The income consumption path (or income expansion path) traces how a consumer's optimal bundle changes as income rises, with prices held constant. An Engel curve graphs the relationship between income and quantity demanded for a specific good. The shape of the Engel curve—upward (normal good) or downward (inferior good)—reveals how tastes change with income.
Draw budget lines at increasing income levels, find optimum at each, and trace the path. Separately graph income vs. quantity for one good to see the Engel curve.
You already know how a consumer finds their optimal bundle using indifference curves: the optimal choice sits at the tangency between the budget line and the highest reachable indifference curve. The price of each good and the consumer's income determine where the budget line sits. The income consumption path answers a simple question: what happens to that optimal bundle as income rises, holding both prices fixed? Each income level produces a parallel budget line shifted outward, and each budget line has its own tangency point. Connecting all those tangency points traces the income consumption path through the indifference map.
The shape of the path depends entirely on how preferences are structured. If the consumer always allocates a constant share of income to each good — as with Cobb-Douglas preferences — the path is a straight ray through the origin, meaning both goods scale proportionally with income. More interesting cases arise when the ratio shifts. If the path veers toward good X as income rises, the consumer is buying proportionally more X — X is a normal good. If the path bends away from good X (the consumer actually buys less of X at higher income), X is an inferior good. Ramen noodles and bus rides are the classic examples: as income rises past a threshold, people switch to more expensive substitutes, and demand for the inferior good falls absolutely, not just proportionally.
The Engel curve takes information from the income consumption path and projects it onto a simpler two-dimensional graph: income on the vertical axis, quantity of one specific good on the horizontal axis. Each point on the Engel curve corresponds to one income level and the optimal quantity of that good at that income. A positively sloped Engel curve confirms a normal good; a negatively sloped portion reveals an inferior good at that income range. Note that a good can switch from normal to inferior as income passes through different ranges — luxury goods show an Engel curve that slopes even more steeply than income (budget share rises), while necessities have a flatter slope (budget share falls as income rises).
The connection to your prerequisite on diminishing marginal utility is this: as you accumulate income and consume more of any good, the marginal utility from additional units declines. At some income level, you may stop wanting more of a cheap staple — not because it made you worse off before, but because the marginal utility of alternatives has overtaken it. This is the microeconomic foundation for inferior goods. The Engel curve also lays the groundwork for income elasticity of demand, which you will study next: it measures the percentage change in quantity demanded per percentage change in income, and its sign directly reflects whether the good is normal or inferior.