Supply and demand handed you a demand curve as a given equation. Consumer theory shows you where that curve actually comes from: a rational buyer with a fixed budget, choosing the bundle of goods that makes them as well off as possible. Three pieces. The budget constraint, Px·X + Py·Y = M, is every combination of goods X and Y a buyer with income M can just afford at prices Px and Py, a straight line whose slope, −Px/Py, is the rate at which the market lets you trade one good for the other. Utility, U(X,Y), is a function representing preferences, higher U means a preferred bundle, and an indifference curve is the set of bundles giving the same utility, sloping down and (ordinarily) bowed toward the origin because buyers are willing to give up less and less of a good the less of it they have (diminishing marginal rate of substitution). The marginal rate of substitution, MRS, is the slope of the indifference curve, how much of Y a buyer would willingly give up for one more unit of X and stay equally happy, and it equals the ratio of marginal utilities, MUx/MUy.
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How does a buyer choose the best bundle?
The buyer's optimum is where the highest attainable indifference curve is tangent to the budget line, the tangency condition, MRS = Px/Py. The economic content is simple once stated: at any point where MRS ≠ Px/Py, the buyer can reshuffle the bundle (buy a bit more of the cheap-relative-to-its-value good, a bit less of the other) and land on a higher indifference curve for the same money, so a rational buyer never stops adjusting until the two slopes match. For the widely used Cobb-Douglas utility function, U = X^a Y^(1−a), this tangency condition solves to a strikingly clean closed form: X* = a·M/Px and Y* = (1−a)·M/Py, the buyer spends a fixed fraction a of income on X and (1−a) on Y, regardless of the prices, a convenient special case Worked Example 3 solves fully and checks against the tangency condition directly.
How do income and price changes affect demand?
Two comparative-statics questions close the picture. First, what happens to demand when income changes, holding prices fixed? For an ordinary Cobb-Douglas buyer, both goods' optimal quantities scale up exactly proportionally with income, a computable fact, not folklore: a 50% rise in income produces exactly a 50% rise in each optimal quantity, meaning both goods carry an income elasticity of demand of precisely 1.0 for this utility form (Worked Example 3 verifies this to the decimal). Second, what happens when one good's price changes, holding income and the other price fixed? The observed change in quantity demanded can be decomposed, in principle, into a substitution effect (the buyer reshuffles toward the now-relatively-cheaper good, holding satisfaction constant) and an income effect (the price change also changes the buyer's real purchasing power, which independently shifts demand). For an ordinary good the two effects reinforce each other on the good whose price fell (both push quantity demanded up); the full algebraic split, the Slutsky equation, belongs to an intermediate microeconomics text (see Reading & Resources) and is not re-derived here, but you should be able to state the direction correctly, name which good the standard "law of demand" can fail for (a Giffen good, where the income effect on a very large budget share of an inferior good perversely dominates the substitution effect, an empirically rare curiosity worth naming once and setting aside), and recognize the two-effect language whenever a model or a colleague invokes it.