Lesson 06: Consumer Choice & Utility Maximization

Warm-Up

First slice of pizza after practice: life-changing. Second: great. Third: good. Fourth: fine, whatever. Fifth: why did I do that? Each slice adds less satisfaction than the one before — that's diminishing marginal utility, and it's not just a pizza fact. It's the reason demand curves slope down and the engine behind one of the most reliable calculation questions on the AP exam: the utility-maximization table.

Core Concept

Utility: measuring satisfaction

Utility is satisfaction, measured in imaginary units called utils.

• Total utility (TU): all satisfaction from consuming a given quantity.
• Marginal utility (MU): the extra satisfaction from one more unit: MU = ΔTU / ΔQ.

Law of diminishing marginal utility: as consumption of a good rises (holding everything else fixed), the marginal utility of additional units eventually falls.

Read the table's structure: - TU rises as long as MU > 0, but at a decreasing rate. - TU is maximized where MU = 0 (slice 5). - MU < 0 → TU falls (you're worse off — a rational consumer stops before this even at a price of zero).

[GRAPH: Two stacked panels sharing X-axis "Quantity". Top: TU curve rising at a decreasing rate, peaking at Q = 5, then declining. Bottom: MU curve declining, crossing zero at Q = 5. Dashed vertical line links TU's peak to MU = 0.]

The utility-maximizing rule

Money is scarce; goods cost different prices. The rational consumer asks not "which good has higher MU?" but "which good gives more utility per dollar?"

Bang per buck = MU / P

Utility-maximizing rule (consumer equilibrium): allocate the budget so that

MUx / Px = MUy / Py     (spending the entire budget)

If MUx/Px > MUy/Py, shift a dollar from Y to X: you gain more than you lose. Reallocation continues — and diminishing MU guarantees the ratios converge — until per-dollar marginal utilities are equal.

Step-by-step table method (exam workhorse)

Movie streams cost $4; arcade visits cost $8. Budget = $28.

Buy in descending order of MU/P, tracking the budget: 1. Arcade 1 (6.0) — $8 spent, $20 left 2. Arcade 2 (5.0, tie) — $16 spent 3. Stream 1 (5.0, tie) — $20 spent (when ratios tie and the budget allows, buy both) 4. Arcade 3 (4.0) — $28 spent → budget exhausted

Final: 3 arcade visits + 1 stream, spending exactly $28. Verify with a swap test: trading arcade 3 (32 utils, $8) for streams 2 and 3 (16 + 12 = 28 utils, $8) would lose 4 utils, so no reallocation improves the bundle. Optimal: 1 stream + 3 arcade = 20 + 120 = 140 utils.

Practical procedure: rank all purchases by MU/P, buy from the top, stop when the budget runs out, and verify with a swap test at the margin.

From diminishing MU to the law of demand

Why do demand curves slope down? If the price of a good falls, its MU/P rises above other goods', so consumers buy more of it until diminishing MU pulls the ratio back into balance. Lower P → higher quantity demanded. Diminishing marginal utility is the micro-foundation of Lesson 3's demand curve. (The income and substitution effects from Lesson 3 are the complementary explanation.)

Worked Examples

Example 1 (easy): TU ↔ MU

Given TU: 0, 15, 27, 36, 42, 45 for quantities 0–5. Find MU of each unit and where diminishing MU begins.

Solution: MU = 15, 12, 9, 6, 3. Diminishing from the 2nd unit onward (15 → 12). MU never hits zero here, so TU rises throughout.

Interpretation: MU is always a difference of adjacent TU values. If given MU and asked for TU, add them up.

Example 2 (medium): Equalize the ratios

Books cost $10, coffees cost $2. Currently MU of the last book = 50 utils and MU of the last coffee = 8 utils. Is the consumer maximizing? If not, what should change?

Solution: Books: 50/10 = 5 utils/$. Coffee: 8/2 = 4 utils/$. Not maximizing — books deliver more per dollar. Buy more books, less coffee. As book quantity rises, MU of books falls (and coffee's rises) until ratios equalize.

Interpretation: Compare per-dollar, never raw MU — the raw comparison (50 vs. 8) is the trap answer.

Example 3 (AP-style): Full budget allocation

Pizza slices $3, sodas $1, budget $12.

Solution: MU/P pizza: 8, 6, 4, 2, 1. MU/P soda: 9, 6, 3, 2, 1. Purchase sequence by descending ratio: soda1 (9.0, $1 spent), pizza1 (8.0, $4 spent), pizza2 & soda2 (tie at 6.0 — buy both, $8 spent), pizza3 (4.0, $11 spent), soda3 (3.0, $12 spent). Result: 3 pizzas + 3 sodas = $12 exactly. TU = (24+18+12) + (9+6+3) = 72 utils. Swap test: no affordable exchange raises utility — dropping soda3 frees only $1, and no $1 purchase beats its 3 utils.

Interpretation: Ties are fine; budget exhaustion + no beneficial swap = maximized.

Common Mistakes

1. Comparing raw MUs instead of MU/P. A 50-util book isn't better than an 8-util coffee if the book costs 25× as much. Per-dollar or nothing.
2. Confusing TU and MU. "Utility is maximized" refers to TU; the rule operates on MU. TU max (free goods) is where MU = 0, not where TU = 0.
3. Stopping diminishing-MU at negative MU. Diminishing MU starts when MU begins falling — usually the 2nd unit — not when it turns negative.
4. Ignoring the budget constraint. The right answer spends (at most) exactly the budget. If your allocation over- or under-spends, re-rank.
5. Assuming equal quantities of both goods. Equilibrium equalizes per-dollar marginal utility, not quantities purchased.

Practice Problems

9. (FRQ-style) Nadia has $11 to spend on smoothies ($3 each) and granola bars ($1 each).

(a) Calculate marginal utility per dollar for the first two units of each good. (b) Determine the utility-maximizing combination. Show that it exhausts the budget. (c) Calculate Nadia's total utility at the optimum. (d) The price of smoothies falls to $1.50. Without recomputing the full table, explain the direction of the change in Nadia's smoothie consumption, referencing the utility-maximizing rule.