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In our video on Maximizing Profit Under Monopoly, we cover how firms can use their market power to raise the price of a good well beyond its marginal cost. A practice question from the Microeconomics final exam asked you to find the total profit of a monopolist under certain conditions. In this Office Hours session, Mary Clare Peate, Marginal Revolution University’s Instructional Designer, helps you solve that problem.

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**Transcript**

I've reviewed the data online. I've talked to a ton of college students. Everyone is missing this one question. It's time to make a video.

Today we're going to answer the following question from our Microeconomics final exam, and that is to find total profit of the monopolist under the following conditions: Demand for this good is marked by P equals 100 minus 2Q, and this monopolist’s fixed cost is 100, and his marginal cost is 20. Now, if you haven't already done so, check out our video Maximizing Profit Under Monopoly. Then, actually try to do this problem by yourself, and then come back and we'll work through this problem together.

Ready? I'm going to quickly recap three important truths about a monopoly. These points are covered in great detail in our monopoly video, but they're worth repeating as they'll form the initial steps for solving our problem today.

Monopoly truth number one: Monopolists have market power. They're a big player in the market. Or in the classic case, they're the only player in the market, which means that the quantity the monopoly produces actually affects the market price. You can think of them as price makers in their market. Monopoly truth number two: When a monopolist is choosing how much to produce such that it maximizes its profits, monopolists behave the exact same way as their competitive counterparts. All firms, even the price-makers of the world, set marginal revenue equal to marginal cost to find that profit maximizing quantity. And monopoly truth number three: Because monopolists are price makers, their marginal revenue is no longer simply the price of the good as it is for a competitive firm. Instead, the monopolist’s marginal revenue varies with the quantity it sells and is less than the market price.

As mentioned, these three truths form the initial steps for solving our problem today. First, we actually need to find the monopolist’s marginal revenue curve. We then set marginal revenue equal to marginal cost, as we always do, to find that profit-maximizing quantity. From there, we use quantity to find the firm's profit-maximizing price. And finally, once we have the monopolist’s price and quantity, we can then find the monopolist’s total revenue and total cost to solve for profit.

Step one is to find the monopolist’s marginal revenue. The shortcut to finding the marginal revenue curve is to simply double the slope of our demand curve, and that's it. One thing to note here, that shortcut only works for linear demand curves. But that makes it sound way fancier than it is. A linear demand curve is literally just a straight line. Now, if you'd like me to derive the marginal revenue curve in a future video, just let me know by voting at the end. In this instance, the slope of our demand curve is 2. Double that to 4 and we arrive at a marginal revenue of 100 minus 4Q.

Step one is complete, and we can now move on to step two, which is to set marginal revenue equal to marginal cost and solve for the profit- maximizing quantity. The monopolist's marginal cost, as you know, is 20. Set that equal to the marginal revenue and solve. I know you can do this math, and I know you're doing this math right now, so I don't really have to go through it. After solving, you'll arrive at a profit-maximizing quantity of 20.

Step two is done, and we can now move on to step three, which is to find the market price. If the monopolist sells 20 units, what is the maximum price it can charge as the price maker? To find out how much consumers are willing to pay given this quantity, we turn back to our demand curve which provides us with a clear relationship between the price and the quantity of a good. Simply plug Q into the demand curve and solve for the maximum price the monopolist can charge. Again, I know you're going through these steps right now, so I don't have to go through each one. We'll eventually solve for a price of 60, and now step three is also done. We now have our monopolist’s profit-maximizing price and quantity.

To find a monopolist’s profit, or any firm's profit for that matter, we need to find how much money this firm is spending and subtract it from how much money this firm is making. Now, if you're an Econ nerd like I am, that's just another way of saying: Find total cost and subtract it from total revenue. Total cost, as you know, is fixed cost plus variable cost. And we know from the initial conditions that fixed costs are 100 and marginal costs are 20. Plug these back into the equation to arrive at a total cost of 500. Total revenue is simply the units sold, or the quantity, times the price. After plugging in our price and quantity, we'll arrive at a total revenue of 1200. And now, all we need to do is subtract our total cost from our total revenue to arrive at the firm's profit of 700.

And that's it! As always, please let me know what other concepts and questions you'd like me to cover. And if you'd like to challenge yourself, we've included some additional questions for you to try at the end. Thanks.