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
1/3
1/3
(b) U = x + y
1 23
1 23
dU = x dx + y dy
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 to the utility of this consumer as you multiply the co
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 = 12 yx + 21x 2
Z Y = 6 y 3 + 6 x 2
z = 7y + 5yx2 - 9x3
Z
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 are local maximum or minimum points or points
of inflect
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
(c) y = 4x1/6 + 3
(d) y = axb+1 + ax
5
dy 4 6
dy
= x
= a
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 + 5yx2 - 9x3
(iii)
Qa = Pa-2.5 Pb-0.2 Pc0.5
2.
Find U /
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 the utility of this consumer as you multiply the consump
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 25,000 , this sum to be received in one year and in
twen
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) What is the slope (gradient) of the indifference curv
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 compounded continually and the interest rate is 8%.
2.
S
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 or minimum points or points of inflection. Are
there an
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
2
1
2
Q=K +L
where Q is output, K is capital and L is
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
2.
(d) y = axb+1 + ax
(f) y =
2
3
3
x
2
3
If a firm's
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):
1
0:2z
P (z)
=
1
0:2z
0:8z 2
P (1)
=
1
0:2
0:8 = 0
0:
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
=
=
=
=
yt 1
L(Lk
yt
1
1
)yt = yt
k
Show that L is a li
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 following formula:
E (st =st
1)
p
=
1
st
st
q
1
1
=1
=0
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 then be obtained by way of the Law of Iterated Expectation
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
gressive coe cient as f"t gt= 1 . However, as the "mutatis
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
0:5
2
=
0
0:25
0
0
Consider the VMA(1) representation (wh
Problem Set 2: Box-Jenkins methodology
1) For an AR(1) process we have:
(0)
=
2
"
=
(0)
2
"
1
2
1
2
Hence,
2
p lim R2 =
For a MA(1) process,
(0)
2
"
(0)
=
2
(1 +
1
1+
=
)
2
"
2
Therefore,
p lim R2
=
1
1+
1
2
2
=
1+
2
We can see that a consistent estimator
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 describes the evolution of yt and the following p 1 ident
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 on yt
unconditional.
1
and
2) For the following model:
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 stationary can be non covariance stationary. Lets construct a pr
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 random variable. In addition, letting vt and ! t represent
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 example to show that Granger causality is not transitive.
3)
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 expectation and variance of st .
2) Compute the expected nu
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 passed (mutatis
+1
+1
mutandis) on to fyt gt= 1 ; i.e. fy
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 yt and xt ?
ii. Under which conditions are yt and xt coi
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 companion form is an np-variate VAR(1) process.
3) Consider
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 series rf in mpeppext.w and the series cons in weldat.w