Business calculus

Factual Questions
1

You wanna know how long it’s been since I took a precalculus class? Yes, that long! :eek: I suppose I could google it, or else try to find my old text book; but someone else may benefit from a public answer.

Let’s say I have a business, manufacturing a product. If I sell the product for less than it costs to produce, I lose money. If I price the product at a million times what it costs to make, I’ll make a helluva profit – if I sell any. I recall that there are two equations that can be graphed, which show at their intersection the optimum price needed to maximise profits. That is, at some point the price is low enough that sales are at their peak and high enough that the profit per unit is greatest.

Can someone remind me of these equations?

Also, how does a business IRL determine how much someone will pay for a product? In a class the numbers are arbitrary. Netflix raised their prices, and met disaster. (What did they lose? Millions of customers?) Setting a price and seeing what happens seems risky. I assume there is an equation/algorithm/model to determine likely results?

2

Yes, there are equations you can graph to determine maximum profit, but the trick is that you don’t actually know what all of them are. This ties into your second question: You can’t know the optimum price unless you know how many people are willing to buy a product at a given price, and there’s no way to mathematically model that a priori. When you’ve got a product that nobody has ever sold before, how could you conceivably know what numbers to attach to it? You can’t prove mathematically that the Segway will be a flop but that the iPod will be a hit.

3

I think you’re thinking of supply and demand curves, though I don’t think there’s a generic form for the equations of the curves; it would seem likely to me that different goods would have different shapes for the curves (due to varying degrees of elasticity, for example.)

4

Where MR = MC. That’s all I got out of my two economics courses. Yeah, helped me a hell of a lot, too!

5

Really, I’m just looking for the general equations for cost vs. profits. As I said, it’s been a while; but ISTR that you can set up an equation that graphs the number of sales vs. the retail price, and you can set up an equation that graphs the cost to make a number of products. Where the graphs meet is the optimum price and profit.

For example, let’s say I buy a Cessna (since I have planes on the brain) for $50,000. No, wait. That gets complicated with the fixed costs and hourly costs and insurance and such. Let’s say I’m making humdaddies. The machine to make them cost $10,000. Materials cost $3.00. Ignore salaries and overhead, and just figure the cost of the machine and materials. I need to make $3.00 times x-number of units plus $10,000 to break even. I know that I can sell a large number of units at $6.00 each, and a smaller number of units at $12.00 each.

That sort of thing. It’s just one of those things that popped into my head and got me curious. If I get more curious I could go figure it out, but I thought someone might have a quick answer. Since this is idle curiosity, feel free to make up numbers. But since it is just idle curiosity, this is just a fun question for math types to spread some knowledge. :wink:

6

Don’t make the mistake of thinking that anything taught in econ 101 is actually applicable to the real world. The fact that you don’t know how your customers will respond to a new product is the most significant problem facing a real firm, and there’s a lot of money spent on figuring out how to handle it.

7

I always liked this article, which starts off very econ 101 - y, then rejects most of that in the face of real world complications.

8

There are a lot of fancy words like “forecast” that people will apply, but the bottom line is that you have to guess. You can look at how similar products are priced, you can do market research, but when it comes right down to it, you have to take your best guess and ship the product.

Huge corporations have more options than small businesses do. If you had a restaurant and developed a new sandwich, you’d need to just take your best shot at what it’s worth and put it on the menu.

A big national chain could pick five different price points, so their restaurants in New York, Chicago, Dallas, Denver, and Seattle each had the sandwich at a different price. They’d compare results from the five locations, and then roll out nationally, where they still might experiment with different rates in different regions, but with less variety (In the first experiment, the most expensive might be triple the price of the least expensive. In the rollout, the price difference between highest and lowest might be 25%). Over time, they’d settle into the most profitable price point based on comparing data from all locations.

9

Your question about humdaddies is more of an accounting question than economics or math. There are lots of equations that can solve the problem… if you know how many units will sell at each price point.

So, again, what you’re asking is unknown. If you want to know whether $12 humdaddies will make you more money than $6 humdaddies, you hire a market research firm, pay them thousands of dollars to do focus groups, simulations, test markets, etc.

If you find a mathematical equation that can predict this ahead of time, then all you’ve found is someone who took observed values and fit an equation to them. There’s no guarantee that new products will fit the same equation.

10

As others have said. There is no equation for sales volume versus sales price to enable you to optimize revenue.

What there can be, *after you run a bunch of test sales, *is an empirical table of how many you sold at each price point you tried. And then armed with that data, you can use math to make a best-fit equation that you can use to project how sales would *probably *fare for prices above, below, or between your empirical data points.
On the cost side, it’s fairly straightforward to generate an equation for costs as a function of volume. Take fixed costs plus per-item costs times desired volume gives total cost. The devil is in the details. Some cost factors are easy to determine, others are much tougher. But …

“Fixed” costs are only really fixed over a narrow range of volume, so you need to understand your constraints thoroughly. Typically your “fixed” costs actually are variable but as a step function of vaolume, not a continuous function of volume.

e.g. Once you buy a machine which can make 1000 widgets per day, your cost for that machine is fixed whether you’re making 20 a day or 600 a day. And it stays fixed until you need to make 1001 widgets. Then you have to buy a second machine and your machine-related costs just doubled. And then they remain flat until you need a 3rd machine. etc.

