Satiation at the 5th unit, where MU = 0 and TU is maximum (46). Note MU falls throughout — 20, 14, 8, 4, 0 — exactly as the Law of DMU predicts.

(a) Yes — that part is correct. TU is maximum exactly where MU = 0. In Table 2.1 that is the 5th unit, where TU peaks at 24.

(b) The advice is wrong. What has been left out is the price.

Consuming up to MU = 0 would be right only if bananas cost nothing. Since they have a price, each banana carries an opportunity cost — the satisfaction forgone from the mangoes that money could have bought. The student is maximising satisfaction from one good while ignoring the budget constraint and the other good entirely.

(c) The condition is the tangency condition:

She stops where the rate at which she is willing to trade mangoes for bananas equals the rate at which the market lets her. At that point MU of bananas is still strictly positive — she would happily eat more, but not at that price.

The general error is important beyond this question: maximising the satisfaction from a single good is not the same as maximising overall satisfaction. The consumer's problem is a problem of allocation across goods under a constraint — which is exactly why Chapter 2 needs both an indifference map and a budget line, not just one of them.

Note the structural echo of Chapter 1: a free good would be consumed to satiation; an economic good never is, precisely because it is scarce and carries an opportunity cost.

1. No. Bundle (10, 8) has more of both goods than (8, 6). Monotonicity requires her to strictly prefer (10, 8). She cannot be indifferent.

2. No — they are not monotonic. The bundles (5, 6) and (6, 6) have the same quantity of good 2, but (6, 6) has one more unit of good 1. Monotonicity would require (6, 6) to be strictly preferred. Since your friend is indifferent, monotonicity is violated.

3. (10, 10) ≻ (10, 9) ≻ (9, 9)

(10,10) beats (10,9): same good 1, more good 2. (10,9) beats (9,9): same good 2, more good 1. And (10,10) beats (9,9) on both counts.

(a) A concave curve satisfies features 1, 2 and 3 — it can still slope downward, higher curves can still mean higher utility, and two such curves still need not intersect. All three of those come from monotonicity, which concavity does not touch.

It violates feature 4, convexity — and therefore violates the Law of Diminishing MRS, which is the assumption feature 4 rests on.

(b) MRS would be increasing. As the consumer acquired more bananas, she would be willing to give up more and more mangoes for each additional banana — the opposite of diminishing MRS.

Economically that is very odd: it says the more bananas she has, the more desperate she becomes for another one, and the less she values the mangoes she still holds.

(c) The tangency would be a minimum, not a maximum.

Such a consumer would push all the way to a corner — spending her entire income on bananas alone or mangoes alone, never a mixture. Concave preferences describe someone who wants extremes, not balance.

Why this matters: convexity is not decoration. It is precisely the assumption that makes the tangency condition identify a maximum, and it is why real consumers buy mixtures of goods rather than everything of one. This is the same logic you will meet in Chapter 4, where P = MC identifies a profit maximum only if MC is rising — a condition on curvature, not on the tangency itself.

1. 4x₁ + 5x₂ = 20

2. Good 1: M/p₁ = 20/4 = 5 units. Good 2: M/p₂ = 20/5 = 4 units.

3. Slope = −p₁/p₂ = −4/5 = −0.8.

4. New line 4x₁ + 5x₂ = 40. Intercepts become 10 and 8 — both doubled. Slope still −4/5, so there is a parallel outward shift.

5. p₂ = 4, so 4x₁ + 4x₂ = 20. The good-1 intercept stays at 5; the good-2 intercept rises from 4 to 5. Slope changes from −4/5 to −1. The line pivots outward around the horizontal intercept and becomes steeper.

6. No change at all. The new constraint is 8x₁ + 10x₂ ≤ 40. Divide throughout by 2: 4x₁ + 5x₂ ≤ 20 — identical to the original. What matters is real income (income relative to prices), not the numbers themselves.

(a) The bundle (6, 8) costs exactly her income:

M = p₁x₁ + p₂x₂ = (6 × 6) + (8 × 8) = 36 + 64 = ₹100

(b) The constraint is 10x₁ + 10x₂ ≤ 40, i.e. x₁ + x₂ ≤ 4.

All 15 available bundles:
(0,0) (0,1) (0,2) (0,3) (0,4)
(1,0) (1,1) (1,2) (1,3)
(2,0) (2,1) (2,2)
(3,0) (3,1)
(4,0)

Costing exactly ₹40 (those with x₁ + x₂ = 4, lying on the budget line): (0,4), (1,3), (2,2), (3,1), (4,0).

(a) Nothing changes at all. The new constraint is:

1.2p₁·x₁ + 1.2p₂·x₂ ≤ 1.2M

Divide throughout by 1.2:

p₁x₁ + p₂x₂ ≤ M

Identical to the original. Both intercepts (M/p₁ and M/p₂) are unchanged, because the 1.2 cancels in each. The slope −p₁/p₂ is unchanged for the same reason.

(b) Her optimal bundle is exactly the same. The budget set has not moved and her preferences have not changed, so the same tangency point is still the best affordable bundle.

