Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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One of the most important parts of economics is knowing the revenues and costs and how they relate to increased production. These can both be modeled by functions. These **cost and revenue** functions can then be manipulated like any other function. The profit is the difference between total revenue and total cost.

There're a lot of applications in Economics for Calculus. Before we get into those applications we have to talk a little bit about some basic terms, cost, revenue and profit. First let's talk about cost, suppose your business manufactures sneakers, let x be the number of pairs that your company makes. A cost function tells you how much it costs to produce x pairs of sneakers so here's the x axis number of pairs of sneakers and here's the cost axis and your function might look something like this. If you produced 0 sneakers you would still have some cost because they're a fixed cost associated with running a business. Like rent for your factory or salaries for your workers and so on. So this would be the fixed cost and then the additional cost is called the variable cost, this is the cost of producing certain number of sneakers so it's variable cost plus fixed cost.

And this model, this linear model assumes that the cost stays constant for example the cost of materials does not go up the more you buy them or go down. This is a more variable model that takes into account the effect of initially buying in bulk, like decrease the cost of materials a little bit so the cost curve will curve down a little bit but then if you buy too much you might create a shortage and the cost might sky rocket. So a cost function can be more complicated it doesn't need to be just linear, and that's cost.

Let's take a look at revenue, if you get p dollars for each pair of sneakers you sell and you sell x pairs , revenue is going to be the price of the sneakers times the number of pairs you sell. This will be the amount of money that you're bringing into your company, p times x that's the revenue. The revenue function could look like this r equals p times x if the price is constant you'll just have this linear function for your revenue. But it is also possible that the price isn't constant that if you sell, if you have too many pairs of sneakers out in the market the price is going to go down. The demand point then goes down so you might have this kind of concave down piece at the end. So you have linear and a non linear revenue function.

And profit is revenue minus cost right, the money that you take into your company minus the money that you spend that's profit. And if you look at your cost function so this is your cost function and your revenue function, the point at which they cross is called the break even point. It's at this point where you're producing just the right number of shoes where cost and revenue are exactly the same. Your profit will be 0 but it's at this point where you can start making a profit. If you produce more than x sub 0 shoes you'll make a profit. Your revenue will be above your cost, if you produce less your revenue will be below. So this is a really important point in Economics the break even point, so we have cost, revenue and profit these are the important ideas that we're going to be talking about in the next few lessons.