Question

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?

Short answer

Answer: A monopoly can determine the optimal total output and allocate it among its plants to maximize profits by following these steps: 1. Understand the concept of marginal cost, which represents the cost of producing one more unit of output. 2. Calculate the Marginal Cost for each plant by finding the marginal cost function (derivative of the total cost function). 3. Determine the monopolist's Marginal Revenue by finding the marginal revenue function (derivative of the total revenue function). 4. Find the optimal total output by setting up an equation where the marginal revenue function equals the sum of the marginal costs functions of each plant, and solve for the quantity of output. 5. Allocate the output among plants by equalizing the marginal cost across all plants, and solve for their corresponding output levels using the individual marginal cost functions. 6. Verify the solution by ensuring the summed marginal cost across all plants equals the marginal revenue and the marginal cost of each plant is equalized at their respective production levels.

Step-by-step solution

Understand the concept of marginal cost

Marginal cost represents the cost of producing one more unit of output. To maximize profits, a firm should produce and sell output up to the point where its marginal cost (MC) is equal to its marginal revenue (MR).

Calculate the Marginal Cost for each plant

For each plant, we need to find its marginal cost function. This is typically done by taking the derivative of the total cost function, as it measures the rate of increase of cost with respect to output. If the cost functions are given, calculate the marginal cost for each plant. If they are not provided, it's important to note that different plants have differing cost structures, and their marginal cost functions will also be different.

Determine the monopolist's Marginal Revenue

A monopolist's marginal revenue function depends on its demand function. If the demand function is given, calculate the marginal revenue function by taking the derivative of total revenue function with respect to the quantity sold. If it's not given, assume that the marginal revenue function is decreasing in the quantity of output sold, as is generally the case for monopolists.

Find the optimal total output

To find the optimal total output, we want to find the quantity of output where the summed marginal cost across all plants is equal to the marginal revenue. To do this, we can set up an equation where the marginal revenue function equals the sum of the marginal costs functions of each plant, and solve for the quantity of output.

Allocate the output among plants

To distribute the output among its plants to maximize profits, the firm should allocate production such that the marginal cost is equalized across all plants. This means that each plant's marginal cost at its allocated quantity should be the same. Using the individual marginal cost functions of each plant, set them equal to each other and solve for their corresponding output levels. This will give us the quantity of production allocated to each plant that maximizes the monopolist's profit.

Verify the solution

To ensure that the solution is correct and that the monopoly is maximizing its profits, confirm that the following conditions are met: 1. The summed marginal cost across all plants is equal to the marginal revenue. 2. The marginal cost of each plant is equalized at their respective production levels.

Key concepts

Marginal Cost

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.

Marginal Revenue

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!

Profit Maximization

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.