Your first coffee of the workday: essential. The second: nice. The third: fine. The fourth: jittery regret. Each cup adds less satisfaction than the one before — that's diminishing marginal utility, and it isn't just a coffee fact. It's the reason demand curves slope down, and the engine behind one of the most reliable calculation questions on the CLEP exam: the budget-allocation table.
Utility is satisfaction, measured in imaginary units called utils.
MU = ΔTU / ΔQ.| Cups of coffee | TU | MU |
|---|---|---|
| 1 | 20 | 20 |
| 2 | 36 | 16 |
| 3 | 46 | 10 |
| 4 | 50 | 4 |
| 5 | 50 | 0 |
| 6 | 46 | −4 |
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 (cup 5). - MU < 0 → TU falls — 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.]
Money is scarce and goods cost different prices, so the rational consumer asks not "which good has higher MU?" but "which good gives more utility per dollar?"
Utility per dollar = MU / P
Utility-maximizing rule (consumer equilibrium): allocate the budget so that
MUx / Px = MUy / Py (while spending the entire budget)
If MUx/Px > MUy/Py, shift a dollar from Y to X: you gain more utility than you lose. Reallocation continues — and diminishing MU guarantees the ratios converge — until per-dollar marginal utilities are equal.
A household's weekly discretionary budget is $12. Workday lunches out cost $3; specialty coffees cost $1.
| Q | MU lunches | MU/P lunches | MU coffees | MU/P coffees |
|---|---|---|---|---|
| 1 | 24 | 8.0 | 9 | 9.0 |
| 2 | 18 | 6.0 | 6 | 6.0 |
| 3 | 12 | 4.0 | 3 | 3.0 |
| 4 | 6 | 2.0 | 2 | 2.0 |
| 5 | 3 | 1.0 | 1 | 1.0 |
Buy in descending order of MU/P, tracking the budget: 1. Coffee 1 (9.0) — $1 spent, $11 left 2. Lunch 1 (8.0) — $4 spent 3. Lunch 2 and coffee 2 (tie at 6.0 — when ratios tie and the budget allows, buy both) — $8 spent 4. Lunch 3 (4.0) — $11 spent 5. Coffee 3 (3.0) — $12 spent → budget exhausted
Final: 3 lunches + 3 coffees, spending exactly $12. TU = (24 + 18 + 12) + (9 + 6 + 3) = 72 utils. Verify with a swap test: dropping coffee 3 frees only $1, and no $1 purchase beats its 3 utils, so no reallocation improves the bundle.
Practical procedure: build the MU/P columns, buy greedily from the highest ratio down, stop when the budget runs out, and verify with a swap test at the margin. The right answer spends (at most) exactly the budget — and equilibrium equalizes per-dollar marginal utility, not the quantities purchased.
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.
Two complementary forces produce the same result when a price falls: - Substitution effect: the good is now cheaper relative to alternatives, so buyers substitute toward it. - Income effect: the same dollars now stretch further — real purchasing power rises — so buyers can afford more.
Diminishing marginal utility plus these two effects are the micro-foundation of the demand curve you drew in Lesson 3.
1. A — MU = ΔTU/ΔQ: the extra satisfaction contributed by one more unit. - B: TU/Q is average utility — the classic look-alike distractor. - C: total satisfaction from all units is total utility. - D: price is what you pay, not the satisfaction you receive; the two are compared, never equated by definition. - E: the first unit's utility is just one particular marginal value, not the definition. - Fix: "Marginal" always means the change from one more unit — a difference, never a total or an average.
2. C — TU peaks exactly where the last unit adds nothing: MU = 0. Beyond that point MU turns negative and TU falls. - A: MU is at its maximum near the first units, long before TU peaks. - B: rising MU would mean TU is accelerating upward, not peaking. - D: negative MU means TU is already falling — past the peak, not at it. - E: MU = P is the optimal-purchase condition when buying at a price, not the TU-maximum condition. - Fix: TU max ↔ MU = 0; TU falling ↔ MU < 0; TU rising ↔ MU > 0.
