Theory of Consumer Behavior

Why does the demand curve slope downward? Consumer theory explains it from the way a consumer with a limited income allocates spending to get the greatest satisfaction.

Two approaches exist: the older cardinal utility analysis and the modern ordinal indifference-curve analysis.

3.1 Cardinal utility analysis

Utility is the want-satisfying power of a good. The cardinal approach (Marshall) assumes utility can be measured in units ("utils").

Total utility (TU) is the total satisfaction from all units consumed; marginal utility (MU) is the extra utility from one more unit: MUₙ = TUₙ − TUₙ₋₁.

Law of diminishing marginal utility: as a consumer takes more units of a good, MU falls; TU rises at a decreasing rate, reaches a maximum when MU = 0, and then falls (MU negative).

Units of apples	1	2	3	4	5
TU	10	18	24	28	30
MU	10	8	6	4	2

Assumptions: utility is measurable in cardinal numbers; the consumer is rational and wants to maximise utility; the marginal utility of money is constant; MU diminishes; utilities of goods are independent; income and prices are given; consumption is in standard units.

Consumer's equilibrium

With one good, the consumer buys until MU of the good = price (in units of the MU of money).

With two or more goods, the equi-marginal principle: spend so that the marginal utility per rupee is equal across goods: MU_x/P_x = MU_y/P_y = … = MU of money, with all income spent.

Derivation of the demand curve (cardinal): with one good and the MU of money fixed, the consumer buys where MU = P. If the price falls from 8 to 6, he buys more units until MU falls to 6, because MU diminishes: price and quantity are negatively related, which is the demand curve.

Criticisms: utility is not truly measurable; the constant MU of money is unrealistic for big price changes; it cannot separate the income and substitution effects, and it neglects inferior and Giffen goods.

3.2 Ordinal utility: indifference curves

The ordinal approach (Hicks and Allen) needs only that the consumer can rank bundles. An indifference curve (IC) joins all combinations of two goods giving the same satisfaction; an indifference map is a set of ICs, higher = preferred.

Properties: (1) ICs slope downward (more of one good needs less of the other); (2) convex to the origin (diminishing MRS); (3) higher ICs mean higher satisfaction; (4) ICs never intersect (it would contradict transitivity); (5) they do not touch the axes in the basic two-goods model.

Marginal rate of substitution: MRS_xy = −ΔY/ΔX along an IC = MU_x/MU_y: the amount of Y given up for one more X. It diminishes as the consumer has more X.

Budget line: all bundles costing exactly the income M: P_x·X + P_y·Y = M; intercepts M/P_x and M/P_y; slope = −P_x/P_y. An income rise shifts it out in parallel; a price change rotates it around the intercept of the unchanged good.

Consumer's equilibrium: the highest reachable IC touches the budget line: tangency with MRS = P_x/P_y (equivalently MU_x/P_x = MU_y/P_y, matching the cardinal rule), and the IC convex at that point (second-order condition).

3.3 Price effect, income effect and substitution effect

Varying one price and joining the equilibria gives the price consumption curve (PCC); its points of (price, quantity) give the demand curve.

Varying income and joining equilibria gives the income consumption curve (ICC): upward sloping for normal goods, bending back for inferior goods; the Engel curve plots quantity against income.

When P_x rises the price effect (total change in X) splits into:

• Substitution effect: X is now relatively dearer, so substitute Y; always negative (moves opposite to price). Hicks measures it by compensating income to keep the consumer on the original IC.
• Income effect: the price rise cuts real income; for a normal good this lowers consumption (reinforcing); for an inferior good it raises it (offsetting). If an inferior good's income effect is stronger than its substitution effect we get the rare Giffen good.

Derivation of the demand curve (Hicksian approach): the demand curve is the locus of price–quantity points from the PCC (the Marshallian demand includes both effects); the Hicksian (compensated) demand holds utility constant and includes only the substitution effect, so it is steeper for normal goods.

The ordinal approach needs fewer assumptions and explains inferior and Giffen goods.

