Sodden jeans and sneakers are not a pleasant sensation. It is a cool, cloudy day for Anaheim and late enough in the afternoon that our clothing takes longer to dry than it would on a warmer day. While we wait, Sean explains that there is a calculus problem in our current plight: The rate at which our clothing dries—that is, the rate of evaporation of water from the fabric—forms an exponential decay curve. It is similar to the rate at which a cup of hot coffee cools until it reaches thermal equilibrium with its surroundings.
The coffee cools off very quickly at first, but as it gets closer to thermal equilibrium, that rate of cooling slows down and eventually levels off. This is because the amount of heat lost is proportional to the temperature of the coffee: It is determined by the ratio of the excess heat to the lower temperature limit—how cool the coffee can get, usually ambient room temperature. So as the coffee cools down and gets close to room temperature, there is less excess heat and thus a smaller ratio between the two variables. And the rate of cooling levels off.
The same thing happens with the evaporation of the moisture in our clothing. Plot the rate of evaporation as a function of time, and you can see this in the resulting curve: There is a steep drop initially, followed by a gradual leveling off. The alert reader will note that because we are dealing with a rate of change, we must be taking a derivative. That means we can find an answer to the question, “How fast is the water in our clothes evaporating at x time?”—a form of the velocity function, similar to determining our instantaneous speed in chapter 2—by finding the slope of the tangent line along that particular point on our curve.
We experience this exponential decay curve firsthand and soon find ourselves wondering, Will we forever be slightly damp? It is beginning to feel that way, and we have dinner reservations in an hour at the Blue Bayou restaurant—the sole fine dining establishment in Disneyland, situated just inside the Pirates of the Caribbean ride. We end up squishing our way over to the gift shops in New Orleans Square in search of a change of clothes, where the sales clerk assures us this happens all the time. They do a brisk business, thanks to Splash Mountain.
And thus we find ourselves, an hour or so later, seated in the Blue Bayou’s fake outdoor grotto in matching Pirates of the Caribbean hooded sweatshirts, my outfit completed by a jaunty newsboy cap with a skull-and-crossbones motif to hide my hopelessly tangled hair. By this time Sean is very much in need of a drink, and the Tinkerbell Fruit Punch with a fairy light garnish simply isn’t going to cut it. Alas, there is no alcohol to be found in the Magic Kingdom, depriving us of a prime opportunity to work out the calculus of inebriation. (Oh yes, it can be done.) We content ourselves with a sugar rush instead and split the signature dessert: a boat-shaped “cookie” with an edible sail featuring the obligatory skull and crossbones. It is a pirate’s life, indeed.
5
Show Me the Money
It is clear that economics, if it is to be a science at all, must be a mathematical science . . . simply because it deals with quantities. . . . As the complete theory of almost every other science involves the use of calculus, so we cannot have a true theory of economics without its aid.
—W. S. JEVONS
Like beauty, an object’s intrinsic value rests in the eyes of the beholder. One man’s priceless treasure is another man’s culinary delight. In seventeenth-century Holland, a hungry sailor mistook a rare tulip bulb that was on display for an onion and stole it from a local merchant. The merchant chased him down Amsterdam’s busy streets, catching up just in time to find the sailor “eating a breakfast whose cost might have regaled a whole ship’s crew for a twelvemonth.” That was the going rate for a single bulb at the height of what is now called tulip mania. Incensed, the merchant had the sailor thrown into prison for his crime.
That is one of the more outlandish anecdotes about tulip mania popularized in the nineteenth century with the publication of Charles Mackay’s Extraordinary Popular Delusions and the Madness of Crowds. Today, the excesses of tulip mania are the stuff of legend, trotted out as a cautionary tale whenever economists find themselves analyzing catastrophic bubble markets.31 Modern economists dispute many details of Mackay’s account, which was based on dubious source material, but it makes for lively reading. And while the sailor’s story might not be true, it epitomizes the kind of irrational exuberance and frenzied overvaluation of assets that so often serve as harbingers of economic disaster.
How did the tulip become such a collector’s item in the first place? Holland is widely known as the land of colorful tulips, and one would think the bulbs would be a cheap commodity. But the bright bell-shaped flower is actually a relative newcomer to the country. In 1593, a Dutch botanist named Carolus Clusius returned from a trip to Constantinople with a few precious tulip bulbs and planted them in his garden, supposedly to study them for medicinal purposes. Then his neighbors broke into the garden and stole some of the bulbs, figuring—correctly—that the exotic flora would bring in a pretty penny. Thus was born the Dutch tulip trade and the onset of a collective mania that drove prices to dizzying heights.
