Consumer Behaviour

Lessons

Senior 4 · Economics

Consumer Behaviour

4 lessons

Lesson 1 of 4

Utility and the law of diminishing marginal utility

25 min

Water or a diamond?

In 1776 Adam Smith set out a puzzle that took economists almost a century to solve. "Nothing is more useful than water," he wrote, "but it will purchase scarce any thing." A diamond, on the other hand, "has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it." Life is impossible without water and perfectly possible without diamonds, so why is the diamond so much dearer?

The answer came in the early 1870s, when three economists working separately, William Stanley Jevons in England (1871), Carl Menger in Austria (1871) and Léon Walras in Switzerland (1874), turned attention from the total satisfaction a good gives to the satisfaction from one more unit of it. That idea, marginal utility, is the starting point of the theory of consumer behaviour, and by the end of this lesson you will be able to use it to resolve Smith's puzzle yourself.

1. The consumer and the assumptions of consumer behaviour

A consumer is a person or household that buys goods and services to satisfy its own wants, as the final user. Consumer behaviour is the study of how consumers decide what to buy, and how much of each good, given their income and the prices they face. The theory rests on these assumptions:

Rationality. The consumer aims to get the greatest possible satisfaction from the income available.
Limited income. Income is fixed in the period studied and is all spent (or saving is treated as another use).
Given prices. One buyer is too small to change market prices, so prices are taken as given.
Known and consistent preferences. The consumer knows the goods available and can rank them; tastes do not change during the period.
Diminishing marginal utility. Each extra unit of a good adds less satisfaction than the one before (the law studied below).
Ceteris paribus. Everything not under study (incomes, the prices of other goods, fashion) is held constant.

Economists have built the theory in two ways, and an A-level answer should name which one it uses.

	Cardinal utility approach	Ordinal utility approach
Associated with	Alfred Marshall, Principles of Economics (1890)	J. R. Hicks and R. G. D. Allen (1934)
Can utility be measured?	Yes, in units called utils, or in the money a consumer would pay	No: the consumer can only rank bundles (first, second, equal)
Main tools	Total and marginal utility; the equi-marginal principle	Indifference curves and the budget line (lessons 2 and 3)
Extra assumption	The marginal utility of money is constant	Preferences are complete, consistent (transitive) and more is preferred to less

Definition

Utility: the satisfaction a consumer gets from consuming a good or service; the want-satisfying power of a good. Utility is subjective (it differs between people), it changes with time and place, and it is not the same as usefulness or goodness: a cigarette has utility for a smoker although it harms his health.

Example: a raincoat has high utility for a moto rider in the rainy season of March to May and much less in the dry season of July.

Definition

Total utility (TU): the whole satisfaction a consumer gets from consuming a given quantity of a good in a period.

Marginal utility (MU): the change in total utility when one more unit of the good is consumed:

MU_n = TU_n - TU_{n-1} \qquad\text{or, for any change,}\qquad MU = \frac{\Delta TU}{\Delta Q}

Util: an imaginary unit of satisfaction used in the cardinal approach.

Example: if your total utility from 2 bottles of water is 30 utils and from 3 bottles is 38 utils, the marginal utility of the 3rd bottle is $38 - 30 = 8$ utils.

2. Total and marginal utility from a schedule

Ingabire comes home hungry from a long Saturday of umuganda and buys brochettes. Her satisfaction from each quantity, in the same sitting, is shown below. (The utils are illustrative: they follow the equation $T U = 20 Q - 2 Q^{2}$ .)

Brochettes eaten, $Q$	Total utility, TU (utils)	Marginal utility, MU (utils)	Average utility, $AU = \frac{TU}{Q}$
0	0	–	–
1	18	$18 - 0 = 18$	18
2	32	$32 - 18 = 14$	16
3	42	$42 - 32 = 10$	14
4	48	$48 - 42 = 6$	12
5	50	$50 - 48 = 2$	10
6	48	$48 - 50 = - 2$	8

Two facts follow from the definitions. Total utility is the sum of the marginal utilities, $TU_n = MU_1 + MU_2 + \dots + MU_n$ : here $18 + 14 + 10 + 6 + 2 = 50$ utils after 5 brochettes. And marginal utility is the slope of the total utility curve, so when TU is at its peak, MU is zero. (If you have met differentiation: $MU = \frac{d(TU)}{dQ} = 20 - 4Q$ , which is 0 at $Q = 5$ .)

