Question

A,B,C refer to the following question
A monopolist is producing iron ore. The quantity Q is measured in tonnes of iron ore per day and can take only non-negative values. Given the current technology, the maximum level of production is Q=180. Suppose that the demand curve facing this monopolist is
Q=200-1/2×P
where P denotes the price of iron ore measured in dollars per tonne, and the total cost of producing iron ore is described by the function:
TC=0.2×Q^2+4×Q+400
The variable TC is measure in dollars per day.
a.
To maximize profits the monopolist will charge a price of
   220 $/tonne
   210 $/tonne
   200 $/tonne
   190 $/tonne
   180 $/tonne
b.
The marginal cost curve cuts the marginal revenue curve when the output level is
   50 tonnes/day
   60 tonnes/day
   70 tonnes/day
   80 tonnes/day
   90 tonnes/day
c.
Suppose that the monopolist takes advantage of the corona virus crisis and makes the government incur all the variable costs of producing iron ore, that is, the total cost function now is TC=400 but the monopolist gets all the money for selling iron ore. How much money the monopolist would be making when total profit is maximized?

   20,000 $/day   
   19,600 $/day
   17,420 $/day   
   17,400 $/day   
   17,300 $/day

Answer 1

a. 220 $/tonne
(Q=200-1/2×P = 200-0.5P
So, 0.5P = 200 - Q
So, P = (200/0.5) - (Q/0.5)
So, P = 400 - 2Q
TR = P*Q = (400 - 2Q)Q = 400Q - 2Q²
MR = d(TR)/dQ = 400 - 2(2Q) = 400 - 4Q
MC = d(TC)/dQ = 2(0.2Q) + 4 = 0.4Q + 4
Monopolist maximizes profit where MR = MC. So,
400 - 4Q = 0.4Q + 4
So, 4Q + 0.4Q = 4.4Q = 400 - 4 = 396
So, Q = 396/4.4 = 90
P = 400 - 2Q = 400 - 2(90) = 400 - 180 = 220)

b. 90 tonnes/day

c. 19,600 $/day
(Profit is maximized when MR = 0
So, 400 - 4Q = 0
So, 4Q = 400
So, Q = 400/4 = 100
P = 400 - 2Q = 400 - 2(100) = 400 - 200 = 200
TR = P*Q = 200*100 = 20,000
Profit = TR - TC = 20,000 - 400 = 19,600)