Utility Maximization

The Optimal Bundle

The consumer reaches the highest indifference curve on their budget constraint:

$$p_x x + p_y y = M$$

where $p_x, p_y$ are prices and $M$ is income. The budget line slope is $-p_x/p_y$.

The optimal bundle occurs where the highest indifference curve is tangent to the budget line, so $MRS = p_x/p_y$, or:

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

Marginal utility per euro is equalised. If $MU_x/p_x > MU_y/p_y$, buy more $x$.

Corner solutions: if MRS always exceeds the price ratio, buy only one good (perfect substitutes).

Cobb-Douglas utility ($U = x^a y^{1-a}$): spend fraction $a$ of income on $x$ and $(1-a)$ on $y$.

Pitfall: Equalising $MU_x = MU_y$ instead of $MU/p$. A good with double the $MU$ is only worth buying if it costs less than twice as much.

Budget Line Shifting

The budget set contains all affordable bundles; rational consumers choose a point on the line itself.

For a price decrease, the new $x$-intercept is $M / p_x'$.

Taxes and subsidies: A per-unit tax on $x$ steepens the line. A lump-sum tax shifts it inward (reducing $M$, slope unchanged).

Insight: A lump-sum tax leaving the same revenue makes the consumer better off than a per-unit tax because it avoids relative price distortion.

Tip: An income change moves both intercepts proportionally (slope fixed); a price change moves one intercept only. This decodes any diagram.