(a) A thali: first helping, highest utility

(b) Marginal utility of five rounds of food

(a) Contour lines on a weather map

(b) Substitutes, complements and the usual convex case

3.5 Eight uses of utility and indifference-curve analysis

1. Explaining why the demand curve slopes down. Because each extra unit gives less satisfaction, a buyer will take more only at a lower price. A mobile-data user who already has plenty of gigabytes will buy another pack only at a discount. This is the root of the law of demand.
2. Budgeting by students and families. The equi-marginal rule says to move money from goods with low marginal utility per rupee to goods with high. A student who compares the value of one more meal, one more recharge and one more book is applying it. The final rupee should give equal satisfaction everywhere.
3. Pricing and bundling by firms. Telecom and streaming companies know that satisfaction falls with each extra unit, so they sell packs and bundles instead of single units. A pack of 10 GB at a fixed price gets more from the buyer than 10 separate sales at one price. Nepal Telecom and Ncell packs follow this idea.
4. Understanding the value of free and cheap goods. Water costs little although it is vital, while a diamond costs much although it is useless for life. The reason is the marginal, not the total, utility. Total utility of water is huge, but its marginal utility is low because there is plenty.
5. Designing taxes and subsidies. The indifference-curve method splits a price change into substitution and income effects. A government can see how much of a fall in demand after a tax is due to dearer goods and how much to lower real income. This helps decide whether to compensate the poor.
6. Studying the effect of a rise in income. The income consumption curve and the Engel curve show how demand for each good grows with income. Rice and salt grow slowly, smartphones and travel quickly, and inferior goods fall. Planners use these curves to forecast demand as Nepal's income grows.
7. Explaining inferior and Giffen goods. The cardinal approach cannot explain why a price rise may increase demand for a very poor household's staple. The ordinal approach shows that a strong income effect can outweigh the substitution effect. This rare case helps us understand the limits of the law of demand.
8. Analysing choices of work and saving. The same budget-line and indifference tools are used for choosing between income and leisure, or between consumption today and tomorrow. A young person comparing a salary job with a degree is making such a choice. The tangency rule gives a clear picture of the best compromise.

Formula and fact sheet

• MUₙ = TUₙ − TUₙ₋₁. Equilibrium: MU_x/P_x = MU_y/P_y = MU_money (equi-marginal rule).
• MRS_xy = MU_x/MU_y. Budget line: P_xX + P_yY = M, slope −P_x/P_y.
• Ordinal equilibrium: tangency, MRS = P_x/P_y.
• Price effect = substitution effect + income effect. Giffen: inferior with dominant income effect.

Chapter summary

• Cardinal analysis explains demand through diminishing marginal utility and the equi-marginal principle.
• Ordinal analysis uses indifference curves and the budget line; equilibrium is tangency (MRS = price ratio).
• Price effects decompose into substitution and income effects; PCC gives demand and ICC gives Engel curves.

Practice

1. From TU = 12, 22, 30, 36, 40, find the MU column and the unit at which MU is lowest.
2. State the law of diminishing marginal utility and its assumptions.
3. P_x = 5, P_y = 10, income 100. Draw the budget line (intercepts, slope). If MU_x/MU_y = 3, is the consumer in equilibrium? What should she do?
4. List the properties of indifference curves and say why they cannot cross.
5. Explain the income and substitution effects of a price fall for a normal good.
6. For U = XY, M = 80, P_x = 2, P_y = 4, find the optimal bundle and utility.
7. What are the criticisms of cardinal utility?

Answers and worked solutions

1. MU = 12, 10, 8, 6, 4; MU is lowest (4) at the fifth unit.

2. As consumption of a good increases, the extra satisfaction from each additional unit falls. Assumptions: standard units, consistent tastes, continuous consumption, constant MU of money, rational consumer.

3. Intercepts: X = 100/5 = 20, Y = 100/10 = 10; slope = −P_x/P_y = −0.5. MRS = 3 versus price ratio 0.5: not in equilibrium; X is worth 3 units of Y to her but costs only 0.5, so she should buy more X and less Y until MRS falls to 0.5.

4. See 3.2. If two ICs crossed, the crossing bundle would lie on both, implying that two different satisfaction levels are equal, violating transitivity.

5. A fall in P_x makes X relatively cheaper (substitution towards X) and raises real income (income effect also raises X for a normal good): both increase consumption.

6. X = M/(2P_x) = 80/4 = 20, Y = 80/8 = 10, U = 200.

7. Utility cannot be measured in numbers; MU of money is not constant; it ignores income and substitution effects; cannot explain Giffen goods.

Model answers: medium (5 marks) and long (10 marks) questions

Q1 (5 marks). State the law of diminishing marginal utility with its assumptions. Using a schedule, show how a consumer reaches equilibrium with two goods.

