Ordinal utility: indifference curves, MRS and preferences

Detail

1. What ordinal utility means

• Utility means the satisfaction a person gets from using a good.
• Cardinal utility (the older approach) says utility can be measured in numbers, called "utils". For example, 1 mango gives 10 utils.
• Ordinal utility says utility cannot be measured. The consumer can only rank bundles as better, worse or equally good.
• A bundle is a combination of goods, for example (2 bananas, 12 mangoes).

• "Ordinal" comes from order: 1st, 2nd, 3rd. Only the order matters. The size of the gap does not.

• J.R. Hicks and R.G.D. Allen fully developed the ordinal approach in Britain in 1934.

• Where the idea came from:
• F.Y. Edgeworth (England, 1881) and Vilfredo Pareto (Italy, 1896–97) first proposed it. [3]
• Eugen Slutsky (Russia, 1915) and Hicks and Allen (Britain, 1934) completed it. [3]
• The main idea is simple. To study a choice between bundles A and B, you only need to know which one the consumer prefers. [3]

• This removed the weak assumption of cardinal theory that utility can be measured. [3]

• The indifference curve as a tool was developed by Francis Y. Edgeworth, an Irish-born British economist. [2]

2. Indifference curve (IC)

• An indifference curve joins all bundles that give the consumer equal satisfaction.
• She is indifferent between any two points on it (A, B, C and D in Fig 2.3). "Indifferent" means she does not prefer one over the other.
• Why it slopes downward:
• She wants one more banana but must stay on the same IC.
• More bananas raise her satisfaction, so she must give up some mangoes to cancel that gain.

• One good goes up and the other goes down, so the curve falls from left to right.

• Britannica describes the classic IC as falling from left to right and convex to the origin. The consumer does not prefer any one point on it over another. [2]

3. Marginal rate of substitution (MRS)

• The MRS is the number of mangoes (Y) she is willing to give up for one extra banana (X), while her total utility stays the same.
• Formula: MRS = |ΔY/ΔX|
• This is the size of the IC's slope. The minus sign is ignored.

• Example: if ΔY/ΔX = −3/1, then MRS = 3.

• Link to marginal utility (MU): the slope of the IC equals MUx / MUy. [2]

• Marginal utility is the extra satisfaction from one more unit of a good.
• Why this holds: along an IC, total utility does not change. So the utility lost from fewer mangoes equals the utility gained from more bananas.
  • ΔY × MUy + ΔX × MUx = 0
  • So |ΔY/ΔX| = MUx / MUy

• Worked example: say MUx (banana) = 6 and MUy (mango) = 2.

  • Then MRS = 6/2 = 3.
  • One extra banana is worth 3 mangoes to her.

• Britannica calls the average slope of the IC over a stretch (an arc) the "marginal rate of substitution between the two commodities". [2]

4. Law of diminishing MRS — Table 2.2

Bundle	Bananas (X)	Mangoes (Y)	Change in Y	Change in X	MRS
A	1	15	–	–	–
B	2	12	−3	+1	3:1
C	3	10	−2	+1	2:1
D	4	9	−1	+1	1:1

• Law of diminishing MRS: as she gets more bananas, she gives up fewer and fewer mangoes for each extra banana (3, then 2, then 1).
• Why:
• Bananas become plentiful, so the MU of one more banana falls.
• Mangoes become scarce, so the MU of each remaining mango rises.

• So MUx/MUy falls, and the MRS falls with it.

• Convexity: a falling MRS makes the IC convex to the origin. This means it bends inward, towards the point (0, 0).

• The curve is steep at the top left and flat at the bottom right.
• This is the most common shape of an IC.

5. Special shapes of ICs

Case	Meaning	Example	MRS	Shape of IC
Normal goods (usual case)	Goods can replace each other, but not perfectly	Bananas and mangoes	Falls	Convex to the origin
Perfect substitutes	Goods that replace each other and give exactly the same utility	₹5 notes and ₹5 coins	Constant (1:1)	Straight line, sloping down
Perfect complements (beyond NCERT)	Goods used only in a fixed ratio	Left and right shoes	0 or undefined (not a normal trade-off)	L-shaped

• Perfect substitutes, Table 2.3: the bundles (1, 8), (2, 7), (3, 6) and (4, 5) are all equally good.
• Each time she gains 1 note, she gives up 1 coin.

• So the MRS stays at 1:1 and the IC is a straight line.

• Perfect complements:

• An extra left shoe without a matching right shoe adds nothing to her utility.
• So the IC has a corner (a kink) at the fixed ratio, and both arms of the "L" are flat or vertical.

