Cardinal utility: total utility, marginal utility and the law of diminishing marginal utility

Detail

1. What is cardinal utility?

• Utility: the satisfaction a person gets from using (consuming) a good or service.
• Cardinal utility analysis: assumes utility can be measured and written as a number.
• Example: "this shirt gives me 50 units of utility."

• The imaginary unit is often called a "util".

• Because utility is a number, we can add it, subtract it and compare how large the gaps are. We can say "bundle A gives 10 utils more than bundle B."

• Ordinal utility is the contrast. It only ranks bundles (1st, 2nd, 3rd). It does not measure them (see section 3 of the parent note).

2. Total utility (TU)

• Total utility (TU): the total satisfaction from consuming a given amount of a good.
• TUₙ = the utility from consuming n units.
• Example: TU₄ = 28 means 4 bananas together give 28 utils.

3. Marginal utility (MU)

• Marginal utility (MU): the change in TU when she consumes one more unit.
• Formula: MUₙ = TUₙ − TUₙ₋₁
• Worked example (bananas):
• 4 bananas give TU = 28 and 5 bananas give TU = 30.

• MU₅ = 30 − 28 = 2 utils.

• TU is the sum of MUs: TUₙ = MU₁ + MU₂ + … + MUₙ

• On a graph, MU is the slope of the TU curve.
• MU > 0 → TU is rising.
• MU = 0 → TU is at its peak.
• MU < 0 → TU is falling.

4. Table 2.1 (Class 12): reading TU and MU together

Units	TU	MU
1	12	12
2	18	6
3	22	4
4	24	2
5	24	0
6	22	−2

• Check the sum rule: TU₆ = 12 + 6 + 4 + 2 + 0 + (−2) = 22. ✔
• Stage I (units 1–4): TU rises at a falling rate.
• MU falls: 12 → 6 → 4 → 2.

• Each unit still adds something, but it adds less than the one before.

• Stage II (unit 5): TU is at its maximum and MU = 0.

• This point is called satiety (the point where she is fully satisfied with the good).

• Stage III (unit 6): MU is negative (−2) and TU falls (24 → 22).

• One more unit now causes disutility (discomfort). Think of being forced to eat a 6th sweet.

• Shape of the curves:

• The TU curve rises, flattens at the top and then falls (like a hill).
• The MU curve slopes downward and cuts the x-axis at the 5th unit.

5. Law of diminishing marginal utility (DMU)

• Statement: the MU of each extra unit of a good falls as she consumes more of it, holding her consumption of other goods constant.
• Reason: once she has some of the good, her desire for more of it grows weaker.
• Class 9 example: the first mango tastes delicious, the second is good, and by the third she is barely interested.
• Other name: Gossen's first law, after Hermann Heinrich Gossen, a German economist.
• His work, published in 1854, stated that the extra satisfaction from one more unit of a good falls as more units are consumed [2].

• Gossen's ideas later became part of the base of modern demand analysis [2].

• Conditions the law needs. These follow from "other things constant":

• The units are the same size and quality (not one small mango and one big mango).
• The units are eaten one after another, without a long gap. A mango tomorrow may taste delicious again.
• Her tastes, her income and the prices of other goods do not change.

6. How DMU explains the downward-sloping demand curve

• Chain of logic:
• Each extra unit is worth less to her (DMU).
• So she is willing to pay less for each extra unit.
• So she buys more units only when the price falls.

• This gives a demand curve that slopes downward (lower price → larger quantity bought).

• Class 12 example:

• At ₹40 per unit she buys 5 units.
• The 6th unit is worth less to her than the 5th.

• So she buys the 6th unit only if the price falls below ₹40.

• Hidden assumption: the marginal utility of money (the satisfaction from one more rupee) stays constant. Only then can we turn MU in utils into a rupee price she is willing to pay. Marshall relied on this assumption [4].

7. Extension: law of equi-marginal utility (Gossen's second law)

• Statement: a consumer gets the most satisfaction when the MU per rupee is the same for every good she buys. This is also called the equimarginal principle or the law of equal marginal utilities [5].
• Condition: MUx/Px = MUy/Py = MU of money
• Adjustment rule: if MUx/Px > MUy/Py, she moves a rupee from y to x. She keeps doing this until the two ratios are equal.
• Worked example:
• Px = ₹10 and MUx = 40 → MUx/Px = 4 utils per rupee.
• Py = ₹5 and MUy = 10 → MUy/Py = 2 utils per rupee.
• A rupee spent on x gives 4 utils. A rupee spent on y gives only 2. So she buys more x and less y.
• DMU makes the gap close:
  • more x → MUx falls
  • less y → MUy rises

• She stops when the two ratios are equal (for example, both at 3).

• At this best point, one rupee of spending gives the same utility whichever good it is spent on [5].

