BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 4
Tony Humm
October 2015 Answers
1.
Find the total differential of:
(a) U = x1/3y2/3
1 2 2
2 1 1
dU = x 3 y 3 dx + x 3 y 3 dy
3
3
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 6
November 2015
Answers
1
1.
(a)
Tony Humm
1
Let U = x14 x22 be a utility function. Find the degree of homogeneity and say what
happens
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 3
Tony Humm
October 2015
1.
Find the first order partial derivatives of the following functions:
(i)
(ii)
z = 3y-2 + 6yx2 + 7x3
Z X = 1
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 2
Tony Humm
October 2015
1.
Identify the stationary points of the following functions. State whether the
stationary points identified a
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 1
Tony Humm
October 2015
1.
Find the ordinary derivatives, (dy/dx) of:
2.
(a) y = -3x-5
(b) y = x4/5
1
dy 4 5
dy
= x
= 15 x 6
dx 5
dx
(
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 3
Tony Humm October 2015
1.
Find the first order partial derivatives of the following functions:
(i)
z = 3y-2 + 6yx2 + 7x3 (ii)
z = 7y
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 6
Tony Humm
November 2015
1
1.
2.
1
(a)
Let U = x14 x22 be a utility function. Find the degree of homogeneity and say what
happens to t
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 5
Tony Humm
November 2015
1.
This wasnt worded as well as it should have been. I intended that students
look for the Present Value of 2
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 4
Tony Humm
October 2015
1.
Find the total differential of:
(a) U = x1/3y2/3
(b) U = x1/3 + y1/3
(c) Q = (KL)1/3
(d) Q = K1/2 + L1/2
(e
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 5
Tony Humm
November 2015
1.
Compare the value of 25,000 due in one year, to the same amount due in 20 years.
Assume that interest is c
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 2
Tony Humm
October 2015
1.
Identify the stationary points of the following functions. State whether the stationary
points identified are local maximum
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 7
Tony Humm
December 2015
1. A competitive firm produces ice cream. Its technology is described by the following
production function:
1
BIRKBECK COLLEGE
DEPARTMENT OF ECONOMICS
GDE/GDF/ESP Problem Set 1
Tony Humm
October 2015
1.
Find the ordinary (first) derivatives, (dy/dx) of:
(a) y = -3x-5
(b) y = x4/5
(c) y = 4x1/6 + 3
(e) y =
3
x
Problem Set 6: Answer key
1) The model is:
yt + xt = u1t
u1t = 0:2u1t
yt + xt = u2t
u2t = u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. The order the integration can be found by noting that u1t is I(1)
Problem set 0: Prior notions
+1
Let fyt gt=
1
be a sequence of scalar bounded random variables.
1)
Let L and 1 be the "lag" and "identity" operators respectively,
dened as follows:
Lyt
Lk yt
L0 yt
L0
Problem Set 8: Answer key
1) We shall begin with computing conditional expectations.
E (st =st
1
= 1)
E (st =st
1
= 0)
= P (st = 1=st
= p
= P (st = 1=st
= 1 q
1
= 1)
1
= 0)
We can combine both in the
Problem Set 7: Answer key
1) Our model is:
yt
"t
=
+ "t
= ut ( 0 +
t=+1
1
fut gt=
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
We shall begin by computing the conditional moments, as the unconditional ones can the
Problem Set 1: Answer Key
1) To obtain the process for yt is very simple:
"t
= yt
=
(yt
=
(1
yt
yt
1
) + !t
) + yt + ! t
+1
So, fyt gt= 1 is also a stationary AR(1) process with the same autore+1
gres
Problem Set 4: Answer key
1) In exercise 3 of Problem Set 3 we had the following process:
yt =
Let
2
0:02
0:03
(L) = I
0:5 0:1
0:4 0:5
+
yt
2
2L ,
1L
1
+
0
0:25
0
0
yt
2
+ ut
where:
1
=
0:5
0:4
0:1
0:
Problem Set 3: Answer Key
1) Lets write the univariate AR(p) process as:
yt = c +
1 yt 1
+ : +
p yt p
+ "t
Like we did in exercise 6 of Problem Set 0, we shall build a system with
the equation that de
Problem Set 7: ARCH processes
1) For the following ARCH(1) model:
yt
"t
t=+1
1
fut gt=
=
+ "t
= ut ( 0 +
i:i:d:
s
2
:5
1 "t 1 )
N (0; 1)
compute the rst through fourth moments for yt both conditional
Problem Set 5: Answer key
1) The claim is not true. A process that is stationary in the strict sense and
that has nite second moments is covariance stationary. But a process
that is strictly stationar
Problem Set 9: Answer Key
A state representation consists of a state equation and an observation
equation. To obtain it, we must bare in mind that the state variable, t , has to
be an unobservable ran
Problem Set 4: VAR Analysis
1) For the bivariate VAR(2) process in exercise 3 of Problem Set 3 compute the
impulse responses s for s= 1,2 as well as the matrix long run multiplier.
2) Provide an examp
Problem Set 8: Markov Chains
For the following Markov Chain process:
+1
fst gt= 1 :
p (st = 1=st 1 = 1) =
p (st = 0=st 1 = 0) =
st 2 S = f0; 1g 8t
p
q
1) Compute the conditional and unconditional expe
Problem Set 1: ARMA processes
1) Consider the following three stochastic processes:
yt
=
+ "t
"t
=
"t
1 + !t
+1
f! t gt= 1
is a White Noise
+1
Show that the autoregressive behaviour of f"t gt= 1 is pa
Problem Set 6: Cointegration
1) Consider the following example:
yt + xt
= u1t
yt + xt
= u2t
where:
u1t
=
0:2u1t
u2t
=
u2t
1
1
+ 0:8u1t
+ 0:5u2t
2
2
+ "1t
+ "2t
i. What is the order of integration of y
Problem Set 3: Introduction to VAR processes
1) Show that the companion form of a scalar AR(p) process is a p-variate
VAR(1) process
2) For a stationary n-variate VAR(p) process:
i. show that its comp
Problem Set 9: Kalman lter
1) Provide the state-space representation for the following (stationary) arma
processes:
a. yt =
+ ut + u t
1
b. yt = c + yt
1
+ ut
c. yt = c + yt
1
+ ut + u t
1
2) For the