3. A — Each additional unit eventually contributes less extra satisfaction than the one before — MU declines as quantity rises. - B: TU keeps rising while MU is positive; it only falls once MU turns negative. - C: MU usually stays positive for many units; diminishing begins when MU starts falling, not when it goes negative. - D: switching goods is a budget-allocation behavior, not the content of the law. - E: utils are a measurement convention; the law is about the pattern of satisfaction, not measurability. - Fix: Diminishing MU = the increments shrink; it says nothing about totals falling or values turning negative.
4. B — Per dollar: coffee = 24/4 = 6 utils/$; lunches = 50/10 = 5 utils/$. Coffee delivers more per dollar, so shift spending toward coffee until the ratios equalize. - A: chooses lunches by comparing raw MUs (50 > 24) — the exact trap the per-dollar rule exists to avoid. - C: the bundle is optimal only when MU/P is equal across goods; 6 ≠ 5. - D: total utilities are irrelevant to the margin — decisions happen unit by unit. - E: equilibrium equalizes per-dollar marginal utilities, not quantities. - Fix: Divide every MU by its price before comparing — per-dollar or nothing.
5. E — MU of the 4th unit = TU(4) − TU(3) = 54 − 45 = 9. - A (54): reports TU at 4 units instead of the marginal difference. - B (12): computes TU(3) − TU(2) — the 3rd unit's MU. - C (15): computes TU(2) − TU(1) — the 2nd unit's MU. - D (6): computes TU(5) − TU(4) — the 5th unit's MU. - Fix: MU of unit n = TU(n) − TU(n − 1); count carefully which adjacent pair you're differencing.
| Q | MU lunches | MU coffees |
|---|---|---|
| 1 | 24 | 9 |
| 2 | 18 | 6 |
| 3 | 12 | 3 |
| 4 | 6 | 2 |
| 5 | 3 | 1 |
The utility-maximizing combination is:
6. D — MU/P for lunches: 8, 6, 4, 2, 1; for coffees: 9, 6, 3, 2, 1. Greedy purchase order: coffee1 (9), lunch1 (8), lunch2 + coffee2 (tie at 6), lunch3 (4), coffee3 (3) → $3+$3+$3 lunches + $1+$1+$1 coffees = $12 exactly. TU = 54 + 18 = 72 utils. - A: spends only $11 and swaps a 12-util lunch for low-value coffees — underspends and misallocates. - B: maxes out lunches while ignoring coffees whose early units yield up to 9 utils per dollar. - C: spends only $11, leaving a dollar that coffee 3 (3 utils) would profitably claim. - E: equalizes MU/P at 2 by buying 4 of each — but that costs $16, violating the $12 budget. - Fix: Rank all units by MU/P, buy from the top, and stop exactly when the budget is gone.
7. E — A lower Px raises MUx/Px above MUy/Py, so the consumer buys more X; diminishing MU then lowers MUx until the per-dollar ratios re-equalize. This is the law of demand emerging from the rule. - A: buying less of the now-cheaper good moves the ratios further apart. - B: Y's consumption adjusts as a consequence of the rule, not "regardless" of MUy. - C: the old bundle no longer satisfies MUx/Px = MUy/Py, so standing pat leaves utility on the table. - D: consumers equate marginal utilities per dollar, never total utilities. - Fix: Price change → per-dollar ratio unbalanced → buy more of the cheaper good until diminishing MU rebalances it.
8. B — Because each extra unit is worth less to the consumer, willingness to pay for extra units falls — so higher quantities are purchased only at lower prices: a downward-sloping demand curve. - A: supply slopes up because of rising marginal cost, a producer-side story. - C: diminishing MU makes quantity respond to price — that's responsiveness, not zero elasticity. - D: the law applies to essentially all goods, normal and inferior alike. - E: this describes a shift, but diminishing MU explains the slope of a single curve, not shifts of it. - Fix: Falling marginal value → falling willingness to pay → downward-sloping demand: slope, not shift.
9. C — "Cheaper relative to beef" is the substitution effect; "budget stretches further" (higher real purchasing power) is the income effect. - A: names the right two effects but reverses their order against the question's clauses. - B: diminishing MU is the background mechanism, not the name of the purchasing-power channel. - D: the relative-price channel is substitution, not income; and diminishing MU isn't either effect. - E: the law of demand is the combined result of these effects, and supply is uninvolved. - Fix: Relative-price story → substitution effect; purchasing-power story → income effect — match the clause, in order.
