Prof. Samuelson has developed an alternative method to consumer behaviour theory that, in theory, does not require the consumer to provide any information about himself.

He published the article named “Consumption Theory in terms of Revealed Preference” in 1938.

If his preferences do not change, this theory, known as the Revealed Preference Theory (RPT), allows us to learn everything we need to know simply by observing his market behaviour, by seeing what he buys at different prices, assuming that his acquisitions and purchasing experiences do not change his preference patterns or purchase desires. Samuelson’s RPT is based on a basic concept. A consumer will choose to buy a certain combination of things because he prefers it to the other options accessible to him or because it is less expensive. Assume we see that, out of two collections of commodities for sale, the consumer picks A but not B.

We cannot therefore conclude that he prefers A to B, because it is equally feasible that he purchases A because it is the less costly collection, even though he would have been delighted with B. However, pricing information may be able to eliminate this uncertainty. If their prices indicate that A is not less costly than B (or that B is not more costly than A), then there is only one reasonable explanation for the consumer’s choice—he bought A because he preferred it.

More broadly, if a consumer purchases some collection of goods, A, rather than any of the alternative collections B, C, and D, and if none of the latter collections is more costly than A, we say that A has been revealed preferred to the combinations B, C, and D, or that B, C, and D have been revealed inferior to A.

As a result, if the customer purchases combination E₁ (x₁, y₁) of the commodities X and Y but does not purchase combination E2 (x2, y2) at the prices (p1x, p1y,) of the items, we may conclude that he prefers combination E1 to combination E2.

P_x¹x₁+ P_y¹y₁≥P_x¹x₂+ P_y¹y₂

The consumer’s price line may be used to find the whole set of combinations of the commodities X and Y to which a specific combination is clearly favoured. Assume that the consumer’s budget line in Figure is L₁M₁ and that he is seen to purchase the combination E₁ (x₁, y₁) that sits on this line. Now, because the costs of all combinations on the budget line are the same as those of E₁, and because the costs of all combinations below and to the left of the budget line are lower than those of E₁, we can say that E₁ is revealed preferred to all combinations on or below the consumer’s budget line.

Again, because the costs of the combinations above and to the right of the budget line are greater than those of E₁, we cannot conclude that the customer prefers E₁ to these combinations when he is ob­served to buy E1. It is important to distinguish between “preference” and “revealed preference” here. Combination A being “preferred” over B suggests that the consumer prefers A over B.

However, the fact that A is “shown preferable to B” implies that A is chosen when B is inexpensive (no-more-expensive). In our consumer behaviour model, we usually assume that consumers choose the greatest combination they can afford, and that the choices they make are preferable to the alternatives. That instance, if (x₁ y₁) is clearly preferred to (x₂, y₂), then (x₁, y₁) is really preferred to (x₂, y₂) (x₂, y₂).

Assume that the consumer is purchasing the combination (x₁, y₁) at the price set (p’_x, P’_y), and that another combination is (x₂, y₂), so that p’x₁ + p’yy₁ = p’_xx₂ + p’_yy₂. Now, if the client purchases the most favoured combination within his financial constraints, we may claim that combination (x1, y1) is strictly preferred to combination (x₂, y₂) (x₂, y₂).

According to revealed preference, a theory proposed by American economist Paul Anthony Samuelson in 1938, consumer behaviour is the best indicator of their preferences if their income and the price of the product are held constant. The revealed preference theory assumes that consumers are rational.

WARP, SARP, and GARP are the three basic axioms of revealed preference.

