The Calculus Diaries - Jennifer Ouellette [48]
TIPTOE THROUGH THE TULIPS
Why did the tulip market go boom, then bust? There were several contributing factors, but it had mostly to do with simple supply and demand. The tulip bulb was a rare commodity from the start, although ordinary bulbs were often sold by the pound. Then some of the tulips contracted a mosaic virus that altered the color of the blooms, streaking their petals with scarlet. Those varieties were even more rare, attracting wealthy collectors and commanding an even higher price. Demand grew so rapidly that the supply of bulbs could not keep pace, and prices rose and rose.
Dutch residents were flush with extra cash after the end of hostilities with Spain. Amsterdam’s merchants were thriving at the center of the lucrative East Indies trade, earning profit margins as high as 400 percent in a single voyage. So the market could absorb—temporarily—the outrageously high prices demanded for tulip bulbs. But no market can sustain that kind of exponential growth rate indefinitely. Eventually the price became so high that very few buyers were able to meet it. Once that first buyer didn’t show up for the sale, a domino effect occurred. Demand dropped suddenly, panic ensued, and the bubble burst, with dire economic consequences for those who had speculated on the market.
Let’s imagine that I am a tulip dealer in seventeenth-century Holland, eager to turn a tidy profit in this burgeoning industry. I am drawn to tulip bulbs because they command a hefty price and there are still a substantial number of buyers willing to pay that price. Also, flowers are pretty. I just have to be careful not to raise the price so much that I chase away prospective buyers; if prices get too high, demand will drop, and my profits will never materialize. Ideally, I want to maximize my profit—which will be the gross revenue I bring in with the sale of my exotic tulip bulbs, less the associated costs I incur to obtain them—and minimize my production costs. Calculus can help me do this.
The cost of producing a given product depends on how many items are produced. If I decide to print flyers advertising my tulip bulbs, there is a basic cost I will incur for setting up the equipment to do so. It’s probably not worth that initial cash outlay to print only a hundred flyers; I’m better off printing twelve thousand flyers and stocking up for the future. Or am I? There might be storage costs to consider, and these must be offset against the money I save by printing more flyers. Perhaps it would be better to make two print runs of six thousand flyers each. I need to strike just the right balance between these two factors.
Assume I have fixed setup costs of $2,000 for the printing press. The cost of storing twelve thousand flyers is minimal—$3 per year—but I still need to factor that into my financial planning. With these two bits of information, I can devise an equation that gives me the total cost of maintaining inventory plus the produced and setup costs. I designate y as the number of print runs, and each run costs $2,000. The number of flyers produced and stored is represented by x. But it’s not going to be $3 constantly; x fluctuates over time, unlike y, which is fixed. My storage space is full after every production run, but as I hand out flyers over time, the number in storage steadily decreases, until all the flyers are gone and my storage costs are back to zero. So I take the average storage cost,