Utility Maximization

The Best Affordable Bundle

The consumer wants the highest indifference curve they can reach — but they face a budget constraint:

$$p_x x + p_y y = M$$

where $p_x, p_y$ are prices and $M$ is income. The budget line has slope $-p_x/p_y$ (the market exchange rate) and intercepts $M/p_x$ and $M/p_y$.

The optimal bundle is where the highest indifference curve is tangent to the budget line. At tangency:

$$MRS = \frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

This is the tangency condition: the consumer's willingness to trade ($MRS$) equals the market's exchange rate ($p_x/p_y$). Equivalently:

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$$

Marginal utility per euro is equalised across all goods — no reallocation of spending can improve utility. If $MU_x/p_x > MU_y/p_y$, the consumer should buy more $x$ and less $y$.

Corner solutions: if the MRS is always above or below the price ratio, the consumer buys only one good. This happens with perfect substitutes when one good is strictly cheaper per unit of utility.

Cobb-Douglas utility $U = x^a y^{1-a}$: the optimal shares are constant — spend fraction $a$ of income on $x$ and $(1-a)$ on $y$, regardless of prices. This makes Cobb-Douglas the workhorse for textbook examples.

Reading the tangency

Common pitfall: Equalising marginal utilities instead of marginal utility per euro. The rule is $MU_x/p_x = MU_y/p_y$ — a good with twice the marginal utility is only worth buying if it costs less than twice as much.

The Budget Constraint in Motion

The budget line shifts when income or prices change — and each shift reveals something about the consumer's response.

Income increase ($M$ rises): the budget line shifts outward (parallel — slope unchanged, because relative prices haven't changed). The consumer can reach a higher indifference curve. The path connecting optimal bundles at different income levels is the income-consumption curve (ICC); plotted as quantity vs income, it is the Engel curve.

Price decrease ($p_x$ falls): the budget line pivots outward from the $y$-intercept (the $x$-intercept moves right; the $y$-intercept stays). The consumer substitutes toward the cheaper good AND has greater real purchasing power.

$$\text{New } x\text{-intercept} = M / p_x' > M / p_x$$

Taxes and subsidies modify the effective budget constraint:

• A per-unit tax on $x$ steepens the budget line (raises $p_x$).
• A lump-sum tax shifts the line inward (reduces $M$, slope unchanged).
• Economic insight: a lump-sum tax raising the same revenue leaves the consumer better off than a per-unit tax, because it doesn't distort relative prices.

The budget set (the triangle below the budget line) contains all affordable bundles. Rational consumers choose a point on the line itself (non-satiation) — interior points waste purchasing power.

Two moves of the budget line

Tip: Watch the intercepts, not the middle: an income change moves both intercepts proportionally (slope fixed); a price change moves one intercept only (slope changes). That single observation decodes any budget-line diagram.