Economics as a field has a lot of math built into it to apply theory to specific problems. I decided to focus specifically on consumer theory because it was an aspect of microeconomics that I feel has one of the most intuitive applications of multivariable calculus and specifically partial derivatives. The main idea behind consumer theory is that each agent has preferences within a certain bundle of goods which can be represented as a vector of inputs. Economists mathematically map preferences using utility functions. Utility is the idea that an agent has preferences over bundles. She can say “I prefer x to y” or “I am indifferent between z and y”. A utility function u(x) assigns a number to every bundle so that if the agent prefers x to y, u(x)>u(y).
Example: Suppose n=1 single good and the agent prefers less to more
The following utility functions represent that preference:
- U1(x)=-x
- U2(x)=5-x2
- U3(x)= 1/1+x
Because each of these functions has lower values for higher inputs of x, they all map out the preference stated above. Within this setting, agents are given a budget constraint which is a function of income y and the prices p of each good. The budget constraint is the set of all goods that are affordable given income and prices. This is the optimization constraint that we are used to in the setting of Lagrange multipliers from class. By maximizing the utility function with respect to the budget constraint, we can solve for utility-maximizing quantities of goods given their prices and an agent’s income.
Preferences can be represented diagrammatically when n=2 goods using indifference curves. Indifference curves are the levels of goods where the agent would be indifferent between different bundles of goods. They are downward-sloping and curved convexly. I then graphed these curves on Mathematica.
