Economics: A Study of Human Behavior through Mathematics
When people hear about economics, often their minds immediately jump to finance. While certainly related, most subfields of economics deal with modeling human behavior. This is especially true in microeconomics, which traditionally focuses on decision making of the “rational” man.
The basic model consists of the following elements. The person, or “agent,” has a set of preferences that, through axiomization, gives rise to a utility function which represents how “happy” the agent is. The agent then tries to maximize his utility given some constraint – commonly her budget. To illustrate, consider an agent whose utility function consists of only apples and bananas, represented by a and b respectively. One possible function is U(a, b) = a + b. Another could be U(a, b) = ab2. A possible constraint could be something like 0.75a + 0.80b ≤ 2.80. This can be interpreted as trying to buy 75 cent apples and 80 cent bananas up to a total cost of 2.80. If we could divide these goods infinitesimally, we can see that given a utility function like those above, it’s in the agent’s best interest to buy as much as possible.
A natural question to ask is how the function is determined. In theory, the exact function form is some-thing that represents the agent’s underlying preferences, which in practice can be quite nebulous. What is important to realize is that as long as the function captures these preferences, the problem is specified well enough. Utility functions are ordinal and not cardinal, in that economists only care about the order when comparing bundles of goods. In other words, given any utility function, the exact utility doesn’t matter, as long as the order of bundles is preserved, which should again reflect the underlying consumer preferences.
To illustrate, we can show without much trouble that utility functions give the same optimal bundle under x all else equal should yield the same bundle. Ultimately, economics translates the question of what constitutes rationality into an applied math question
of optimization. A classic problem would look something like the following: