Single Period Choice: Break Even Problem

Okay, now this allows us to move along to some of the basic single period choice problems. What is nice about these problems is that it allows you to be fairly effective in a business context knowing relatively little. You are also going to be valuable on a budget committee because it allows you to ask some fairly sophisticated questions. One of the questions you can ask about is break even.

What you’re looking for here is, I’m trying to find the volume index, or rather, the volume such that net benefits are zero. That’s what we’ve defined as the volume index. So, what you can think about it as is that you’re trying to find “q”. So, it’s the total revenue is just going to be equal to the total cost. So, that’s what we’re looking for in the volume index, we’re trying to locate these two things.

The best example I have is actually for the most part a real life example, and this comes from a graduate student of mine that I had, now doing rather well, his name was Serin Dutebar, and what he did is that he ended up going down to the San Antonio River Walk, and he came across a guy that was operating a sunglass, little cart, that was out there on the walk. And Serin being the kind of guy that he was, went up and started a conversation with the guy, and started to collect a little bit of information about his business while he was there. One of the pieces of information that Serin collected from the guy was that he was actually charged \$5000 a month for his space there on the river walk. So I’ve expressed that as a fixed cost that’s right there. So he’s charged \$5000 no matter how many pairs of sunglasses he goes ahead and sells.

Well, the other thing you can know about this, knowing about the variable cost associated with the sunglasses. And if any of you has ever priced out wholesale sunglasses, as counterfeit as possible, you know that these things probably cost you, at most, \$10 each. And so I’ve expressed the variable cost in terms of quantity, which we always do, and that number there is average variable cost. That’s that alpha there that we’ve been talking about before. And so that’s the variable cost measure that we have there.

Now, those of you who have ever gone ahead and tried to sell things to people with significant amounts of money, you know it’s pretty difficult to sell them something at a low price, rather than at a high price, because for some reason, people with a lot of money visualize higher prices as being indicative of quality. So you can sell them the same sunglasses, but you’ll never sell them to them for \$20. It’s always to best to try to sell them the sunglasses for something that’s a little more than \$20. I’m going to suggest a price, something along the lines of \$120. And so what we have here is the ability to calculate what volume you have to sell per month, how money sunglasses per month in order to make it so that the revenue is equal to the cost that you incurred.

So, we’re going to go ahead and take that price, which is \$120, we’re going to multiply it by the quantity. There’s your total revenue which is right there. And on the other side of the equal sign what we’re going to put is our fixed cost, which is \$5000 per month, AND your variable cost, which is right there, which is going to be \$10 times that quantity which we are actually selling. Now we’re going to do a little bit of algebra right here, and we’ll go ahead take the \$120 and subtract 10, we have to put 110q over here, and 5000 right over here. And we’re going to solve for q, and the volume index right there as being 5000 divided by 110, and since you’re not here, I’m going to have to go ahead and do this math all by myself, which is a really, really, really, strange thing for me to have to do here, but, let’s take it as 5000, and divide by 110, and it turns out to be 45.45. In other words, in order to break even and make it so that the total revenue is equal to the total cost, the guy only has to sell about 45 pairs of sunglasses per month, in other words, he’s got to pay himself a pair and a half per day.

Should this guy worry about this at all? No, he’s very likely to sell at least a pair and a half per day. The rest of the story goes on where Serin interviews the gentleman, and he says, “No, no, no, no, no, don’t worry about me, I pull \$120,000 a year out of this place,” and points over to the BMW that he has sitting over in the corner and he says, “That’s what I pay for with this stuff over here.” And again, remembering, this guy only works a short season, it’s not supposedly even an entire year, but there’s an example of break even. Now, when you get numbers that are that small, you know you don’t have to worry about things, you can go back to sleep, and that’s what I mean about the break even kind of problem.

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