ECON7150 Assignment 2
All answers to this assignment must be handwritten in a clear manner, scanned as
one pdf document (scanners are available at the Library) and submitted ONLY
ONCE via Turnitin between 18 Oct from 2 pm and 20 Oct 12 by 12 pm. Multiple
submissions are NOT allowed, so make sure before submitting. If you email me
because you cannot resubmit again, it is highly likely that by the time this gets
sorted, it will be a late submission and you will incur a penalty mark.

You can also handwrite on your ipad, scan it and upload as a single pdf document. If
you do not have access to a flatbed scanner you can use a phone app such as
“Adobe Scan" or Microsoft Office Lens".

The first page of the assignment should contain Assignment Number, followed by
your given and family name, and lastly, your student number.

Please make sure you don’t wait until the last minute to upload your assignment as
this takes up time. So be mindful that you need to start uploading a few hours before
the deadline in case things go wrong or you forget a page here and there. Also,
make sure that you wait until the upload is complete and check that your assignment
has successfully uploaded on Turnitin.

Several students in the past claimed that they submitted on Turnitin and were
shocked to receive no marks because their assignment wasn’t actually uploaded. So
this is a warning that it is your responsibility to ensure that you recheck and see that
the assignment is actually there for marking.

The total mark of 54 will be converted to make up 30% of the final exam mark.
The questions are based on material covered in lectures 3 to 10.
Make sure you show all steps, key formulae, and workings clearly.

Question 1 (6 marks)
Classify the stationary points of 2 3 31 12 3( , )
y yf x y x e x ye= − − .

Question 2 (8 marks)
The function g is defined by 3 2 2( , ) 3g x y x x y= + − − on the domain D given by points
in the xy-plane satisfying 2 2 1x y+ ≤ and 0x ≥ .
(a) Find the stationary points of the function g, and classify them.
(b) Find the global extreme points of g in D.

Question 3 (10 marks)
Suppose a consumer who can borrow or lend at an annual rate of interest r ≥ 0
anticipates receiving positive income y1 this year and y2 next year. (The consumer
ignores the future more than one year ahead.) The same consumer chooses the
levels of consumption c1 this year and c2 next year in order to maximize the utility
function
1 2 1 2 1 2( , ) ln ln ( 0, 0)U c c c c c c= + > >

subject to the budget constraint

2 21 11 1
c yc y
r r
+ = +
+ +


(a) Write out the Lagrangian for the constrained maximization problem.
(b) Show that the Lagrangian is concave as a function of (c1, c2).
(c) Write out the first-order conditions for a constrained maximum.
(d) Find the utility maximizing expenditures in both periods, as well as the Lagrange
multiplier λ, all as functions of the parameter triple (r, y1, y2).
(e) Find the utility maximizing expenditures in both periods for the special case
when the interest rate r = 0, and also when r = 10%.

Question 4 (5 marks)
A consumer spends a positive amount m in order to buy x units of one good at the
price of 6 per unit, and y units of a different good at the price of 10 per unit. The
consumer chooses x and y to maximize the utility function U(x, y) = (x + y)(y + 2).
Suppose that 8 ≤ m ≤ 40.
(a) Find the optimal quantities x* and y*, as well as the Lagrange multiplier, all as
functions of m.
(b) Find the maximum utility value as a function of m.


Question 5 (9 marks)
You are not asked to solve for the values for x and y. Just do the computations given
that the matrices have these constants in them.

a) �
2 + 10 −37 −2 + 3 1
−2 −5 4 � + �−5 + −7 −5−8 −9 4−7 −7 − 4 0 � (3 marks)
b) �−3 7 + 2 2 � � 8 2 − 3 − 3 −5 � (6 marks)

Question 6 (6 marks)

A firm’s marginal revenue as a function of the number of units sold, q, is given by

2( ) 12 8 3R q q q′ = + −
Find an expression in terms of q only for the firm’s
(a) total revenue function
(b) average revenue function
(c) inverse demand function (express p in terms of q)
(d) What is the elasticity of demand when q = 3?

Question 7 (10 marks)
(a) Evaluate the following integrals:
(i)
3
21
ln
e x dx
x∫
(ii)
315
20

1
x dx
x +∫

(b) Find the integral
2( )
n mx xI dx
x
−
= ∫ , where m and n are natural numbers.


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