by Jasper Gilley
The house in which I grew up has a driveway, which slopes down from the house to the street at a very modest incline of perhaps 1 or 2 degrees below the horizontal. Ordinarily, this is almost unnoticeable, but a few times every year, someone would use a garden hose at the top of the driveway. On such occasions, the water would travel down the ~30 feet of driveway at an average rate of something like 2 inches/second, taking sometimes meandering and sometimes direct paths to get to the drain at the curb. Observing this process was inevitably a source of infinite entertainment for me, noting the water’s progress and sometimes aiding it in its quest.
As it turns out, mathematicians, physicists, and computer scientists are equally infinitely entertained by observing and modeling analogous such processes, which are confusingly termed in their disciplines gradient descent. Whether the gradient being descended is a map of differential temperature zones, a literal map of mountains’ elevations, or a map of mechanical strain as a function of applied stress, gradient descent is a powerful method for optimizing systems.
A sidenote: the mathematical construct used to analyze the “slope” of multidimensional functions is simply called the gradient. If you remember high school calculus, the gradient is literally just the multidimensional analog of the derivative. That is, if the derivative is the slope of a single-variable function, like so:
…then the gradient is the slope of a multidimensional function, like so:
This means that the seemingly complicated mathematical process of gradient descent can be visualized in an extremely intuitive way:
This is literally just the same process as water flowing down the driveway. The best part about the gradient operator, though, is its symbol, which is known as the Nabla symbol:
You’d be lying if you said that wasn’t the most elegant mathematical operator you’ve ever seen.
Anyway, one of the fields to which the gradient is most applicable is thermodynamics. (The term thermodynamics literally means “changing temperatures” – thermo = temperature, dynamics = the study of changing systems – which necessitates the existence of temperature differences, which can be expressed either by the derivative or by the gradient, depending on the system’s dimensionality.) When temperature differences exist, work can be done, via what is known as a “heat engine”:
How this diagram should be viewed is that the heat engine exploits the difference in temperature between the “hot reservoir” and the “cold reservoir” to generate some form of energy, which is in the diagram the blue right outflow. However, not all of the temperature difference can be converted to energy – by necessity, some of the temperatures end up equalizing (this is formally known as the Second Law of Thermodynamics.)
Here’s the interesting part: in economics, a firm can be viewed as exactly analogous to a heat engine.
Let me break this down. Let’s say you’re a consumer looking to buy an iPhone. You have determined that you’d rather have an iPhone than an extra $500 in your bank account (or $1000, if you’re in the market for an iPhone X.) So in the diagram above, you’re an individual with too much cash. You pay Apple $500 (the red arrow), which Apple counts as revenue. Apple then distributes about $250 to a large number of people in China (this is the Expenses arrow), who build your iPhone and ship it to you. (Essentially, these people have determined that they’d rather have about $2.50 than an hour of their life, so they’re the people with too little cash.) Apple then pockets the other $250 (the Profit arrow.)
If you just consider this one exchange, you’re looking at a single-variable difference in supply and demand. So it could be effectively analyzed using single-variable calculus (e.g., the derivative.) Of course, thousands of people are doing the same thing as you every minute, which is why Apple has about $285 billion laying around. If you imagine every individual having their own unique point in xy-space, you might begin to be able to visualize it more like this:
The pink points are people who are buying iPhones, and the blue points are the people who are producing them. Apple simply exists to facilitate cash exchange between the two groups of people. What is really interesting, of course, is that this map is constantly changing as people transact. So immediately after buying an iPhone, your own pink point goes back down to 0, but as your iPhone gradually becomes obsolete, your point begins creeping back up.
You can also look at it in the reverse way:
That is, the individual with too much supply (in the case of Apple, of labor, or any of the materials needed to make an iPhone, like glass, steel, or microchips) gives said supply to the firm, which uses its unique advantages (which I have termed proprietaries) to bring the product to the consumer. Proprietaries essentially are whatever makes a business more valuable than its competitors: usually some combination of unique technology, network effects, economies of scale, and/or branding. It’s incredibly important to note that in the above diagram, the blue consumer gets more product than the red producer put in, since proprietaries add to the value of the end product. That is, in the case of an iPhone, the producer gives $250 worth of product to Apple, and Apple gives $500 worth of product to you. So Apple’s proprietaries (namely, mostly branding and economies of scale) are worth exactly $250 per unit sold. Or, they’re worth $285 billion in total, which is obviously a lot.
Looking at economic transactions through a thermodynamic lens also yields an explanation of what is sometimes known as Schumpeter’s Law of Creative Destruction. As transactions occur, variance in the gradient field of supply and demand tends to cancel out, as discussed above. But a firm’s intrinsic value derives from its providing a conduit for supply/demand value equalization. This explains why firms inevitably have a limited lifespan – the process of their cashing in on value alters the value differential landscape, eventually destroying future value, unless the firm adapts to accommodate the landscape. (Viewed through this lens, being an entrepreneur is nothing more than being able to recognize supply/demand value differentials and creating a channel to equalize them.)
What’s really interesting is the manner in which the proprietaries are created. Namely, the cash value of creating the proprietaries is not equal to the cash value of the proprietaries themselves! So when Steve Jobs (or some unknown designer) designed the Apple logo, for instance, they didn’t do anything other than some clicking in Adobe Illustrator. Even when you factor in the cost of paying that designer and the cost of Adobe Illustrator, there’s no way it adds up to the literal monetary value that that logo has provided Apple over the years. So that designer literally created value ex nihilo! This might be the first time the analogy between firms and heat engines breaks down, because a fundamental tenet of all physics is that energy (the analog to value) can neither be created or destroyed. Barring hitherto-undiscovered quantum mechanical subtleties making the conservation of energy not exactly true, this is a prime example of the fact that humans are not quite the same as hydrogen atoms. Perhaps the real question, however, should be why humans aren’t the same as hydrogen atoms – because for most of human history, economic growth was stagnant, and the analogy would have applied all but perfectly.