1. We can begin by examining the two good, single consumer case. Each consumer starts with a budget constraint, representing how one's income is spent on a set of goods and services. We'll assume that there are only two goods to consider in the typical consumer budget and that all of this consumer's income is spent on these goods.
The Budget Constraint is:
We can take this equation, rearrange it to get:
What can we say about the rearranged budget constraint equation? First, we may notice that this rearranged budget constraint is an equation for a line (with a negative slope P2/P1 and vertical intercept I/P1). Intuitively, we may recognize that the ratio of prices represents a comparison of the cost to consumers of one unit of each good. Therefore, in a sense, we can say that P2/P1 is the ratio of the marginal cost of goods 1 and 2 respectively. Recalling our macroeconomic discussion of price indexes, we see that I/P1 is a measure of our good 1 purchasing power (i.e. how much of good 1 our income can buy). If P1 falls, I/P1 gets bigger - which means that we can buy more of good 1.
2. While the budget constraint represents how much a consumer is able to spend, we also need to know how much a consumer wants to spend on each good. That is, we need some information about this consumer's preferences regarding each good.
This information is found in an indifference curve. Indifference curves are drawn with two basic ideas in mind: (a) within certain limits, consumers always prefer more of everything to less (e.g. I'd prefer receiving 3 boxes of Cocoa Puffs and 2 boxes of Honeycomb to 2 and 1 box respectively); and (b) it is possible to derive the same satisfaction out of a variety of potential purchase combinations (e.g. when considering a potential cereal purchase, a consumer may be indifferent between buying 3 boxes of Cocoa Puffs and 2 boxes of Honeycomb versus 2 and 3 boxes respectively).
Therefore, by considering one's preferences, we see that consumers make purchasing decisions which depend upon the satisfaction (more formally, the utility) derived from a particular good. Each unit consumed (e.g. each box of cereal) in a given time period yields some sort of satisfaction. When we examine the amount of satisfaction derived from each unit consumed, we are considering something called marginal utility (MU). The slope of the indifference curve may be expressed as a ratio of the marginal utilities associated with each good (MU2/MU1). Rather than write this ratio, however, we can simplify things by calling it the marginal rate of substitution between goods 1 and 2 (MRS).
Where does equilibrium occur? Equilibrium occurs where the slopes of the indifference curve and budget constraint are equal. Mathematically, this occurs where MRS = P2/P1. This is an equilibrium point because at this point there is no reason to move away. The marginal rate of substitution can also be thought of as a ratio of marginal benefit that each good provides our consumer. Therefore, equilibrium in this setting involves equating the marginal benefit for two goods with their marginal cost. In simpler terms, we're saying that our consumer is getting out of each good exactly what they're worth.
We can demonstrate equilibrium graphically as well (see the graph below). Consider two different indifference curves: IC (the red curve) and IC' (the blue curve). Every point on IC (and IC') represents a different potential purchase of goods 1 and 2. As mentioned above, on each indifference curve our consumer is indifferent about purchasing any of the potential combinations along that curve. Consequently, along a particular curve, the only difference between each point is the amount of goods 1 and 2 that are purchased. The consumer is just as satisfied with any of the points on a given curve. Two things will determine which point gets selected: the consumer's income and the price of each good.
To find out where the equilibrium is, if one exists, we want to see if there is one point that is always prefered to every other point. We can begin by starting at a specific point (the one we pick isn't important). To keep things simple, we'll continue to assume that our consumer spends their entire income on these two goods. Start at point B, at the top of the Budget Constraint. Based on our discussion above, we know that points A and B provide this consumer with equal levels of satisfaction. That is, the consumer is indifferent between points A and B.
Although this consumer is indifferent between points A and B, this is not the case with points A and C. Point C is clearly better than point A for one important reason. At point C, our consumer gets more of both goods. As mentioned above, the basic idea behind these indifference curves (where each good's MU is greater than zero) is that "more is better". When comparing two points, like A and C, this is always true. When you get more of one good but less of the other, it may be true but not necessarily so (e.g. our consumer is not better off when moving from A to B).
Thus far we know that our consumer is indifferent between A and B,
but prefers C to A. Therefore, logic dictates that our consumer must
also prefer C to B. No matter which point we start with, our result would
be the same. In the end we realize that, if "all roads lead to point
C", point C must be the equilibrium.