The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by Jennifer Ouellette

  House hunting is the ultimate experiment in comparison shopping: weighing different variables and seeking the optimal combination of those factors. In a sense, we are doing conceptual calculus. Mathematicians merely take this process to the next level by quantifying everything and organizing that data into an equation. In principle, we can turn our house-hunting experience into a multivariable optimization problem, similar to what we did to determine the optimal price for our tulip bulbs in order to maximize profit. We just need to find some way to quantify our subjective criteria.

  Because we need a continuous curve, we’ll assume we have an infinite number of houses to choose from. Anyone who has undertaken serious house hunting knows it can feel like infinity sometimes. Calculus will help us narrow the search by optimizing our happiness with our final choice. For simplicity’s sake, we will restrict our variables to two easily quantifiable qualities: square footage (q) and walkability (w), the latter based on an online “walkability score” algorithm. That gives us our function: f(w,q). The “curve” for this will look much different from the graph for a function with a single variable: It will be a contoured surface floating above a plane.

  Think of a map that only shows your location with two intersecting points: latitude and longitude, or the place where Wilshire Boulevard meets Figueroa Street in downtown Los Angeles. What is lacking is the altitude. Bringing in a second variable to our optimization problem is like adding altitude to a map, so we can tell not just where we are, but the elevation of that particular spot. Not only do we have the x and y axes on our Cartesian grid—representing walkability and square footage, respectively—we also have a third, the z axis, jutting out at an angle.

  If we merely consider square footage and walkability, what stops us from increasing those two variables to infinity to gain optimal happiness? Clearly we need some kind of constraint, and we find it in the price. We do not have an infinite amount of funds, so we need to build a third aspect into our “happiness function”: cost. We can assume that cost depends directly on size and walkability. One of the first steps prospective home buyers take is determining their price range. Go beyond that price range, and our happiness will start to decrease again, even though square footage and walkability continue to increase. If we can’t afford it, we won’t be as happy.

  We plot happiness as a function of our two variables (w,q) to get a nice smooth curvy plane that goes up, peaks, and descends after the peak. Then it is simply a matter of taking a derivative of each variable separately—this is called partial differentiation, or taking a partial derivative—and finding the value that sends both to zero. That will be the point(s) on our curve where the slope of the tangent plane is zero (horizontal). Wherever the tangent planes are flat is where we will find our optimal solution. That is where we will find maximum happiness with our choice. We find we must indeed make a trade-off between walkability and square footage. The price per square foot is significantly higher in very walkable locations, so we can’t afford as much square footage in prime areas and still stay near the peak of our happiness curve. Similarly, beyond a certain point, too little square footage will also decrease our happiness. Finding the “sweet spot” on our multivariable curved surface enables us to narrow our options down from infinity to three:

  Option 1: This is a three-bedroom, three-bath “architectural” townhouse featuring bamboo floors and cabinetry, and a wall of windows bathing the main loft area in sunlight. There are ample closets and a private two-car garage. The location isn’t as walkable as we would like, but the price per square foot is below market rate, so we would get a lot of space for the money.

  Option 2: This is a three-bedroom, three-bath condominium. The interior features dark woods, and Asian influences abound. It is slightly smaller, but there are many closets, and there’s a large balcony off the dining room. The drawbacks are the tandem parking spots in the communal garage (side by side is preferable) and the location, which is not very walkable.

  Option 3: This is a two-bedroom, two-bath unit with den in a prime location with excellent walkability. The design features Mediterranean influences, with a balcony, spacious living room and open kitchen, and luxurious baths. The price per square foot is significantly higher, so we can only afford one of the smaller units. There are fewer closets, and the parking spaces in the shared garage are tandem.

  All else being equal, how do we find the optimal choice among them? Calculus is less helpful here. Ultimately, one’s choice of home is an emotional, subjective decision. But we do engage in an approximation of an optimization problem whenever we comparison shop; it’s one way to bring some rationality to the process. Yet even then, our choice of how much to weight a given variable is highly subjective. Dutch psychologist Ap Dijksterhuis studies how house hunters are often subject to “weighting mistakes.” Given the choice between a larger home in the suburbs with a longer commute, and a smaller, more expensive home in a central location, most home buyers opt for the larger home. They underestimate the negative impact of a long commute on overall quality of life over time.

  DOWN THE RABBIT HOLE

  In the end, we choose Option 1. We trade our former prime downtown location for extra space, a shorter commute for Sean, and a private garage. Now the nail-biting anxiety sets in as we try to lock in our mortgage rate. The rates change literally every day. Two days after our offer is accepted, we get a nasty surprise: There is a new 1 percent hike in mortgage interest rates for condominium units. So that 5.25 percent interest rate we used to calculate our estimated monthly payments will be 6.25 percent instead.

  The earliest recorded mortgages date back to 1190 in England, when landowners would sell their land for a set fee, with no interest. Whatever the land produced would enable the buyer to pay the seller. Mort comes from the Latin word for “death,” while gage means a pledge to forfeit an asset for nonpayment of a debt. The modern concept is not much different: We want to buy a house, but we don’t have enough cash in hand to pay the full price, so we put down the cash we have and borrow the rest, using the house as collateral. There is a monthly payment, determined by the interest rate (usually fixed) and the lifetime of the loan (typically thirty years). At the end of that time, we will have paid off the principal loan plus the accumulated interest.

