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Here is an elaborated discussion on the revealed preference approach to consumer behaviour.
Introduction to the Revealed Preference Approach:
As we already know, what preference could tell us about a consumer’s behaviour. Paul A. Samuelson has invented the revealed preference theory in 1938 to predict a consumer’s preferences from observing his actual behaviour assuming that his preferences remain unchanged during the observation period. Since then the topic has assumed considerable importance in the theory of consumer demand.
The revealed preference approach is no doubt a major breakthrough in the theory of demand, because it made possible the establishment of the ‘law of demand’ directly (on the basis of two revealed preference axioms) without the use of indifference curves and all the restrictive assumptions on which the indifference curve approach is based.
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As for the ordering of consumer’s preference, the revealed preference approach has the advantage over the indifference curve approach of establishing the existence and the convexity of the indifference curves (it does not accept them axiomatically). However, if we accept the revealed preference approach the indifference curves become redundant in the derivation of the demand curve.
We know that indifference curves are derived by asking the consumer to choose from among various market baskets or combinations of commodities. But there is no guarantee that consumers can or will give reliable answers to direct questions about their preferences.
According to the revealed preference approach, a consumer’s indifference curves can be derived from observing a consumer’s actual behaviour in the market place and without any need to inquire directly about his preferences. For example, if a consumer prefers a basket X rather than Y, even though X is not cheaper than Y, we can infer that the consumer prefers X to Y.
Assumptions of the Revealed Preference Approach:
The theory of revealed preference is based on the following assumptions:
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1. Rationality:
The consumer is assumed to behave rationally, that is, he prefers bundles of goods that include more (larger quantities) of the two goods (x1 and x2).
2. Unchanged Taste:
The taste of the consumer does not change over the observation period.
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3. Consistency:
The consumer’s tastes are consistent so that if he purchases basket X rather than Y, he will never prefer Y to X. Symbolically if X > Y, Y > X.
4. Transitivity:
The consumer’s tastes are transitive, so that if he prefers X to Y and Y to Z, he will prefer X to Z. Symbolically, if in any particular situation X > Y and Y > Z, then X > Z.
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5. The Revealed Preference Axiom:
The rational consumer, by choosing a collection of goods in any one budget situation, reveals his preference among all other alternative bundles available under the budget constraint. The chosen ‘basket of goods’ maximises his utility. The revealed preference for a particular collection of goods implies (axiomatically) the maximisation of utility of the consumer.
6. Price Effect:
The consumer can be induced to purchase any basket of commodities if its price is sufficiently lowered.
The Basic Idea of Revealed Preference Approach:
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Here we assume, for the sake of simplicity, that the underlying preference of the consumer is strictly convex. This assumption is not really necessary for the development of the revealed preference approach but it enables us to derive indifference curves from the actual choice of the consumer in the market place.
In Fig. 10.1 we show two bundles (x1, x2) and (y1, y2) and both are affordable. If (x1, x2) is better than (y1, y2) it is quite obvious that (x1, x2) is the optimal bundle. Although both are available at the given budget (x1, x2) is preferable. With the budget line AB, the chosen bundle (x1, x2) is revealed preferred to the bundle (y1, y2) which could have been chosen.
This argument holds for any other bundle on the budget line or inside the budget line other than the chosen bundle.
Since it could have been bought with the given budget but was not, then (x1, x2) must be better. In other words, all of the bundles in the shaded area in Fig. 10.1 below the budget line are revealed worse than the chosen bundle.
The reason is that these could have been chosen with the existing budget constraint but were rejected in favour of (x1, x2). So here we make the assumption that there is a uniquely demanded bundle for each budget.
