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Assignment 2

Due June 3 2022 at 4 PM

Instructions:

? All answers to the assignment must be neatly hand written.

? Scan your assignment and save it as one 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”.

? 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. (18 marks) Suppose that there are two firms that are both producing the same good. Their

production levels are q1 and q2 respectively. If they produce at these levels, the price that

they will be able to charge is 100? 6(q1 + q2). Each firm can produce at a marginal cost of 1

(their total costs are equal to the quantity that they produce).

a) (3 marks) Write down the profit maximization problem for firm 1, assuming that they take

the production levels of firm 2 as a given. In other words, write down firm 1’s problem as

if it were treating q2 as a constant.

b) (6 marks) Find the level of production q1 which solves the problem in part a) in terms of

q2. Be sure to show that this level of production satisfies both the first order and second

1

2order conditions. This is a single variable optimisation problem.

c) (6 marks) Complete the same exercise for firm 2, solving for the optimal level q2 that

maximises profits for firm 2 while treating q1 as a constant. This is a single variable

optimisation problem.

d) (3 marks) Find the levels of q1 and q2 which satisfy the conditions you found in parts b)

and c).

2. (16 marks) Define A =

?????

3 5 8

2 w ?1

4 0 4

?????, B =

?????

10 ?10 x

2 ?6y 10

5 ?4 5

?????, and C =

?????

5 5 ?5

?4 2 5z

1 ?2 5

?????.

a) (4 marks) For what value 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.

3. (12 marks) Consider the function f(x1, x2) = x1e

?x1(x22 ? 9x2).

a) (6 marks) Find all stationary points of f(x1, x2).

b) (6 marks) Classify any points you found in part a).

4. (15 marks) A firm produces x units of good X and y units of good Y . Good X sells for $300

per unit and good Y sells for $400 per unit. The firm’s cost of producing x and y, in dollars,

is

C(x, y) = 10x2 + 2xy + 8y2

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.

c) (6 marks) Use the second order conditions to show that the solution from part b) is a

maximum.

5. (15 marks) Suppose that a decision maker is choosing how much of goods 1 and 2 to consume.

The quantities of each good are denoted by x1 and x2 respectively. Good 1 has a unit cost of

3 and good 5 has a unit cost of 2. The consumer has a total budget of 7. The overall utility

function of the decision maker is given by 2

√

x1 + 3x2. Thus, the decision maker’s problem is

max

x1,x2

2

√

x1 + 3x2

subject to 3x1 + 2x2 = 7

(a) (3 marks) Write down the Lagrangian for this problem.

3(b) (6 marks) Write down the first order conditions for the decision maker’s problem and

find the values of x1 and x2 which satisfy those first order conditions.

(c) (6 marks) Check the second order conditions for the solution you found in part (b). Do

these values of x1 and x2 solve the decision maker’s problem?

6. (15 marks) Suppose that in a market inverse supply is given by PS(Q) = 5Q2 + 2Q+ 30 and

inverse demand is given by PD(Q) = 1713? 12Q.

a) (3 marks) Find the equilibrium price and quantity supplied.

b) (6 marks) Compute consumer surplus.

c) (6 marks) Compute producer surplus.

7. (9 marks) Compute the following

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