And once you bought the first factory building which holds the widget-making machines you’ve got a fixed cost for the building … until you need 15 machines and the building only has room for 14. So now you need another building and again your “fixed” costs take a step increment with sales volume.

G&A expenses tend to behave like machinery & building expenses: a fixed number with a very small per-volume component, but with periodic significant step increments as volume goes up.

Because each component of machinery, building, G&A, etc., has its own scaling points and behavior, a total-factor cost curve in a complex (i.e. non-textbook) business will tend to behave a lot like a pure variable cost with a pretty small fixed component and a few percent of bumpiness in the slope as we grow the volume.

For a simple business willing to be constrained by its larger fixed-cost increments, costs behave much more as a fixed nut plus a very small variable component. e.g. a single retail location which provides a personal service is almost pure fixed costs.

11

Are you planning to make like Howard Hughes and start your own aircraft manufacturing business to compete with Cessna? Or open a retail dealership? If you’re in the San Fernando Valley, you could call it Casa de Cessna. (Inside joke for Asimovian and anyone else who knows the L. A. area.)

12

That’s certainly true in reality. However for the sake of completeness, what Johnny L.A. is thinking of from basic econ is that firms should set production levels at a point where marginal cost equals marginal revenue. (As Jinx said).

You can plot revenue as dollars in total revenue vs units sold and cost as dollars in total cost vs units made. The derivative of each of those are the marginal revenue and marginal cost. That is, the additional revenue or additional cost incurred by selling or building one more unit.

In basic economics this is profit maximizing production because if you think about it conceptually, building one more unit and every additional unit thereafter would cost more incrementally than it would bring in. Building less than MR=MC would also make no sense because if you build additional units you would bring in more revenue than the increased cost. Thus, the only logical conclusion is to produce at MR=MC.

Of course, again, everyone who is saying this isn’t useful in the real world is correct. But Johnny L.A., that is the concept you were thinking of.

13

Wow, that was a great read. I highly recommend it for everyone in this thread, most especially the OP.

14

Ha! No, I was just trying to remember the pairs of equations used back in precalculus.

I believe your understanding is correct. Only I’m not looking at from an Economy standpoint; I’m trying to remember it from a Math standpoint. As I recall, A=Pe[sup]rt[/sup] is involved.

Thanks for the link, TheonX I haven’t time to read it now, but I’ll try to get to it when I can.

15

Johnny L.A. - That is the formula for continuously compounded interest.

I scanned this quickly, unfortunately I’m too tired and/or lazy to do the math now, but I’m pretty sure this can be solved with algebra, as parabolas, without having to use calculus. Calculus would be easier (I mean, if you know if it’d be quicker), but unless my tired mind deceives me, you can solve this with about Algebra II level equations.
…Ok I read a little bit more, yes, you’re talking about maximizing price/profit, that’s be an a parabola that opens upside down A ‘U’ shape), and the maximum price/profit would be the peak of the graph (the maximum). These would be quite easy to graph, if you have what it costs to make the product, and what you will sell it for. It could be solved in a few minutes by hand (or 30 seconds with a graphing calculator).

16

You may be thinking of Break-Even Point calculations. This only looks at the supply side of the equation, not demand.

The only things I can think of using exponents are various calculations for value, such as NPV and IRR. None of those match the equation that you’re thinking of exactly. NPV and IRR are not really used to set prices, but to determine which of multiple courses of action are the most profitable (or, at least, which are profitable above some minimum).

17

In math terms, f(x) is some function representing the total revenue a firm earns given sales of quantity x. g(x) is another function representing the total cost a firm incurs for producing quantity x.

To find the profit maximizing quantity and price, find f’(x) and g’(x) and solve a system of equations for x. The solution, x will be profit maximizing quantity and f’(x) or g’(x) will be profit maximizing price.

18

A=PErt is continuous growth (continuously compounded interest) and is taught in business calculus, but is not related to maximizing profit.

For finding the best price in business calculus (which breaks down once you try to apply this to the real world):

Usually given: price demand equation solved for price p(x) and Cost C(x)

[ol]
[li]Find Revenue as R(x)=x*p(x)[/li][li]Take Revenue and Cost based on units sold (x) and create a single Profit graph for them P(x)=R(x)-C(x).[/li][li]Find the best x value (demand) based on the maximum value for P(x) (besides the eyeball test, you can use first and second derivatives to find the exact value)[/li][li]Plug in the demand (x) into the price-demand equation to find the price you need to charge.[/li][/ol]

Again, this is dependent on a price-demand equation which is mostly guess work.