Financial Maths, Consider a consumer who faces the following constrained optimisation problem

Feb 15, 2024

Financial Maths, Consider a consumer who faces the following constrained optimisation problem

Financial Maths

1
BUSN2060 22S1 ASSIGNMENT 3
DUE DATE: 20 MAY 2022 (4PM AWST/6PM AEST)
WORTH 10% OF YOUR MARK FOR THE COURSE
INSTRUCTIONS
Answer all questions. Be sure to show your working. Your answers may be typed or hand written, but they must be uploaded digitally via the link provided on Blackboard.
Please clearly show your workings for each question.
The assignment will be marked out of total of 50 marks.
QUESTION ONE (20 MARKS)
Consider a consumer who faces the following constrained optimisation problem:

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,
( , ) =
s.t. 10 + 4 = 100,
where is the total utility of the consumer, is the quantity consumed of good x, and
is the quantity consumed of good y.
a. State the objective function for this constrained maximisation problem. [1
mark]
b. State the constraint for this constrained maximisation problem. [1 mark]
c. Rewrite this constrained optimisation problem in terms of a Lagrangian
function. [2 marks]
d. Solve this constrained optimisation problem using the Lagrange method. [6
marks]
e. Express the dual optimisation problem for this consumer. [Hint: What is the
objective function and the constraint for the dual of this optimisation
problem?] [4 marks]
f. Solve the dual of the constrained optimisation problem expressed in part e
using the Lagrange method. [6 marks]
2
QUESTION TWO (20 MARKS)
Consider a perfectly competitive market. The demand relationship in this market is
represented as = 25 −
2
and the supply relationship is represented as = 1 + 2
.
a. Find the equilibrium output, Qe, and equilibrium price, Pe. [4 marks]
b. Use integration to determine the consumer surplus at equilibrium. [5 marks]
c. Use integration to determine the producer surplus at equilibrium. [5 marks]
d. Use integration to determine the dead weight loss when a quota is set at =

  1. [6 marks]
    QUESTION THREE (10 MARKS)
    Consider a profit-maximising monopolist with a demand function = 274 −
    2
    and
    marginal cost function = 4 + 3 .
    a. Find the level of output, Q, and price, P, that maximises profit for this
    monopolist. [4 marks]
    b. Use integration to determine the consumer surplus if the monopolist
    maximises profit. [6 marks]
    [ENDS]

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