Department of Economics
Trent University
ECON3250H – Mathematical Economics
Fall Semester, 2022
Pete
orough Campus
Assignment #3
Due in Class November 21, 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. (2 marks) Define definiteness of a quadratic form ?.
2. (24 marks) Consider unconstrained optimization of multi-variable functions:
a. State Young’s Theorem as it relates to ?(?1, … , ??).
. Determine the definiteness of the quadratic form ? = 3?2 + 4?? − 2?? − ?2.
c. Define concavity and convexity as they relate to ?(?1, … , ??).
d. State the Absoluteness and Uniqueness of Extrema Theorem as it relates to
?(?1, … , ??).
e. Define convexity as it relates to a set ?. Illustrate the set {(?, ?): ?2 ≤ ? ≤ √?} in the
?? plane and prove that it is convex.
f. Find and classify the extrema of ?(?, ?, ?) = ?2 + ?(? − 2) + 3(? − 1)2 + ?2.
3. (8 marks) Consider the optimization of ? = ?(?) subject to the constraints ??(?) = ?? for
? ∈ {1,… ,?}, where ? = (?1, … , ??) are choice variables, the ?? are parameters and ? > ?.
Let the Lagrangian be ℒ(?), the Lagrange multiplier for constraint ? be ??, the bordered
Hessian be �̅�, the optimal choice variables be ?∗ = (?1
∗, … , ??
∗) and the optimized objective
function be ?∗ = ?(?∗).
a. State the Lagrangain form of the optimization problem and its first-order and
second-order conditions for a maximum as well as for a minimum.
. Consider the case of ? = 2 and ? = 1, whereby the objective function is ? = ?(?, ?),
the sole constraint is ?(?, ?) = ?, the Lagrangian is ℒ(?, ?), the sole Lagrangian
multiplier is ?, the optimal choice variables are ?∗ = ?∗(?) and ?∗ = ?∗(?), the
optimal Lagrangian multiplier is ?∗ = ?∗(?) and the optimized Lagrangian is ℒ∗ =
ℒ∗(?∗, ?∗). Prove that
?ℒ∗
??
= ?∗ and very
iefly interpret this result.
4. (8 marks) A competitive firm uses labour ? and capital ? to produce output according to the
production function ?(?, ?) = ???? where ? > 0, ? > 0 and 0 < ? + ? < 1. The prices for
output, labour and capital are ?, ? and ?, respectively.
a. Derive the factor demand functions ?∗(?, ?, ?) and ?∗(?, ?, ?), where dependence
on the parameters ? and ? is suppressed for simplicity of notation.
. Use the Hessian of the optimization problem to verify that the above-noted factor
demand functions indeed maximize profit.
5. (8 marks) A consumer lives for two periods, 1 and 2, for which consumption is denoted as ?
and ?, respectively in units of expenditure (i.e. consumption is measured in the same units
as expenditure and income). The consumer seeks to maximize his or her utility, which is
given by ?(?, ?) = ? ln ? + ln? where ? > 0 is a parameter. The consumer has exogenous
income of ? > 0 in period 1, has no exogenous income in period 2 and is able to bo
ow and
lend income across the two periods at the interest (i.e. discount) rate ? > 0.
a. Derive the demand functions ?∗(?, ?, ?) and ?∗(?, ?, ?).
. Use the bordered Hessian of the optimization problem to verify that the above-
noted demand functions indeed maximize utility.