10. D — Chips deliver 8 utils per dollar versus fruit's 5, so moving dollars from fruit to chips raises total utility until the ratios equalize. - A: exhausting the budget is necessary but not sufficient — the per-dollar ratios must also be equal. - B: shifts in the wrong direction, away from the higher-yielding good. - C: equal quantities is never the criterion; equal MU per dollar is. - E: she can raise utility with the same income, simply by reallocating. - Fix: Budget spent + unequal MU/P = not done yet; shift dollars toward the higher ratio.
11. E — At a zero price, a rational diner consumes every plate with positive MU and stops when the next plate's MU would be negative — that is, when it would reduce total utility. - A: TU stays well above zero at the optimum; it's the margin that hits zero. - B: MU is at its maximum at the first plate — stopping there wastes every remaining positive-MU plate. - C: TU rising is the reason to continue, not to stop. - D: comparing price to total utility mixes levels with margins; with price zero the relevant test is MU ≥ 0. - Fix: At price zero, consume while MU > 0 and stop at MU = 0 — never let MU go negative.
12. A — Per dollar: coffees = 15/3 = 5 utils/$; meals = 40/10 = 4 utils/$. Coffees yield more per dollar, so reallocate toward coffees. - B: falls for the raw-MU trap — 40 > 15, but the meal costs more than three times as much. - C: positive MUs don't mean optimal; the per-dollar ratios (5 vs. 4) are unequal. - D: buying more of both violates the fixed budget. - E: reducing both and comparing total utilities abandons both the budget logic and the marginal rule. - Fix: Optimality is MU/P equality — whenever the ratios differ, shift dollars toward the higher one.
1. A — MU = ΔTU/ΔQ: the extra satisfaction contributed by one more unit. - B: TU/Q is average utility — the classic look-alike distractor. - C: total satisfaction from all units is total utility. - D: price is what you pay, not the satisfaction you receive; the two are compared, never equated by definition. - E: the first unit's utility is just one particular marginal value, not the definition. - Fix: "Marginal" always means the change from one more unit — a difference, never a total or an average.
2. C — TU peaks exactly where the last unit adds nothing: MU = 0. Beyond that point MU turns negative and TU falls. - A: MU is at its maximum near the first units, long before TU peaks. - B: rising MU would mean TU is accelerating upward, not peaking. - D: negative MU means TU is already falling — past the peak, not at it. - E: MU = P is the optimal-purchase condition when buying at a price, not the TU-maximum condition. - Fix: TU max ↔ MU = 0; TU falling ↔ MU < 0; TU rising ↔ MU > 0.
3. A — Each additional unit eventually contributes less extra satisfaction than the one before — MU declines as quantity rises. - B: TU keeps rising while MU is positive; it only falls once MU turns negative. - C: MU usually stays positive for many units; diminishing begins when MU starts falling, not when it goes negative. - D: switching goods is a budget-allocation behavior, not the content of the law. - E: utils are a measurement convention; the law is about the pattern of satisfaction, not measurability. - Fix: Diminishing MU = the increments shrink; it says nothing about totals falling or values turning negative.
4. B — Per dollar: coffee = 24/4 = 6 utils/$; lunches = 50/10 = 5 utils/$. Coffee delivers more per dollar, so shift spending toward coffee until the ratios equalize. - A: chooses lunches by comparing raw MUs (50 > 24) — the exact trap the per-dollar rule exists to avoid. - C: the bundle is optimal only when MU/P is equal across goods; 6 ≠ 5. - D: total utilities are irrelevant to the margin — decisions happen unit by unit. - E: equilibrium equalizes per-dollar marginal utilities, not quantities. - Fix: Divide every MU by its price before comparing — per-dollar or nothing.
5. E — MU of the 4th unit = TU(4) − TU(3) = 54 − 45 = 9. - A (54): reports TU at 4 units instead of the marginal difference. - B (12): computes TU(3) − TU(2) — the 3rd unit's MU. - C (15): computes TU(2) − TU(1) — the 2nd unit's MU. - D (6): computes TU(5) − TU(4) — the 5th unit's MU. - Fix: MU of unit n = TU(n) − TU(n − 1); count carefully which adjacent pair you're differencing.
