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Econ 106 Mathematical Economics11st Semester, AY 2013-2014
First Long Exam [100 points]8:30-10 a.m., January 28, 2014
Instructions
1. On every sheet of yellow paper, write your name and student
ID.
2. The use of mobile phones or any other electronic devices
during the exam is not allowed.
3. You have an hour and a half to complete this two-page,
closed-book exam
4. Refrain from talking to your seatmate.
A reminder: Cheating, of any form, does not pay.
All the best.
1. [18 points] Given the technology matrix for a three-sector
economy:
A=
24 a11 a12 a13a21 a22 a23
a31 a32 a33
35 ;
(a) [5 points ] State the conditions for (I A) to satisfy the
Brauer-Solow condition. Write theseconditions in terms ofaij; i;j =
1; 2:
(b) [7 points] Show in detail how you would solve for the
required output vector x =
24 x1x2
x3
35 given
the nal demand vector d =24 d1d2
d3
35 : Note: let L (I A)1 so that (I A)1 L =24 L11 L12 L13L21 L22
L23
L31 L32 L33
35 :
(c) [6 points] Given
(I A)1 L=
24 L11 L12 L13L21 L22 L23
L31 L32 L33
35 ;
and
d=
24 0d2
0
35 ;
what will be x1 andx3 in terms ofLij; i;j = 1; 2; 3?
2. [12 points] Consider the matrix below:
B =
24 1 0 00 0 0
0 0 1
35 :
1 S. Daway
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(a) [4 points] What conditions should be met in order for you to
use either the eigenvalues or principalminors method to determine
the deniteness ofB ?
(b) [8 points] Determine the deniteness ofB : Show your
work.
3. [15 points] Prove the following for any n n matrix A:
(a) [8 points] IfA is nonsingular, then = 0 cannot be an
eigenvalue ofA:
(b) [7 points] If is an eigenvalue ofA, then2 is an eigenvalue
ofA2:
4. [30 points] Consider the utility function:
U(x1; x2) = x1x
2; ; >0;
whereU() is twice continuously dierentiable; and x1 andx2 are
consumption goods.
(a) [5 points] Determine the degree of homogeneity ofU() : Show
your work.
(b) [5 points] Derive @U(x1; x2) =@x1 and @U(x1; x2) =@x2: What
is the economic interpretation ofthese?
(c) [5 points] Derive the second partial derivatives ofU(x1;
x2)(i.e., @2U(x1;x2)
@x21
and @2U(x1;x2)
@x22
):What
restrictions should be placed on and forU(x1; x2) to exhibit
diminishing marginal utility?
(d) [5 points] Log-linearize U(x1; x2) : LeteU(ex1; ex2) ln
U(x1; x2) : What restrictions should beplaced on x1 andx2?
(e) [5 points] By how much willeU(x1; x2)change ifdx1 = dx2 =
2?(f) [5 points] What will be the slope of an indierence curve
ofeU(x1; x2) that passes through the
point(2; 1)?Hint: solve for dx2=dx1 holdingeUconstant.
5. [25 points] Given y = f(x) = ex2
;
(a) [5 points] Find dy=dx:
(b) [5 points] Find d2y
dx2:
(c) [15 points] Take a quadratic or second-order Taylors
approximation off(x) at x0 = 1: Write itin the formax2 +bx+c:
BONUS
[5 points] Free for all:
Given q(x1; x2) = 4x21 6x1x2 x
22; nd a symmetric matrix that yields q(x1; x2) :
[2 points] For those who helped in making the lantern:
Giveny = f(x) = ln eex
; nddy=dx:
[2 points] For those who joined the Lantern Parade:
Given y = f(x) = elnx; nddy=dx:
[2 points] Only for those who attended the latest
Kapekonomiya:
Given y = f(x) = 3x2
; nddy=dx:
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