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Cardinal Utility, DMU and Consumer Equilibrium ISC 12

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EconomicsISC Class 12 · Demand

Cardinal Utility Analysis: TU, MU, Diminishing Marginal Utility and Consumer's Equilibrium

Updated 17 October 2026 · Fact-checked

In the cardinal approach, satisfaction is counted in utils. Marginal utility (MU) falls as more units are consumed, which is the law of diminishing marginal utility. A consumer buying one good is in equilibrium when MUx = price x MU of money. With two goods, equilibrium needs MUx/Px = MUy/Py (law of equi-marginal utility), with the whole income spent.

Notice that TU is highest exactly where MU touches zero; after that MU is negative and TU falls.

Illustration drawn for QuestionBuddy.

Understand Consumer's Equilibrium: Cardinal Utility Analysis

1. Utility, total utility and marginal utility. Utility is the want-satisfying power of a good. In the cardinal approach it is assumed to be measurable in imaginary units called utils. Total utility (TU) is the satisfaction from all units consumed. Marginal utility (MU) is the addition to TU from consuming one more unit: MUn = TUn - TUn-1 (or MU = change in TU / change in quantity).

2. Relationship between TU and MU.

  • While MU is positive, TU rises (but at a decreasing rate when MU is falling).
  • When MU = 0, TU is at its maximum (the satiety point).
  • When MU is negative, TU falls.
  • TU is the sum of the MUs: TU = MU1 + MU2 + ... + MUn.
  • Diagram: the TU curve rises, flattens at its peak and then turns down; the MU curve slopes downward, cuts the quantity axis exactly below the peak of TU, and goes below the axis afterwards.

3. Law of Diminishing Marginal Utility (DMU). *Statement:* as a consumer consumes more units of a good, the marginal utility from each additional unit goes on decreasing, other things remaining the same. For ISC, only the assumptions of the law are taught (criticisms are not required):

  • Utility is measurable in cardinal numbers (utils).
  • Units are identical in size, quality and taste, and are consumed in quick succession (no long gap).
  • The consumer is rational and the taste, income and prices stay unchanged.
  • The MU of money remains constant.
  • Consumption is of a suitable standard unit (not a tiny fraction of a unit).

Reading the schedule and the diagram. In a schedule, the MU column falls down the rows; in the diagram the MU curve is downward sloping, convex to the origin, and the portion below the axis shows negative MU. The reason is that each unit satisfies a less intense want than the one before.

4. Consumer's equilibrium for one commodity. The consumer compares the utility given up (the money paid) with the utility gained. Let the price be P and let MUm be the marginal utility of one rupee (assumed constant). The consumer keeps buying as long as MUx/P is greater than MUm and stops where MUx/P = MUm, that is MUx = P x MUm. If the MU of money is taken as one util per rupee, the condition is simply MUx = Px. If MUx > P x MUm, buy more; if MUx < P x MUm, buy less. In a diagram the equilibrium is at the point where the downward-sloping MU curve cuts the horizontal price line.

5. Law of equi-marginal utility (two or more goods). *Statement:* a rational consumer with a limited income spreads the expenditure over goods so that the marginal utility of the last rupee spent on each good is the same. *Conditions of equilibrium:*
1. MUx/Px = MUy/Py (= MU of money).
2. The whole income is spent: Px.Qx + Py.Qy = M.
3. MU falls as consumption increases (the second order condition), so the consumer does not drift away from the point.
If MUx/Px > MUy/Py, the consumer moves a rupee from y to x; MUx falls and MUy rises until the ratios are equal.

Link with the demand curve. When the price of a good falls, MUx/Px rises above MUm, so the consumer buys more units until MUx falls to the new price line. This gives the downward-sloping demand curve (see the page on the law of demand).

Consumer's Equilibrium: Cardinal Utility Analysis: key facts

Marginal utility
MUn = TUn - TUn-1 = change in TU / change in units
Total utility
TU = sum of MU1 to MUn
Equilibrium, one good
MUx / Px = MU of money (MUx = Px when MU of money = 1 util)
Equi-marginal law
MUx / Px = MUy / Py, with Px.Qx + Py.Qy = income
Satiety
MU = 0 when TU is maximum

How to answer Consumer's Equilibrium: Cardinal Utility Analysis questions

Utility numericals have two forms: finding MU or TU from a table, and finding how many units to buy. Build the table first.