For a long time, the concept of utility was used to explain consumer behaviour, particularly consumer choice. Utility in economics refers to how much happiness or pleasure customers gain from purchasing a product, service, or experiencing an event. However, utility is extremely difficult to calculate in absolute terms, and by the turn of the twentieth century, economists were protesting about the widespread dependence on utility. Replacement theories were suggested, but all were equally condemned, until Samuelson’s “Revealed Preference Theory,” which proposed that consumer behaviour was based on observable behaviour rather than utility, and that it relied on a limited set of largely accepted assumptions. Revealed preference is an economic theory about an individual’s consumption patterns that says that observing their purchase activity is the best approach to gauge consumer preferences. The revealed preference theory assumes that consumers are rational. In other words, they will have evaluated a variety of options before choosing the best purchase selection for them. As a result, if a consumer selects one choice from the collection, that option must be the preferred option. According to revealed preference theory, the preferred option might change based on price and financial limitations. A schedule of a given population’s favourite products under a diverse schedule of price and budget limitations may be generated by studying the preferred preference at each point of constraint. According to the theory, given a consumer’s budget, they would choose the same bundle of products (the “preferred” bundle) as long as it stays affordable. They will only convert to a less priced, less appealing bundle of products if the favoured bundle becomes unaffordable. The primary goal of revealed preference theory was to build on Jeremy Bentham’s notion of marginal utility. Because utility, or enjoyment from a good, is difficult to quantify, Samuelson set out to find a mechanism to do so. Since then, a number of economists have developed on revealed preference theory, which remains a key explanation of consumer behaviour. The theory is notably valuable in giving a mechanism for experimentally examining consumer choice.

Consider Consumer X, who buys a pound of peaches as an illustration of the linkages discussed in revealed preference theory. According to revealed preference theory, consumer X prefers that pound of peaches above all other things that cost the same or less than that pound of peaches. Because customer X favours that pound of peaches over all other goods they can afford, they will only buy something else if the pound of peaches becomes unaffordable. If the pound of peaches gets too expensive, consumer X will switch to a less desirable alternative product.

Three Axioms of Revealed Preference:

As revealed preference theory evolved, economists recognised three fundamental axioms: the weak axiom, the strong axiom, and the generalised axiom.

1.Weak Axiom of Revealed Preference (WARP):

This axiom asserts that given money and price, if one product or service is purchased instead of another, we will always make the same decision as consumers. The weak axiom also argues that once we buy one thing, we will never buy another product or brand unless it is cheaper, more convenient, or of higher quality (i.e. unless it provides more benefits). According to the weak axiom, as consumers, we will buy what we enjoy and our decisions will be consistent.

2.Completeness / Strong Axiom of Revealed Preference (SARP): 

This axiom asserts that in a two-dimensional world with just two commodities to select from, strong and weak acts are proved to be comparable.

3.Generalized Axiom of Revealed Preference (GARP):

This axiom applies when we acquire the same degree of benefit from more than one consumption bundle for a given level of income or price. In other words, this axiom takes into consideration when there is no single bundle that optimises utility.

Criticisms of Revealed Preference Theory

According to some economists, revealed preference theory contains too many assumptions. For example, how can we be certain that consumer preferences stay consistent over time? Isn’t it feasible that a certain activity at a specific moment indicates a portion of a consumer’s preference scale at that time? For example, if just an orange and an apple were available for purchase, and the buyer chose the apple, we may safely conclude that the apple is clearly favoured over the orange.

There is no evidence to support the assumption that a preference remains constant from one point in time to the next. There are several options in the actual world. It is hard to say which product or set of products or behavioural alternatives were rejected in favour of purchasing an apple.

1. According to Stanley Wong, revealed preference theory was an unsuccessful research programme. Samuelson proposed revealed preference theory as an alternative to utility theory in 1938, and the proven equivalence of the two theories in 1950 as a justification rather than a rebuttal of his stance.

2. If just an apple and an orange are available, and an orange is chosen, it is safe to conclude that an orange is favoured over an apple. In the real world, when a customer purchases an orange, it is hard to determine which item or set of goods or behavioural alternatives were foregone in favour of purchasing an orange. In this sense, preference is not revealed at all in the sense of ordinal utility.

3. According to the revealed preference theory, the preference scale remains consistent over time. If this is not the case, all that can be said is that an activity at a certain time indicates a portion of a person’s preference scale at that time. There is no reason to believe that it remains consistent from one moment in time to the next. Theorists of “revealed preference” presume consistency in addition to consistent behaviour (“rationality”). Consistency refers to a person’s ability to maintain a transitive order of rank on his preference scale (if A is preferred to B and B is preferred to C, then A is preferred to C). However, the revealed preference technique is based on a different assumption: that an individual retains the same value scale across time. While the former may be considered unreasonable, there is nothing illogical about someone’s value scales shifting with time. It is stated that no meaningful theory can be based on the assumption of constancy.

