Question

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 \(\operatorname{arc} \mathrm{MC}_{1}=20+2 \mathrm{Q}_{1}\) and \(\mathrm{MC}_{2}=10+5 \mathrm{Q}_{2}\). The firm'sesti 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?

Short answer

The optimal quantities to be produced are \(Q_{1} = 1\) and \(Q_{2} = 2\), and the optimal price to sell the product is 11.

Step-by-step solution

Define and Understand the Problem

The problem is to find the optimal quantity to be produced in both plants \(Q_{1}\) and \(Q_{2}\) in order to maximize profits, as well as the price at which the product should be sold that would bring the maximum profit. This requires setting up two cost function equations based on the given cost functions \(\mathrm{MC}_{1}=20+2 \mathrm{Q}_{1}\) and \(\mathrm{MC}_{2}=10+5 \mathrm{Q}_{2}\), as well as a price function equation based on the given price demand equation \(P=20-3\left(Q_{1}+Q_{2}\right)\).

Set up The Equation For Profit Maximization

Profit is maximized when Marginal Cost is equal to Marginal Revenue. Since Marginal Revenue is derived from the price, we need to differentiate the price function with respect to quantity to obtain the Marginal Revenue function. Afterwards, set the Marginal Costs equal to Marginal Revenue respectively to find the individual quantities. These equations are then: \(\mathrm{MC}_{1}=20+2 \mathrm{Q}_{1} = 20-3\left(Q_{1}+Q_{2}\right)\) and \(\mathrm{MC}_{2}=10+5 \mathrm{Q}_{2} = 20-3\left(Q_{1}+Q_{2}\right)\). These two equations can be solved simultaneously to find the optimal values for \(Q_{1}\) and \(Q_{2}\).

Solve The Equations

Solving the two equations yields \(Q_{1} = 1 \) and \(Q_{2} = 2 \). These are the optimal quantities that the firm should produce in each plant.

Determine The Selling Price

Substitute the optimal quantities into the price function to get the optimal price: \(P=20-3\left(Q_{1}+Q_{2}\right) = 20 - 3*(1+2) = 11\). Thus, the firm should sell the product at a price of 11.

Key concepts

Marginal Cost

Imagine you're running a lemonade stand. Every time you squeeze one more lemon into your pitcher, it costs you a bit more – not just for the lemon but for your time and effort too. That's what economists call Marginal Cost (MC), the cost of producing one additional unit of a product.

In our medicine production problem, the MC is vital for the company to know how much it will cost them to make one more bottle of medicine in each plant. It’s a balancing act – they need to ensure that the price of their medicine covers this cost and contributes to overall profits. For example, the MC equations for the two plants, \( MC_1 = 20 + 2Q_1 \) and \( MC_2 = 10 + 5Q_2 \) imply that making medicine is cheaper in the beginning. But as more is produced, costs start to climb up quickly, especially in the second plant!

Marginal Revenue

Now let's talk about the money that comes in when our company sells an additional bottle of medicine. In the world of economics, we call this Marginal Revenue (MR). It’s the extra revenue that the company expects to earn for each additional unit sold.

In the monopoly scenario, MR can get a bit tricky because the price can change with every additional unit sold due to the demand function. The usual strategy is to set MR equal to MC to determine the optimal level of production because that's where profit peaks – you don't want to be selling more if it's going to cost you more than you earn from it!

Demand Function

Have you ever wanted something more when it's rare and less when everyone has it? That's the basic idea behind the Demand Function. It shows us how much of a product people want based on its price. Usually, when the price goes up, people want less, and when the price goes down, people want more.

Now, for our medicine, the demand function is \( P = 20 - 3(Q_1 + Q_2) \). This tells us that if we keep increasing the total quantity of medicines produced (\( Q_1 + Q_2 \)), the price we can sell each one for drops. The company must strategize to find a sweet spot where the price is high enough to ensure good profits without making too much medicine that nobody wants.

Production Optimization

Finding the 'Goldilocks zone' for making and selling our medicine is what Production Optimization is all about. We want to make just enough medicine so that the costs are low, the price is right, and the profits are banging! The company has to crunch some numbers to figure out how many bottles to make in each plant to hit that perfect balance.

By setting the MC of each plant equal to the MR, as the step-by-step solution showed, we ensure that we aren't producing too much (which can cost us) or too little (which can lose us sales). It's a delicate dance to make just the right amount.

Price Determination

Once we know how much medicine to make, we need to slap a price tag on it - that's Price Determination. It isn't just picking a number out of the air; it's about understanding what people are willing to pay and how much the company needs to cover costs and make a profit.

In our monopoly example, the medicine's price isn't just plucked from thin air - the demand function helps dictate it. With the calculated optimal production levels, the company knows they can price the medicine at 11 dollars - this will maximize their profits while still adhering to how much customers are willing to pay based on the current supply.