Suppose a monopoly produces its output in several different plants and that these plants have differing cost structures. How should the firm decide how much total output to produce? How should it distribute this output among its plants to maximize profits?

Imagine you're running a lemonade stand. Each glass of lemonade you sell costs a bit more to make because you need more lemons and sugar. In economics, we call the extra cost of making 'one more' glass of lemonade the marginal cost (MC).

For a company producing goods, understanding marginal cost is crucial. It's like knowing exactly when each squeeze of a lemon becomes more expensive. Firms calculate MC by looking at the change in total cost when they make one additional unit. Mathematically, it's the derivative of the total cost function with respect to quantity. If lemon costs double suddenly, the marginal cost of your lemonade shoots up.

In real industries, especially with multiple production plants like in our exercise, each plant will have its own MC owing to different technologies, workers, or even local lemon prices! So, each plant's MC curve might look different, and it's essential for the company to calculate each one to decide how many glasses of lemonade to make at each plant for the lowest overall cost.

Why Marginal Cost Matters

If a firm keeps track of MC, it can ensure it's not paying more to make a glass of lemonade than what it will sell for. In the case of a monopoly, which our exercise describes, this becomes even more critical because the firm has more control over prices and output.

After figuring out the cost of that extra glass of lemonade, we also need to think about the extra income, or marginal revenue (MR), it brings in. It's the revenue you earn from selling 'one more' glass. For our lemonade stand monopoly, it means how much cash you'll get for every additional glass sold.

In monopoly situations, unlike perfect competition, each additional glass sold doesn't always earn the same amount. Why? Because to sell more, you might have to lower the price a little bit each time, which means MR will decrease as you sell more. So, to figure out MR, you need to look at how total revenue changes with every additional unit sold, and for this, you would take the derivative of the total revenue function with respect to quantity.

Just like marginal cost, marginal revenue is also a piece of the profit-maximizing puzzle. It tells the monopoly exactly how much output to produce to make the most money. By plotting the MR curve alongside the MC curve, the point where they intersect is where the magic happens - maximizing profit!

So, we've got our lemonade cost and income sorted per glass, but how do we make sure we're earning the most profit? That’s what profit maximization is all about. It's the sweet spot where each additional glass of lemonade makes just as much extra income as it costs to make.

For our monopolistic lemonade stand, profit maximization occurs where the MC of making one more glass equals the MR from selling it. If MC is higher than MR, making more lemonade would mean losing money on every additional glass. But if MR is higher than MC, there's money left on the table - we could have made and sold more lemonade to increase profit.

In the textbook exercise, we've got several plants to think about. We have to find the total quantity where MC across all plants equals the MR, then allocate that quantity among the plants so that the MC at each plant is the same. This way, the monopoly uses its resources most efficiently and doesn't waste money making lemonade where it's too pricey or miss out on selling where it can make more.

Checking the Profit-Maximizing Level

It’s like a lemonade symphony where all plants play in harmony. The company needs to verify that the sum of MCs equals MR, and each plant is producing just enough so that their last glass is as costly as it is at every other plant. This is how a monopoly can turn lemons into the most profitable lemonade.