Budgets and marginal utility · printable packet

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1. What you will learn

Analyze budgets and marginal utility using explicit assumptions, calculated results and a stated limit of the model.

2. Starting point

Opportunity cost compares a choice with its next-best feasible alternative. A budget adds a spending constraint: price times quantity must be added across purchases. Fractions let us compare benefits obtained from the same amount of spending.

3. Terms and units

4. A model of one consumer's choices

A consumer-choice model asks how a person can select among affordable bundles. A bundle specifies quantities, not merely a list of product names. Two servings of soup and one bread roll form a different bundle from one soup and two rolls. The relevant alternatives depend on preferences, prices and the available budget. The model keeps these components separate so that a change in price is not automatically described as a change in taste.

Utility is a representation of the consumer's preferences. In a numerical classroom example, utility units make marginal comparisons possible. They are not money and do not provide a scientific ruler for comparing the happiness of different people. Assigning one person's meal twenty utility units and another person's meal ten does not establish that the first person is twice as happy. Even for one person, a utility representation needs care: the meaningful economic prediction concerns the ordering of feasible choices and the trade-offs represented by the model, not a claim that satisfaction is directly observable on a universal scale.

Our tabular examples supply utility values as assumptions. That makes the exercise determinate: the learner works out what follows from those values. In an actual investigation, preferences would need to be inferred from choices, reports or other evidence, and might change with the situation. A hungry consumer may rank another meal differently from a consumer who has just eaten. Holding circumstances fixed is therefore part of the comparison, not a statement that preferences never respond to circumstances.

The model also distinguishes a prediction about chosen purchases from a judgment about a person's character. A consumer choosing the greatest modeled benefit from a budget need not be selfish in everyday language. The person's preferences may include sharing, supporting a cause or helping a friend. The optimization describes a relationship among preferences and constraints. It does not tell the learner what preferences are admirable, nor does it establish that every real decision follows the model exactly.

Term	What it means
Utility	A model's representation of a consumer's satisfaction or preferences.
Total utility	The utility obtained from the whole chosen bundle.
Marginal utility	The change in total utility from one additional unit, holding other consumption fixed.
Budget constraint	The spending limit set by income and prices.
Marginal utility per dollar	Additional utility divided by the price of the additional unit.
Diminishing marginal utility	A pattern in which successive units add less utility, with relevant circumstances fixed.

Another way: Draw the affordable set before choosing

Let A and B denote quantities of two goods. If A costs two dollars and B costs one dollar, a twelve-dollar budget permits bundles satisfying 2A + B <= 12. The inequality includes bundles that leave some money unspent. The boundary 2A + B = 12 contains bundles that exhaust the budget. Setting B to zero gives A = 6; setting A to zero gives B = 12. These intercepts answer how much of a single good could be bought if the consumer bought none of the other.

The boundary's slope describes the market trade-off. Buying one more A requires two extra dollars. If the budget is already exhausted, those dollars can be obtained by giving up two B units. Thus the opportunity cost of an A, expressed in B, is the price ratio 2/1 = 2. The opportunity cost of a B is half an A. These are affordability trade-offs set by prices, separate from how much the consumer personally values either good.

An increase in the budget with unchanged prices shifts both intercepts outward proportionately. A rise in the price of A with budget and B's price unchanged lowers the A intercept but leaves the B intercept fixed. The budget boundary rotates inward. Calling this merely 'having less money' misses a useful distinction: the nominal budget is unchanged, but its purchasing possibilities have narrowed. The changed relative price also changes the opportunity cost of A in terms of B.

Divisibility matters. If both goods are divisible, every nonnegative point below or on the boundary is feasible. If goods must be purchased in whole units, only certain grid points are feasible. The maximum single-good quantity may then be the integer part of budget divided by price, with some money left over. An example that explicitly permits divisible quantities should not silently round the answer down. An example requiring whole units should not recommend half a ticket. Always identify the feasible set before searching it for the best bundle.

Another way: Marginal utility compares additions

Suppose a consumer's total utility from successive servings is 0, 12, 21, 27 and 30. The marginal utilities are 12, 9, 6 and 3 because each is the difference between adjacent total-utility entries. Total utility rises throughout the sequence even though marginal utility falls. Diminishing marginal utility therefore does not mean that the consumer dislikes an additional serving. It means that each addition contributes less than the preceding one. Marginal utility becomes negative only if the additional unit lowers total utility.

The word marginal refers to an increment at the current position. If the consumer already has two servings, the next serving adds six utility units, not the average utility of all three servings. Using total utility instead would attribute benefits from earlier purchases to the present increment. Likewise, comparing the first unit of one good with the fifth unit of another requires knowing that those are actually the next available units at the current bundle.

When prices differ, comparing marginal utility alone is insufficient. An A unit offering twelve utility units at a price of two dollars provides six units per dollar. A B unit offering five utility units at a price of one dollar provides five units per dollar. Under the stated next-unit comparison, A offers the larger benefit per dollar. This ratio places both alternatives on a common spending basis. Its units are utility per dollar, not dollars per utility unit; reversing it would reverse which ratio should be preferred.

The ratio comparison does not by itself prove that a whole bundle is optimal. After buying A, its next marginal utility may fall, and the available budget is smaller. A complete solution must update the relevant marginal values or examine all feasible bundles. With indivisible purchases, even a greedy sequence based on the next ratio can fail to find the best combination because an expensive item may use up funds that could finance a better collection of smaller items. A table of affordable bundles can settle that discrete problem directly.

Another way: Equal marginal returns and their qualifications

For divisible goods at an interior optimum, with a smooth diminishing-marginal-utility model and a binding budget, marginal utility per dollar is equal across goods. If A's ratio were higher, a small amount of spending could move from B to A and increase utility without changing total expenditure. Such a profitable reallocation contradicts the claim that the original bundle was best. At the optimum there is no remaining feasible small reallocation that raises utility under the stated assumptions.

