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?
c)Now assume that firm A is only active in Tanzania where it competes with firm B. Firm B cost function is CB(QB) = 40Q. Suppose that both compete in quantities . What is the equilibriam quantities and price? Show your work clearly step by step in each part.

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

Given:
Demand function: P(Q) = 100 - 10Q
Cost function: TC(Q) = 10Q^2

To find MR, we need to take the derivative of the demand function with respect to Q:
MR = d(P(Q))/dQ

MR = d(100 - 10Q)/dQ
MR = -10

To find MC, we need to take the derivative of the cost function with respect to Q:
MC = d(TC(Q))/dQ

MC = d(10Q^2)/dQ
MC = 20Q

Now, equate MR and MC to find the profit-maximizing quantity:
-10 = 20Q
Q = -10/20
Q = -0.5

Since quantity cannot be negative, we ignore the negative solution. Therefore, the profit-maximizing quantity is Q = 0.5.

To find the price, substitute the quantity (Q) into the demand function:
P(Q) = 100 - 10*(0.5)
P(Q) = 100 - 5
P(Q) = 95

The profit-maximizing quantity is 0.5, and the price is 95.

b) Under third-degree price discrimination, Firm A can charge different prices for the two markets (Tanzania and Kenya). To find the optimal price and quantity for each market, we need to maximize profit separately for each market by setting MR equal to MC.

For the Tanzania market:
Demand function: P(Q) = 100 - 10Q
Cost function: TC(Q) = 10Q^2

MR = -10 (same as before)
MC = 20Q (same as before)

Set MR equal to MC:
-10 = 20Q
Q_Tanzania = -10/20
Q_Tanzania = -0.5 (ignore the negative solution)
Q_Tanzania = 0.5

Substitute the quantity back into the demand function to find the price:
P_Tanzania = 100 - 10*(0.5)
P_Tanzania = 100 - 5
P_Tanzania = 95

For the Kenya market:
Demand function: P(Q) = 50 - 20Q
Cost function: TC(Q) = 10Q^2 (same as before)

MR = -20 (different from before)
MC = 20Q (same as before)

Set MR equal to MC:
-20 = 20Q
Q_Kenya = -20/20
Q_Kenya = -1 (ignore the negative solution)
Q_Kenya = 1

Substitute the quantity back into the demand function to find the price:
P_Kenya = 50 - 20*1
P_Kenya = 50 - 20
P_Kenya = 30

So, the optimal quantity and price for the Tanzania market are Q_Tanzania = 0.5 and P_Tanzania = 95, while for the Kenya market, they are Q_Kenya = 1 and P_Kenya = 30.

c) Now, suppose Firm A competes with Firm B in the Tanzanian market, and both firms compete in quantities. Firm B's cost function is CB(QB) = 40Q.

To find the equilibrium quantities and price, we need to set the total quantity supplied (Q_total) equal to the total quantity demanded (Q_demand).

Q_total = Q_A + Q_B

Where:
Q_A is the quantity supplied by Firm A
Q_B is the quantity supplied by Firm B

Q_demand is the quantity demanded in the market given by the demand function.

Demand function: P(Q) = 100 - 10Q (same as before)

First, let's find the equilibrium quantity where Q_A + Q_B = Q_demand:
Q_A + Q_B = 100 - 10Q
Q_A + Q_B = 100 - 10(Q_A + Q_B)
Q_A + Q_B = 100 - 10Q_A - 10Q_B
11Q_A + 11Q_B = 100
Q_A + Q_B = 100/11

Now, let's find the equilibrium price by substituting the equilibrium quantity back into the demand function:
P = 100 - 10(100/11)

Simplifying, we get:
P = 100 - 1000/11
P = (1100 - 1000)/11
P = 100/11

So, the equilibrium quantities are Q_A = Q_B = 100/11, and the equilibrium price is 100/11.