
Solutions
14.06 2004  Midterm
Prof: Marios Angeletos
Question 1 (Continuous Time): (a) The household takes , wt , rt and Tt as given and selects a
consumption path to solve
Z
max
et u (ct ) dt
0
subject to
= (1 ) (rt kt + wt ) + Tt ct kt
kt
This is a s
14.30 PROBLEM SET 7
TA: Tonja Bowen Bishop
Due: Tuesday, April 25, 2006
Note: The rst three problems are required, and the remaining three are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the grader. Alt
14.30 PROBLEM SET 3
TA: Tonja Bowen Bishop
Due: Tuesday, March 8, 2006 , by 4:30 p.m.
Note: The rst four problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the
14.30 PROBLEM SET 5
TA: Tonja Bowen Bishop
Due: Tuesday, April 4, by 4:30 p.m.
Note: The rst three problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the grader
14.30 PROBLEM SET 1  SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
For any A and B , we can create two sets that are disjoint and
exhaustive on B : A \ B and AC \ B . Thus
Pr (B ) = Pr (A \ B ) + Pr AC \ B
Because A
B , we know that A = A \ B . S
14.30 PROBLEM SET 4
TA: Tonja Bowen Bishop
Due: Tuesday, March 21, by 4:30 p.m.
Note: The rst three problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the grade
14.30 PROBLEM SET 8
TA: Tonja Bowen Bishop
Due: Tuesday, May 2, by 4:30 p.m.
Note: The rst three problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the grader.
14.30 PROBLEM SET 3  SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
E (ag1 (X) + bg2 (X) + c) =
Z
1
(ag1 (x) + bg2 (x) + c) fX (x) dx
Z
Z 1
Z 1
=
ag1 (x) fX (x) dx +
bg2 (x) fX (x) dx +
cfX (x) dx
1
Z1
Z11
Z 1
1
= a
g1 (x) fX (x) dx + b
g2 (x) fX
14.30 PROBLEM SET 2
TA: Tonja Bowen Bishop
Due: Tuesday, February 28, 2006 , by 4:30 p.m.
Note: The rst four problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from
14.30 PROBLEM SET 9
TA: Tonja Bowen Bishop
Due: Never! These are all practice problems. Please note that these
problems are mostly theoretical and NOT necessarily representative of what
will be on the exam. In recitation we will cover more practical/stand
14.30 PROBLEM SET 8 SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
Suppose that the sample statistics for a random sample of 10 observations
from a N ; 2 population are the following:
X = 5
S2 = 4
a.
We know that
p X
n
t9
S
Since n is not very large,
14.06 Final
Spring 2004
You have 1h30min. Please answer each question in separate booklets. Each question counts for
a total of 60 points. Allocate your time eciently. If you can not answer some part analytically, but
you can still give the intuition for
14.06 Midterm
Spring 2004
You have 1.5 hours.
Total points: 100.
luck!
You must answer all questions. No books or notes are allowed.
Answer question 1 in either continuous or discrete time. Good
Question 1 (Continuous Time)
Consider the neoclassical growt
14.30 PROBLEM SET 6 SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
e
e
a.
Let X denote the sum of the weight of the 100 sampled coins: X =
100
P
e
Xi . Now, X must be distributed normally, because it is a linear combina
i=1
e
tion of independent norm
Final Exam Answers
Course 14.06 Intermediate Applied Macroeconomics
1. Question #1
(a) The firms problem is as follows:
max = At Kt L1 rt K t wt Lt
t
K ,L
The FOCS can be written as:
rt = At kt 1
wt = (1 ) At kt
Plugging for At , we have:
rt = g1 kt 1
t
w
14.06 Macroeconomics Spring 2003
Final Exam Solutions Part A (True, false or uncertain) 1. Because more capital allows more output to be produced, it is always better for a country to have more capital stock. False. A per capita capital stock above the go
14.30 PROBLEM SET 1
TA: Tonja Bowen Bishop
Due: Tuesday, February 21, 2006 , by 4:30 p.m.
Note: The rst four problems are required, and the remaining two are practice problems. If you choose to do the practice problems now, you will receive feedback from
14.30 PROBLEM SET 5  SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
The joint pdf of X and Y will be equal to the product of the marginal
pdfs, since X and Y are independent.
fX,Y (x, y) = fX (x) fY (y)
1 2
1 2
1
1
= e 2 x e 2 y
2
2
1 1 (x2 +y2 )
e 2
14.30 PROBLEM SET 6
TA: Tonja Bowen Bishop
Due: Thursday, April 13, by 4:30 p.m.
Note: The rst three problems are required, and the fourth is a practice
problem. If you choose to do the practice problems now, you will receive
feedback from the grader. Alt
14.30 PROBLEM SET 9 SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
For parts (a)(g), please refer to the relevant sections in the handouts/textbooks.
(h) False. With the knowledge of the distribution of a statistic under
the null hypothesis, we can c
14.30 PROBLEM SET 4  SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
The distribution of Y jX is the distribution of + X + ", where
" is the only random variable and + X is xed, call it C. Then, the
1
distribution of C + e is that of e , shifted by
14.06 Macroeconomics
Spring 2003
Midterm Exam Solutions
Part A (True, false or uncertain)
1. An economy that increases its saving rate will experience faster growth.
Uncertain. In the Solow model an economy that increases its saving rate will tem
poraril
14.12 Game Theory Final (Answers)
12/21/2007
Prof. Muhamet Yildiz
Instructions. This is an open book exam; you can use any written material. You have two
hour and 50 minutes. Each question is 25 points. Good luck!
1. There are two siblings, who have inher
14.30 PROBLEM SET 2  SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
There are 12 5 dierent ways for the team to accumulate a 732
7
3
7
3
2
record, and the probability of each of these outcomes is 1 1 1 . So
3
2
6
7 1 3
2
55
the probability of th
14.30 PROBLEM SET 7 SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
For the statistic to be unbiased we must have
!
n
X
ci Xi
= E
i=1
= E (Xi )
n
X
ci
i=1
n
X
=
ci
i=1
So the statistic will be unbiased if and only if
n
P
ci = 1.
i=1
b.
Given that al