5. A drug company has a monopoly on a new patented medicine. The product can be made in either of two plants. The costs of production for the two plants are MC1 = 20 + 2Q1, and MC2 = 10 + 5Q2. The firm’s estimate of the demand for the product is P = 20 - 3(Q1 + Q2). How much should the firm plan to produce in each plant? At what price should it plan to sell the product?
q1=48
q2=90
p=500
To find out how much the firm should plan to produce in each plant and at what price it should plan to sell the product, we need to determine the optimal production level for each plant.
Step 1: Set the marginal cost (MC) equal to the marginal revenue (MR) to find the profit-maximizing level of output for each plant.
For Plant 1:
MC1 = MR1
20 + 2Q1 = ∂(20 - 3(Q1 + Q2))/∂Q1
20 + 2Q1 = -3
2Q1 = -23
Q1 = -11.5
For Plant 2:
MC2 = MR2
10 + 5Q2 = ∂(20 - 3(Q1 + Q2))/∂Q2
10 + 5Q2 = -3
5Q2 = -13
Q2 = -2.6
Step 2: Since quantities cannot be negative in this context, we need to constrain the production levels to be non-negative.
Q1 ≥ 0 and Q2 ≥ 0
Step 3: Check if the constrained solution satisfies the MC = MR condition.
For Plant 1:
MC1 = 20 + 2Q1 = 20 + 2(0) = 20
MR1 = ∂(20 - 3(Q1 + Q2))/∂Q1 = -3
20 ≠ -3 (Not satisfied)
For Plant 2:
MC2 = 10 + 5Q2 = 10 + 5(0) = 10
MR2 = ∂(20 - 3(Q1 + Q2))/∂Q2 = -3
10 ≠ -3 (Not satisfied)
Step 4: Since the MC = MR condition is not satisfied, we need to reconsider the production levels.
We cannot satisfy the MC = MR condition with the given cost and demand functions. This implies that there is no profit-maximizing level of output for each plant.
Therefore, we cannot determine the exact quantities and price the firm should plan to produce and sell the product with the given information.
To determine how much the firm should plan to produce in each plant and at what price it should plan to sell the product, we need to find the quantity and price that maximize the firm's profit.
Step 1: Find the total cost functions for each plant.
The total cost for each plant can be obtained by integrating their respective marginal cost functions.
TC1 = ∫(20 + 2Q1)dQ1 = 20Q1 + Q1^2
TC2 = ∫(10 + 5Q2)dQ2 = 10Q2 + (5/2)Q2^2
Step 2: Express the firm's profit function.
The profit function, Π, is given by the formula:
Π = TR - TC
To find the total revenue function, TR, we need to multiply the price (P) by the total quantity produced (Q1 + Q2):
TR = P * (Q1 + Q2)
Plugging in the given demand function, P = 20 - 3(Q1 + Q2), we have:
TR = (20 - 3(Q1 + Q2))(Q1 + Q2)
TR = 20Q1 + 20Q2 - 3Q1^2 - 3Q1Q2 - 3Q2^2
Step 3: Maximize profit.
We need to find the values of Q1 and Q2 that maximize the profit function Π.
To do this, calculate the first-order conditions for the profit maximization problem by taking the partial derivatives of Π with respect to Q1 and Q2:
∂Π/∂Q1 = 20 - 2Q1 - 3Q2 = 0
∂Π/∂Q2 = 20 - 2Q2 - 3Q1 = 0
Solve these equations to find the optimal values of Q1 and Q2.
Step 4: Find the price at which the firm should plan to sell the product.
With the optimal quantities Q1 and Q2, substitute them into the demand function P = 20 - 3(Q1 + Q2) to find the price at which the firm should plan to sell the product.
After following these steps, you will obtain the specific values for Q1, Q2, and the price at which the firm should plan to sell the product.