a drug company has a monopoly on a new patented medicine. the product made in either of two plants the cost of production for the two plants are MC1=10+2Q1 and MC2=25+5*Q2. the firms estimate of demand for the product is p=2000-3(Q1+Q2). how much should the firm plan to produce in each plant? at what price should in plan to sell the product?
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To determine how much the firm should produce in each plant and at what price to sell the product, we need to find the profit-maximizing quantities and prices. Here are the steps to solve this problem:
Step 1: Set up the profit equation.
The profit is calculated as the difference between the total revenue and the total cost. The total revenue is given by the formula: Total Revenue = Price × Quantity. We can rewrite the demand equation in terms of just Q1 by rearranging it as Q2 = (2000 - 3Q1 - p)/3.
Step 2: Calculate the total cost for each plant.
The total cost for each plant can be calculated by integrating the marginal cost equation. Integrating MC1 = 10 + 2Q1 gives us the total cost for Plant 1, while integrating MC2 = 25 + 5Q2 gives us the total cost for Plant 2.
Step 3: Express the profit equation in terms of Q1.
Using the total cost equations and the total revenue equation, we can express the profit equation solely in terms of Q1 by substituting the value of Q2 obtained in Step 1.
Step 4: Maximize the profit.
To maximize the profit, we need to find the value of Q1 that maximizes the profit equation from Step 3. Differentiate the profit equation with respect to Q1 and set the derivative equal to zero to find the critical point.
Step 5: Calculate Q2 and the price.
Using the value of Q1 obtained in Step 4, we can then calculate Q2 by substituting it back into the Q2 equation from Step 1. Finally, substitute the values of Q1 and Q2 into the demand equation to find the price.
By following these steps, you should be able to determine how much the firm should produce in each plant and at what price to sell the product.