Economic applications (§4.4)

In word problems in economics or finance, a few quantities arise regularly, which you should know about.

First, quantity in this context has a specific meaning: the amount of goods or services made and/or sold in a given period of time. Quantity is thus measured in such units as pounds per week, items per year, or litres per hour. Quantity is variously denoted q or x.

Next, price (or unit price) is the amount of money received for a given amount of goods or services. So price is measured in units such as dollars per pound or euros per item. Price is denoted p, a lowercase Pee.

Then revenue is the amount of money received for goods or services in a given period of time. Revenue is measured in dollars per week, euros per year, etc. Revenue is denoted R, and we have this equation:

R = qp, or R = xp.
(Notice that the units make sense in this equation; amount over time, multiplied by money over amount, becomes money over time.)

In this terminology, cost is completely different from price. Like revenue, cost is measured in units of money over time. But while revenue is the money that comes into the business, cost is the amount of money that the business has to spend (in a given period of time) in order to produce and distribute their goods and services.

Finally, profit is the amount of money that the business makes and keeps in a given period of time. Unlike everything else here, it makes sense for profit to be negative. Profit is denoted P, an uppercase Pee, and we have another equation:

P = R − C.

In business, you generally want to maximize profit: make it not only positive but as large as possible. Even if you don't want to maximize profit as normally measured (because you care about something else besides money), economists typically try to calculate whatever else you care about and still say that you maximize profit (in a generalized sense).

So if you can find a way to express profit as a function of some other quantity whose range of possible values you already understand, and if this is a quadratic function, then you know how to maximize that! To do this, you'll need two more equations, typically one that relates price to quantity (expressing how much consumers are willing to buy at a given price) and another that relates cost to quantity (expressing how much it costs to produce and distribute a given amount). If both of these equations are linear, then profit will be a quadratic function of price, and you can see what price you must set to maximize profit.

The problems in the book are less complicated; they leave out cost and profit and ask you to maximize revenue. This is not really what you want to do in business, but it makes for a simpler problem.


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