  1. 1For MU from TU, subtract each TU from the next one; the first MU equals the first TU.
  2. 2For one good, convert the condition to MUx = Px x MU of money and find the unit whose MU equals that value (or the last unit whose MU is at least that).
  3. 3For two goods, write MU per rupee (MU divided by price) for every unit of both goods.
  4. 4Pick units in descending order of MU per rupee until the income is spent; the last units of the two goods should have equal MU per rupee.
  5. 5Check three things: the ratio equality, that the cost of the bundle equals income, and total utility if asked.

Consumer's Equilibrium: Cardinal Utility Analysis in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Slices of sweets at a party

Suppose Meera eats gulab jamuns at a family party. The first one gives her 20 utils of satisfaction, the second 16, the third 12, and by the sixth she feels she has had enough.

Why does she stop around the sixth piece even though more are free? What happens to total utility if she is pushed to eat a seventh?

Try it first, then show the solutionSolution
  1. MU falls as she eats: 20, 16, 12, ... down to zero.
  2. At the sixth piece MU is about zero, so TU stops rising: she is at satiety.
  3. A seventh piece gives negative MU, so TU would fall.

Answer: She stops because the extra satisfaction has fallen to zero; forcing more makes TU fall, which is the law of diminishing marginal utility in action.

Memory hook: The last bite is never as good as the first.

Tricks to remember Consumer's Equilibrium: Cardinal Utility Analysis

Short memory aids. Say each one aloud once, then use it on a question the same day.

  • TU peaks where MU touches zero

    Think of walking up a hill: you stop climbing (MU = 0) exactly at the top (TU maximum); after the top you go down (MU negative).

    Use it when: Relationship questions and diagrams.

  • MU per rupee

    Divide the util by the price and compare 'bang per rupee'. Equal bang per rupee on every good means no reason to shift money.

    Use it when: Equi-marginal law.

  • Equilibrium needs two ticks

    Tick 1: MUx/Px = MUy/Py. Tick 2: money all spent.

    Use it when: Numericals; missing tick 2 loses a mark.

Consumer's Equilibrium: Cardinal Utility Analysis: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: MU at the point of maximum TU · 1 mark · MCQBoard level

When total utility is maximum, marginal utility is:
(a) maximum
(b) positive and falling
(c) zero
(d) negative

Try it first, then show the solutionSolution
  1. TU stops rising when the last unit adds nothing, so MU = 0.

Answer: (c) zero

Example 2: Finding MU from a TU schedule · 3 marks · Short answerBoard level

A consumer's total utility from 1 to 7 units of a good is 20, 36, 48, 56, 60, 60, 54 utils. Calculate marginal utility and state at which unit satiety is reached. Is the law of DMU followed? (3 marks)

Try it first, then show the solutionSolution
  1. MU1 = 20.
  2. MU2 = 36 - 20 = 16; MU3 = 48 - 36 = 12; MU4 = 56 - 48 = 8; MU5 = 60 - 56 = 4.
  3. MU6 = 60 - 60 = 0; MU7 = 54 - 60 = -6.
  4. MU series: 20, 16, 12, 8, 4, 0, -6.

Answer: MU is 20, 16, 12, 8, 4, 0, -6. MU = 0 at the 6th unit, where TU stays at its maximum of 60 (satiety). MU falls steadily, so the law of DMU holds; the 7th unit gives negative utility.

Example 3: Equilibrium for one commodity · 4 marks · Short answerHarder board level

The marginal utilities of successive units of a good are 30, 24, 18, 12 and 6 utils. The price is Rs 3 per unit and the marginal utility of one rupee is 4 utils. How many units will the consumer buy? Show why he buys the 3rd unit but does not buy the 5th. (4 marks)

Try it first, then show the solutionSolution
  1. Equilibrium: MUx = Px x MU of money = 3 x 4 = 12 utils.
  2. Unit 3: MU = 18 > 12 (MU per rupee 18/3 = 6 > 4), so buying adds more than the money is worth: buy.
  3. Unit 4: MU = 12 = 12 (12/3 = 4 = MU of money): equilibrium.
  4. Unit 5: MU = 6 < 12 (6/3 = 2 < 4): the satisfaction is worth less than the Rs 3 given up, so do not buy.

Answer: The consumer buys 4 units. The 5th unit is rejected because its MU of 6 utils is below the 12 utils the Rs 3 is worth.

