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ECON7150
Assignment
Instructions:
• All answers to the assignment must be neatly hand written. This can be done on paper or
directly onto a file using a tablet.
• Save your assignment as one pdf document. If it is written on paper, you should scan it
into a single file. If you do not have access to a flatbed scanner you can use a phone app such
as “Adobe Scan” or “Microsoft Office Lens”.
• Submit your assignment on Blackboard through Turnitin.
• Make sure you show all steps, key formulae, and workings clearly. Final solutions should
be simplified as much as possible and either highlighted or circled. Round to the nearest
hundredth if necessary.
• 100 Marks - 30% of overall assessment
1. (15 marks) Suppose that a decision maker has $100. They need to choose how many kilos of
chocolate (c) and vegetables (v) they will purchase. Chocolate costs $2 per kilo and vegetables
cost $1 per kilo. They are choosing c and v to maximise the utility function
U(c, v) =
5
α
cα +
1
α
vα
where α is a constant and 0 < α < 1.
a) (5 marks) Formally write down the decision maker’s optimisation problem and the asso-
1
2ciated Lagrangian.
b) (5 marks) Write down the first order conditions and find the values of c and v that satisfy
those conditions. Your answer should be a function of α.
c) (5 marks) Check the second order conditions for the solution(s) you found in part (b). Do
these values of c and v solve the decision maker’s problem?
2. (16 marks) Define A =
5 w 1
w −1 3
−2 −3 4
, B =
−1 2x y
y − x 4 1
3y − 1 1 5
, C =
1 0 z − 2
3 −1 z
7 3 2
a) (4 marks) For what value(s) of w is the matrix A singular?
b) (4 marks) Treating x and y as constants, find the determinant of the matrix B.
c) (8 marks) Treating z as a constant, find the inverse of the matrix C. [Hint: check your
answer by plugging in simple values for z and making sure that the product of C and its
inverse is the identify matrix.]
3. (12 marks) Consider the function f(x, y) = (y − 3)ey2−4x2 .
a) (6 marks) Find all stationary points of f(x, y).
b) (6 marks) Classify any points you found in part a) as either saddle points, local maxima,
or local minima.
4. (15 marks) A firm produces q1 units of good 1 and q2 units of good 2. The firm is a price
taker and faces prices $100 per unit of good 1 and $50 per unit of good 2. The firm’s cost of
producing q1 and q2 in dollars is
C(q1, q2) = 5q
2
1 + 2q1q2 + 4q
2
2
a) (3 marks) Write down the firm’s profit maximization problem in terms of x and y.
b) (6 marks) Use the first order conditions and find the values of x and y that solve the firm’s
problem. [Note: For this problem, we are not assuming that the quantities need to be a
whole numbers. They can be any positive real number.]
c) (6 marks) Use the second order conditions to show that the solution from part b) is a
maximum.
5. (18 marks) Consider the problem
max − 3x+ 4y
subject to 5x2 + 2y2 = 10
(a) (3 marks) Write down the Lagrangian for this problem.
3(b) (9 marks) Write down the first order conditions for the optimisation problem and find
all values of x and y that satisfy those first order conditions.
(c) (6 marks) Check the second order conditions to classify the solution(s) you found in part
(b). Do these values of x and y solve the problem?