Utility Maximization Rule
What is Utility Maximization Rule?
The utility-maximization rule says consumers maximize satisfaction by equalizing the marginal utility per dollar spent across all goods.
A consumer is in equilibrium when the last dollar spent on each good yields the same marginal utility. If one good gives more marginal utility per dollar, the consumer shifts spending toward it until the ratios are equal, subject to the budget.
Utility Maximization Rule: a worked example
Mateo has $10. Tacos cost $2 and smoothies cost $3. His marginal utility from tacos runs 20, 16, 12, 8 for the first four; from smoothies it runs 30, 24, 18, 12. Convert each to marginal utility per dollar. Tacos give 10, 8, 6, 4 utils per dollar. Smoothies give 10, 8, 6, 4 utils per dollar. Buy in order of the highest per-dollar figure: the first taco and first smoothie at 10 each cost $5, then the second taco and second smoothie at 8 each cost another $5. That exhausts the $10 with 2 tacos and 2 smoothies, and the rule holds because 16 ÷ 2 = 8 and 24 ÷ 3 = 8. Total utility is 20 + 16 + 30 + 24 = 90 utils, higher than any other affordable bundle.
The mistake students make with utility maximization rule
The frequent error is setting marginal utilities equal instead of marginal utilities per dollar. A student hunts for a number appearing in both columns, finds 12 in each, and reports 3 tacos and 4 smoothies. The shortcut is tempting because the rule gets summarized as equal marginal utility. That bundle fails twice over. It costs 3 × $2 + 4 × $3 = $18 against a $10 budget, and a 12 util smoothie at $3 returns 4 utils per dollar while a 12 util taco at $2 returns 6, so the last dollars are not doing equal work. Divide every marginal utility by its own price first, then match the ratios and check that the bundle spends the budget exactly.
Utility Maximization Rule questions
What should a consumer do if MUx/Px is greater than MUy/Py?
Shift spending toward good X. A higher marginal utility per dollar on X means every dollar moved from Y to X buys more satisfaction than it gives up. As the consumer buys more X, the marginal utility of X falls, and as they buy less Y, the marginal utility of Y rises. The two ratios close the gap until they are equal, which is where total utility stops improving and the budget is fully spent.
Does the utility maximization rule require spending the entire budget?
Yes, the equal-marginal-per-dollar condition only identifies the optimum when the whole budget is spent. Leftover money could always buy another unit with positive marginal utility, which would raise total satisfaction. On an exam, a bundle that satisfies the ratio test but leaves $4 unspent is not the answer. Check two boxes every time: the marginal utility per dollar matches across goods, and quantity times price summed over all goods equals income exactly.
What happens to the optimal bundle when the price of one good rises?
Raising the price of good X lowers MUx/Px immediately, so that good now delivers fewer utils per dollar than good Y. The consumer buys less X, which pushes the marginal utility of X back up, and buys more Y until the ratios match again at the new, smaller affordable set. That adjustment is where the downward sloping demand curve comes from in consumer choice theory.
Formula / Example
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