• Link: this is the cardinal version of the ordinal optimum MRS = p₁/p₂ (section 5).

8. Extension: consumer surplus (Marshall)

• Origin: Alfred Marshall, Principles of Economics (1890). This book also introduced elasticity of demand and the representative firm [8].
• Definition: the extra benefit a buyer gets over what she pays. It is the gap between the price she would pay rather than go without the good and the price she actually pays [4].

• Consumer surplus = what she is willing to pay − what she actually pays

• On a graph: the area below the demand curve and above the market price.

• This area measures surplus only if we assume the MU of money is constant and money can stand in for utility [4].

• Worked example:

• She would pay ₹60 for the 1st unit and ₹50 for the 2nd. The market price is ₹40.

• Surplus = (60 − 40) + (50 − 40) = 20 + 10 = ₹30.

• Its use in measuring welfare loss (deadweight loss) is covered in the market-equilibrium and price-controls topic.

9. Extension: the diamond-water paradox

• The puzzle (Adam Smith, Wealth of Nations, 1776):
• Water is essential for life but cheap.
• Diamonds are not essential but are very costly.

• Smith thought about this puzzle but could not solve it [3].

• The solution: separate TU from MU.

• Price follows marginal utility and scarcity. It does not follow total utility.
• Water: it is plentiful, so the MU of one more litre is tiny. Its TU is huge, because the first few units keep us alive [3].

• Diamonds: they are scarce, so the MU of one more diamond is high. Weight for weight, the marginal value of diamonds is greater than that of water [3].

• Who solved it: the marginalists, whose work is called the "marginal utility revolution":

• W. S. Jevons, The Theory of Political Economy (1871, England) [6]
• Carl Menger (1871, Vienna) [6]
• Léon Walras (1874, Switzerland) [6][7]

• Menger's key idea: goods have value because they serve many uses, and these uses are not equally important [6].

• Marshall's synthesis: Marshall brought time into the analysis. This let him combine the classical cost-of-production view of value with the marginal-utility view [8].

• Policy link: the IMF has argued that water can be free where supply is plentiful compared to demand. Where growing use meets a limited supply, water should carry a positive price [9] (IMF First Deputy MD speech, 2015).

10. Weakness of the cardinal approach

• In real life nobody measures utility in numbers. At most we rank bundles.
• The approach assumes the MU of money is constant. In reality, one more rupee means more to a poor person than to a rich person.
• It studies one good at a time ("other goods constant"). It cannot easily handle substitutes and complements.
• These drawbacks led to the ordinal approach, which uses indifference curves (section 3).

Prelims Hooks

• MUₙ = TUₙ − TUₙ₋₁ and TUₙ = ΣMU. MU is the slope of the TU curve.
• When MU = 0, TU is at its maximum (satiety). When MU < 0, TU falls. Trap: "TU is maximum when MU is maximum" is wrong.
• If MU is falling but still positive, TU rises at a diminishing rate. It does not fall.
• The law of DMU is Gossen's first law (1854) [2]. The law of equi-marginal utility is Gossen's second law.
• Consumer equilibrium (cardinal): MUx/Px = MUy/Py = MU of money [5].
• Consumer surplus comes from Alfred Marshall, Principles of Economics (1890). The same book introduced elasticity of demand and the representative firm [8].
• Diamond-water paradox: posed by Adam Smith (1776). It was solved through marginal, not total, utility [3]. The marginal revolution: Jevons and Menger (1871), Walras (1874) [6][7].
• DMU holds only when other goods' consumption, tastes and income are held constant.
• DMU is the reason in utility analysis for the downward-sloping demand curve.

Mains Points

• Ethical case for progressive taxation and transfers:
• Money itself has diminishing MU.
• ₹1,000 taken from a rich person costs them little satisfaction. ₹1,000 given to a poor household adds a lot.

• This supports progressive income tax and targeted cash transfers (DBT). The limit: it needs interpersonal comparison of utility, which is exactly what the ordinal approach rejects.

• Pricing essential resources (water, power, electricity for farms):

• Low marginal prices for water encourage over-use.
• The IMF view is that scarce water needs a positive price [9].

• A practical balance is tiered tariffs:

  • a free or low-cost "lifeline" block protects the first units, which have high MU and are essential for life
  • higher prices on extra units discourage waste

• Consumer surplus as a welfare measure:

• It lets policymakers value, in rupees, the gains from public goods, subsidies or lower prices. Examples: GST rate cuts, highways, digital payments at zero price.

• The main caveat: it assumes the MU of money is constant [4].

• Limits of cardinal theory:

• Utility cannot be measured, which pushed economics towards ordinal theory and revealed-preference methods.
• Behavioural economics also shows that real consumers do not always equalise MU per rupee (habits, addiction, limited attention). This matters when designing "nudge" policies.