How to Use the Utility-Maximizing Rule
Buy the bundle where marginal utility per dollar is equal across goods, MUx ÷ Px = MUy ÷ Py, and the entire budget is spent.
The Utility-Maximizing Rule formula
Calculator
Test a bundle against both conditions of the rule: equal utils per dollar, and the whole budget spent.
The budget the bundle has to fit. In the example, a student with $12.
Tacos at $2 in the worked example.
How many units of X the consumer is buying.
MUx at this bundle. The third taco is worth 12 utils.
Drinks at $3 in the worked example.
How many units of Y the consumer is buying.
MUy at this bundle. The second drink is worth 18 utils.
Both goods return 6 utils per dollar, so moving a dollar between them gains nothing. The bundle spends the budget exactly, so the second condition holds too.
- Marginal utility per dollar, good X
- 6
- Marginal utility per dollar, good Y
- 6
- Cost of the bundle
- $12
- Budget condition
- Whole budget spent
The last dollar spent on good X returns 6 utils.
The last dollar spent on good Y returns 6 utils.
PxQx + PyQy comes to $12 against an income of $12.
How to calculate Utility-Maximizing Rule, step by step
- 1Compute MU per dollar for each good. Divide each good's marginal utility by its price to get MUx ÷ Px and MUy ÷ Py.
- 2Spend on the higher ratio. Buy the good with the larger MU per dollar first, since that dollar buys more satisfaction.
- 3Recompute after each purchase. Marginal utility falls as you consume more of a good, so the ratios change after every unit.
- 4Stop when the ratios are equal. Keep going until MUx ÷ Px = MUy ÷ Py, the point where moving a dollar between goods gains nothing.
- 5Confirm the budget is spent. Equal ratios alone are not enough; the bundle must also satisfy PxQx + PyQy = Income.
Worked example: Utility-Maximizing Rule
A student has $12, tacos cost $2 and drinks cost $3. Buying 3 tacos and 2 drinks costs (3 × $2) + (2 × $3) = $6 + $6 = $12, spending the whole budget. At that bundle the third taco has MU = 12, so 12 ÷ 2 = 6 utils per dollar, and the second drink has MU = 18, so 18 ÷ 3 = 6 utils per dollar. Both conditions hold, so utility is maximized.
Utility-Maximizing Rule questions
Why do both conditions have to hold?
Equal MU per dollar only fixes the mix of goods, while spending the entire budget fixes the size of the bundle. A bundle that leaves income unspent can always be improved by buying more.
What do you do if MUx ÷ Px is greater than MUy ÷ Py?
Buy more of good x and less of good y. As x consumption rises its marginal utility falls, and as y consumption falls its marginal utility rises, pushing the two ratios toward equality.
Does the rule work with more than two goods?
Yes, the condition extends to every good in the bundle: MUa ÷ Pa = MUb ÷ Pb = MUc ÷ Pc, with the budget still fully spent.
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