Maximize utility by equalizing marginal utility per dollar

Compare what the last dollar buys, not marginal utility alone.

At an interior optimum, MU_X/P_X=MU_Y/P_Y. If the ratio for X is higher, move spending toward X and away from Y. Consumption changes alter marginal utilities until the ratios equalize. Equal marginal utilities are not required unless prices are equal.

The ratio is the added utility from the next dollar spent. If the last dollar on X yields 8 utils while the last dollar on Y yields 5, shifting a dollar from Y toward X can increase utility. Diminishing marginal utility then lowers X’s ratio as more X is consumed, while consuming less Y can raise Y’s marginal utility. Reallocation continues until no affordable shift improves the total.

The complete interior condition is (MU_X)/(P_X)=(MU_Y)/(P_Y)=marginal utility of income, with the budget exhausted when all goods are desirable and divisible. Equalizing MU_X and MU_Y would ignore that one unit may cost several times as much as the other.

The next dollar

The next sandwich provides 30 utils and costs $6, or 5 utils per dollar. The next smoothie provides 18 utils and costs $3, or 6 utils per dollar. The consumer can raise total utility by shifting the next available dollars toward smoothies until diminishing marginal utility changes the comparison.

Because units may be indivisible, a shift often requires releasing enough dollars from one good to buy a unit of another. If sandwiches cost $6 and smoothies $3, giving up one sandwich frees funds for two smoothies. Compare the utility lost from the sandwich with the utility gained from the two successive smoothies, not only one smoothie.

With indivisible units, exact equality may be impossible. List affordable bundles or select units in descending marginal-utility-per-dollar order while respecting changing marginal utility and the budget. The best bundle is the affordable one with greatest total utility, not necessarily one that spends every cent if remaining money cannot purchase another unit.

With an $11 budget, the first book and first meal cost exactly $11 and yield 61 utils. Buying only a first and second book is unaffordable at $12. Ranking individual ratios suggests the first book then first meal, but the final bundle should always be checked against all affordable combinations when units and prices are lumpy.

Individual demand can be derived by finding the utility-maximizing quantity at each possible price. When X becomes cheaper, its marginal utility per dollar rises initially, encouraging more X. Repeating the optimization at different prices traces a downward-sloping individual demand curve.

Corner solutions are possible. If one good’s marginal utility per dollar remains below the other’s over every affordable unit, the consumer may buy none of it. The equal-ratio rule describes an interior optimum where positive amounts of both goods are consumed. A test item that provides an unaffordable or dominated bundle should not force an equality that the discrete schedule cannot achieve.

Repair an unequal allocation

A consumer’s last dollar on X yields 9 utils and the last dollar on Y yields 4. Holding the budget fixed, spending should shift toward X. “Buy more of both” does not preserve the constraint, and “choose Y because its total utility is larger” compares a total with a marginal per-dollar decision.

Check the direction of reallocation in words: move a dollar away from the lower ratio and toward the higher ratio. Many distractors reverse this direction or compare marginal utilities without prices. Write the two ratios side by side before deciding.

Do not compare marginal utilities without prices

A 30-util item is not automatically the better purchase than an 18-util item. The exam tests the marginal utility obtained per dollar spent.