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(s) of the function and use the Hessian matrix to deter
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(T ) s
2
T
0.
0
Find the impact on the value function
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 Function Theorem tells us the slopes of the objective fun
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, into vectors in m.
(T:n m.)
Given vectors x and y in n
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
x1 a11 a12 x10 a11x10 a12 x2
.
x Ax 1
0
0
0
x2 a
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 ,., xn )
g ( x1 , x2 ,., xn )
*
0 , for i = 1, 2, , n;
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 profit is
*(w,p) = (x*(w,p);w,p) = pf(x*(w,p) wx*(w,p),
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 argument,
evaluated at (x1, x2, ., xn), is
f i ( x1 ,
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 ) T 1QT 1 .
In summation form, we have
L t 0 [ t qt Pt (
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 ( s0 , s1 , , sT ). Bellmans
*
*
*
Principle of Optimality
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 left-hand side is now the derivative of eatx(t), so the
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. For any gamble G, there exists a constant, w [0,1], s
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 ].
2
1
],
1
i) What are the steady-state populations?
0
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 + 6x 2y. Use the Hessian to
determine whether the critic
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 subject to
the constraint x 10 y. Explain why this bund
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 corresponding
2 3
quadratic form and use the Hessian matrix to de
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(s) of the function and use the Hessian matrix to deter
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 of the matrix A c a b .
b c a
If a = b = 4, for what
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 minima.
Find the global minimum of the function g(x,y) = x2
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 an expression for u(w). (Hint: Let z(w) = u(w) and
sol
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 minima.
Find the global minimum of the function g(x,y) = x2
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 to spend on these two activities. He also has 10 hours
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 of the matrix A c a b .
b c a
If a = b = 4, fo
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 to spend on these two activities. He also has 10 hours
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 an expression for u(w). (Hint: Let z(w) = u(w) and
sol
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 distributed on
[0,1]. The seller holds an all-pay auction
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 = p and pY = 1,
and she has a budget of m. (Assume that m
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 = p and pY = 1,
and she has a budget of m. (Assume that m
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 distributed on
[0,1]. The seller holds an all-pay auction
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 corresponding
2 3
quadratic form and use the Hessian matrix to