The Calculus Diaries - Jennifer Ouellette [51]
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