Department of Economics
Trent University
ECON3250H – Mathematical Economics
Fall Semester, 2022
Pete
orough Campus
Assignment #2
Due in Class October 31, 2022
General Information and Instructions:
Worth 10% of the course grade, this assignment has 5 questions for a total of 50 marks. Marks for
each question are as indicated and are evenly divided among the parts of the question. Answers to
the questions must be word-processed or hand-written and they must be legible, orderly and
concise; written explanations, where requested, should have no more than a very
ief paragraph of
content. While students may work together on the assignment, they must develop, write up and
submit answers independently. On the due date noted above, answers to the assignment must be
submitted in a single hard copy package at the beginning of class.
Questions:
1. (18 marks) Consider general models and implicit functions:
a. State the three conditions of the Implicit Function Theorem for ? functions of the
form ??(?1, … , ??; ?1, … , ??) where ?? is endogenous for all ? ∈ {1,2,… , ?} and ?? is
exogenous for all ? ∈ {1,2,… ,?}. That is, for a point (?10, … , ??0; ?10, … , ??0), state
the three conditions by which there exists a neighbourhood around this point for
which ?? = ?
?(?1, … , ??) is an implicitly defined and continuously differentiable
function, and where ??(?1, … , ??; ?1, … , ??) = 0 is an identity, for all ? ∈ {1,2,… , ?}.
. Assuming the conditions refe
ed to in Part (a) hold, take the total differential of
??(?1, … , ??; ?1, … , ??) = 0 for each ? ∈ {1,2,… , ?} to derive the Implicit Function
Derivative Rule
???
???
=
|??|
|?|
for all ? ∈ {1,2,… , ?} where ? is the Jacobian matrix
? =
[
??1
??1
⋯
??1
???
⋮ ⋱ ⋮
???
??1
⋯
???
??? ]
and ?? is the matrix ? wherein the ?
?ℎ column is replaced with the negative of the
gradient vector [
??1
???
, … ,
???
???
]
?
.
c. Apply the concepts involved in Parts (a) and (b) to the basic national income model
? = ? + ?0 + ?0
? = ? + ?(? − ?)
? = ? + ??
to verify the comparative static
??
??0
=
?(1 − ?)
1 − ?(1 − ?)
0
where ?, ? ∈ (0,1), ?, ? > 0 and ?0 and ?0 are exogenous. You may assume without
verification that the conditions of the Implicit Function Theorem are satisfied for the
equili
ium point of the model.
2. (12 marks) Find the critical and inflection points of ?(?) = 2 sin3 ? + 3 sin? for ? ∈ [0, ?],
making sure to classify the critical points.
3. (4 marks) Apply Taylor’s Theorem to prove that (i) sin ? = ? −
?3
3!
+
?5
5!
−
?7
7!
+
?9
9!
−
?11
11!
+ ⋯
and (ii) ln(? + 1) = ? −
?2
2
+
?3
3
−
?4
4
+
?5
5
−
?6
6
+ ⋯.
4. (4 marks) Let the exponential ?(?) be the inverse of the logarithm ?(?). Use the properties
of logarithms to prove that (i) ?(? + ?) = ?(?)?(?) and (ii) ?′(?) = ?(?)/?′(1).
5. (12 marks) Suppose a developer owns a parcel of land worth ?(?) = ?0?
2√? (measured in
dollars) when developed at its highest and best use at time ? ≥ 0 (measured in years from
today), where ?0 > 0 is the value of the parcel today and ? > 1 is a parameter. Assume the
only cost to the developer of owning the parcel is foregone interest, at an annual rate of ?
0, on the capital tied up in the parcel. Letting ?(?) be the present value of the parcel, the
developer seeks to maximize ?(?) by choosing the time at which the parcel is developed at
its highest and best use.
a. Derive the optimal time ?∗ at which to develop the parcel at its highest and best use,
and then check the second-order condition to confirm that ?∗ indeed provides for a
elative maximum.
. Use the result of Part (a) to evaluate ?∗ for the case of ? = √?
5
and ? = 5%, where ?
is Euler’s number.