Budget Line and Consumer Equilibrium

An indifference map by itself only shows a consumer's preferences; it says nothing about what the consumer can actually afford. Affordability is shown by the budget line (also called the price line or price-income line) — the locus of all combinations of the two goods that a consumer can purchase by spending the whole of a given money income at given prices.

If a consumer has income $M$ to spend on goods $X$ and $Y$ priced at $P_x$ and $P_y$ respectively, the budget line is given by: $P_x X + P_y Y = M$.

Rearranged in the form of the $Y$-intercept, this is $Y = M/P_y - (P_x/P_y)X$, so the budget line's slope is $-P_x/P_y$, the negative of the ratio of the two prices, which represents the rate at which the market allows the consumer to exchange one good for the other. A change in income, prices unchanged, shifts the budget line outward (income rises) or inward (income falls) without changing its slope. A change in the price of either good, income unchanged, rotates the line around the intercept on the other good's axis, changing its slope.

Consumer equilibrium under the ordinal approach occurs at the combination of $X$ and $Y$ where the budget line is tangent to the highest indifference curve that the consumer's income allows the consumer to reach. At the point of tangency, the slope of the indifference curve (the marginal rate of substitution) equals the slope of the budget line (the price ratio): $MRS_{xy} = P_x/P_y$.

A second condition is also required for the equilibrium to be a stable, satisfaction-maximising one: the indifference curve must be convex to the origin at the point of tangency, so that $MRS_{xy}$ is diminishing in the neighbourhood of the equilibrium point. Any combination on a lower indifference curve leaves the consumer with satisfaction that could be increased at no extra cost by moving along the budget line towards the tangency point, and any combination on a higher indifference curve is simply unaffordable, so the tangency point is the unique best attainable choice. …