One recorded list of items traded for a single tulip bulb included a bed, some clothing, and a thousand pounds of cheese, but prices rapidly escalated beyond such humble items. In 1624, a buyer offered 3,000 guilders (equivalent to a year’s earnings) to a man in Amsterdam in exchange for a dozen specimens of the rarest tulip, known as Semper Augustus and identifiable by its blue-black petals accented with streaks of crimson and a sprinkling of white. A sale of forty bulbs for 100,000 guilders was recorded in 1635. The most expensive bulbs were far too valuable to be planted, so instead it became the fashion among their (once) wealthy owners to display the plain bulbs—well away from the gaze of famished sailors.
Speculators were desperate to cash in on the gravy train, mortgaging whatever they could to raise capital to invest in a few “starter bulbs,” in hopes of jump-starting a lucrative business in the tulip trade. One transaction records the trade of a farmhouse in 1633 in exchange for three rare bulbs. There was even a thriving futures market for tulip bulbs, with business often being conducted in local taverns; at the height of the frenzy, one bulb changed hands ten times in a single day. But the tulip bubble burst almost as quickly as it formed. One day a buyer didn’t show up with the cash, and panic set in and spread. Within days, bulbs that had sold for staggering sums were now “worth” roughly one-hundredth of their former value.
Such are the harsh realities of supply and demand. Those Dutch tulip speculators might have benefited from a spot of calculus. (Unfortunately it hadn’t been invented yet.) The tools of calculus are particularly well suited to the financial sector, which deals heavily in rates of change: inflation, interest rates, mortgage rates, and the impact of supply and demand on pricing can all be described by functions linking one feature to another. We can use the derivative to determine the rate at which one factor changes relative to another, and we can employ the integral to determine the cumulative effect of any ongoing process. In the case of tulips—or any product, for that matter—supply and demand are interdependent quantities: A change in one affects the other, and choices about production and supply affect the profit you make. The integral comes into play when calculating interest, whether accumulating interest on a savings or retirement account or calculating the interest on a mortgage loan.
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. Amster
dam’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, which will be half of $3: $1.50.32
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 new market early may reap enormous profits, but as more and more people enter the fray over time and prices go up and up, there are fewer and fewer buyers. Eventually the market will hit a peak and collapse—and the “decay” will be steep and sudden. That’s what happened with tulip mania. What happens when the bubble mentality comes to real estate and literally hits people where they live?
HOME SWEET HOME
It’s a bit disheartening to tour a foreclosed home; a sense of loss seems to permeate the space. Despite being only four years old, the town house we are touring has seen better days: The floors are scuffed, the window screens are torn, and the previous owners have absconded with the appliances as compensation for losing their home. While the unit is spacious, the interior feels cramped and dingy on this overcast afternoon, particularly since the electricity has been turned off. But the building is in a prime location, a few blocks from many of our favorite shops and restaurants.
Two years after moving to Los Angeles, we have joined the ranks of nervous house hunters, cautiously dipping a toe into the volatile Southern California real estate market to test the waters. We are in no hurry. Our rental apartment is sufficient for the short term: It’s in a very walkable location in downtown Los Angeles, with free parking in the garage across the street, and a friendly full-time concierge named Mike. But we are running out of space, having merged two households when we got married. Most of our books are in storage. The dining-room table is strewn with Sean’s books and physics papers, while the second bedroom performs double duty as my office and a cramped guest room. And there isn’t nearly enough closet space.
It is both the best and worst of times to buy. We have been patiently waiting for prices to drop to m
ore affordable levels, and in the wake of the economic collapse of September 2008, housing values are plummeting. The median home price in California dropped 41 percent—more than double the 16 percent decline in median home prices for the United States as a whole—between February 2008 and February 2009, according to the California Association of Realtors, as a tidal wave of foreclosures drove down values. Nobody knows how much farther prices will fall, and that uncertainty means everyone is a wee bit skittish, particularly the banks: Loans are much harder to come by, even for highly qualified buyers. The process is fraught with anxiety—starting with the search for a suitable home.
Every prospective home buyer knows there is no such thing as the perfect place. The exercise of touring several homes helps us get a sense of the market, what we can afford, and the factors that matter most to us. We know we need three bedrooms (or two bedrooms with a den), with parking for two cars. We prefer central locations with shops and restaurants within easy walking distance. Such areas tend to have higher housing costs, so we know we will have to make a trade-off between square footage and prime location. We like to entertain, so a spacious living area with open kitchen is desirable. And we don’t want to do any heavy remodeling. Can we find the optimal combination of our desired features—within our price range?
One unit has an awkward layout. Another boasts a dramatic curving staircase in the foyer, but there is no extra space for an office—a prime consideration for a professional writer. I like a faux-Spanish townhouse, but it is less to the taste of my more modernist spouse. One home has bizarre blue plastic kitchen cabinetry; in another, the bathtubs are freakishly small for my six-foot-one-inch spouse; and yet another suffers from cheap flooring and astronomical home association fees. All are preferable to the dingy “penthouse” unit with stained carpets, chipped tiles, and a “rooftop deck” lined with sticky tar paper. As for that first battered foreclosure, we decide the living/ dining area is too cramped for our needs.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 11