Okay, so today was my first day of Math 144--Business Calculus, and
although I have (much to my own dismay and disbelief) decided to major
in Accounting, I am a horrible math student.  I think the reason I am
a terrible math student is because I'm not content with knowing 'how,'
I want to know 'why'.  I used to be an English major until I somehow
found myself in a bookkeeping position and started learning about
accounting, and I guess for me it's like this:  You don't read great
literature to memorize the names of the characters or even the plot
line, you read it because you want to grok it.  I am sure that this
course could teach me a lot that I could use in business and
accounting, but I don't grok it and I am really trying to.

So, here's my specific question:  I understand linear functions, with
a dependent and an independent variable.  That's a piece of cake. 
Variable costs in accounting can be plotted on a line using a linear
function.  No problem.  But the prof said today that MOST functions in
business are non-linear, which blew my mind.  In my four years of
bookkeeping experience, I've never had to find the square of a single
dollar figure.  I want to know just where this squared figure comes
from (as in 2x^2 + 3x + 2000).  I notice it seems to be tied to the
revenue function and the revenue portion of the profit or break-even
functions (at least so far in one day of classes).  I want to know why
revenue isn't a straight line and just what the squaring of this
figure (which also tends to be negative so far) actually represents. 
Is it somehow related to the "normal distributions" I kept hearing
about in my last ungroked math class (Stats)?  I guess this may sound
really naive, but again, I just don't grok it.  I grok adding,
subtracting (debiting, crediting), even dividing and multiplying can
give me something tangible to consider (a fraction, a percentage,
etc.)  What does this mysterious little exponent yield?

Maybe this is a really dumb question, but I would really like to
understand this better.  I hope that you will find the question
stimulating, because I can't offer much in the way of payment.  Thanks
in advance for trying.

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Clarification of Question by hopelessnerd-ga on 06 Jun 2005 21:22 PDT

This is not so much part of my real question, but I do want to correct
something.  I said that the squared figure tends to be negative in
these revenue functions, but what I meant was that the result of the
term tends to be negative like -2x^2.  Most of the problems I have
seen so far involving these revenue functions, this term winds up
being negative which makes me wonder just why this mysterious little
^2 term is taking away from my revenue.  But that's really just a
small detail.

---

Clarification of Question by hopelessnerd-ga on 10 Jun 2005 21:41 PDT

Okay, this is not to say that the comments posted so far have not
given me plenty to think about, but maybe my own fuzzy-headedness is
making the question more complicated than it needs to be.  So, here is
a concrete example of what I am talking about.  Maybe someone can
provide me with some concrete reasoning behind it.

In accounting, we calculate total costs using the linear function
TC=(Variable Costs)*(# of Units)+(Fixed Costs).  Obviously, the more
units we produce, the more we have to pay for variable costs (labor,
direct materials, machine hours, etc.) while fixed costs (rent,
advertising, administrative salaries, etc.) remain fixed, thus the
name.  If I am using the correct mathematical lingo, this is a linear
function with a y-intercept equal to fixed costs (i.e. even at zero
units of production, fixed costs will still be incurred).  This I get.
 Now comes this:

"A medium sized company produces a quantity of 9 units (q) per week.  
The company's total fixed cost (FC) = £10,000 and its total variable
cost (VC) = (-1,000q + 200q2)."

Now, all I want to know is where on earth does this strange
configuration for variable costs come from?  What are the terms
representative of?  My prof says they represent nothing tangible and
that they are part of a function the business has gained from
scrutinizing the cost behavior over time.  He says some businesses
actually obtain these functions from similar industries who have
discovered that this is the way costs behave.  I think that's the
biggest cop-out I've ever heard (and believe me, I've heard a lot of
cop-outs from math teachers who INSIST that we should not question
why, but just learn to plug the numbers in by rote).  Which is, I
might add, my biggest problem with math.

