Optimal Extraction of Exhaustible Resources
Chs. 17 & 18 in Neher
Optimal Extraction Rates
b = stock of a nonrenewable resource
Owner problem is
s
M AX
<x>
Z
T
(U (x)
C(x; b)e
rt
dt
0
S:T:
x
b =
0
b
x
0
b(0) = b0 b(T ) = bT
given
A second stage of the opt
Monopoly Market Model
We will suppose now that a monopoly controls the extraction of the deposit of the
nonrenewable resource. For the monopolist total revenue is Rq qU q. The
monopolists optimal extraction plan then satisfies
R q C q
mt
mt
A m for t 0,
Dynamic Fishery Economics
CH. 10. A SCHOOLING FISHERY
It is assumed that production function has the form
x = F (a) so total cost = wa = C(x) where C 0 (x) > 0; C 00 (x) > 0
The optimization problem is
M AX
<x>
Z
T
[px
C(x)]e
rt
dt
0
S:T:b = G(b)
x
x
0
b(
THE MINE
We will rst try to characterize how a nonrenewable resource might be exploited under the conditions of a competitive market. These conditions are
(i) Producers should be wealth maximizers
(ii) Producers make their extraction decisions assuming th
NOTES:
Grades will be recorded by question, so that if you all blew a question I can throw it out.
Typically 2 points per section of a question.
For numerical calculation questions, if you write down the correct calculation OR the
correct answer, you g
1
Let
Getting an estimate of V (b)
Y = X + ";
E [X "] = 0K ;
2
E ["0 ] =
IN ;
0
and let X be full rank with a rank of K.
We know that
V ( b ) = 2 (X 0 X)
1
;
2
and this is ne if we know , but if we don where do we get an estimate of
t
it, and what does th
Limited Dependent Variables
1.
2.
What if the left-hand side variable is not a continuous thing spread from minus infinity to
plus infinity? That is, given a model Y = f ( X , , ) , where
a.
Y is bounded below at zero, such as wages or height;
b.
Or, Y is
ECON 836: Lecture 1
1
Introduction
1. Suppose there is a relationship which holds for each observation (say each
year)
Y = f (X; ; ")
where Y is an N -vector of the dependent variable, X is an NxK matrix of
independent variables, is a vector of parameters
1
E cient OLS
1. Consider the model
Y = X +"
E [X 0 "] = 0K
E ["0 ] =
= 2 IN :
X is full-rank and contains 1N . This is OLS happyland! OLS is BLUE
here.
2. So, you get an estimated parameter vector
1
^OLS = (X 0 X)
3. Its bias is
h
E ^OLS
i
h
= E (X 0 X)
Lecture 5: Systems of Equations
1.
(SUR) So, youve got a model with more than one equation: eg, consumer demand is
about modelling how all expenditure shares vary with prices, expenditure and
demographic characteristics.
a.
Yi j = X i j j + i j , j = 1,.,
1
Endogeneity
1. Formally, the problem is that, in a model
Y = g(X; ) + ";
the disturbances are endogenous, or equivalently, correlated with the regressors, as in
E[X 0 "] 6= 0
In the Venn Diagram (Ballentine) on page 167 of Kennedy, we get a picture
of t
1
Panels
Panel data are no dierent from regular data except that they have an extra
subscript. That is each panel datum has an i subscript (i = 1; :; N ) which
indexes the unit number and a t subscript (t = 1; :; T ) which indexes the time
period.
1. In f
Lecture 7: Some GLS Approaches to Time-Series
1.
Time-series analysis is about dealing with a subscript that has additional meaning than
just name of the observation.
a.
Consider Yt = X + t . Here, the subscript is t to indicate that we are thinking
in te
Econ 836 Midterm Exam
1.
[14 points] Consider the following code and output from a log-wage regression using 2006 Census data
on male residents of Toronto. The first line sets the line delimiter to ";".
.
use "C:\DATA\2006 Census\pumf2006.dta", clear;
.g
ECON 836 Midterm Spring 2010
1.
[4 points] Suppose you have a panel of countries as in Islam's growth model. Assume that the true
2
2
model is Yit = X it + i + it , where E [ X i ' it ] = E [i it ] = 0 and E (i ) = 2 , E ( it ) = 2 .
a. Suppose you run re
Econ 836 Final Exam
1) [4 points] Let
Y = X + ,
X = w + u,
2
w ~ N (0, w I N ),
2
u ~ N (0, u I N ),
2
~ N (u , I N ),
O
where X is a just one column. Let denote the OLS estimator, and define residuals e as
O
e Y X.
Suppose finally that there exists a va