Question

Explain with rules

Consider a monopolist who encountered a constant average and marginal cost of $5 and a linear demand function given by P-20-2Q, where P is the price the monopolist charges and O is the quantity consumers purchase. To obtain the optimal quantity and price, the monopolist needs to obtain the marginal revenue function, which has the same intercept as the demand function but twice as steep. 1. Obtain the monopolist's MR function, optimal output and price. 2. Without computing the optimal output and just uses the marginal revenue and demand function, we can also determine the range of output which the monopolist will operate and the range that it will never operate. Establish the range of output and price that the monopolist will operate.

Answer 1

1.

Monopolist is the one who sells differentiated goods at elastic portion where MR is greater than 0
1. Here we have demand function, P = 20-2Q
2. To get MR function we need multiply the demand function by Quantity
TR= P*Q=(20-2Q)*Q = 20Q-2Q^2
3. Now differentiating TR function, we get MR function as MR= 20-4Q
MC=5
4. Setting MC=MR
5=20-4Q
4Q=20-5=15
Q=15/4=3.75
5. P= 20-2*3.75= 12.5

2.

Now to find the range of output and prices we need to set MR=0, as a monopolist would operate at elastic portion
So, MR=20-4Q=0
4Q=20
Q=5
For Q=5, P=20-2*5=20-10=10
Hence the monopolist will operate at price between 10 and 20 and Q between 0 and 5

Q	P	TR	MR	MC	TC	Profit
0	20	0	20	5	0	0.00
1	18	18	16	5	5	13.00
2	16	32	12	5	10	22.00
3	14	42	8	5	15	27.00
4	12	48	4	5	20	28.00
5	10	50	0	5	25	25.00
6	8	48	-4	5	30	18.00
7	6	42	-8	5	35	7.00
8	4	32	-12	5	40	-8.00
9	2	18	-16	5	45	-27.00
10	0	0	-20	5	50	-50.00