The Algebra of Revealed Preference:
Suppose the consumer purchases the bundle (x1, x2) at prices (p1, p2) with his income m. That (y1, y2) is affordable at those prices and income simply means that (y1, y2) satisfies the budget constraint;
p1y1 + p2y2 < m ………. (1)
Since (x1, x2) is actually demanded at the given budget, and the budget is exhausted, we have;
p1x1 + p2x2 = m ……….. (2)
Moreover since (y1, y2) is affordable at the budget (p1, p2, m) we get, by combining (1) and (2), the following inequality:
p1x1 + p2x2 > p1y1 + p2y2 (3)
Suppose inequality (3) is satisfied and (y1, y2) is actually a different bundle from (x1, x2). In this case (x1, x2) is said to be directly revealed preferred to (y1, y2). Thus revealed preference approach enables us to compare the bundle that is actually demanded at a particular budget and the bundles that could have been demanded at that budget.
The implication of the statement that X is revealed preferred to Y is that, when both were affordable, X was chosen. This point is quite obvious from inequality (3).
From Revealed Preference to Preference:
The basic logic of revealed preference is that people choose the best things they can afford. The actual choices they make are preferred to the potential choices, i.e., that could have been made with the same budget. If, for instance, (x1, x2) is directly revealed preferred to (y1, y2), then (x1, x2) is, in fact, preferred to (y1, y2). This point is highlighted by the following basic principle of revealed preference.
The basic principle of revealed preference:
Let (x1, x2) be the chosen bundle at fixed prices (p1, p2) and let (y1, y2) be some other bundle such that p1x1 + p2x2 > p1y1 + p2y2. Then if the consumer ends up choosing (x1, x2), the most preferred bundle then he must have (x1, x2) 
(y1, y2).
If the consumer chooses the best bundle X = (x1, x2) and not Y = (y1, y2) when both were affordable revealed preference implies ‘preference’. He arrives at his preference through ranking. In this case X is ranked ahead of Y and thus ‘chosen over Y’.
Thus the basic principle of revealed preference can be stated as follows:
If a bundle X is chosen over a bundle Y, then X must be preferred to Y.
This basic principle enables us to use actual choices of a representative consumer in the market place to predict his (her) underlying preferences.
The central point of the revealed preference approach is plain and simple- if we observe that one bundle is chosen when another is affordable, then the first is preferred to the second.
Let us suppose that (y1, y2) is a demanded bundle at prices (p’1, p’2) and that (y1, y2) is itself revealed preferred to some other bundle (z1, z2) at those prices. That is;
p’1y1 + p’2y2 > p’1z1 + P’2z2
Then (x1, x2) 
(y1, y2) and (y1, y2) 
(z1, z2). The transitivity assumption of the revealed preference approach enables us to conclude that (x1, x2) 
(z1, z2).
This point is illustrated in Fig. 10.2 where (x1, x2) must be better than (z1, z2) for the consumer who made the actual choice. In this case the bundle (x1, x2) is indirectly revealed preferred to the bundle (z1, z2). If a bundle is directly or indirectly revealed preferred to another bundle then the first bundle is revealed preferred to the second.
Just by looking at a consumer’s actual choice we gain some insight about his underlying preferences. Fig. 10.2 contains several observations on demanded bundles at two different budgets. Suppose m corresponds to the budget line AB and m’ the budget line CD.
Since (x1, x2) is revealed preferred, either directly or indirectly, to all of the bundles in the shaded area, (x1, x2) is, in fact, preferred to those bundles by the consumer who made these choices. This implies that the true indifference curve through (x1, x2) has to lie above the shaded area. This point may now be proved.
Drawing Indifference Curves from Actual Preference:
The cardinal utility approach was rejected by J.R. Hicks and R. G. D. Allen on the ground that utility is not measurable in numbers. Similarly whether the consumer possesses indifference curves is questionable. However, H. S. Houthakker has proved that a consumer who always conforms to the axioms of revealed preference must possess an indifference map.
Let us suppose two bundles Y and Z are revealed preferred to X as shown in Fig. 10.3. Since averages are preferred to extremes all of the weighted averages of Y and Z are preferred to X as well. And if preferences are monotonic, then all the bundles that have more of both goods (x1, x2) than X, Y and Z — or any weighted averages — are also preferred to X.
