problem 1: A firm has the given total revenue (TR) function:

TR = (4Q + 2) e^4Q

Where Q is the Quantity.

Find out the firm’s marginal revenue function?

problem 2: A firm has the given inverse demand function:

P = [300/(Q - 4)] – 3

Where Q is the Quantity and P is the Price.

a) Find out the firm’s marginal revenue function.
b) Find out the level of output where the firm maximizes its total revenue.

problem 3: David has £5000 which he wishes to save for six years. Bank A offers him an interest rate of 4% per annum compounded monthly. Bank B offers him an interest rate of 3.95% per annum compounded constantly.

a) Compute how much David would have in each bank after six years.
b) Which Bank would give David the most interest? How much more interest?

problem 4: Categorize the four stationary points of the function:

z = 2x³ + y³ - 18 x – 12y + 50

according to whether they define a maximum, minimum or the saddle point.

problem 5: A firm’s total revenue (TR) is given by pq, where p is the price and q is the quantity sold.

Suppose the firm is initially selling 1000 units of its product at a price of 100 Euros. Then it increases the price by 10% which results in a reduction of 5% in quantity sold.

a) Use the total differential of the TR function to approximate the change in total revenue.
b) What percentage error results from using the differential compared with the true answer?

problem 6: The inverse demand and supply functions for a product are given as:

P_d = 900 – 0.25Q²_d

P_s = 100 + 10Q_s

Where P is the price, Q is the quantity and the subscripts d and s indicate demand and supply, respectively.

a) Compute the equilibrium price and quantity.

b) By using the definite integral, compute the consumer and producer surpluses at the equilibrium position.

c) Describe your answers to part (a) and (b) on an appropriate diagram.

d) The government introduces a specific tax of 59. What will be the price and quantity at the new equilibrium?

e) Show the new equilibrium price and quantity on your diagram drawn for part (c).

problem 7:

a) A firm manufactures and sells a product that has the following demand function:

Q = 180 - 4P

Where P is the price, Q is the quantity. It too faces the given total cost (TC) function:

TC = 0.033Q² + 15Q + 150

Where Q is the quantity produced.

i) Compute the firm’s profit function.

ii)  By using your profit function find outd in part (i) find the level of output where the firm maximizes its profit.

iii) How much profit will the firm make if it sells the profit maximizing level of output?

b) The firm is considering manufacturing a second product in its factory alongside the first. The demand functions for the two products are:

Q_d1 = 180 – 4P₁
Q_d2 = 90 – 2P₂

Where the subscripts 1 and 2 refer to product 1 and product 2 correspondingly. The firm now faces a total cost function:

TC = Q + 150

Where Q = Q₁ + Q₂

i) Compute the new total profit function for the firm as a function of Q₁ and Q₂.

ii) By using your new profit function find outd in part (b)(i) find the level of output for each product at the profit maximizing point and show your outcome is a maximum.

iii)  Find the price charged for each product and the profit the firm would earn if adopting a profit maximizing strategy.