Reading the diagram. Each marginal utility belongs to the step from one quantity to the next, so it is plotted at the middle of that step (the 3rd brochette at $Q = 2.5$ ). The MU points lie on one straight line, and that line reaches zero at $Q = 5$ , directly below the top of the TU curve.

Worked example 1: from total to marginal utility and back

Habimana drinks cups of passion-fruit juice on a hot afternoon (illustrative utils).

Cups	1	2	3	4	5	6	7
TU (utils)	30	54	72	84	90	90	84

(a) Marginal utility of each cup. $MU = 30,\ 24,\ 18,\ 12,\ 6,\ 0,\ -6$ utils (for example, the 4th cup: $84 - 72 = 12$ ).

(b) Point of satiety. TU stops rising at 90 utils; the 6th cup adds nothing ( $M U_{6} = 0$ ). Habimana is fully satisfied with 5 or 6 cups and would not choose a 7th even if it were free, because it lowers TU by 6 utils.

(c) Check by adding. $30 + 24 + 18 + 12 + 6 = 90$ utils, the TU of 5 cups. ✔

(d) Pattern. MU falls by 6 utils with each cup: diminishing marginal utility, at a constant rate.

3. The law of diminishing marginal utility

The law: as a consumer takes more and more units of a good in a given period, the extra satisfaction from each additional unit eventually falls, other things being equal. Marshall stated it in 1890 as the basic "law of satiable wants": any one want can be satisfied, so the more of a thing you have, the less an extra unit adds.

Marginal utility is…	Total utility is…	Ingabire's brochettes
positive but falling	rising, at a slower and slower rate	1st to 5th brochette
zero	at its maximum (point of satiety)	at 5 brochettes
negative	falling (disutility)	6th brochette

Assumptions of the law. The law holds only if: (1) the units are homogeneous (identical in size and quality); (2) they are of a reasonable size (one grain of rice or one shoe is not a sensible unit); (3) consumption is continuous, without a long gap (lunch today and lunch tomorrow are separate wants); (4) the consumer's tastes, income and habits do not change; and (5) the consumer is rational.

Apparent exceptions. Several cases seem to break the law, but on inspection most of them break one of its assumptions instead.

Hobbies and collections. A stamp or coin collector may value each new item more as the set nears completion. But each item is different, so the units are not homogeneous.
Addictive goods. For a person addicted to alcohol or tobacco, each dose seems to raise the craving. Here the consumer's tastes change as he consumes, and the choice is not fully rational.
Money. Some people seem to want more money the more they have. Money is a claim on all goods, not one good, and most economists still hold that the utility of an extra Frw 10,000 is lower for a rich person than for a poor one.
Units that are too small, and fashion or prestige goods. The second shoe of a pair adds more than the first; and a prestige good may gain utility as others admire it. Neither is a reasonable, unchanging unit.

Importance of the law. It explains why the demand curve slopes downwards (section 4), why consumers spread their income over many goods rather than one, the idea of consumer surplus (lesson 4), and the argument for taxing higher incomes at higher rates. It also resolves the water–diamond puzzle.

Smith's puzzle, solved. Price depends on marginal utility, not total utility. Water is plentiful, so the last litre a person uses (for washing a car, say) adds very little satisfaction and commands a low price, even though the total utility of water is enormous. Diamonds are so scarce that the marginal utility of one more is high. A person lost in a desert, with no water at all, would gladly swap a diamond for a litre.

Warning

Falling MU is not falling TU. As long as marginal utility is positive, total utility is still rising, only more slowly. Total utility falls only when marginal utility becomes negative. A common exam slip is to say "the law of diminishing marginal utility means the consumer gets less satisfaction".

