Econ 4111
Professor: John Nachbar
Spring 2017
Homework 1
Answers
1. Let denote ( or ( and ) ( or ) and ).
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and or ( and ) or ( or ) and
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Econ 4111
Professor: John Nachbar
Spring 2017
Homework 2
Answers
1. (a) f 1 (B) is the disk, including the boundary, centered at the origin, of
radius 1.
(b) f 1 (B) is the half-space, not including the boundary, lying to the southwest of the line given b
Econ 4111
Professor: John Nachbar
Spring 2016
Homework 3
Answers
1. (a) h : R R is defined by h(x) = 2x x2 .
(b) x h1 (C) iff g(f (x) C iff f (x) g 1 (C) iff x f 1 (g 1 (C), which
is to say x f 1 g 1 (C).
P 2 2
P
2. No, f violates the second norm property
El Saving screenshot.
horizontal
parallel to the ground
it 49
imaginary number
any number of the form a + bi, where a
and be are real numbers, b is not equal
to zero, and I is the square root of i
fr 4
independent variable
the input values of a function
a All 6| 12:07 AM
real income effect
as prices rise and your income stays the
same, quantity demanded for all goods
or services goes down.
if? 49
diminishing marginal utility
assuming all things stay the same with
each additional unit consumed, we
recie
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inferior goods
goods that people buy less of as income
increases
if? 4
2 factors influencing supply
incentive for greater prot and market
entry
13:7 49
what is the largest factor influencing
elasticity
time
if: 49
in the longe run supp
q
only shows how we respond to changes
in price
demand curve
a 49
market demand
sum of all individual demands in the
market
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2 causes in shifts
income and expectations
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income
ex) normal gods and inferior
71: I19
normal goods
Quiz 2: Comparative Advantage
Econ 1011
Dr.SudeshnaC.Bandyopadhyay
1.
If Scout has an absolute advantage over Dill,
A. Scout has more money than Dill.
B. the problem of scarcity applies to Dill but not to Scout.
C. the problem of scarcity applies to Scout
Engineering economics
Engineering economics is the application of economic
techniques to the evaluation of design and engineering alternatives.
1 The role of engineering economics is to assess
the appropriateness of a given project, estimate its value,
an
Econ 4111
Professor: John Nachbar
February 9, 2012
Set Theory
1
Overview
This is an informal introduction to Set Theory, which is somewhat ironic because
set theory is, by its nature, a highly formal subject. I try to convey the main line
of development w
Econ 4111
Professor: John Nachbar
September 30, 2011
Metric Spaces
1
Metric Spaces Basics.
1.1
Metric spaces.
A metric space (X, d) consists of a set of points, X together with a distance function,
or metric, d : X X R. The interpretation is that d(a, b)
Econ 4111
Professor: John Nachbar
August 31, 2011
Logic and Proofs
1
Sentential Connectives and Tautologies
Formal logical statements are built up out of sentential connectives, such as and
and implies, and quantiers, such as there exists. To understand t
Econ 4111
Professor: John Nachbar
October 5, 2010
Compactness and Completeness in RN .
1
R is complete.
Theorem 7, the Heine-Borel theorem, states that a set in RN is compact i it is
closed and bounded. Theorem 7 is immediate if I can show (a) that RN is
Econ 4111
Professor: John Nachbar
Spring 2012
Test 2
Answers
1. If C is compact and f : C R is continuous then f (C) is compact (by a
previous theorem), hence closed and bounded. Since f (C) is bounded and a
subset of R, it has a least upper bound, say y
Econ 4111
Professor: John Nachbar
December 14, 2010
Compactness
1
Introduction.
An important fact about metric spaces is that the following ve properties of a set
C are equivalent.
1. C is compact: for any set O of open sets with the property that
C
O
OO
Econ 4111
Professor: John Nachbar
Spring 2012
Test 1
Answers
1. By LUB, there is a least upper bound x. By the denition of least upper
bound, for any > 0, there is a T such that xT > x (otherwise, x < x
is an upper bound). Since the sequence is weakly inc
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 8
Answers
1. Suppose a b, b S c, and c S a. Note that this is complete but not transitive.
Let B = cfw_B1 , B2 with B1 = cfw_a, b and B2 = cfw_a, b, c. Then the implied
choice function has C(B1 ) = c
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 7
Answers
1. (a) The solution is x = (1, 1).
(b) No, because
(c)
g(x ) = (0, 0).
f (x ) = (1, 1), so there is no such that
f (x ) = g(x ).
(d) The Slater condition is that there is a point interior to
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 6
Answers
1. (a) The constraint set is lens-shaped and lies along the 45 degree line, with
one end at (0,0) and the other at (1,1).
(b)
maxx
f (x) = x1 + x2
s.t. g1 (x) = (x1 1)2 + x2 1 0
2
g2 (x) = x
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 5
Answers
1. Let x be any element of S. Then a x and x b. By transitivity, a b.
2. (a) By contraposition. If p is rational and p + x = r is rational then x =
r + (p), which is the sum of two rationals
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 4
Answers
1. (a) Consider any closed set C. If f 1 (C) = then I am done, because is
closed. Otherwise, consider any sequence cfw_xt in f 1 (C) and suppose
that xt x. I must show that x f 1 (C). Since
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 1
Answers
1. Let denote ( or ( and ) ( or ) and ).
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
F
F
F
F
and or ( and ) or ( or ) and
T
T
T
T
T
F
T
T
T
T
T
T
T
T
T
F
F
F
F
T
F
T
T
F
F
F
T
T
F
F
F
F
T
F
T
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 3
Answers
1. Let O be an open set containing x . Then there is an > 0 such that
N (x ) O. Since x is a limit point, N (x ) A contains a point other than
x . Since N (x ) A O A, the result follows.
Si
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 2
Answers
1. (a) y f (A1 A2 ) i there is an x A1 A2 such that y = f (x). For such
x, either x A1 or x A2 , hence either y f (A1 ) or y f (A2 ), hence
y f (A1 ) f (A2 ), as was to be shown.
(b) Conside