The region 0ARD consists of all the bundles to which X is revealed preferred because all these bundles cost less than X. So all the bundles in the upper shaded area are better than X, just as all the bundles in the lower shaded area are worse than X, according to the preferences of the consumer who made the choices. Then the true indifference curve through X must lie somewhere between the two shaded bundles.
The consumer’s indifference map would be constructed quite accurately (the ‘true’ indifference map could be approximated as closely as is desired) by confronting him with various approximately chosen price sets and observing his purchases. If the consumer does not confirm to the axioms, he said to be ‘irrational’.
His consistent actions mean that he does possess an indifference map and the shape of his utility function cannot be determined by observing his behaviour.
The Strong Axiom of Revealed Preference (SARP):
The WARP suggests that if the consumer has consistent preferences then if X is preferred to Y and Y is preferred to Z, then Y will never be preferred to X. The SARP requires the same condition to hold for indirect revealed preference.
This axiom is stated as follows:
Statement of the Axiom:
If (x1, x2) is revealed preferred to (y1, y2) (either directly or indirectly) and (y1, y2) is different from (x1, x2), then (y1, y2) cannot be directly or indirectly revealed preferred to (x1, x2). The observed behaviour of an optimising consumer must satisfy the SARP.
Thus if a consumer is optimising and (x1, x2) is revealed preferred to (y1, y2), then (y1, y2) cannot be revealed preferred to (x1, x2), because there is a logical contradiction in this case.
Since the underlying preferences of the consumer must be transitive, the revealed preferences of the consumer have to be transitive, two. Thus for a rational consumer the SARP is a necessary condition of optimising (welfare-maximising) behaviour.
This implies that if a consumer always chooses the best thing he can afford, then his actual (observed) behaviour must satisfy SARR Alternatively stated, any behaviour satisfying the SARP will surely be generated by the welfare-maximising behaviour of a rational consumer.
The reason is that if the actual choices satisfy SARR we can always find nice, well-behaved preferences that could have generated the actual (observed) choices. Thus SARP is also a sufficient condition for welfare maximisation. This implies that if actual choices satisfy SARP, then it is possible to find preferences for which actual behaviour is the same as the optimising behaviour.
Checking SARP:
In Table 10.4 bundle 1 is directly preferred to bundle 2 since there is a star in row 1, column 2 and bundle 2 is directly revealed preferred to bundle 3 since there is star in row 2, column 3. Therefore bundle 1 is indirectly revealed preferred to bundle 3, and we indicate this by putting a star (in brackets) in row 1, column 3.
Negative Substitution Effect:
Substitution effect refers to the change in demand when prices change but a consumer’s real income or purchasing power is held constant, so that he can still buy the original bundle, i.e., the bundle he used to buy before the price change. That substitution effect is always negative can be proved by using the axioms of revealed preference.
Let(x1, x2) be an actually demanded bundle at some prices (p1, p2) and let (y1, y2) be an optimal bundle at some other prices (p’1, p’2). The consumer’s income is assumed to be such that he is indifferent between (x1, x2) and (y1, y2) so that neither bundle is revealed preferred to the other.
From the definition (axioms) of revealed preference, it follows that the following two inequalities are not true:
This is a general result regarding how the quantity demanded changes when prices change if income is so adjusted as to make the consumer neither better off nor worse off (i.e., to keep him on the same budget line). In this case we are changing only p1 keeping p2 constant.
Therefore p2 = p2 and we arrive at the following condition:
(p’1 – p1) (y1 – x1) < 0
This inequality simply suggests that the change in quantity demanded must have the opposite sign from that of price change. Thus the substitution effect is always negative.
The Superiority of Revealed Preference Approach to IC Approach:
Utility functions are conventional numerical representations of preferences. However, neither they nor a rational consumer’s preferences are directly observable.
This subjectivity of the foundations of consumer theory stimulated interest in the development of a theory of demand based on observable and measurable phenomena, viz., the bundles actually bought by a consumer and the prices and incomes at which they were bought.
The emphasis in Samuelson’s revealed preference approach is on assumptions about the consumer’s actual behaviour in the market place which can be observed rather than on his preferences, which cannot.