4. From marginal utility to the demand curve and consumer equilibrium

Suppose utility is measured by the most a consumer would pay for each extra unit. Ingabire's marginal utility in francs (illustrative) is: 1st brochette Frw 1,200, 2nd Frw 1,000, 3rd Frw 800, 4th Frw 600, 5th Frw 400. A rational consumer keeps buying while the marginal utility of a unit is at least its price, and stops where

M U_{x} = P_{x}

So at a price of Frw 800 she buys 3 brochettes; if the price falls to Frw 600 she buys 4; at Frw 400, 5. Because marginal utility falls, a larger quantity is bought only at a lower price: diminishing marginal utility is the reason the individual demand curve slopes downwards.

More than one good. With several goods and a fixed income, the consumer is in equilibrium (has no reason to change his spending) when the last franc spent on each good brings the same marginal utility, and all income is spent:

\frac{MU_x}{P_x} = \frac{MU_y}{P_y} = \dots = MU_m \qquad\text{and}\qquad P_x X + P_y Y = M

where $M$ is income and $M U_{m}$ is the marginal utility of a franc. If $\frac{MU_x}{P_x} > \frac{MU_y}{P_y}$ , a franc moved from Y to X adds more utility than it loses, so the consumer buys more X and less Y until the ratios are equal.

Definition

Equi-marginal principle (law of equal marginal utility per franc): a consumer with a fixed income maximises total utility by spending it so that the marginal utility per franc is the same for every good bought.

Example: if the last Frw 100 spent on airtime gives 5 utils and the last Frw 100 spent on bread gives 8 utils, moving Frw 100 from airtime to bread raises total utility by about 3 utils.

Worked example 2: equilibrium with two goods

Mutoni has Frw 1,000 to spend on bottles of juice (X, Frw 200 each) and mandazi (Y, Frw 100 each). Her marginal utilities are (illustrative):

Unit	1	2	3	4	5	6
$M U_{x}$ (utils)	80	60	40	30	20	10
$\frac{MU_x}{P_x}$	0.40	0.30	0.20	0.15	0.10	0.05
$M U_{y}$ (utils)	50	40	30	20	10	5
$\frac{MU_y}{P_y}$	0.50	0.40	0.30	0.20	0.10	0.05

Step 1: find bundles with equal ratios. 0.30 = 0.30 at $X = 2, Y = 3$ ; 0.20 = 0.20 at $X = 3, Y = 4$ ; 0.10 = 0.10 at $X = 5, Y = 5$ .

Step 2: test each against the budget.

200(2) + 100(3) = 700 \qquad 200(3) + 100(4) = 1{,}000 \qquad 200(5) + 100(5) = 1{,}500

Only $X = 3,\ Y = 4$ spends exactly Frw 1,000. The first leaves Frw 300 unspent and the last is unaffordable.

Step 3: total utility. $T U = (80 + 60 + 40) + (50 + 40 + 30 + 20) = 180 + 140 = 320$ utils.

Check: another affordable bundle, $X = 2,\ Y = 6$ (also Frw 1,000), gives $140 + 155 = 295$ utils, which is less. Equal marginal utility per franc really does give the most satisfaction.

Tip

"MU is the slope of TU." Steep TU, big MU; flat top of TU, zero MU; TU turning down, negative MU. For two goods, compare MU per franc, never MU alone: a good with a bigger MU may still be the worse buy if it is much dearer.

Important

Discuss. Many governments tax higher incomes at higher rates, arguing that Frw 10,000 taken from a rich person costs her less utility than Frw 10,000 taken from a poor person. Use the law of diminishing marginal utility to support this argument, then give one reason why an economist might say it cannot be proved.

tutor2u, "Total and Marginal Utility I A Level and IB Economics", a short revision clip. Watch how total utility changes as marginal utility falls, and the link drawn between diminishing marginal utility and the downward-sloping demand curve; compare both with the brochette diagram above.

