, is a function that maps vectors in n, say,
Topics in linear algebra handout
The number in state 1 next period is the number in state 1 this year who remain ( a11x10 )
0
plus the number in state 2 this period who move to state 1 ( a12 x2 ):
1
0
Consider the general problem:
max xX f(x) subject to g(x) c.
If x* is an interior solution of the maximization problem, there exists * such that
*
*
*
*
*
*
*
*
*
L( x1 , x2 ,., xn , * ) f ( x1 , x2 ,
The Envelope Theorem
Consider a perfectly competitive firm with the production function f. The firms profit is
(x;w,p) = pf(x) wx,
with p the price of output and w the price of the input. The maximal
Functions with n variables handout
Consider a function of n variables, f(x1, x2, ., xn). Suppose that the function is
continuous, without any kinks. The partial derivative of f with respect to the ith
Optimal Control
The Lagrangian for the problem can be written as
L q0 P0 (q0 ) 0 (Q Q0 )
q1P (q1 ) 1 (Q0 q0 Q1 )
1
2 q2 P2 (q2 ) 2 (Q1 q1 Q2 )
.
T qT P (qT ) T (QT 1 qT 1 QT )
T
T 1 (QT qT QT 1 )
Dynamic Programming
Consider the problem of maximizing
T
t 0
f ( st , ct , t ) and think of a candidate solution
*
*
*
*
*
with control variables ( c0 , c1* , , cT ) and associated state variables ( s
Differential Equations
Consider the linear first-order differential equation
x(t ) ax(t ) b ,
with a 0. Multiplying through by e at makes the differential equation
e at x(t ) ae at x(t ) be at .
The l
Uncertainty
Axiom 1: Completeness. For any two gambles G and G*, either G G* or G* G (or both).
Axiom 2: Transitivity. For any gambles G, G*, and G*; if G G* and G* G*, then G
G*.
Axiom 3: Continuity
Economics 425
Problem Set #1
Due: Tuesday, Sept. 20, 2016
1. People migrate between two states according to the transition matrix = [
0
where 0 < a < 1 and 0 < b < 1. The initial populations are [ 10
Economics 425
Problem Set #2
Due: Tuesday, Oct. 4, 2016
1. This question asks you to find the maxima and/or minima of two functions.
i) Find the critical points of the function f(x,y) = 3x2 2xy + y2 +
Consider the problem:
max x,y f(x, y) subject to g(x,y) = c.
Let (x*,y*) be an interior solution to the problem. Then, the constraint curve is tangent to a level
curve of f at (x*,y*). The Implicit Fu
Economics 425
Final Exam
Wednesday, Dec. 12, 2012
Part A. Answer three questions. Each question is worth 10 points.
1. Consider selecting c(t) to maximize
T
c(t )dt subject to s(t ) c(t ) , s(0) 0, s
Economics 425
Midterm Exam
Wednesday, Oct. 24, 2012
Part A. Answer three questions. Each question is worth 10 points.
1. Find the bundle (x,y) that maximizes the function f(x,y) = 49 x2 y2 + 10x + 8y
Economics 425
Problem Set #1
Due: Wednesday, Sept. 19, 2012
1. Consider the matrix
and the vector
. Show that the product Ax is a
linear combination of the columns of A.
a b c
2. Find the determinant
Economics 425
Problem Set #2
Due: Wednesday, Oct. 10, 2012
1. Find the maxima and/or minima of the function f(x,y) = x2 2xy + 2y2 2x. Use the Hessian
to determine whether the points are maxima or mini
Economics 425
Problem Set #4
Due: Wednesday, Dec. 5, 2012
1. Let u(w) be the von-Neumann-Morgenstern utility function when wealth is w. Let
A(w) = u(w)/u(w)
Suppose that A(w) = a > 0 for all w. Derive
Economics 425
Problem Set #3
Due: Wednesday, November 14, 2012
1. An athletic center rents out a hockey rink for $20 per hour and a basketball court for $10 per
hour. Preston has a voucher worth $150
Economics 425
Problem Set #3
Due: Wednesday, November 14, 2012
1. An athletic center rents out a hockey rink for $20 per hour and a basketball court for $10 per
hour. Preston has a voucher worth $150
Economics 425
Problem Set #4
Due: Wednesday, Dec. 5, 2012
1. Let u(w) be the von-Neumann-Morgenstern utility function when wealth is w. Let
A(w) = u(w)/u(w)
Suppose that A(w) = a > 0 for all w. Derive
Economics 425
Final Exam
Thursday, December 15, 2011
Part A. Answer three questions. Each question is worth 10 points.
1. A seller faces two bidders whose valuations are independently and uniformly di
Economics 425
Final Exam
Wednesday, Dec. 15, 2010
Part A. Answer three questions. Each question is worth 10 points.
1. Kate has the utility function u(x,y) = ln(x)+y. The prices of the goods are pX =
Economics 425
Final Exam
Wednesday, Dec. 15, 2010
Part A. Answer three questions. Each question is worth 10 points.
1. Kate has the utility function u(x,y) = ln(x)+y. The prices of the goods are pX =
Economics 425
Final Exam
Thursday, December 15, 2011
Part A. Answer three questions. Each question is worth 10 points.
1. A seller faces two bidders whose valuations are independently and uniformly di
Economics 425
Midterm Exam
Wednesday, Oct. 27, 2010
Part A. Answer three questions. Each question is worth 10 points.
1 2
1. Consider the matrix A =
. Find the stationary point(s) of the correspond
Economics 425
Midterm Exam
Wednesday, Oct. 26, 2011
Part A. Answer three questions. Each question is worth 10 points.
1. Consider the function f(x,y) = (x2)2 + (y x)2 + 2.
a) Find the stationary point
Economics 425
Midterm Exam
Wednesday, Oct. 27, 2010
Part A. Answer three questions. Each question is worth 10 points.
1 2
1. Consider the matrix A
. Find the stationary point(s) of the correspondin