This is why a pay rise that merely matches inflation makes nobody better off — a result you can now prove rather than merely assert.

(c) A parallel inward shift.

New constraint: 1.2p₁·x₁ + 1.2p₂·x₂ ≤ 1.1M. Divide by 1.2:

p₁x₁ + p₂x₂ ≤ (1.1/1.2)M = 0.9167M

This is the original prices with an income of about 91.7% of M. So:

Slope unchanged (both prices moved by the same proportion, so the ratio p₁/p₂ is untouched) · both intercepts fall by the same factor → a parallel inward shift.

Not a pivot. A pivot requires the relative price to change — one price moving differently from the other. Here both moved together, so only the real income fell. She is unambiguously worse off, and will end on a lower indifference curve.

(a) The price ratio is p₁/p₂ = 8/4 = 2.
Since MRS (4) > price ratio (2), she is not in equilibrium.

(b) She is willing to sacrifice 4 mangoes for one extra banana, but the market only requires her to sacrifice 2. The banana is worth more to her than it costs. So she will buy more bananas and fewer mangoes.

As she does so, by the Law of Diminishing MRS her MRS falls (more bananas → each worth less; fewer mangoes → each worth more). She keeps adjusting until MRS = 2, at which point equilibrium is reached.

(a) Yes — their MRS must be equal.

Each consumer optimises where MRS = p₁/p₂. They face the same prices, so the same price ratio. Therefore:

Despite different incomes, different quantities and different tastes, at the margin they value the two goods in exactly the same ratio.

(b) It certainly does not mean they are equally satisfied.

Anil is on a much higher indifference curve — his larger budget set lets him reach one Bela cannot afford. Their curves are tangent to different budget lines with the same slope.

Why the result is interesting anyway: it means no mutually beneficial trade is left between them. If Anil valued bananas more highly at the margin than Bela did, they could both gain by trading. Equal MRS means that opportunity is exhausted — which is a key ingredient of what makes a competitive market efficient.

(c) Because cardinal utility invites a comparison that is meaningless.

The cardinal approach would tempt you to say "Anil gets 500 utils and Bela 200, so Anil is 2.5 times happier". But utility has no natural unit, and there is no way to compare one person's utils with another's — an interpersonal comparison that no evidence can settle.

The ordinal approach never makes that claim. It only ever says each consumer ranks her own bundles, and the result above uses nothing more: it compares a slope — which is observable from choices — not a level of feeling. That is precisely why the ordinal approach is regarded as the more defensible of the two.

1. Movement along the curve — downward (extension of demand). The good's own price changed.
2. Rightward shift. Coffee is a substitute; its price rose, so consumers switch to tea. Demand for tea rises at every price.
3. Leftward shift. Sugar is a complement; costlier sugar makes tea-drinking more expensive overall, so tea demand falls at every price.
4. Rightward shift. Tea is normal, so higher income raises demand at every price.
5. Rightward shift. A favourable change in tastes and preferences.

Only item 1 involved the price of tea itself — and only item 1 is a movement. Everything else shifts the curve.

The student has forgotten the phrase "other things remaining equal". The Law of Demand describes a movement along a demand curve. If something else also changed, the curve itself moved, and what we observe is the net effect of two separate changes.

1 · Income rose (normal good). Higher income shifts the demand curve rightward. If that shift outweighs the price rise, the quantity bought rises. → shift (plus a movement along the new curve).

2 · The price of a substitute rose by more. If coffee rose sharply, demand for tea shifts rightward, and more tea is bought despite tea's own price rising. → shift.

3 · Tastes changed in the good's favour. A favourable report, a change of season, a fashion — the curve shifts rightward. → shift.

The genuine exception — a Giffen good. For a strongly inferior good, a price rise makes the consumer poorer in real terms, and because the good is inferior, that income effect pushes her to buy more of it. If the income effect outweighs the substitution effect, quantity demanded rises with price. → movement along an upward-sloping demand curve.

This is the crucial distinction. In cases 1–3 the law holds and something else moved. In the Giffen case the law itself fails — the demand curve is genuinely positively sloped.

The methodological lesson. To test the Law of Demand you cannot simply observe price and quantity over time — too many things move at once. You must isolate the effect of price with everything else held constant. This is why "ceteris paribus" is not a formality but the substance of the claim.

(a) Consumer 2 leaves the market above p = 15; consumer 1 above p = 20. So there are three price ranges:

For 0 ≤ p ≤ 15 (both buy): (20 − p) + (30 − 2p) = 50 − 3p
For 15 < p ≤ 20 (only consumer 1): 20 − p
For p > 20: 0

Check at p = 15: d₁ = 5, d₂ = 0, market = 5. And 50 − 3(15) = 5. ✓

(b) All 20 consumers are identical, so simply multiply one consumer's demand by 20:

D(p) = 20 × (10 − 3p) = 200 − 60p  for 0 ≤ p ≤ 10/3
= 0  for p > 10/3

Add the two quantities at each price (horizontal summation):

Market demand falls as price rises — the market demand curve, like the individual ones, is downward sloping.

