How does marginal utility relate to indifference curves in microeconomics?

The importance of indifference curve analysis to neoclassical microeconomic consumer theory can hardly be overstated. Until the early 20th century, economists had been unable to provide a compelling case for the use of mathematics, particularly differential calculus, to help study and explain the behavior of market actors. Marginal utility was seen as undeniably ordinal, not cardinal, and therefore incompatible with comparative equations. Indifference curves, somewhat controversially, filled that gap.

After the subjectivist revolution in the 19th century, economists were able to deductively prove the importance of marginal utility and highlight the law of diminishing marginal utility. For example, a consumer chooses product A over product B because he expects to gain more utility from product A; economic utility essentially means satisfaction or removal of discomfort. His second purchase necessarily brings less expected utility than the first, otherwise he would have chosen them in reverse order. Economists also say the consumer is not indifferent between A and B due to the fact he ended up choosing one over the other.

This kind of ranking is ordinal, such as first, second, third, etc. It cannot be converted into cardinal numbers such as 1.21, 3.75 or 5/8 because utility is subjective and not technically measurable. This means mathematical formulas, being cardinal in nature, do not apply cleanly to consumer theory.

Though notions of indifference bundles existed in the 1880s, the first treatment of actual indifference curves on a graph came with Vilfredo Pareto's book "Manual of Political Economy" in 1906. Pareto also authored the concept of Pareto efficiency.

Indifference bundle theorists said that consumer economics did not need cardinal numbers; comparative consumer preferences could be demonstrated by pricing different goods in terms of each other or bundles of each other.

For example, a consumer might prefer apples to oranges. However, he might be indifferent between having one set of three oranges and two apples or another set of two oranges and five apples. This indifference demonstrates equal utility between sets. Economists can calculate the marginal rate of substitution between different goods.

Using this, an apple can be expressed in terms of fractions of oranges and visa versa. Ordinal utility can then, on the surface at least, give way to cardinal numbers. Through this, microeconomists derive some minor conclusions, such as the existence of optimal sets given budget constraints, and some major conclusions, including that marginal utility can be expressed in magnitudes through cardinal utility functions.

This argument rests on a few assumptions that not all economists accept. One such assumption is called the continuity assumption, which states that indifference sets are continuous and can be represented as convex lines on a graph.

Another assumption is that consumers take prices as exogenous, also known as the price-taking assumption. This is one of the most important assumptions in general equilibrium theory. Some critics point out that prices are necessarily determined dynamically by both supply and demand, which means consumers cannot be taking exogenous prices. Consumers' decisions presuppose the very prices their decisions affect, making the argument circular.