The demand for detergent in Tanzania is characterized by the following functuon P(Q) = 100 - 10Q. Firm A supplies detergent and has a cost function given by TC(Q) = 10Q²

a) assuming that firm A is a monopolist in Tanzania what is the profit maximizing quantity and price?
b) Now assume that firm A can produce for two different countries Tanzania and Kenya. Demand in kenya is given by P(Q) = 50 - 20Q. Assume firm A is a monopolist for both markets, what is the optumal price and quantity for each markets under third degree of price discrination? Show your work clearly

a) To find the profit-maximizing quantity and price for firm A as a monopolist in Tanzania, we need to determine the quantity at which marginal revenue (MR) equals marginal cost (MC).

1. Calculate the total revenue (TR) function:
TR = P(Q) * Q = (100 - 10Q) * Q = 100Q - 10Q²

2. Calculate the marginal revenue (MR) by taking the derivative of the total revenue function with respect to Q:
MR = dTR/dQ = 100 - 20Q

3. Set MR equal to MC to find the profit-maximizing quantity:
MR = MC
100 - 20Q = d(TC)/dQ = d(10Q²)/dQ = 20Q

100 - 20Q = 20Q
100 = 40Q
Q = 2.5

4. Substitute the value of Q into the demand function to find the corresponding price:
P(Q) = 100 - 10Q
P(2.5) = 100 - 10(2.5) = 100 - 25 = 75

Therefore, the profit-maximizing quantity for firm A as a monopolist in Tanzania is 2.5 units, and the corresponding price is 75.

b) To find the optimal price and quantity for each market under third-degree price discrimination, we need to analyze the demand functions for Tanzania and Kenya separately.

For Tanzania:
1. Follow the same steps as in part a to find the profit-maximizing quantity and price for the Tanzanian market.

For Kenya:
1. Calculate the total revenue (TR) function for Kenya:
TR(Kenya) = P(Kenya) * Q = (50 - 20Q) * Q = 50Q - 20Q²

2. Calculate the marginal revenue (MR) for Kenya by taking the derivative of the total revenue function with respect to Q:
MR(Kenya) = dTR(Kenya)/dQ = 50 - 40Q

3. Set MR(Kenya) equal to MC to find the profit-maximizing quantity for Kenya:
MR(Kenya) = MC
50 - 40Q = d(TC)/dQ = d(10Q²)/dQ = 20Q

50 - 40Q = 20Q
50 = 60Q
Q = 50/60 = 5/6

4. Substitute the value of Q into the demand function to find the corresponding price for Kenya:
P(Kenya) = 50 - 20Q
P(Kenya) = 50 - 20(5/6) = 50 - 50/3 = 100/3

Therefore, the profit-maximizing quantity for firm A in Kenya is 5/6 units, and the corresponding price is 100/3.