A two product monopolist faces the demand and cost functions as below:
Q1=40-2(P1)-(P2) Q2=35-(P1)-(P2) C=(Q1)^2+2(Q2)^2+10
a) Find the profit maximizing levels of output and the price charged for each product.
no answer...
To find the profit-maximizing levels of output and the price charged for each product, we need to determine the first-order conditions of the monopolist's profit maximization problem.
The monopolist's profit can be expressed as:
Profit = Total Revenue - Total Cost
Total Revenue is the product of the price of each product and the quantity sold:
Total Revenue = P1 * Q1 + P2 * Q2
Total Cost is given by the cost function:
Total Cost = C = Q1^2 + 2 * Q2^2 + 10
To maximize profit, the monopolist needs to set the level of output (Q1 and Q2) and the prices (P1 and P2) such that the derivative of profit with respect to each variable is equal to zero.
Taking the first derivative of profit with respect to Q1, we get:
d(P1 * Q1 + P2 * Q2 - Q1^2 - 2 * Q2^2 - 10)/dQ1 = P1 - 2Q1 = 0
Solving this equation, we find:
P1 = 2Q1
Similarly, taking the first derivative of profit with respect to Q2, we get:
d(P1 * Q1 + P2 * Q2 - Q1^2 - 2 * Q2^2 - 10)/dQ2 = P2 - 4Q2 = 0
Solving this equation, we find:
P2 = 4Q2
To find the profit-maximizing levels of output, we substitute the expressions for P1 and P2 into the demand functions:
Q1 = 40 - 2(P1) - (P2)
=> Q1 = 40 - 2(2Q1) - 4Q2
=> Q1 = 40 - 4Q1 - 4Q2
=> 5Q1 + 4Q2 = 40 (Equation 1)
Q2 = 35 - (P1) - (P2)
=> Q2 = 35 - (2Q1) - 4Q2
=> 5Q1 + 5Q2 = 35 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns (Q1 and Q2). We can solve this system of equations to find the values of Q1 and Q2.
By solving these equations simultaneously, we find:
Q1 = 4
Q2 = 3
Substituting these values of Q1 and Q2 into the expressions for P1 and P2, we get:
P1 = 2Q1 = 2(4) = 8
P2 = 4Q2 = 4(3) = 12
Therefore, the profit-maximizing levels of output are Q1 = 4 and Q2 = 3, and the prices charged for each product are P1 = 8 and P2 = 12.