I want to understand why the quantity of units should be squared,
multiplied (by WHAT?) and added (or subtracted) from the quantity
multiplied again (by WHAT).  Assume I am a small business owner who
has decided that perhaps one of them there newfangled non-linear
functions would better represent his costs.  How would I go about
finding such a function, and what is the logic behind it?  I don't
think it does me any good as a business owner to know how to maximize
or minimize the function without knowing how to find a fitting
function in the first place.

Thanks for taking the time to humor me.

I'm not totally sure this is what you mean, but an simple example I
can think of is that of revenue from sales, related to the price of
the goods. Note, I'm not an economist, so this is in simple language
and there may be better technical terms to illustrate this argument.

If something is cheap, then people may well buy a lot of it, but your
gross income from these sales is low.

Similarly, if you price too high, people may only buy a few of your
product resulting again in a low income.

Naturally what you need to aim for is a happy medium where you sell
lots of stuff at a price that people are willing to pay. You can model
your income approximately by a curve of the form y(100-y) (where the
100 is arbitary here, and the y is the selling price [max 100]).

Note that when y is small, you reach the form of a small number times
~ 100, making a small number. When y is large then again you get ~100
times a small number. However when y is in the middle of this range,
say 50 you then get ~50 x ~50 which gives a max income of around
~2500.

The formula you've got above can be rewritten as 100y - y^2, which is
where the negative exponent of y^2 can come from.

Any help?

S

Hopelessnerd --

Indeed most business functions are non-linear -- and thank goodness
for it or we'd be living in a rudimentary pre-industrial economy.  Can
you imagine a farm where planting twice as much corn would cost twice
as much?

You're almost certainly familiar with "learning curve" effects in
which the more we learn in a process, the more efficient we become. 
The detailed statistical understanding started with ship & aircraft
production in World War II -- when it was discovered that every time
production doubled, costs dropped:
http://ax.losangeles.af.mil/se_revitalization/aa_functions/manufacturing/Attachments/18.%20The%20Learning%20Curve.htm

---

You're unclear with the formula, which I believe is y = -2x^2 + 3x +
2000, indicating that there's a fixed cost even when revenues (x) = 0.
 As x rises, it appears that whatever y is declines:

x = 1, y = 2001
x = 2, y = 1999
x = 3, y = 1991
x = 4, y = 1980 
.
.
.
x = 25, y = 825

What could a function like this decribe?  A non-linear function like
this could well be the INCREMENTAL cost/unit of sales: even with no
sales the firm has a base cost for production/research/overhead.  When
unit #1 is sold, costs actually rise slightly -- perhaps only from the
cost of shipping to that first unit to a customer.  But then they'll
fall consistently.

At some point in this model you actually see negative numbers.  When
above about x = 32 the function goes negative.  Is it possible to sell
100 or 1,000 of something and have your PER UNIT costs actually go
down?  Ask Henry Ford about the Model T, which started at $850 in 1909
when several hundred were shipped -- and was priced at $260 by 1925,
when 2 million of the cars were made.

---

I've made some gross asumptions here, not knowing precisely what your
formula's meant to represent.  And y =  -2x^2 + 3x + 2000 produces big
negative y's at low numbers -- so it starts to look like the old adage
about "Doing more and more with less and less.  Pretty soon we'll be
doing everything with nothing."

Often these non-linear relationships are good on a mathematical range
-- frequently limited by time.  Another well-known non-linear
relationship is Moore's Law, in which the size of a semiconductor
doubles every 18-24 months:
http://www.intel.com/research/silicon/mooreslaw.htm

Though it's merely a different way of looking at the "learning curve,"
this time based on semiconductor technology, even Gordon Moore sees
limits to the application of the "law" with his name:
http://www.pcw.co.uk/vnunet/news/2127129/gordon-moore-calls-law

Best regards,

Omnivorous-GA

Despite the nonlinearity of reality, much progress can be made by
relentlessly pounding round pegs into linear holes... such an
intention underlies calculus.

As far as bookkeeping goes, yes, a ledger (say of accounts payable) is
ideally a simple matter of adding and subtracting, although interest
expense starts to bring in more interesting math.