Introduction. Utility is the want-satisfying power of a good. Marginal utility (MU) is the extra utility from one more unit, MU_n = TU_n − TU_(n−1).

Marshall's law of diminishing marginal utility says that, as a consumer takes more units of a good, the MU of each further unit falls, other things equal.

Schedule (Figure 3.1). TU for 1 to 5 apples is 10, 18, 24, 28, 30, so MU is 10, 8, 6, 4, 2. TU rises at a decreasing rate and is maximum when MU = 0; beyond that MU is negative.

Assumptions. Utility is measurable in utils; units are of standard size and consumed in a short time; tastes, income and prices are constant; the marginal utility of money is constant; the consumer is rational.

Two-goods equilibrium (Figure 3.2). The consumer spends so that MU_x/P_x = MU_y/P_y. For income NPR 20, P_x = 4 and P_y = 2, buying in order of MU per rupee gives 3 units of X and 4 of Y, total cost 2 + 2 + 4 + 2 + 4 + 2 + 4 = 20.

At that bundle MU_x/P_x = 8/4 = 2 = MU_y/P_y = 4/2. Total utility is (16 + 12 + 8) + (10 + 8 + 6 + 4) = 64, and no other affordable bundle does better.

Conclusion. Diminishing MU explains the downward demand curve, and the equi-marginal rule tells how to share a limited income.

Q2 (5 marks). What is an indifference curve? Explain its properties. Why is the cardinal approach criticised?

Introduction. An indifference curve (IC) is the locus of all combinations of two goods that give the consumer the same satisfaction. A set of such curves is an indifference map. The ordinal approach of Hicks and Allen needs only that the consumer can rank bundles.

Properties.

1. Downward sloping: to have more of one good the consumer must give up some of the other.
2. Convex to the origin: the marginal rate of substitution (MRS = MU_x/MU_y) diminishes as X increases.
3. Higher curves mean more satisfaction: a curve farther from the origin has more of both goods.
4. Never intersect: at a crossing point two different satisfaction levels would be equal, which breaks transitivity.
5. Do not touch the axes in the basic two-goods model, because the consumer buys some of both.

Criticism of cardinal utility. Satisfaction cannot truly be measured in utils; the marginal utility of money is not constant when a price changes much; and the method cannot separate the income and substitution effects or explain inferior and Giffen goods.

Conclusion. ICs give a more realistic theory because they need only ranking, not measurement.

Q3 (10 marks). Explain consumer's equilibrium using indifference curves and a budget line. Decompose the effect of a price rise into the substitution and income effects with the Hicksian method, using a numerical example.

Introduction. A rational consumer chooses the best bundle she can afford. The budget line P_x·X + P_y·Y = M shows all affordable bundles; its intercepts are M/P_x and M/P_y and its slope is −P_x/P_y. The indifference map shows her tastes (Figure 3.3).

Equilibrium. The consumer reaches the highest IC that touches the budget line. At that point the line is tangent to the curve, so MRS = P_x/P_y, which is the same as MU_x/P_x = MU_y/P_y (the cardinal rule). The curve must be convex there (second-order condition).

Example. U = X·Y, M = 120, P_x = 4, P_y = 2. The optimum is X = M/(2P_x) = 15, Y = M/(2P_y) = 30, U = 450 (point A).

Price effect. P_x rises to 8. The new optimum is X = 120/16 = 7.5, Y = 30, U = 225 (point B). The price effect on X is 7.5 − 15 = −7.5.

Hicksian decomposition.

1. Compensate the consumer so that she can still reach U = 450 at the new prices: X = √(450 × 2/8) = 10.61, Y = √(450 × 8/2) = 42.43 (point C). Expenditure is 8 × 10.61 + 2 × 42.43 = 169.71, which is 49.71 more than her income.
2. Substitution effect (A to C) = 10.61 − 15 = −4.39. It is always negative, because X is now relatively dearer.
3. Income effect (C to B) = 7.5 − 10.61 = −3.11. It is negative because X is a normal good and real income has fallen.
4. Check: −4.39 + (−3.11) = −7.5.

Good type	Substitution effect	Income effect	Total effect
Normal	negative	negative (reinforces)	demand falls
Inferior	negative	positive (offsets)	falls, by less
Giffen	negative	positive and larger	demand rises

Conclusion. Tangency gives the optimum, and the Hicks method shows how much of a price effect is a pure change of relative price and how much a loss of real income.