The Calculus Diaries - Jennifer Ouellette [49]
I end up with a total cost of ($2,000y) + 1.5x. Multiply x and y—the number of flyers I produce with each print run times the number of print runs—and I get the total number of flyers printed over the course of one year: 12,000. I can simplify my equation by eliminating y entirely, because it is equivalent to 12,000 over x. This means I can rewrite the total cost equation as $2,000 times 12,000 over x, plus 1.5x, to get my “cost function,” and once I have that, it’s a relatively straightforward process to determine how often I should order a print run. I just need to minimize the sum of the storage costs plus the setup costs. I can find that “sweet spot” on the graph by setting the derivative of the cost function equal to 0 and then figuring out what value of x gives that answer. In this case, my best bet would be to make three print runs of 4,000 flyers each over the course of a year.
Now we estimate the expected revenue based on how much of a product I produce. How do I price my tulip bulbs in order to maximize my profit? Tulip bulbs incur a lot of initial costs, unless I opt for the sneaky alternative of stealing them from my globe-trotting neighbor, Carolus Clusius. Even then, my theft would yield a very limited supply. But it might bring in sufficient revenue to finance my little start-up venture. It takes about seven years to grow tulips from seeds: There would be costs associated with renting a greenhouse, buying fertilizer, watering the seeds, and so forth, over the seven-year incubation period for producing the tulip bulbs. And each bulb can produce only a few clones before expiring, so there will always be a limited supply of bulbs. (Only bulbs produce genetically identical offspring; seeds introduce genetic variability.)
Let’s assume that my fixed cost will be $100,000 and that it costs around $30 per bulb on top of that to “make” my product (the bulbs). So my function for cost is $100,000 + 30q, where q stands for the quantity of bulbs. The change in cost is called the marginal cost; it measures the incremental expense of producing one more tulip bulb. Then there is the marginal revenue, the rate at which the revenue increases with the production of one extra bulb—in other words, it’s a derivative.
Starting with an estimated production of 20,000 bulbs, I can determine a maximum and minimum price ( p), where at a given price, approximately 20,000 − 50p bulbs will be sold. At a maximum price of $400, there would be no buyers, and if I gave away the bulbs for free, all 20,000 bulbs would find a home with a buyer—if someone who pays nothing can be described as a buyer. If I sold them for $100, however, 15,000 bulbs would be sold, according to my spiffy formula (100 × 50 = 5,000, which we then subtract from the 20,000 total bulbs). So my revenue R is equivalent to the price per bulb multiplied by the number of bulbs I sell, or $1.5 million.
We want to set the marginal cost equal to the marginal revenue. That’s where the maximum profit will be. If the marginal revenue is greater than the marginal cost at a particular production level, then growing one more tulip means the increase in revenue will be greater than the increase in cost, and I make more profit. If the marginal cost is greater than the marginal revenue, I will also increase my profit, this time by growing fewer bulbs, because I will reduce my costs more than I will reduce my revenue. The answer: I should grow 9,250 bulbs and sell them at $215 each in order to maximize my profits.
That’s roughly how the market should work under ideal conditions, but we do not live in a simple world. Something has gone seriously amiss when a rare tulip bulb possesses more value than a farmhouse. The exponential decay curve decreases rapidly initially and then gradually slows its rate of change; the exponential growth curve exhibits similar behavior in reverse. But when a bubble forms, the result is a so-called boom-and-bust curve: Growth starts out increasing exponentially but peaks and collapses quite suddenly. Those who enter a hot