4. Because it is impossible to define or assess preferences apart from’revealed preferences,’ some scholars regard the idea as a tautological fallacy. Amartya Sen’s critiques in a series of articles, including “Behavior and the Concept of Preference” (Sen 1973), “Rational Fools: A Critique of the Behavioural Foundations of Economic Theory” (Sen 1977), “Internal Consistency of Choice” (Sen 1993), “Maximization and the Act of Choice” (Sen 1997), and his book “Rationality and Freedom” (Sen 2002).

Assumptions:

We can construct a strong theory of consumer demand using the simple principle of Revealed Preference. The following assumptions will be made:

(i)The consumer purchases and utilises only two products (X and Y). These commodities’ quantities x and y are continuous variables. In other words, the consumer chooses only one combination of goods at a given price line.

(ii) Both of these products are of the MIB (more-is-better) variety. This assumption is also known as the monotonicity assumption. This assumption suggests that the consumer’s ICs are negatively sloped.

(iii) Consumer preferences are strictly convex. This assumption means that the consumer’s ICs are convex to the origin, which suggests that just one point (the point of tangency) on the consumer’s budget line is achieved, which he chooses above all other affordable combinations.

This is a critical assumption. Based on this assumption, we will obtain a one-to-one relationship between the consumer’s price-income situation or budget line and his equilibrium choice—for any particular budget line of the consumer, we will obtain one and only one equilibrium combination of goods, and for any combination to be an equilibrium one, we will obtain one and only one budget line.

(iv) The fourth assumption of the RP theory is referred to as the Revealed Preference weak axiom (WARP). In this case, we suppose that if a consumer picks the combination E₁(x₁, y₁) over another affordable combination E₂(x₂, y₂) in a given price-income condition, he would never choose E₂ over E₁ if E₁ is cheap.

In other words, if a combination E₁ is revealed preferred to E₂, E₂ can never be revealed preferred to E₁. In another words, the consumer’s choices are consistent which means that if a combination of goods at point A is preferred to the combination by point B in a situation, point B will not be preferred to A in another situation.

(v) The RP theory’s fifth assumption is known as the Revealed Preference strong axiom (SARP). According to this assumption, if the consumer exposes the combination E₁ as preferred to E₂, E₂ to E₃,…, E_k-1 to E_k under multiple price-income conditions, then E₁ would be disclosed preferred to E_k and E_k would never (under no price-income scenario) be shown preferred to E1.

(vi) The consumer’s taste do not change.

(vii) The consumer is a rational consumer i.e. the consumer maximize his total utility with the given resources. It follows that in one set of market conditions, he selects one combination of goods and his preference for good changes under market conditions.

(viii) Income elasticity of demand is always positive i.e. more units of a commodity are demanded in case the income of a consumer increases and less of it when income falls. This means that elasticity of demand of a consumer should neither be zero nor should it be negative as it happens in the case of an inferior good or Giffen good.

(ix) It is also assumed that a consumer definitely reveals his preference when he makes a choice e.g. if he prefers a combination of goods at point A to all other combinations at different points, he would prefer point A and reject all other points which means that choice reveals preference i.e. by choosing one combination and rejecting others, the consumer definitely shows his preference. This is called “strong ordering’’. It is opposite to “weak ordering” represented by different combination of goods on the same indifference curve as they yield the same level of satisfaction. Since “strong ordering’’ is assumed in the revealed preference theory. It, therefore, rules out the possibility of “weak ordering” as explained by the indifference curve technique. Thus Samuelson does not regard indifference curve as an operationally significant concept.

Revealed Preference—Direct and Indirect:

If Revealed Preference is restricted to only two commodities combinations, E₁ and E₂, and if, in a certain price-income situation, E₁ (x₁,y₁) is disclosed preferred to combination E₂ (x₂, y₂), then E₁ is said to be directly revealed preferred to E₂.

