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 \(\mathrm{MC}_{1}=20+2 Q_{1}\) and \(\mathrm{MC}_{2}=10+5 Q_{2}\). The firm's esti- mate of demand for the product is \(P=20-3\left(Q_{1}+Q_{2}\right)\) How much should the firm plan to produce in each plant? At what price should it plan to sell the product?

In the fascinating world of economics, understanding the concept of marginal cost is like holding a key to unlock the door of efficient production. Let's dive into the notion of marginal cost, which reflects the cost of producing one additional unit of a good. In our scenario, a drug company faces different marginal costs for the same product across two plants.With the formulas given as \( MC_1 = 20 + 2Q_1 \) for the first plant and \( MC_2 = 10 + 5Q_2 \) for the second, we see that the costs are not just about the quantity produced but also include fixed components, likely inherent to each plant's operational dynamics.
In practical terms, if plant 1 decides to produce one more unit of the medicine, the cost will increase by \(2, while at plant 2, it would spike by \)5. These costs play a pivotal role when the company strategizes on how to distribute production between the two plants to minimize costs and maximize profits. By equating marginal cost to marginal revenue, which is the additional income from selling one more unit, the firm can determine the most efficient quantity to produce.

Price determination is akin to setting the sails for a ship; it's about finding that sweet spot where the wind (demand) meets the vessel (product) just right. The firm is tasked with establishing a price that balances its objectives with market dynamics. Our monopoly drug company does this using the demand estimate \( P = 20 - 3(Q_1 + Q_2) \), a classic inverse relationship, signifying that higher production leads to a lower price.
This is crucial because it dictates how much the consumers are willing to pay as more units become available. The equation implies that for each additional unit produced by the combined plants, the selling price will drop by $3. Therefore, setting the right price is about striking a balance; too high and sales dwindle, too low and profits fade. It is a strategic play where the firm seeks the highest price it can set while still maximizing sales volume and revenue.

Profit maximization is the endgame for any business endeavor. It is the quest to achieve the highest possible profit from the operations. In our example, the drug company wants to find the optimal production level where profits soar. It does this by equating the marginal cost of each plant to the price to ensure no money is left on the table.
After establishing the equations from the given marginal costs and demand, we solve them simultaneously to pinpoint the quantities \( Q_1 \) and \( Q_2 \) that will yield maximum profit. And since the last piece of the puzzle is the optimized price, by substituting these quantities back into the demand equation, we discover the magic number—the optimal price at which to sell the product.
This optimized price balances demand with production cost, ensuring that the firm captures the most revenue while ensuring consumers still value the product highly. It's a meticulous blend of economics and business strategy that ensures the company's success.