Cardinal utility: law of diminishing marginal utility and law of equi-marginal utility

1. Utility and the cardinal approach

Utility is the want-satisfying power of a commodity. It is subjective (it differs between persons), relative (to time and place), and ethically neutral (a cigarette has utility for a smoker). It is not the same as usefulness or pleasure.

The cardinal approach, developed by Jevons, Menger, Walras and above all Marshall, assumes utility can be measured in numbers called utils, and that the marginal utility of money is constant, so that utility can be measured by the price a consumer is willing to pay.

• Total utility (TU): the sum of utility from all units consumed.
• Marginal utility (MU): the addition to total utility from one more unit. MU_n = TU_n − TU_n−1.

2. Law of diminishing marginal utility

Marshall: “The additional benefit which a person derives from a given increase of his stock of a thing diminishes with every increase in the stock that he already has.” As a consumer takes more units of a good in succession, the utility from each additional unit falls. The idea was stated earlier by H.H. Gossen and is called Gossen’s first law.

Relation of TU and MU (describe the two curves): while MU is positive, TU rises; when MU is zero (6th glass), TU is at its maximum, the point of satiety; when MU becomes negative, TU falls. The MU curve slopes downward and cuts the quantity axis at 6.

Assumptions: units are of a standard size (a glass, not a sip); consumption is continuous, without a time gap; the consumer’s tastes, income and the price of the good do not change; the consumer is normal and rational.

Exceptions and limitations:

• Hobbies and collections: each additional rare stamp or coin may give more satisfaction.
• Music, poetry, reading: a second hearing may give more pleasure than the first.
• Money: a miser may want more money the more he has, though for most people the MU of money also diminishes, slowly.
• Drunkards and addicts: utility from liquor appears to rise with consumption.
• Too small units: the first spoonful of water to a thirsty man may raise MU.
• Measurement: utility is a psychological state and cannot really be measured in numbers. This is the main criticism, and led Hicks and Allen to the ordinal approach (next note).

Importance: it is the basis of the law of demand (the consumer buys more only at a lower price), explains the diamond-water paradox (water has high total but low marginal utility), justifies progressive taxation (the MU of money is lower for the rich), and explains why variety in consumption is preferred.

3. Law of equi-marginal utility

Also called the law of substitution, the law of maximum satisfaction and Gossen’s second law. A consumer with a limited income buying several goods gets maximum satisfaction when the marginal utility of the last rupee spent on each good is the same. In equilibrium:

MU_x / P_x = MU_y / P_y = … = MU of money

and the whole income is spent. Marshall: “If a person has a thing which he can put to several uses, he will distribute it among these uses in such a way that it has the same marginal utility in all.” If the last rupee spent on X gives more utility than the last rupee on Y, the consumer gains by moving a rupee from Y to X. As he buys more X its MU falls and as he buys less Y its MU rises, until the two are equal.

4. Worked budget

Income Rs. 11. Price of X = Rs. 2, price of Y = Re. 1.

The ratios are equal at 6 when the consumer buys 3 units of X and 5 units of Y. Spending: 3 × 2 + 5 × 1 = Rs. 11, the whole income. Total utility = (20 + 16 + 12) + (10 + 9 + 8 + 7 + 6) = 48 + 40 = 88 utils.

Check against other combinations costing Rs. 11: 2X + 7Y gives 36 + 49 = 85; 4X + 3Y gives 56 + 27 = 83. Both are lower, so 3X + 5Y is the equilibrium.

Described as a diagram: draw the MU_x/P_x and MU_y/P_y curves side by side, both sloping down; a horizontal line at 6 cuts them at 3 units of X and 5 units of Y.

5. Limitations and applications

Limitations: utility is not measurable; consumers do not calculate; goods are often indivisible (a car, a house); the MU of money is not constant; habit, fashion and ignorance interfere; it does not apply where goods are complements consumed together.

Applications: consumption (allocating income), production (a firm combines factors so that the marginal product per rupee is equal across factors, the least-cost combination of Unit III), exchange, distribution of saving between present and future, public finance (a government spends so that the marginal social benefit of each head is equal, Dalton’s maximum social advantage), and even the allocation of a student’s time between subjects.

Principles of Economics covers the same laws in its note on utility and equi-marginal utility.

How this is examined

“Explain the law of equi-marginal utility” was Q.1 in both January 2026 papers (16 marks in the 100-mark paper, 10 marks in the 80-mark paper); “Briefly explain the law of diminishing marginal utility” was Q.1 for 16 marks in February 2025. A full answer: meaning of utility, statement (with Marshall’s words), assumptions, the schedule, the diagram described, the equilibrium condition, limitations and applications. A schedule with figures that add up is what separates a 12 from a 16.