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?
To find the profit-maximizing quantity and price for Firm A as a monopolist in Tanzania, we need to understand the relationship between price, quantity, and profit. Let's start with part (a).
a) Profit is maximized when marginal revenue (MR) equals marginal cost (MC).
First, let's find the marginal revenue (MR) function. The demand function given is P(Q) = 100 - 10Q. The marginal revenue is the derivative of the total revenue function, which is the product of price (P) and quantity (Q):
TR(Q) = P(Q) × Q = (100 - 10Q) × Q = 100Q - 10Q²
Taking the derivative of TR(Q) with respect to Q, we get the marginal revenue (MR):
MR = d(TR)/d(Q) = 100 - 20Q
Next, let's find the marginal cost (MC) function. The cost function given is TC(Q) = 10Q². Taking the derivative of TC(Q) with respect to Q, we get the marginal cost (MC):
MC = d(TC)/d(Q) = 20Q
To find the profit-maximizing quantity and price, we set MR equal to MC:
MR = MC
100 - 20Q = 20Q
Simplifying the equation, we get:
100 = 40Q
Q = 100/40
Q = 2.5
Now, substitute the value of Q back into the demand function to find the price (P):
P(Q) = 100 - 10Q
P(2.5) = 100 - 10(2.5)
P(2.5) = 100 - 25
P(2.5) = 75
Therefore, the profit-maximizing quantity (Q) for Firm A as a monopolist in Tanzania is 2.5 units, and the corresponding price (P) is 75.
b) Moving on to part (b), under third-degree price discrimination, Firm A can charge different prices for each market (Tanzania and Kenya). We need to find the optimal price and quantity for each market.
In Tanzania, the demand function remains the same: P(Q) = 100 - 10Q.
Let's find the optimal quantity and price for Tanzania using the same steps as in part (a):
MR = MC
100 - 20Q = 20Q
Simplifying the equation, we get:
100 = 40Q
Q = 100/40
Q = 2.5
Now substitute the value of Q back into the demand function to find the price (P):
P(Q) = 100 - 10Q
P(2.5) = 100 - 10(2.5)
P(2.5) = 100 - 25
P(2.5) = 75
Therefore, for Tanzania, the optimal quantity (Q) is 2.5 units, and the corresponding price (P) is 75.
In Kenya, the demand function is given as P(Q) = 50 - 20Q. Following the same steps, we can find the optimal quantity and price for Kenya:
MR = MC
50 - 20Q = 20Q
Simplifying the equation, we get:
50 = 40Q
Q = 50/40
Q = 1.25
Now substitute the value of Q back into the demand function to find the price (P):
P(Q) = 50 - 20Q
P(1.25) = 50 - 20(1.25)
P(1.25) = 50 - 25
P(1.25) = 25
Therefore, for Kenya, the optimal quantity (Q) is 1.25 units, and the corresponding price (P) is 25.
In summary, under third-degree price discrimination, the optimal price and quantity for Firm A in Tanzania are 75 and 2.5 units, respectively. In Kenya, the optimal price and quantity are 25 and 1.25 units, respectively.