Example 4: Equi-marginal utility with two goods · 5 marks · Long answerHarder board level

A consumer has Rs 13. Price of x is Rs 2 and of y is Rs 1. MU of successive units: x = 16, 14, 12, 10, 8, 6, 4, 2; y = 9, 8, 7, 6, 5, 4, 3, 2. Find the equilibrium bundle, check the conditions and find the total utility. (5 marks)

Try it first, then show the solutionSolution
  1. MU per rupee of x (MU/2): 8, 7, 6, 5, 4, 3, 2, 1. MU per rupee of y (MU/1): 9, 8, 7, 6, 5, 4, 3, 2.
  2. Spend the rupees in order of the highest MU per rupee: y1 (9, Re 1), x1 (8, Rs 2), y2 (8, Re 1), x2 (7, Rs 2), y3 (7, Re 1), x3 (6, Rs 2), y4 (6, Re 1), x4 (5, Rs 2), y5 (5, Re 1).
  3. Cost so far: 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 = Rs 13, so the income is exhausted. Bundle: 4 units of x and 5 of y.
  4. Check: Rs 2 x 4 + Rs 1 x 5 = 8 + 5 = Rs 13. MUx/Px = 10/2 = 5 and MUy/Py = 5/1 = 5: equal.
  5. TU = (16 + 14 + 12 + 10) + (9 + 8 + 7 + 6 + 5) = 52 + 35 = 87 utils.
  6. Cross-check: the bundle 5x + 3y also costs Rs 13 but gives 60 + 24 = 84 utils, less than 87.

Answer: Equilibrium bundle is 4 units of x and 5 of y, MUx/Px = MUy/Py = 5 and the whole Rs 13 is spent; TU = 87 utils.

Example 5: Deriving a demand point from utility · 4 marks · Long answerHOTS / competency

Using the data of the previous example (income Rs 13, Py = Re 1), find how many units of x the consumer buys when the price of x falls from Rs 2 to Re 1. What does this show about the demand curve? (4 marks)

Try it first, then show the solutionSolution
  1. New MU per rupee of x (MU/1): 16, 14, 12, 10, 8, 6, 4, 2. MU per rupee of y: 9, 8, 7, 6, 5, 4, 3, 2.
  2. Choose the 13 best rupees: x: 16, 14, 12, 10, 8, 6, 4 (7 units, Rs 7); y: 9, 8, 7, 6, 5, 4 (6 units, Rs 6). Total Rs 13.
  3. Check: MUx/Px = 4/1 = 4 and MUy/Py = 4/1 = 4: equal at the new equilibrium.
  4. Demand for x rises from 4 units at Rs 2 to 7 units at Re 1.

Answer: At Rs 2 the consumer buys 4 units of x and at Re 1 he buys 7 units: price and quantity demanded move in opposite directions, so the utility approach yields a downward-sloping demand curve.

Where students lose marks in Consumer's Equilibrium: Cardinal Utility Analysis

The slips examiners see most often, and what to write instead.

  • Mistake: Stating the equilibrium condition without MU of money

    Treating MUx = Px as universal.

    Wrong working

    Price Rs 3, MU of money 4 utils: buy till MU = 3.

    Right working

    Buy till MU = 3 x 4 = 12 utils.

    Do this: Write MUx/Px = MUm first, then substitute.

    Usually costs: 1-2 marks.

  • Mistake: Ignoring the budget in a two-goods numerical

    Stopping when ratios match.

    Wrong working

    With income Rs 15, Px = Rs 3, Py = Rs 2: ratios are equal (7 = 7) at 2 units of each, so that is equilibrium.

    Right working

    That bundle costs 2 x 3 + 2 x 2 = Rs 10, so Rs 5 is unspent; he continues to 3 units of each (cost Rs 15) where the ratios are 6 = 6.

    Do this: Always verify total expenditure equals income.

    Usually costs: 2 marks.

  • Mistake: Confusing DMU with equi-marginal utility

    Both laws involve marginal utility, so students often mix them up.

    Wrong working

    Stating 'a consumer spends income so that MU is equal in all uses' as the law of DMU.

    Right working

    DMU: MU falls with more units of one good. Equi-marginal: MU per rupee is equalised across goods.

    Do this: Learn the one-line statements separately.

    Usually costs: 2-3 marks.

  • Mistake: Taking MU from the TU column without subtracting

    Using TU values as MU.

    Wrong working

    MU of the 3rd unit = 48.

    Right working

    MU of the 3rd unit = 48 - 36 = 12.

    Do this: MU is the difference between consecutive TU values.

    Usually costs: 2 marks.

Exam tips

  • Present each law under four heads: statement, assumptions, schedule or example, diagram or explanation.
  • For DMU learn the assumptions list; criticisms are not required by the ISC syllabus.
  • Draw TU and MU one below the other so that MU = 0 lines up with the peak of TU.
  • In every two-goods numerical, show the per-rupee table, the budget check and the final TU.
  • Use 'rational consumer', 'limited income' and 'MU of the last rupee' in the equi-marginal statement.

