William
Avering
Blezz Shoes
Beckinforth, PU
22222
January 22, 2008
Calculus Student
Franklin &
Marshall College
Lancaster, PA
17604
Dear Calculus
Student:
I’m not really
sure whether I ought to write to you. I mean, I really need help,
and your incredibly kind and generous professor said that that you
could give me an answer. On the other hand, I hate to disturb
busy people, and (after all), what’s done is done. But on the
other hand--or, rather, back on the first hand actually--I would be
very relieved to have an answer to this question because it’s really
been bugging me. So I’ll go ahead and ask you, even though
probably it’s just too much bother.
Oh, geez.
So, a couple of
years ago, my grandfather (George Blezz) set me up in business in his
shoe factory, Blezz Shoes. I started off in sandals, but that
part of the business seemed too open-ended for me; I worked in boots
for a while but I didn’t really get a kick out of that, either; I
puttered around in the loafer division for a few months, but I didn’t
have a lot of energy for that. My latest stint has been in the
flip-flops division, and surprisingly enough, that worked really
well! It just seemed right for me, if you know what I mean.
My part of the
job is manufacturing the rubber for the soles of the flip-flops.
The market for this rubber isn’t particularly steady--you might say
it’s highly elastic--and so following the price of rubber was an
important part of what I did. I wound up, after exactly one year
on the job, with a huge batch of rubber, which you can see from the
chart below I could sell for a pretty decent price:
$36/bucket! January 1, 2008 was my lucky day!
Or maybe it
wasn’t. Now, the tricky part of this is that I didn’t just sell
the rubber; I also sold the machine that made the rubber. (See, I
knew this story would be really complicated. Sorry.) My
grandfather made a deal two years ago with a business leader named Eve
L. Vellen that she would sell us this machine for a half-million
dollars on January 1, 2007 (same day I started in flip-flops).
The other part of the deal is that at sometime in the future, at some
time before January 31, 2008, he would sell her back both the rubber
machine and all the rubber we had made with it since we bought the
machine. We’d sell the machine for a fixed price ($567,234), and
we’d sell her all the rubber at market value.
If we had sold
everything back to Vellen on July 18th (the 200th day of the year), the
machine would have sold for just the same price. But not only
would we have sold the rubber for a lot less (only $24/bucket), but we
would have made a lot less rubber by that date. That would have
been a double whammy! So we brought in a lot more money by
waiting, both by making more rubber and by selling it all for a higher
price.
Price of rubber
in dollars per bucket, graphed against day of the year, 2007.
But did I wait
too long? or not long enough? I don’t know! See, the
amount of rubber we made wasn’t particularly steady. It wasn’t
like we made 100 buckets a day, every day. No, sometimes we’d
turn the machine up high, and it would just crank the stuff out.
But then I’d worry that we were going to blow the motor, and I’d turn
it down. But then I’d worry that we weren’t making enough rubber,
and I’d crank the motor back up again. Sometimes we’d have lots
of supplies and lots of people to staff the machine, and then I’d see
everyone bumping into one another and I’d order half the people
home. Then we’d be understaffed and I’d have to give the crew
overtime to keep the machine going. You can see that in July I
sort of freaked out about the low price of rubber: I turned the
machine up as high as it would go--more than 200 buckets per day!--and
nearly ruined it. By the end of the year, everyone knew the
end was coming, and output went way down, to barely a tenth of what we
were producing in July.
Output
of the
rubber machine, in buckets/day, graphed against day of the year, 2007.
Part of me
thinks we would have gotten more money from Vellen if we had sold early
in December, when the price was nearly as high and we were cranking out
rubber at over 150 buckets per day. Another part of me is glad I
didn’t hang on longer, with production low and the price of rubber
falling. And still another part of me doesn’t even want to know
what the best answer would have been; as I said before, what’s done is
done and if I messed up, I’m not sure I want to know.
But since I’ve
written this long letter already, I might as well go ahead and ask what
you think. Knowing what you know about math, when do you think I
should have sold my grandfather’s machine and all the rubber?
Would selling earlier (or later) have made a difference, and if so, how
big a difference?
My grandfather’s
getting a new rubber machine on February 1, so if you’re going to
answer the question, it’d be nice to have it by then. Or maybe it
would be better to have it earlier, so I could think about what you
said. On second thought, February 1 ought to be fine; I can read
it then. February 1. Thanks.
Sincerely,
W. Avering
William Avering
Flip-flop Manager
Blezz Shoes
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