Furthermore accounting is much more than bookkeeping.  A specialized
kind of this is done by actuaries, who are required to certify the
funding levels of qualified retirement plans under "federal" law in
many countries.  The math here involves the probabilities of death of
participants at varying ages and is therefore quite nonlinear.

regards, mathtalk-ga

The economic side of this has been well stated in the above comments,
but it was written for an economist and not everyone will understand
it.  Here is the skinny:

If a Toyota plant is set up to make approximately 500 cars in a month,
then it has the employees, storage space, tools. . . to produce 500
cars.
If you ask this plant to produce only 10 cars, then the cost to
produce each car will be very high (which will make your revenues very
low or even negative).
If you ask this plant to produce 500 cars, then the cost to produce
each car will be low.
If you ask this plant to produce 2000 cars, then they will have to
hire alot of temporary workers, buy alot of machinery, rent storage
space. . . and the cost to produce each car will be very high.

This is a fairly typical example that shows in the short run that
costs are very non-linear.  But notice in the long run that many of
these expenses can be levelled out and you could have a much more
linear cost per car when looking at the long run (as long as you know
how many cars you'll be producing in the long run).

I applaud your effort to really understand what's going on, rather
than simply accepting it and plugging things in mindlessly.  The
problem is that this effort is wasted on economics, because, as your
professor told you, there is no real answer to your question.  You are
trying to be rigorous about something that by its nature is fuzzy. 
There is no real meaning to the negative linear term and postive
quadratic term in the example you provided.  This function is simply
chosen because it is the simplest one imaginable for which the cost
per unit has a local minumum--that is, for tc = 10000 -1000q + 200q^2,
the cost per unit (tc/q) is less for q=7 than for any other number of
units.  This behavior may be regarded as 'realistic' because, as
jack_of_few_trades points out, as you increase production, at first
efficiency goes up, but eventually you get swamped and efficiency goes
down again.  The reasons why this happens are many and complicated. 
Often, changes happen in discrete jumps--the whatsits you have to buy
are cheaper if you order at least 100 cases, or you move into a higher
tax bracket, or you have to rent another building.  There is no way
that all these factors are going to combine together to give you a
smooth simple function for total cost.  It is likely that you could
find some complicated function with lots of terms that would fit the
actual data better in any particular situation.  However, the
quadratic function has the advantage that it's simple and easy to
understand and manipulate (differentiate, etc) but it's nonlinear
which allows for things like a local minimum in cost per unit.

Actually though, I really don't like the fact that the linear term in
your example is negative.  There is a good reason for it to be
positive: surely there are some costs associated with production that
are at least almost linear.  If the linear term were positive, you
could interpret it as representing these costs, and the quadratic term
as representing the effect of getting 'swamped' (there still wouldn't
be any reason beyond simplicity for this term to be quadratic--though
I suppose the first term in the Taylor series expansion of most other
nonlinear terms you might choose to add would be quadratic.  Actually,
that might pacify you a bit when it comes to WHY tc = a + bq + cq^2
--- ANY (differentiable) function can be expanded about a point to
yield a + bq + cq^2 + dq^3 + .... This is called a Taylor series
expansion, and as long as you are pretty close to the point about
which you did the expansion, the terms generally get smaller and
smaller.  If you just keep the first three, you have your quadratic
function).  Besides, having a negative linear term means that total
cost initially decreases with increasing units produced.  That is, it
is cheaper to make 2 units than to make 1.  This seems unrealistic.  I
cannot conceive of a reasonable case in which, if you wanted 1 unit,
the cheapest course of action would be to make 2 and discard 1.

I agree with you that if you know how your total costs actually
behave, it doesn't do you any good to fit a quadratic function (you
can just look at the actual data and determine what's best) and if you
don't know how your total costs behave, you don't know how to choose
the coefficients in the quadratic function, and again this stuff
doesn't do you any good.  I don't believe that companies actually make
decisions based on quadratic models.  The utility of studying these
quadratic functions is more in giving you, the student, an
understanding of the mathematical concepts relevant to business.

That's my 2 bits anyway.  I should warn you that I know a bit about
math, but nothing about business...