Samuelson’s revealed preference approach not only eschews such unobservable magnitudes as the marginal utility of money, it does not even follow the cardinal and/or ordinal approaches in deriving hypotheses about behaviour from assumptions about a preference ordering or a utility function.
Though it is known as ‘revealed preference theory’, the term may initially appear to be a misnomer. The reason is that no assumptions are made about preferences as such.
Its only axioms are about behaviour and what is most remarkable about this approach is that the fundamental results of demand theory can be derived from only two such axioms:
Axiom R-1 (Choice):
From any set of alternatives, the consumer makes a choice.
Axiom R-2 (Consistency):
If X is chosen from a set of alternatives that includes Y (which is different from X), then any set of alternatives ‘from which X’ is chosen must not contain X. A violation of Axiom R-2 would occur if X were chosen when Y could have been chosen (the consumer revealed a preference for X over Y) and Y were chosen when X could have been chosen.
The superiority of revealed preference approach lies in the fact that nothing is said about non-saturation, continuity or convexity. The whole approach is based on the actual choice of a consumer, in the market place.
Moreover, the revealed preference approach can be extended to cover more than two commodities. So it is more realistic than the indifference curve approach which is based on the assumption that the consumer buys only two goods which are substitutes of each other.
Finally, although the axioms of revealed preference say nothing about non-saturation or convexity (let alone strict convexity) the corner solution and bliss do not upset the fundamental results of demand theory. Corner solutions cause no difficulty.
It is possible to show that the utility-maximising theory of the consumer and the revealed preference theory are equivalent: all the predictions derived from the assumption about preferences on the basis of the Hicks-Allen approach can also be derived from the assumptions about behaviour made by Samuelson.
A consumer who satisfies the preference assumptions will also satisfy these behavioural assumptions. In a like manner, if the consumer satisfies the behavioural assumptions, it is possible to derive indifference curves which have all the usual properties. And so he can be thought of as acting as if he possessed preferences satisfying the preference assumptions.
Quasi-Substitution Effect:
As in the case of the indifference curve approach here we assume that there are only two commodities (x1 and x2) and the whole budget is spent. See Fig. 10.6. From the initial budget line AB the consumer chooses X. Then p1 falls and the budget line becomes AC. Since Samuelson did not assume the existence of the indifference curves, it is not possible to define a substitution effect in the true sense.
Instead Samuelson defines quasi-substitution effect. The consumer’s budget is reduced so much that he is just able to buy the same amount of each good as before the fall in p1. Thus we draw a new budget line A’C’ passing through X and parallel to AC. We then define the movement (if any) from X along A’C’ as the quasi-substitution effect.
It is possible, from a knowledge of quantities and prices, to calculate precisely the change in the budget required for the quasi-substitution effect. This is not possible, without a precise knowledge of the indifference map, for the substitution effect.
The Homogeneity Property of the Demand Function:
The existence of demand functions and their homogeneity property can also be derived from axioms R-1 and R-2 alone. These two purely behavioural axioms of revealed preference theory imply zero-degree homogeneity of demand because demand functions are homogeneous by nature (unlike production functions which are homogeneous by assumption).
Let the consumer be faced with the usual budget constraint P.X < m where P.X = p1x1 + p2x2. This defines a set of alternatives from which the consumer, by Axiom R-1, makes a choice X. If m and P remain unchanged, or change in the same proportion, the set of alternatives remains unchanged and continues to contain X.
To choose any alternative other than X (and, therefore, a different quantity of any commodity) would violate the consistency Axiom R-2. This is how Samuelson proves that demand functions are homogeneous of degree zero in prices and income.
Final Comments:
Thus we see that the revealed preference approach is, no doubt, a major advancement to the theory of consumer demand. It provides a direct way to derive the demand curve, without requiring the use of the concept of utility.
The theory can prove the existence and convexity of the indifference curves under the axiom (assumption) than the cardinal (utility) and ordinal (indifference curve) approaches. It has also provided the basis for the construction of index numbers of the cost of living and their use for judging changes in consumer welfare in situations where prices change.
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