Indifference curves are a graphical representation of how much value an individual receives from various combinations of consumption. We measure value through the catch-all term “utility”, an concept for the value, well-being, satisfaction, benefit, etc. that someone receives. As I have expressed before, utility is a relatively abstract concept; people constantly face trade-offs between current satisfaction and later success or happiness. We can examine utility from a short or long-term perspective in most cases, pondering the existential question of which happiness is the most important—the current, tangible feeling or that of later, but not-assured, joy.
In this case, however, we will be sticking to the neo-classical interpretation of utility, where we assume people are totally rational and weigh all their decisions fully, implying that they will aim to make choices that are both responsible and maximize their utility in the long run. With this definition of utility in mind, we can tackle the concept of an indifference curve.
An indifference curve aims to display all the various combinations of consumption for two goods that will give an individual the same level of utility. Because the combinations all give you the same level of satisfaction (utility), you won’t care which one you pick—that is, consumers will be indifferent to the various combinations along the indifference curve.
An individual doesn't just have one indifference curve - they will have an identically shaped curve at each different level of income. These different curves account for the fact that incomes and budgets can change, and a new income may mean there is no "optimal" point on the original indifference curve.
So, for a given person, a graph of indifference curves for different levels of income could look like this:
The indifference curves pictured above are just one example of how indifference curves can look. The slope of an indifference curve (how steep/flat it is) will be different depending on the individual’s preferences. My favorite real-world example for this is the tradeoff between hours worked and leisure time - depending on if a person loves or hates work, their curve will be sloped differently.
The slope of the indifference curve is known as the marginal rate of substitution, or MRS. This term seems a lot more complex than it is, really, the MRS documents the marginal utility the individual will gain (or lose) from moving up or down on the indifference curve. Marginal utility is simply the benefit the person gains from consuming an additional unit of a good. You will recall the indifference curve plots 2 different goods, so the marginal rate of substitution is telling us “how much does this person prefer good X to good Y? What is the value they get from an additional unit of this good?”
The MRS is always negative for the same reason that indifference curves are convex: diminishing marginal utility. For all goods, the amount of satisfaction you receive for each additional until you consume is decreasing. A hungry person eating a cheeseburger finds the first one amazing, the second one filling, the third one hard to eat, and the fourth one nausea-inducing (trust me, I would know). The value they receive from each one goes down, and the same idea applies to all goods. Even our example person who hates work will stop valuing leisure after a while, when he has no money! So, you can see how the marginal rate of substitution would be negative—the utility a person receives from each additional unit is decreasing. The convexity of the curve is also due to this assumption—as you get further out towards the extremes of consumption, the slope is skewed towards one variable. For example:
It should be noted that all the scenarios laid out here for slope and typical shapes of curves are just the straightforward scenarios in economics that aren’t very applicable to real life. Indifference curves can be useful for modeling consumer reactions to income, welfare, or wage changes (read: all labor-related) but are otherwise hard to create because most people are spending their income or time on more than 2 things. The traditional methods of teaching indifference curves rely on examples like “popcorn versus candy when you have $20 at the movies”, or “spending your income on hot-dogs versus hamburgers”, both of which seem like awful decisions. Nevertheless, the concepts of indifference curves, preferences, diminishing marginal utility, and the marginal rate of substitution are important in theoretical economics.
Indifference curves capture consumer behavior in scenarios where they are choosing between 2 goods with all their income/time (realistic, right?). The slope of the curve shows how much benefit they get from consuming more or less of a good, and the curve is convex because of diminishing marginal utility—the concept that each additional unit consumed of a good is less valuable to the person. This is not very applicable in the real world outside of labor-market issues.