✦

Practice

Define utility, total utility and marginal utility. Why is utility not the same as usefulness?
State four assumptions of the cardinal utility approach.
Copy and complete: cups of tea 1 to 6 give total utility 25, 45, 60, 70, 75, 75 utils. (a) Find the marginal utility of each cup. (b) At what quantity is the point of satiety reached? (c) What would MU be for a 7th cup if TU fell to 72?
A consumer's total utility is $T U = 30 Q - 3 Q^{2}$ . (a) Find TU for $Q = 1$ to $5$ . (b) Find the marginal utility of each unit. (c) At what quantity is TU greatest?
State the law of diminishing marginal utility and explain two of its assumptions.
"A stamp collector's marginal utility rises with each new stamp, so the law is wrong." Evaluate this statement.
Gatete has Frw 650 to spend on X (Frw 100) and Y (Frw 50). $M U_{x}$ : 50, 40, 30, 20, 10. $M U_{y}$ : 30, 25, 20, 15, 10, 5. Find his equilibrium bundle and his total utility.
Using the law of diminishing marginal utility, explain why water is cheaper than gold although water is more necessary for life.

Show answersHide answers

Utility is the satisfaction from consuming a good. TU is the whole satisfaction from a given quantity; MU is the change in TU from one more unit. Utility is subjective and morally neutral: a harmful good (a cigarette) can still give utility, and a useful good can give none to someone who does not want it.
Any four: the consumer is rational; utility can be measured in utils; income is limited; prices are given; the marginal utility of money is constant; tastes are constant; MU diminishes; ceteris paribus.
(a) 25, 20, 15, 10, 5, 0. (b) At the 6th cup: MU = 0 and TU is at its maximum, 75 (TU is already 75 at 5 cups). (c) $72 - 75 = - 3$ utils.
(a) 27, 48, 63, 72, 75. (b) 27, 21, 15, 9, 3. (c) TU is greatest at $Q = 5$ (75 utils); a 6th unit gives $T U = 180 - 108 = 72$ , so $M U_{6} = - 3$ . (Calculus: $30 - 6 Q = 0$ gives $Q = 5$ .)
As more units of a good are consumed in a period, the extra satisfaction from each additional unit falls, other things equal. Any two assumptions with a reason, e.g. homogeneous units (a better-quality unit could give more utility); continuous consumption (after a gap the want returns).
The units are not homogeneous: each new stamp is a different item that may complete a set. The case breaks an assumption of the law rather than disproving it; for identical copies of the same stamp, MU would fall.
$\frac{MU_x}{P_x}$ : 0.5, 0.4, 0.3, 0.2, 0.1. $\frac{MU_y}{P_y}$ : 0.6, 0.5, 0.4, 0.3, 0.2, 0.1. At $X = 4,\ Y = 5$ both ratios equal 0.2 and spending is $400 + 250 = 650$ . $T U = (50 + 40 + 30 + 20) + (30 + 25 + 20 + 15 + 10) = 140 + 100 = 240$ utils.
Price reflects marginal, not total, utility. Water is plentiful, so the marginal unit adds little satisfaction and fetches a low price, although its total utility is huge. Gold is scarce, so the marginal unit has high utility and a high price.

Quick check

Key points

Utility is satisfaction; total utility is the sum of marginal utilities, and $MU_n = TU_n - TU_{n-1}$ .
The law of diminishing marginal utility: extra units add less and less satisfaction. TU peaks where MU = 0 and falls when MU is negative.
A consumer buys a good until $M U = P$ ; falling MU explains the downward-sloping demand curve.
With several goods, equilibrium is $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ with all income spent.

Sources and credits

Quotation: An Inquiry into the Nature and Causes of the Wealth of Nations, Adam Smith (1776), Book I, Chapter IV; public domain, via Project Gutenberg.
Photo: Rwanda Brochettes, Schlawgclart, Wikimedia Commons, CC0 1.0.
Video: Total and Marginal Utility I A Level and IB Economics, tutor2u, embedded from YouTube.

Written by Ganzaa: the text, the utility diagram (drawn to scale from $T U = 20 Q - 2 Q^{2}$ ), the tables, the worked examples and the practice items. All utils and prices in the examples are illustrative.

1Water or a diamond?

21. The consumer and the assumptions of consumer behaviour

32. Total and marginal utility from a schedule

43. The law of diminishing marginal utility

54. From marginal utility to the demand curve and consumer equilibrium

Water or a diamond?

1. The consumer and the assumptions of consumer behaviour

2. Total and marginal utility from a schedule

3. The law of diminishing marginal utility

4. From marginal utility to the demand curve and consumer equilibrium