  It’s instructive to crunch the numbers and see firsthand why a mere 1 percent hike in the interest rate makes a significant difference on one’s monthly payments. Let’s round down the respective rates to 5 percent and 6 percent to make our calculations easier. If we took out a modest $100,000 mortgage at 5 percent, our payment would be $536.82 per month, compared to a $599.55 monthly payment at 6 percent interest. This assumes the interest is charged yearly. According to Mark Chu-Carroll, a computer scientist who blogs at Good Math, Bad Math, even this trivial difference can result in a higher monthly payment. We would only pay $525 per month at the 5 percent rate if the interest were calculated monthly. Just an extra $62 per month over 30 years adds up to roughly $22,320 in additional interest.

  Fortunately, our story has a happy ending: We are able to negotiate our original estimated interest rate with one lender. It helped that we had a sufficient down payment. Early in the 1900s, aspiring homeowners in the United States were required to have a 50 percent down payment on a five-year mortgage. Because very few people could meet those conditions, fewer than 40 percent of the population owned their own homes, compared to nearly 70 percent today, when 20 percent down is more common for a thirty-year mortgage. How long would it take to save 20 percent of a $300,000 home? That is $60,000—not an easy sum to accrue on a standard living wage. But it is a simple matter to figure out how much we’d need to save each month to reach that goal within five years. We simply divide $60,000 by five to get our answer: $12,000 a year, or $1,000 per month. No calculus required.

  However, that would only be the case if I took that money and stuffed it under a mattress. Common sense would dictate I deposit the funds in an interest-bearing account. Let’s be optimi
stic and assume that account yields 5 percent interest. How does that change the time needed to save a down payment? I am depositing $1,000 per month, but the money that’s been in the account for two years will have earned more money than the money I’ve just deposited.33 So I have to add the $1,000 I deposit that first month and calculate how much interest will accrue in five years, and then do the same for the second monthly deposit, and the third, and the fourth, and so on.

  Now it is a matter of adding together lots of smaller sums—or of taking an integral. Even though I am making monthly deposits, from a calculus standpoint, the funds are accumulating every instant. So for every interval of Δt (for time), we have accumulated $12,000 × Δt dollars, which stays in the bank for however much of that five-year period is left (5 years − t) and earns 5 percent interest. At the end of that five-year period, I will have saved $60,000 plus a bit extra in accrued interest—which I hope will be sufficient to cover the potentially exorbitant closing costs.

  What if you don’t have the required 20 percent deposit on a home—a common problem for those living in areas with especially high housing costs? Traditionally, you would be out of luck; no bank would approve your mortgage. There is very good reason for this. That down payment gives you equity in the house, the difference between your home’s assessed value and the amount of money you still owe the bank. But then alternative types of mortgage loans became increasingly available, some allowing borrowers to take out a mortgage with as little as 5 percent down. The trade-off for the lower down payment is usually higher interest rates and thus higher monthly payments.

  Then someone had the brilliant notion of offering adjustable rate mortgages (ARMs), in which the interest rate fluctuates over time, resetting to a new (higher) rate every few years. We have already seen that a small increase in the interest rate on a mortgage can make a huge difference in the monthly payment. The impact is even more dramatic with an ARM. Say you took out a loan of $100,000 at an adjustable rate over thirty years. You could easily afford the monthly payments at the introductory “teaser rate,” which could be as low as 1.2 percent for the first two to five years: roughly $331 per month. But then the interest rate would reset and jump to 7 percent, and suddenly you would be paying $617 a month. Unless you had a corresponding increase in income, you would quickly fall behind in your payments. Worse, some of those ARMs were interest-only loans, for which people would pay just the interest and the principal never decreased.

  Millions of people took on these risky loans; given the above, it’s fair to ask, what the hell were they thinking? Chances are, they weren’t doing the math. Or perhaps they believed that the value of their houses, and hence their equity, would continue to skyrocket, and they could sell their homes at a tidy profit before the interest rates reset.

  But nothing can expand forever—except, perhaps, the universe—and those homeowners were gambling that they could get out before the market softened or collapsed outright. Several economists warned that the bubble would burst, but their dire predictions did little to dampen the enthusiasm at the height of the housing frenzy. All the classic bubble conditions were present: high demand, limited supply, and an influx of ready cash as banks relaxed their lending standards and made millions of subprime loans to borrowers who—in retrospect—should never have received loan approval because they couldn’t afford the payments once the interest rates reset. When those buyers began to default en masse, the result was a record number of foreclosures.

  VIRTUAL WEALTH

  The fallout from Holland’s tulip mania crash was limited to a few overly enthusiastic traders and wealthy collectors. That’s because the Amsterdam Stock Exchange back in 1630 had the good sense not to get involved with the rampant speculation in tulip bulbs, marginalizing the economic impact when the bubble burst. Most Dutch traders were able to negotiate settlements for their debts, although the price of bulbs continued to fall for decades after the crash. Financial ruin hit those who had invested elsewhere while relying on the profit they expected to make on their tulip bulbs to pay those debts—profit that never transpired.