6. Monotonic preferences

• Monotonic preferences: of two bundles, she prefers the one with more of at least one good and no less of the other. In short, "more is better".
• Examples:
• (10, 10) is preferred to (10, 9), and (10, 9) is preferred to (9, 9).
• A monotonic consumer cannot be indifferent between (10, 8) and (8, 6). The first bundle has more of both goods, so it must be better.

• If a friend is indifferent between (5, 6) and (6, 6), the friend's preferences are not monotonic. The extra unit of good 1 added nothing.

• Why it matters: monotonic preferences are why ICs slope downward and why a higher IC is better.

7. Indifference map

• An indifference map is a family of ICs (IC₁, IC₂, IC₃ …) that together show all of a consumer's preferences.
• With monotonic preferences, a higher IC (further from the origin) is preferred.
• The map shows only the ranking. IC₃ is better than IC₂, but the map cannot say by how much.

8. Properties of indifference curves

1. They slope downward from left to right. - More bananas must be balanced by fewer mangoes. - If she got more bananas without giving up mangoes, she would move to a higher IC.

2. A higher IC gives higher utility. - Table 2.4: A (1, 10), B (2, 10) and C (3, 10) all have the same mangoes but more and more bananas. - So C > B > A, and each one lies on a higher IC. - This holds only while MU is positive.

3. Two ICs never intersect. Proof by contradiction (Fig 2.8): - Suppose IC₁ and IC₂ cross at point A. - A = B, because both are on IC₁. A = C, because both are on IC₂. So B = C. - But B has more mangoes than C and the same bananas. With monotonic preferences, B must be better than C. - B = C and B > C cannot both be true, so the curves cannot cross. - The proof uses transitivity: if A = B and A = C, then B = C.

4. ICs are usually convex to the origin, because of diminishing MRS (see Section 4).

9. Other ways to show preferences

• Utility function: preferences written as a function U(x₁, x₂) that gives higher numbers to preferred bundles.
• It is an alternative to the indifference map.
• Only the ranking of the numbers matters, not their size.

• Worked example: take U = x₁ × x₂.

  • (2, 6) → 12, (3, 4) → 12, (4, 4) → 16.
  • So she is indifferent between (2, 6) and (3, 4), and prefers (4, 4) to both.
  • U = 10 × x₁ × x₂ gives 120, 120 and 160. The ranking is exactly the same, so it describes the same preferences.

• Revealed preference: introduced by Paul Samuelson in 1938. It works out preferences from what people actually buy. [4]

• Samuelson wanted a theory of consumer behaviour that did not depend on utility. [4]
• He said his method rested on observable behaviour and on very few, widely accepted assumptions. [4]
• People's preferences are revealed by what they buy when their income and prices change. [4]
• Rule: if she buys bundle A when bundle B was also affordable, then A is revealed preferred to B.

Prelims Hooks

• The ordinal utility / indifference curve approach was completed by Hicks and Allen (1934). Its roots go back to Edgeworth (1881) and Pareto (1896–97), with further work by Slutsky (1915). [3]
• MRS = |ΔY/ΔX| = MUx/MUy. It is the size of the IC's slope, ignoring the minus sign. [2]
• A convex IC comes from diminishing MRS, not from diminishing total utility. This is a common trap.
• Perfect substitutes → constant MRS → straight-line IC. Perfect complements → L-shaped IC.
• If a consumer is indifferent between (10, 8) and (8, 6), her preferences are not monotonic.
• Two ICs cannot intersect. The proof uses monotonicity + transitivity.
• A higher IC = higher utility, but the map shows only the ranking, not how much better.
• Revealed preference = Paul Samuelson, 1938. It is based on actual purchases, not on measuring utility. [4]
• The indifference curve as a tool was developed by F.Y. Edgeworth. [2]

Mains Points

• Ordinal is more realistic than cardinal:
• Consumers cannot say "this mango gives 10 utils". They can say "I prefer A to B".
• So ordinal theory builds demand on weaker and more believable assumptions.

• Revealed preference goes further and uses only data you can observe. [3][4]

• The MRS as a policy tool:

• Diminishing MRS explains why people like a mix of goods.

• This helps explain why subsidies on one food item (for example, cereals under the PDS) may not fully shift diets. Households still trade off towards pulses, milk and vegetables as they get more cereals.

• Where the model breaks down:

• Real choices can break the assumptions. Examples: addiction (non-convex preferences), goods that give negative MU such as pollution, and habit or social pressure (non-transitive choices).

• This is why behavioural economics matters for "nudge" policies, such as default options in savings and pension schemes.

• Revealed preference in data work:

• Official consumption surveys record what households actually spend, not their stated utility.
• Welfare comparisons and index numbers built from these surveys rest on the logic of revealed preference.