(a) %ΔQ = (20 − 25)/25 × 100 = −20%
%ΔP = (5 − 4)/4 × 100 = +25%

Since 0.8 < 1, demand is inelastic at this price.

(b) Here a = 10, b = 3. First find the quantity:
q = 10 − 3(5/3) = 10 − 5 = 5

|e_D| = 1 — this is the midpoint of the demand curve. Check: midpoint price = a/2b = 10/6 = 5/3 ✓

(a) From e = %ΔQ / %ΔP:
%ΔQ = e × %ΔP = (−0.2) × 5 = −1%
Demand falls by 1%.

(b) %ΔQ = (−0.2) × 10 = −2%.
%ΔE ≈ %ΔP + %ΔQ = 10 + (−2) = +8%
Expenditure rises by about 8%. This is the inelastic case — price rose, and since quantity barely responded, expenditure moved in the same direction as price.

(c) Price fell and expenditure rose — they moved in opposite directions, so demand must be elastic (|e| > 1).

Quantitatively: %ΔQ ≈ %ΔE − %ΔP = 2 − (−4) = +6%
e = 6 / (−4) = −1.5  ⇒  |e| = 1.5, confirming elastic demand.

(a) Demand is inelastic, so |e| < 1 and %ΔQ is smaller than %ΔP. Say e = −0.4: a 20% price rise cuts consumption by only 0.4 × 20 = 8%.

By the elasticity–expenditure rule, for an inelastic good expenditure moves in the same direction as price:

%ΔE ≈ %ΔP + %ΔQ = 20 + (−8) = +12%

Consumption falls 8%; total spending on cigarettes rises 12%.

(b) Yes in the short run — and this is exactly why tobacco is taxed so heavily. Consumption falls (the health goal) while revenue rises (the fiscal goal). Inelastic demand is what makes both possible simultaneously.

So the two goals are compatible in the short run but increasingly in conflict over time. A tax that succeeds completely at its health objective destroys its own tax base.

(c) The colleague has it exactly backwards for revenue purposes.

With elastic demand, expenditure moves opposite to price. Raising the price of an elastic good makes consumers cut back more than proportionately, so total spending — and hence the tax base — falls. It is a poor revenue source.

The general rule for revenue: tax goods with inelastic demand — necessities, goods with few substitutes, addictive goods. This is precisely why salt, fuel, alcohol and tobacco have been taxed by governments for centuries.

But note the normative sting. "Inelastic demand" often means "necessity", and taxes on necessities fall hardest on the poor, who spend a larger share of income on them. The positive analysis says such a tax raises revenue efficiently; whether it is fair is a separate, normative question — the Chapter 1 distinction again.

1. The budget set is the collection of all bundles that a consumer can buy with her income at the prevailing market prices — every bundle (x₁, x₂) satisfying p₁x₁ + p₂x₂ ≤ M. It includes bundles costing less than her income as well as those costing exactly her income.

2. The budget line consists of all bundles that cost the consumer exactly her entire income: p₁x₁ + p₂x₂ = M. It is the boundary of the budget set, with vertical intercept M/p₂, horizontal intercept M/p₁ and slope −p₁/p₂.

3. Every bundle on the budget line costs exactly M. If the consumer wants more of one good, she must spend more on it — and since her income is fixed and fully spent, she must spend less on the other good, and so buy less of it. An increase in one quantity therefore necessarily accompanies a decrease in the other, which is precisely what a downward slope means.

Formally: the slope is −p₁/p₂. Since both prices are positive, the slope is negative.

10. A consumer's preferences are monotonic if and only if, between any two bundles, the consumer prefers the bundle which has more of at least one of the goods and no less of the other good as compared to the other bundle. In short: more is better.

Monotonicity is what makes an indifference curve slope downward, makes higher indifference curves represent greater satisfaction, and forces the consumer's optimum to lie on (not below) the budget line.

21. Price elasticity of demand measures the responsiveness of the quantity demanded of a good to a change in its price. It is defined as the percentage change in demand divided by the percentage change in price:

It is negative (price and quantity move oppositely) but conventionally reported as an absolute value, and it is a pure number, independent of units. Demand is elastic if |e| > 1, inelastic if |e| < 1 and unitary elastic if |e| = 1.

17. A normal good is one for which demand increases as the consumer's income increases, and decreases as income decreases — demand moves in the same direction as income. Examples: milk, branded clothing, cars.

18. An inferior good is one for which demand decreases as the consumer's income increases, and increases as income decreases — demand moves in the opposite direction to income. Examples: coarse cereals, low-quality food items, second-hand clothing.

A good may be normal at low incomes and inferior at higher incomes.

19. Substitutes are goods that can be used in place of each other to satisfy the same want. The demand for a good moves in the same direction as the price of its substitute. Example: tea and coffee — if the price of coffee rises, consumers shift to tea, so the demand for tea rises.

20. Complements are goods that are consumed together to satisfy a want. The demand for a good moves in the opposite direction to the price of its complement. Example: tea and sugar — if the price of sugar rises, the demand for tea falls.