The condition is MUa/Pa = MUb/Pb. It is not MUa = MUb unless the prices are also equal. If A costs twice as much as B, its marginal utility must be twice B's at such an interior optimum. The absolute numbers depend on the utility representation, but the spending comparison consistently uses both marginal utilities and both prices. The budget equation is a second requirement: equal ratios alone do not establish that the bundle is affordable or that all desired spending opportunities have been considered.

An optimum can occur at a corner, where the consumer buys none of a particular good. In that situation, the usual interior equality need not hold. The consumer may prefer spending all available money on A even after comparing the next affordable increments of B. A zero purchase is not automatically a calculation error. Similarly, if extra consumption gives no benefit or money itself has value for another use, exhausting the budget is not automatic without the relevant preference assumptions.

Price changes can affect the consumer through more than one channel. A cheaper A changes the relative price, encouraging substitution toward it compared with other goods. It also changes what the existing budget can buy. This purchasing-power effect may increase demand for a normal good or reduce demand for an inferior good. These are effects within an economic model, not labels of product quality or the buyer's social status. Later lessons will distinguish demand shifts caused by income changes from movements along demand caused by the good's own price.

To report a solution responsibly, state its scope. 'The next A has the higher marginal utility per dollar at this bundle' is a precise conclusion from two supplied ratios. 'The consumer should spend the whole budget on A' is a much stronger statement requiring information about later units and feasible alternatives. An answer should not silently move from the first statement to the second. Economic reasoning is strongest when the verbal conclusion matches exactly the comparison the arithmetic has established.

5. Planning refreshments with a fixed voucher

A fictional student receives a twelve-dollar refreshment voucher that can purchase fruit cups at two dollars each or bread portions at one dollar each. The voucher cannot be converted into cash, and the student is choosing only today's own meal. At the current bundle, the next fruit cup adds twelve utility units and the next bread portion adds five. The fruit ratio is twelve divided by two, or six utility units per dollar. The bread ratio is five divided by one, or five. A small divisible-spending version of this comparison favors shifting spending toward fruit at this point.

Now suppose cups and portions are indivisible, and the student has only one dollar left. The higher fruit ratio cannot make an unaffordable cup feasible. A bread portion is the only additional purchase available. Alternatively, suppose two dollars remain but the next two bread portions each add five units while the fruit cup adds only nine. Then spending those two dollars on bread adds ten units, exceeding the fruit cup's nine. The appropriate comparison uses the actual remaining alternatives, not a ratio remembered from an earlier bundle.

The planner can prepare a table of feasible combinations with their total utilities and choose the highest supplied value. This discrete check complements the marginal rule rather than contradicting it: the usual equality condition was derived for small feasible adjustments under stronger assumptions. The resulting recommendation is conditional on the student's supplied preference table, the voucher rules and the stated prices. It does not establish that everyone should choose the same meal or that the student's utility can be compared numerically with a classmate's. The useful product of the model is a clear explanation of how preferences and a constraint jointly determine a choice.

6. Check the tempting shortcut

Higher total utility is not the same as higher marginal utility, and higher marginal utility is not the same as higher marginal utility per dollar. A next-unit ratio comparison also does not identify a complete optimum without later marginal values, affordability and any indivisibility restrictions.

7. In the fictional Alder model, a consumer has a budget of 6 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 14 utility units and the next unit of B provides 6 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

8. In the fictional Birch model, a consumer has a budget of 12 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 16 utility units and the next unit of B provides 7 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

9. In the fictional Cedar model, a consumer has a budget of 18 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 18 utility units and the next unit of B provides 8 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

10. In the fictional Dune model, a consumer has a budget of 24 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 20 utility units and the next unit of B provides 9 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

11. Guided practice

In the fictional Elm model, a consumer has a budget of 30 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 22 utility units and the next unit of B provides 10 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

12. Guided practice

In the fictional Dune model, a consumer has a budget of 24 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 20 utility units and the next unit of B provides 9 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

13. Guided practice

In the fictional Fern model, a consumer has a budget of 36 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 24 utility units and the next unit of B provides 11 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

14. Practice

In the fictional Grove model, a consumer has a budget of 42 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 26 utility units and the next unit of B provides 12 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

15. Practice

A consumer has a budget b dollars and a single good costs p dollars per unit, where b and p are positive and units are divisible. Construct the maximum affordable quantity using b and p.

16. Somewhere new

A visitor has a refreshment voucher that cannot be saved or exchanged for cash. The visitor's recorded next-unit preferences and the posted prices are supplied below. Separate the next-spending comparison from the intercept describing how much one good alone the voucher could buy. In the fictional Island model, a consumer has a budget of 54 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 30 utility units and the next unit of B provides 14 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

17. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

18. Test question

In the fictional Juniper model, a consumer has a budget of 60 dollars. Good A costs 2 dollars and good B costs 1 dollar. At the current bundle, the next unit of A provides 32 utility units and the next unit of B provides 15 utility units. Utility is a within-person classroom index, not comparable across people. Calculate the next-unit utility per dollar for A, for B, and the maximum affordable A units if no B is bought. These are marginal comparisons, not a claim that all later units have the same utility.

19. What you can do now

Reconstruct the model without the worked example. Explain each requested measure's units and identify an assumption that the conclusion depends on.

Working for the steps left to you

	Calculated value
A marginal utility per dollar
B marginal utility per dollar
A-only budget intercept

	Calculated value
A marginal utility per dollar
B marginal utility per dollar
A-only budget intercept

	Calculated value
A marginal utility per dollar
B marginal utility per dollar
A-only budget intercept