However, if preferences are examined for more than two combinations and preferences are created by Revealed Preference transitivity, it is an instance of indirectly revealed preference. For example, if E₁ is disclosed preferred to E₂,…, E_k-1 to E_k, we may say that E₁ is indirectly revealed preferred to E_k using SARP.

Importance of the strong axiom of revealed preference:

Now consider the importance of the strong axiom of revealed preference (SARP). If the consumer shows a combination E₁ (x₁, y₁) as preferable to another combination E₂ (x₂, y₂), and if E₂ (x₂, y₂) is shown preferred to E₃ (x₃, y₃), then E is always revealed preferred to E₃.

This is referred to as the transitivity of revealed preferences. If the consumer maximises utility, then the transitivity of revealed preferences leads to the transitivity of preferences—if E₁ is preferred to E₂ and E₂ to E₃, then E₁ is preferred to E₃. However, this is required to verify that the ICs are not overlapping and that the non-interesting ICs are required to arrive to the utility-maximizing solution. It is obvious that if any of the WARP and SARP are broken, the customer will be unable to maximise utility.

Fundamental Theorem of Revealed Preference Theory:

Samuelson states Fundamental Theorem of Revealed Preference Theory as, “Any good (Simple or Composite) that is known always to increase in demand when money income always rises most definitely shrink in demand when its price alone rises.”

It means that income elasticity of demand of a good is always positive which price elasticity of demand is negative. To prove the fundamental theorem of revealed preference theory. Let us take price effect on the behaviour of a consumer. We take two cases.

• Effect of rise in the price of good X
• Effect of fall in the price of good X

• Effect of rise in the price of good X

It can be explained with the help of a figure given below:

In the above figure, we can see that consumer’s income in terms of Good Y is OM and it is ON in terms of Good Thus, MN is the price line which shows all the combinations of two goods that a consumer can buy in a price-income situation. Let us assume that the consumer chooses a combination of two goods X&Y determined by point A on the price line MN which gives him maximum level of satisfaction.

By choosing a combination at only point A the consumer is revealed to have preferred this combination to all other combinations available to him in or on the triangle OMN.

Now supposing that price of good X rises and that of good Y remains constant. Thus the price line moves downward and become MR. This means that equilibrium point A goes beyond the reach of the consumer. In order to enable the consumers to achieve the equilibrium point A, the consumer is given some extra amount of money. For this, we draw a line LK, parallel to MR which pass through point A. This means that the consumer is given RK extra amount of money in terms of good X to enable him to achieve point A which is also there on the line LK. As the consumer has attained the equilibrium point A again, he will not prefer combination lower than this. That is , he will not prefer any point on AK part of the price line LK as these points were available to him before, because they are lying within the area of his choice i.e. OMN and were rejected by him in favours of .

Point A hence, he will now choose point A or any other point in the shaded area MLA. However, in the presence situation, he will choose a point on LA part of price line LK. If he selects point A, it means that he is making the same combination of good X and Y as before. But, if he selects any point of LA part of price line LK, it would mean that he is buying less of good X and more of good Y. This shows that the substitution effect of a rise in the price of good X as some units of good Y have been substituted for some units of good X after its price has gone up.

 Thus when an extra amount of money is given to the consumer to compensate for a rise in the price of good X, he either buys the same quantity of good X or less of it in view of rise in its price. Conversely, when the extra amount of the money is not given to him, he will buy any combination on a point left of A on the price line MR. In case , he will definitely buy less of good X as income elasticity of demand for good X is positive. Since, with a rise in the price of good X, its demand has gone down, and it is, hence, proved that when Y elasticity of demand is positive, price elasticity is negative.

• Effect of fall in the price of good X:

Fundamental Theorem can also be established in case when the price of good X falls. For this, let us make a diagram.

In the diagram, we take MN as the original price line .Supposing that the consumer reveals his preference for the combination of good X and good Y at point A to all other combination in or on the triangle OMN. Now we assume that the price of good X falls and as a result the price line moves from MN to MR.

