Question

3. The inverse market demand for mineral water is P = 200-10Q, where Q is the total market output and P is the market price. Two firms, A and B, have complete control over the supply of mineral water and both have zero costs. a. Operating independently, how would each firm determine the quantity to be produced? Will this quantity maximize the profits of both firms?

Answer 1

Answer: The inverse demand function is the same as the average revenue function, since P = AR. We have two firms in the question A and B, whose cost is equal 0. The inverse market demand for mineral water is P = 200-10Q.

a. Since demand function for two is same and also the cost is zero, therefore to find the quantity of each . First, multiply each side of the inverse demand function by Q. This gives us the TR function

i.e. P=200-10Q

PQ = 200Q - 10Q²

Next, take the derivative with respect to Q to get the MR function:

dTR/dQ = MR

i.e. dTR/dQ = 200 - 20Q

(or you can use the rule that for any linear demand curve P = a – bQ the marginal revenue curve is MR = a –2bQ. )

TR is maximized when MR equals zero. Therefore, set the MR function equal to zero and solve for Q:

i.e. MR = 200-20Q =0 => 200 = 20Q => Q= 200/20 = 10

therfore q for each of them is 10 units.

b.) To check whether this quantity will maximize the profits of both firms or not, set MR equal to MC and solve for Q: 200-20Q = 0 which gives us Q = 10. To find price, plug Q=10 into the inverse demand function and solve for P.

P = 200 - 10Q = 200-10*10 = 100. Profit= (P)(Q)=(100 * 10)= $1,000.

Also the second derivative should be less than 0

i.e. dTR²/ dQ² = -20 <0 hence this quantity maximizes the profits of two firms.