  That was the problem with the housing bubble: People speculated on the market, tapping into the equity on their homes to finance other projects—a new car, a lavish vacation, a kitchen remodel, an investment in a second rental property, or a vacation home. When the market crashed and their home values plummeted, those home owners found themselves owing more to the banks than their homes were worth. They had negative equity. Furthermore, investment banks had packaged those mortgages into complicated financial instruments that were sold to investors around the world, so when the waves of foreclosure hit, the massive losses incurred over a short period of time brought the global economy to its knees.

  Economists are going to be analyzing this housing market crash for decades before they fully understand how and why it happened. But anyone observing the virtual economy in the online game Second Life could glean some valuable insights, according to Cornell University economist Robert Bloomfield. He believes virtual economies like those in Second Life can provide useful simulations of the patterns of free markets—and the consequences of failing to self-regulate. In Second Life, players can buy virtual currency with their real-world dollars—250 “Linden dollars” roughly corresponds to one U.S. dollar. They buy and sell goods and services and engage in online investment schemes without all the pesky regulations hampering the free market in “meat space.”

  And therein lay the problem. In 2007, an in-game virtual investment bank, Ginko Financial, collapsed. The bank promised investors a whopping 40 percent return on their Linden money and made loans to other players at equally exorbitant rates. When those players failed to repay their virtual loans, investors panicked and made a run on Ginko to withdraw their funds, quickly outstripping the bank’s reserves. Nor were the losses purely virtual, since Linden dollars were purchased with real currency: Investors collectively lost the equivalent of 750,000 U.S. real-world dollars.

  Second Life creator Linden Lab responded by banning any virtual banks promising interest rate returns on deposits to investors. One year later, in the wake of the mortgage meltdown, revered financial titan Alan Greenspan reluctantly came to a similar real-world conclusion: Lending institutions cannot be trusted to regulate themselves—not because the free market doesn’t work, but because certain unscrupulous people cheated and “gamed” the system. It is human nature that is at fault, more than free-market economics. It makes a strong case for factoring irrational human behavior into any viable economic model. In fact, the burgeoning new field of behavioral economics focuses on studying how and why human beings don’t always act in their best self-interest.

  The parallels to our real-world economy are admittedly imperfect, but the economic lessons drawn from Second Life are compelling, because it is a model built from actual human behavior—raw data—not a programmed computer simulation. People do not always behave rationally (or nobly), and many economic theories fail to take this into account. Jonah Lehrer, author of How We Decide, asserts that the problem lies less with the actual models and more with the human brain. “People love models, especially when they’re big, complex, and quantitative. Models make us feel safe,” he writes. “They take the uncertainty of the future and break it down into neat, bite-sized equations. But we become so focused on the predictions of the model that we stop questioning the basic assumptions of the model. Instead, confirmation bias seeps in and we devote way too much mental energy to proving the model true.”

  Models can still yield intriguing insights. Reginald Smith, an analyst with the Bouchet-Franklin Research Institute in Rochester, New York, decided to map the spread of the collapse from its start in the housing markets of California and Florida in 2007 through October 2008. He found that the problems first emerged in housing stocks, then spread to finance stocks and mainstream banks before hitting the broader stock market in general. While his analysis didn’t shed much light on the why of the collapse, he noticed that his data b
ore a strong resemblance to a different kind of model: that used by scientists to chart the spread of forest fires, fashion trends, . . . and disease. Mathematically speaking, the credit crisis looks like an epidemic, wiping out wealth the way the Black Death decimated the population of medieval Western Europe.

  6

  A Pox upon It

  Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio.

  —THOMAS ROBERT MALTHUS

  It is a truth universally acknowledged that a young man in possession of good fortune must be in want of a well-stocked arsenal—at least if you’re one of the eligible young men who populate the blood-soaked satire Pride and Prejudice and Zombies . Author Seth Grahame-Smith invented an alternate history for Jane Austen’s much-beloved novel, in which a mysterious plague sweeps through the peaceful village of Meryton, turning residents into the walking dead, famished for fresh brains upon which to feast. Under those circumstances, any woman who can wield a weapon as well as a witty bon mot is doubly attractive.

  In bizarro Meryton, Elizabeth Bennett and her four sisters make up an elite zombie-fighting unit, well versed in the usual feminine accomplishments: music, needlepoint, watercolors, and of course, martial arts and weapons training. Their mission: wiping out the undead menace while finding suitable wealthy husbands. The very first ball at Netherfield is overrun by “unmentionables” who feast with abandon on the hapless guests, “sending a shower of dark blood spouting as high as the chandeliers.” Female characters debate whether or not it is “unladylike” to carry a musket (Elizabeth favors a katana, or samurai sword), and couriers routinely get eaten by zombies while relaying messages between houses. The local militia comes to town to exhume and destroy dead bodies, hoping to control the outbreak. And Elizabeth must defeat Lady Catherine de Bourgh and her merry band of ninjas to win the right to marry Darcy.