Utility Analysis

A student has a fixed amount of pocket money and a dozen things she could spend it on. The older of the two main theories of consumer choice answers "how does she choose?" by assuming she can count her satisfaction.

An example. A student rates her first chai of the day at 10 utils and a samosa at 15. The approach accepts such numbers, adds them up, and looks for the affordable mix with the largest total.

W. S. Jevons, Carl Menger and Léon Walras built the theory in the 1870s. Alfred Marshall gave it its textbook form in Principles of Economics (1890), using money as the measuring rod: what a person will pay shows the utility a thing gives them.

Its assumptions.

• Rational consumer: she seeks the most total utility from a given income at given prices.
• Cardinal measurement: utility can be counted in utils, or in the rupees she would pay.
• Divisible goods: she can buy a little more or a little less.
• Diminishing marginal utility: each extra unit of a good adds less satisfaction than the one before.
• Constant marginal utility of money: each rupee is worth the same to her, so money can measure utility.
• Independent utilities: one good's utility does not depend on the others, so utilities can be added.

Don't confuse cardinal with ordinal. Cardinal numbers measure (10 utils is twice 5 utils). Ordinal numbers only rank (first, second). The ordinal approach, in the next chapter, asks only which bundle a consumer prefers. Nor confuse utility with usefulness. Utility is whatever satisfies a want: a cigarette has utility for a smoker, though it does them harm.

Where it breaks. Nobody can measure satisfaction in utils, or compare one person's utils with another's. The marginal utility of money is not constant: as money runs low, each remaining rupee matters more. Utilities are not independent: tea and sugar go together, tea and coffee replace each other. J. R. Hicks and R. G. D. Allen (1934) developed the ordinal approach to do without measurement.

For exams. Expect to list the assumptions, and to name the one the ordinal approach drops: cardinal measurement. The approach is also called Marshallian utility analysis.

Water keeps people alive, yet a litre costs very little. Nobody needs a diamond, yet one costs lakhs. Adam Smith pointed to this puzzle, now called the paradox of value. The gap between total and marginal utility explains it.

An example. After a cricket match on a hot afternoon, a player drinks three glasses of water. Suppose the first gives 20 utils, the second 12 and the third 4.

• TU after one glass is 20; after two, 20 + 12 = 32; after three, 32 + 4 = 36.
• Working back: MU of the third glass = 36 − 32 = 4 utils.

How they move together. As consumption rises, MU falls. TU follows from MU in three steps:

MU is the slope of the TU curve. The next concept shows both curves together.

The paradox solved. Water's total utility is enormous. But most people have plenty, so the last litre adds little: its MU is low. Diamonds are scarce, so the last one still adds a lot. What a buyer pays for one more unit follows its marginal utility, not its total utility.

Don't confuse falling MU with falling TU. While MU is falling but still above zero, TU keeps rising, only more slowly. TU falls only once MU turns negative.

Where it breaks. TU and MU in utils are teaching numbers; nobody reports satisfaction that way. The idea survives in money terms: what a buyer would pay for one more unit.

In practice. Managers decide at the margin: is the next unit, hour or rupee worth its cost? The total already spent does not answer that.

For exams. Expect to work out an MU column from a TU column, or the reverse, and to state the three TU–MU rules. The water–diamond paradox is a common short question.

The first roti at lunch after a long morning is very welcome. By the fifth, you are pushing the plate away. Much of demand theory rests on this everyday experience.

The law is also called Gossen's first law, after H. H. Gossen, who stated it in 1854.

An example. Read the table as a hungry person eating rotis at lunch, with satisfaction in illustrative utils.

Check any row: TU is the running sum of MU. The third roti: 12 + 8 + 5 = 25. The sixth: 28 + (−2) = 26.

Why it happens. A particular want can be satisfied: hunger eases with each roti. And goods are imperfect substitutes, so more rotis cannot meet thirst.

The law holds only if:

• the units are identical in size and quality;
• the units are of a sensible size (one roti, not one crumb);
• consumption is continuous, without a long gap between units;
• tastes, income, habits and fashion do not change meanwhile.

Don't confuse diminishing marginal utility with diminishing returns. Diminishing marginal utility is about a consumer's satisfaction from more units. The law of diminishing returns is about a producer's extra output when more of one input is added to fixed amounts of others.

Where it breaks. Textbooks list apparent exceptions: collections (the stamp that completes a set), addictive goods such as alcohol, money, and rare items. Most break an assumption rather than the law. The final stamp is not an identical unit; addiction changes tastes as consumption goes on.