At this line the consumer is better than off then before. If he is to purchase the original combination of good X and good Y as represented by the point A, we have to take away some amount of money from him so that he neither is better off nor worse off than before. For this, we draw a price line LK parallel to MR which shows that the amount of money with drawn is PK in terms of good X. Since, point A is also on the price line LK, the combination represented by point A becomes available to the consumer.

But owing to reduction in his money income, he cannot purchase any combination to the left of point A i.e. on LA part of the price line LK because these combinations were available to him before  as his area of choice was OMN, but were  rejected by him in favour of A. Hence, he will choose point A or any other point the the right o A i.e. in the shaded area ANK or on AK part of the line LK in the given price-income situation. If he choose A, it means that he is choosing the same combination as was available to him before. And, if he chooses a combination to the right of A, it will mean that he purchases more of good X and less of good Y.

 We see that even when consumer’s income is reduced, he buys either the same quantity of good X or more of it at a lower price. If no money is taken away for him, he will be on price line MR and in this case, he will definitely buy more of good X on a point to the right of point A as his income has gone up. That is, when income elasticity of demand for good X is positive, price elasticity of demand for the product is negative. Thus, the fundamental theorem is again established.

Derivation of an Indifference curve from Revealed Preference Theory:

It is possible to derive an indifference curve from the revealed preference of a consumer. See the following figure.

In this figure, the consumer reveals his preference for the combination of goods X and Y at point A on the price line MN. An Indifference curve IC passes through it.

It can be observed that point B is preferred to every point on or below the price line MN as this point lies above and to the right of A in the area CAD. This is so because point B contains more of the goods than point A.

It follows that the indifference curve passing through A must lie below the area CAD and above the price line MN. This will prove that at point A the indifference curve must have a negative slope; and to be convex to the origin, it must be above the line MN both to the left and right of A which is called the “Zero of ignorance.”

In order to find out the exact location of indifference curve in the zone of ignorance, we take a point R on the price line MN. This point will be revealed inferior to point A because point R is on a lower price line ST which forces the consumer to buy a combination at A. Since point R is revealed inferior to A, every point on or below the line ST will also be revealed inferior to A. Thus the triangle RNT is taken away from the zone of ignorance. By the same reasoning, the entire portion below A on the line MN can be eliminated.

Similarly, we can chop off all the points above and to the left of A. Let us draw another price line LK. Supposing the consumer chooses point H on the price line MN. This point also revealed inferior to point A. So the triangle LHM will be taken out of the zone of ignorance.

Hence, by chopping off the area above and below, the price line MN step by step we can arrive at the exact curve IC through point A.

Derivation of demand curve from Revealed Preference Theory:

We can easily derive demand curve from the revealed preference of a consumer. This is shown in the figure below:

In the figure, MN is the original price line on which the consumer reveals his preference at point A. With a fall in the price of good X, the price line extends to be MR. An increase in the real income of the consumer resulting from a fall in the price of X, is taken away from him in the form of KP quantity of X. As a consequence, LK becomes the new price line which is parallel to MR and passes through A. Now OLK becomes the triangle of consumer’s choice.

Since the consumer had revealed his preference for point A on the original price line MN, therefore, all points lying on MA part of the line MN will be in consistent with the consumer’s behaviour as he cannot have less of good X when its price falls. Thus, he will either choose a combination in the shaded area AKN. If the money taken from the consumer is returned to him. He will definitely buy more of X. That is, he will prefer a point to the right of A and on the line MR e.g. point S.

Since we take money on the Y-axis, the price of good X can be calculated by dividing money income by the total number of units of good. Hence, when the price of good X is OM/ON= OP_0, the quantity demanded  is OQ₀ .As the price of good X falls to OM/OR=OP₁ , its quantity demanded increase to OQ₁. In the lower diagram, the price-quantity combinations are joined which make the demand curve DD.

Merits (Superiority) of Revealed Preference Theory:

It is said that Samuelson’s revealed preference theory is superior to the Marshallian utility analysis and Hicksian Indifference curve technique. To establish this viewpoint, following merits of revealed preference theory are presented.

