Diminishing marginal utility and consumer equilibrium
Each extra unit gives less extra satisfaction. A consumer is in equilibrium when the last rupee spent on every good gives the same marginal utility.
The cardinal utility approach
Utility is the satisfaction a consumer gets from a good. The cardinal approach (associated with Alfred Marshall) assumes utility can be measured in units called utils. It also assumes a rational consumer who wants maximum satisfaction from a limited income, and a constant marginal utility of money. Total utility (TU) is the satisfaction from all units consumed. Marginal utility (MU) is the change in TU from one more unit: MU = ΔTU ÷ ΔQ. The law of diminishing marginal utility says that as a person consumes more units of a good, in a short period and with tastes unchanged, the MU of each extra unit falls.
Worked example 1: diminishing marginal utility
A worker drinks cups of plain tea during a long shift. Total utility after 0 to 7 cups is: 0, 20, 36, 48, 56, 60, 60, 56 utils. Find the marginal utility of each cup.
- Cup 1: 20 − 0 = 20. Cup 2: 36 − 20 = 16. Cup 3: 48 − 36 = 12.
- Cup 4: 56 − 48 = 8. Cup 5: 60 − 56 = 4.
- Cup 6: 60 − 60 = 0. TU is at its maximum (point of satiety).
- Cup 7: 56 − 60 = −4. MU is negative, so TU falls.
MU = 20, 16, 12, 8, 4, 0, −4. MU falls with each cup; TU is highest (60) when MU = 0.
Worked example 2: equilibrium with two goods (equi-marginal principle)
A student has Rs. 500. Good X costs Rs. 100 and good Y costs Rs. 50. MU of units 1-5 of X: 500, 400, 300, 200, 100. MU of units 1-6 of Y: 300, 250, 200, 150, 100, 50. Find the best combination.
- MU per rupee for X (÷ 100): 5, 4, 3, 2, 1. For Y (÷ 50): 6, 5, 4, 3, 2, 1.
- Spend each rupee where MU per rupee is highest, until income runs out.
- At X = 3 and Y = 4: MUx/Px = 300 ÷ 100 = 3 and MUy/Py = 150 ÷ 50 = 3. Equal. ✓
- Spending = 3 × 100 + 4 × 50 = 300 + 200 = Rs. 500. All income used. ✓
- TU = (500 + 400 + 300) + (300 + 250 + 200 + 150) = 1,200 + 900 = 2,100 utils.
- Check other affordable bundles: X = 2, Y = 6 gives 900 + 1,050 = 1,950; X = 4, Y = 2 gives 1,400 + 550 = 1,950. Both lower.
Buy 3 units of X and 4 units of Y. MU per rupee is equal (3 = 3) and TU is at its highest, 2,100 utils.
Worked example 3: deriving the individual demand curve
In example 2, the price of X falls from Rs. 100 to Rs. 50. Income (Rs. 500), Py (Rs. 50) and the MU schedules stay the same. How much X is bought now?
- New MU per rupee for X (÷ 50): 10, 8, 6, 4, 2. For Y it is still 6, 5, 4, 3, 2, 1.
- X is now better value, so the consumer buys more of it.
- At X = 5 and Y = 5: MUx/Px = 100 ÷ 50 = 2 and MUy/Py = 100 ÷ 50 = 2. Equal. ✓
- Spending = 5 × 50 + 5 × 50 = 250 + 250 = Rs. 500. ✓
- Two points on the demand curve for X: (Rs. 100, 3 units) and (Rs. 50, 5 units).
When Px falls from Rs. 100 to Rs. 50, quantity of X rises from 3 to 5. Joining such points gives a downward-sloping demand curve.
From marginal utility to the demand curve
Assumptions and limits of utility analysis
- Units must be of standard size and consumed continuously, with no change in taste or income.
- Utility cannot really be measured in numbers; people can only rank their choices.
- The marginal utility of money is assumed constant, which is unrealistic when prices or income change a lot.
- It cannot separate the income effect from the substitution effect of a price change.
- These weaknesses led to the indifference curve (ordinal) approach.
Buffet restaurants and hotel offers such as 'second item half price' work because of diminishing marginal utility: the next plate or the second item gives less satisfaction, so it must be cheaper to attract the buyer. Marketers at firms like Cargills use the same idea for multi-buy discounts.
Short answer: “State the law of diminishing marginal utility.” As a consumer takes more units of a good in a given period, other things equal, the marginal utility of each extra unit falls.
Structured question: “With Rs. 500, Px = Rs. 100 and Py = Rs. 50 and given MU schedules, find the consumer's equilibrium combination.” Compute MU per rupee for each good; the combination X = 3, Y = 4 equates MUx/Px = MUy/Py = 3 and spends all Rs. 500.
Essay: “Explain consumer equilibrium for two goods using marginal utility analysis, and derive the individual demand curve.” State the law of DMU with a table, the equi-marginal condition, work a numerical case, show how a price fall raises quantity, and note the limits of the approach.
Equal bang for every buck: MUx/Px = MUy/Py. If one ratio is higher, buy more of that good until the ratios meet.