In practice. Each extra unit is worth less to a buyer, so a seller usually must cut the price to sell more. That is the root of the downward-sloping demand curve, and of "second unit at half price" offers. Textbooks also cite the law to support progressive income tax, though that assumes satisfaction can be compared across people.

For exams. State the law with its assumptions, read the table and graph (TU is highest where MU = 0), and explain the exceptions.

MU above zero: TU rises. MU at zero: TU peaks. MU below zero: TU falls.

A student has ₹100 for a week of canteen snacks: chai at ₹10 a cup, samosas at ₹20 each. Spending it all on samosas, or splitting it equally, would leave some satisfaction unclaimed.

The budget must also be fully spent: P₁Q₁ + P₂Q₂ + P₃Q₃ = income. The law is also called Gossen's second law, the law of substitution, or the law of maximum satisfaction.

The example worked out. The student's marginal utilities, in illustrative utils, are:

1. Divide each MU by the price, to get MU per rupee.
2. Spend each next amount where MU per rupee is highest. In order: 1st chai and 1st samosa (5), 2nd chai (4), 2nd samosa (3.5), 3rd chai (3), then 4th chai and 3rd samosa (2).
3. Running total: ₹10, ₹30, ₹40, ₹60, ₹70, ₹80, ₹100. The budget is used up.
4. Equilibrium: 4 chai (₹40) and 3 samosas (₹60). MU per rupee is 2 for both.
5. Total utility: chai 50 + 40 + 30 + 20 = 140; samosas 100 + 70 + 40 = 210; together 350 utils.

Check it. The only other way to spend exactly ₹100 here is 2 chai and 4 samosas. That gives 90 + 230 = 320 utils, which is 30 fewer.

If MU per rupee is higher for one good, the consumer buys more of it, and its MU falls until the ratios are equal. That point is consumer equilibrium in the cardinal approach: no further reallocation raises total utility.

Don't confuse equal MU per rupee with equal MU or equal spending. At equilibrium the last samosa gives 40 utils and the last chai only 20. But the samosa costs twice as much, so each gives 2 utils per rupee.

Where it breaks. Some goods are indivisible: you cannot buy a little more of a scooter. Consumers rarely calculate; habit and impulse intervene. And utility cannot be measured.

In practice. Managerial economics uses the same rule as the equi-marginal principle. Spread any limited resource, such as an advertising budget or sales staff's time, so the last rupee or hour earns the same extra return in every use.

For exams. Expect the formula, a table like the one above to solve, and the limitations.

You would happily pay ₹1,500 for a pair of shoes, and the shop asks ₹1,200. You walk out with something worth ₹300 more to you than it cost.

One unit. Willingness to pay is ₹1,500 and the price paid is ₹1,200, so the surplus is ₹1,500 − ₹1,200 = ₹300. If the price rises to ₹1,300, the surplus falls to ₹200.

Several units. A fruit seller offers mangoes at ₹20 each. A buyer would pay ₹50 for the first, ₹40 for the second, ₹30 for the third and ₹20 for the fourth: diminishing marginal utility, in rupees.

1. She buys until what she would pay for the next mango equals the price: 4 mangoes.
2. Total utility in money: ₹50 + ₹40 + ₹30 + ₹20 = ₹140.
3. Total expenditure: 4 × ₹20 = ₹80.
4. Consumer surplus: ₹140 − ₹80 = ₹60.

Unit by unit gives the same answer: ₹30 + ₹20 + ₹10 + ₹0 = ₹60. If the price fell to ₹10 and she would pay ₹10 for a fifth mango, she would buy 5. Her surplus would rise to ₹100 (₹150 − ₹50).

On a smooth demand curve, consumer surplus is the area below the demand curve and above the price line.

Don't confuse consumer surplus with a discount. A discount is measured from the printed price. Consumer surplus is measured from what you yourself would have paid. Nor confuse it with producer surplus: the seller's gain, the price received above the lowest price they would accept.

Where it breaks. It assumes utility can be measured in money and that the marginal utility of money is constant. For necessities such as salt or life-saving medicine, people would pay almost anything, so the surplus cannot be measured sensibly. Substitutes change what a buyer would pay. J. R. Hicks later restated the idea using indifference curves, which avoids measuring utility.

In practice. Sellers try to capture surplus by charging more to buyers who will pay more. Indian Railways' Tatkal quota charges extra for tickets booked close to the travel date. Governments add up users' surplus when weighing a public project, such as a new bridge, against its cost.

For exams. Expect Marshall's definition, a sum like the mangoes, the demand-curve area, and the criticisms.