1. This theory is behaviouristic because it presents the consumption theory from the actually observed behaviour of the consumer in the market. Compared to it, both Marshallian utility analysis and Hicksian indifference curve technique are introspective as both these theories give psychological explanation of the consumer’s behaviour. Since revealed preference theory studies the actual behaviour of a consumer and not his imaginary behaviour. It is therefore, more realistic, objective and scientific than the earlier theories.
2. In the Indifference curve technique, it is assumed that the curve is continuous as it depicts all conceivable combinations of goods X and Y which a consumer can buy and some of which may be unrealistic. Contrary to it, Samuelson believes in discontinuity as, according to him, the consumer can have only one combination in view of his revealed preference. Samuelson’s approach is so realistic that Hicks in his “Revision of Demand Theory”, dropped the assumption of continuity and replaced it by “strong and weak ordering.”
3. Marshallian and Hicksian theories are based on the assumption that the consumer always behaves rationally and maximizes his satisfaction with the given income. It is hard to find the application of the principle in actual practice because consumers so often behave irrationally. Samuelson’s theory, on the other hand, completely dispenses with this assumption and uses instead the consistency principle to present the consumer’s behaviour which is more realistic.
4. This theory provides the basis for welfare economics in terms of an observable behaviour of the consumer.

Demerits of Revealed Preference Theory:

Along with merit, this theory has also demerits which are given below.

1. This is based on “strong ordering’’ for revealed preference and as such neglects “indifference’’ in the behaviour of a consumer. It may be true that a consumer does not reveal his indifference on a budget line by choosing a set of goods at a particular point say A but it is possible that there are points on each side of point A towards which the consumer may be different. If this is to be true, then Samuelson’s “Fundamental Theorem” falls down.
2. Samuelson’s “Fundamental Theorem” is conditional and not universal. It is based on the assumption that positive income elasticity implies negative price elasticity of demand for a product. We know that price effect can be divided into income effect and substitution effect but here it is not possible here to separate income effect from substitution merely by observation. It is so because if the income effect is not positive, price elasticity will be indeterminate. On the other hand, if the income elasticity is positive, the substitution effect following a change in price will be indeterminate. Thus, the substitution effect cannot be distinguished from the income effect in the revealed preference theory.
3. One glaring defect in Samuelson’s Theory is that it altogether excludes the study of Giffen Paradox which is theoretically conceivable. Samuelson’s theory considers only positive income elasticity of demand while in the Giffen case income elasticity of demand is always negative. Hence, the revealed preference theory fails to a distinguish between powerful negative income effect with a weak substitution effect( as in the case of a Giffen good) and a weak negative income effect with a powerful substitution effect( as in the case of an ordinary inferior good). In this case, Samuelson’s theory seems inferior to Hicksian analysis.
4. The conviction that “choice” reveals “preference’’ is hard to be established. Choice may always not reveal preference because choice requires rational behaviour which is not always there. Since a consumer does not always act rationally, his choice may be accurately reveals his preference for a certain set of goods. Thus, revealed preference theory does not show up the observed consumer behaviour.
5. The revealed preference approach can be used to derive the individual demand curve of a consumer while it falls in drawing market demand schedule. It is so because with a fall in the price of good X, prices of other goods are likely to be affected and this may result in the redistribution of real income in the community. This may lead to an upward sloping demand in the market at some price levels. In this regard too. Hicksian approach seems superior to revealed preference theory, for it can derive both the individual and market demand curves from the price consumption curve.
6. The revealed preference theory fails to analyse consumer’s behaviour in the choices involving risk and uncertainty. There can be cases where chances of occurring 50/50. In such cases, the consumer’s preference cannot be said to be based on observed market behaviour.

Conclusion:

It appears from the above analysis that revealed preference theory is not an improvement over indifference curve technique. Some weakness on the theory are so conspicuous that they overcome its merits which make it somewhat unrealistic. However, it can be said safely that Samuelson’s behaviouristic ordinal utility analysis is an alternative to Hicksian introspective ordinal utility analysis.