Practice questions on Consumer's Equilibrium: Cardinal Utility Analysis

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

  1. Q1 · 1 mark · MCQBoard level

    The utility of the additional unit of a good consumed is called:
    (a) total utility
    (b) average utility
    (c) marginal utility
    (d) money utility

    Hint

    Think of the word 'additional'.

    Answer

    (c) marginal utility

  2. Q2 · 3 marks · Short answerBoard level

    A consumer's TU from 1 to 5 units is 15, 27, 36, 42 and 45 utils. Calculate the MU of each unit. State whether the law of DMU operates and give one assumption of the law. (3 marks)

    Hint

    Subtract successive TUs.

    Answer

    MU = 15, 12, 9, 6, 3. They keep falling, so the law of DMU operates (TU rises at a falling rate). One assumption: utility is measurable in utils and the MU of money is constant (or units are identical and consumed continuously).

  3. Q3 · 4 marks · Short answerHarder board level

    The MU of successive units of a good is 44, 36, 28, 20, 12 and 4 utils. Price is Rs 4 and the MU of one rupee is 5 utils. How many units will the consumer buy? What if the price falls to Rs 2 (MU of money unchanged)? (4 marks)

    Hint

    Required MU = price x MU of money.

    Answer

    At Rs 4: required MU = 4 x 5 = 20, the MU of the 4th unit, so he buys 4 units. At Rs 2: required MU = 2 x 5 = 10. The 5th unit has MU 12 > 10 (buy) and the 6th has MU 4 < 10 (do not buy), so he buys 5 units. The fall in price raises the quantity bought from 4 to 5.

  4. Q4 · 4 marks · Short answerHarder board level

    Price of x is Rs 3 and of y is Rs 2; income is Rs 15. MU of x = 24, 21, 18, 15, 12, 9 and MU of y = 16, 14, 12, 10, 8, 6 for successive units. Find the equilibrium combination and total utility, showing that both conditions are met. (4 marks)

    Hint

    MU per rupee: x = MU/3, y = MU/2.

    Answer

    MU per rupee: x = 8, 7, 6, 5, 4, 3; y = 8, 7, 6, 5, 4, 3. Rising rupee by rupee: (x1, y1) = Rs 5, (x2, y2) = Rs 10, (x3, y3) = Rs 15. Bundle: 3 units each. Check: MUx/Px = 18/3 = 6 = MUy/Py = 12/2 = 6; expenditure 3 x 3 + 3 x 2 = Rs 15. TU = (24 + 21 + 18) + (16 + 14 + 12) = 63 + 42 = 105 utils.

  5. Q5 · 3 marks · Short answerHOTS / competency

    A consumer finds that MUx/Px = 5 while MUy/Py = 3. Without any calculation of income, explain how he should reallocate spending, what happens to the two ratios, and where he finally stops. Why is this movement only possible because of the law of DMU? (3 marks)

    Hint

    Move money from the good with the lower ratio.

    Answer

    The last rupee on x yields 5 utils but on y only 3, so he should shift money from y to x. As he buys more x, MUx falls (DMU) so MUx/Px falls; as he buys less y, MUy rises so MUy/Py rises. He stops when the two ratios are equal. If MU did not diminish, the ratios would never meet and he would spend everything on x.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

  1. 1.MU when TU is maximum?Show answer

    Zero.

  2. 2.MU when TU is falling?Show answer

    Negative.

  3. 3.Equilibrium condition for one good?Show answer

    MUx/Px = MU of money (MUx = Px when MU of money is one util).

  4. 4.Equi-marginal law condition?Show answer

    MUx/Px = MUy/Py with all income spent.

  5. 5.What is a util?Show answer

    An imaginary unit used to measure satisfaction in the cardinal approach.

Consumer's Equilibrium: Cardinal Utility Analysis: frequently asked questions

What is the difference between TU and MU?

TU is the total satisfaction from all units consumed; MU is the extra satisfaction from the last unit.

Which assumptions of DMU does ISC require?

Only the assumptions are taught: measurable utility, identical units consumed continuously, rational consumer, unchanged tastes, income and prices, constant MU of money.

Why is the MU curve downward sloping?

Because each additional unit satisfies a less urgent want, so MU falls as consumption rises.

How is DMU different from the law of equi-marginal utility?

DMU concerns one good over successive units; equi-marginal utility concerns allocating a fixed income across goods to equalise MU per rupee.