6. D — MU/P for lunches: 8, 6, 4, 2, 1; for coffees: 9, 6, 3, 2, 1. Greedy purchase order: coffee1 (9), lunch1 (8), lunch2 + coffee2 (tie at 6), lunch3 (4), coffee3 (3) → $3+$3+$3 lunches + $1+$1+$1 coffees = $12 exactly. TU = 54 + 18 = 72 utils. - A: spends only $11 and swaps a 12-util lunch for low-value coffees — underspends and misallocates. - B: maxes out lunches while ignoring coffees whose early units yield up to 9 utils per dollar. - C: spends only $11, leaving a dollar that coffee 3 (3 utils) would profitably claim. - E: equalizes MU/P at 2 by buying 4 of each — but that costs $16, violating the $12 budget. - Fix: Rank all units by MU/P, buy from the top, and stop exactly when the budget is gone.
7. E — A lower Px raises MUx/Px above MUy/Py, so the consumer buys more X; diminishing MU then lowers MUx until the per-dollar ratios re-equalize. This is the law of demand emerging from the rule. - A: buying less of the now-cheaper good moves the ratios further apart. - B: Y's consumption adjusts as a consequence of the rule, not "regardless" of MUy. - C: the old bundle no longer satisfies MUx/Px = MUy/Py, so standing pat leaves utility on the table. - D: consumers equate marginal utilities per dollar, never total utilities. - Fix: Price change → per-dollar ratio unbalanced → buy more of the cheaper good until diminishing MU rebalances it.
8. B — Because each extra unit is worth less to the consumer, willingness to pay for extra units falls — so higher quantities are purchased only at lower prices: a downward-sloping demand curve. - A: supply slopes up because of rising marginal cost, a producer-side story. - C: diminishing MU makes quantity respond to price — that's responsiveness, not zero elasticity. - D: the law applies to essentially all goods, normal and inferior alike. - E: this describes a shift, but diminishing MU explains the slope of a single curve, not shifts of it. - Fix: Falling marginal value → falling willingness to pay → downward-sloping demand: slope, not shift.
9. C — "Cheaper relative to beef" is the substitution effect; "budget stretches further" (higher real purchasing power) is the income effect. - A: names the right two effects but reverses their order against the question's clauses. - B: diminishing MU is the background mechanism, not the name of the purchasing-power channel. - D: the relative-price channel is substitution, not income; and diminishing MU isn't either effect. - E: the law of demand is the combined result of these effects, and supply is uninvolved. - Fix: Relative-price story → substitution effect; purchasing-power story → income effect — match the clause, in order.
10. D — Chips deliver 8 utils per dollar versus fruit's 5, so moving dollars from fruit to chips raises total utility until the ratios equalize. - A: exhausting the budget is necessary but not sufficient — the per-dollar ratios must also be equal. - B: shifts in the wrong direction, away from the higher-yielding good. - C: equal quantities is never the criterion; equal MU per dollar is. - E: she can raise utility with the same income, simply by reallocating. - Fix: Budget spent + unequal MU/P = not done yet; shift dollars toward the higher ratio.
11. E — At a zero price, a rational diner consumes every plate with positive MU and stops when the next plate's MU would be negative — that is, when it would reduce total utility. - A: TU stays well above zero at the optimum; it's the margin that hits zero. - B: MU is at its maximum at the first plate — stopping there wastes every remaining positive-MU plate. - C: TU rising is the reason to continue, not to stop. - D: comparing price to total utility mixes levels with margins; with price zero the relevant test is MU ≥ 0. - Fix: At price zero, consume while MU > 0 and stop at MU = 0 — never let MU go negative.
12. A — Per dollar: coffees = 15/3 = 5 utils/$; meals = 40/10 = 4 utils/$. Coffees yield more per dollar, so reallocate toward coffees. - B: falls for the raw-MU trap — 40 > 15, but the meal costs more than three times as much. - C: positive MUs don't mean optimal; the per-dollar ratios (5 vs. 4) are unequal. - D: buying more of both violates the fixed budget. - E: reducing both and comparing total utilities abandons both the budget logic and the marginal rule. - Fix: Optimality is MU/P equality — whenever the ratios differ, shift dollars toward the higher one.