Econ 503
Professor John Nachbar
Fall 2007
Midterm
You have until 12 PM.
You can appeal to known results (e.g., the Implict Function theorem) but the
appeal must be explicit.
If you get stuck, move on. If part (a) of a question asks you to prove somethi
Suggested Solutions to Assignment II
Econ 503B
Carmen Astorne-Figari and Feng Dong
Washington University in St. Louis
December 8, 2010
1
Local Non-Satiation (LNS)
"
Fix an arbitrary bundle x = (x1 ; :; xl ) 2 <l . Pick up an arbitrary " 2 <+ . Dene x = (x
Econ 503
Professor: John Nachbar
Fall 2010
Homework 5
Answers
1. (a) Suppose that there exists a concave, monotone, continuously dierentiable, non-satiated utility function u(x) which rationalizes the data.
Then (pt , xt ) is the result of the following u
Econ 503
Professor: John Nachbar
Fall 2010
Homework 1
Answers
1. Suppose that C is generated by a complete, transitive preference relation W .
Consider any chain x1 , . . . , xk such that there is a B1 with x1 , x2 B1 , a
B2 with x2 , x3 B2 , and so on to
Econ 503
Professor: John Nachbar
Fall 2010
Midterm
Answers
1. HW 1, Q4.
(a) By contraposition. Suppose a C(B1 ). I will show a violation of WA.
/
If a C(B1 ) then there is a b = a such that b C(B1 ). Since a B1 ,
b RS a. On the other hand, b B2 since B1 B
Econ 503
Professor: John Nachbar
November 5, 2009
Midterm
Answers
1. See HW1 Q2.
2. First note that f i (p , ei ) = 0 implies [p p ] f i (p , ei ) = 0 for all
p RN . That is, p p is a compensated price change. Thus by
+
LODC for individuals, we know that
Econ 503 Professor John Nachbar Fall 2007
Midterm Answers
This test was harder than I intended, with even questions out of the homework (1 and 2) causing significant problems for many people. I usually try to write exams where a 65, roughly, is a B, repre
Econ 503 Professor: John Nachbar Fall 2008
Midterm Answers
1. (a) Dp2 1 > 0. To see this, note that by Slutsky Dp2 1 = Dp2 h1 - Dm 1 x2 . Since vDp hv < 0 except if v is collinear with p, the diagonal of Dp h is negative. Since Dp hp = 0, and there only t
Econ 503 Professor John Nachbar Fall 2005
Midterm Answers
Grading. I graded on a 100 point scale, 20 points for each of the five questions; points were divided roughly equally across subsections. The top score was 99. The average was 62. The distribution
Econ 503 Professor John Nachbar Fall 2006
Midterm Answers
1. (a) i. No. In B1 , a is revealed (strictly) preferred to b but in B2 , b is revealed strictly preferred to a. ii. Yes. iii. Yes. i. No (if WA fails then SA fails). ii. Yes. iii. No. In B1 , a is
Econ 503
Professor: John Nachbar
Fall 2010
Homework 2
Answers
1. Suppose aW b and bSc. Note that bSc implies bW c and hence by transitivity
aW c. We rule out the case cW a by contraposition. If cW a, then together
with aW b and transitivity we get cW b, a
Suggested Solutions to Assignment I
Econ 503B
Carmen Astorne-Figari and Feng Dong
Washington University in St. Louis
Novermber 20, 2010
1
The Utility Possibility Set (UPS)
Denote that
xi = (x1 ; x2 )T ; i = F; J
i
i
eF + eJ
= (2; 0)T
y = (y 1 ; y 2 )T = (
Econ 503
Professor: John Nachbar
Fall 2010
Homework 4
Answers
1. MWG 6.C.16.
(a) If the individual owns the lottery, his random wealth is (w + G, w + B).
Thus the minimal selling price Rs is dened by
p u(w + G) + (1 p) u(w + B) = u(w + Rs ).
(b) If the in
Econ 511
Professor: John Nachbar
Fall 2010
Homework 3
Answers
1. Fix any x RN . By non-satiation there is an a RN + such that a S x .
+
Take any > 0. Choose any (0, 1) suciently small that x N (x ). Set
x = a + (1 )x . By strict convexity, for any (0, 1),
Econ 503
Professor John Nachbar
Fall 2007
Homework 1
Answers
1. Suppose that C is generated by a complete, transitive preference relation R.
Consider any chain x1 , . . . , xk such that there is a B1 with x1 , x2 B1 , a
B2 with x2 , x3 B2 , and so on to B
Econ 503
Professor John H. Nachbar
Fall 2007
Homework 3
Answers
1. (a) By strict quasiconcavity, there is a demand function, rather than a demand correspondence (the solution to the utility maximization problem
is unique).
Dene by (p) = (p, 1). Consider a
Econ 503
Professor John H. Nachbar
Fall 2007
Homework 4
Answers
1. By Slutsky, Dp (p , m ) = Dp h(p , m ) Dm (p , m )x . Pre and post multiplying by p , since Dp h(p , m )p = 0, x p = p x = m (Walrass Law),
and p Dm (p , m ) = 1 (dierentiate Walrass Law w
Econ 503
Professor John H. Nachbar
Fall 2007
Homework 7
Answers
1. IIA follows from the fact that Pairwise Majority only considers pairs, so it
automatically honors IIA. Unanimity is also immediate, since if a Ri b for
every i, then trivially a majority w
Econ 503
Professor John H. Nachbar
Fall 2007
Homework 5
Answers
1. (a) Multiplying p by > 0 multiplies the objective function by . This is an
increasing transformation of the objective function with no change in the
constraint, so the solution is unchange