Revealed Preference Theory of Samuelson: Concept, Assumptions & Explanation

Introduction

Revealed preference theory was propounded by Paul A. Samuelson in 1947, and it was based on the consumer’s preferences for different goods and services which are available in the market. He has derived the demand curve of a consumer based on the consumer’s budgetary constraints and his preferences revealed in the market without involving any ordinal or cardinal measurement of utility.

Revealed Preference Theory

i.e. baskets of different goods which a consumer buys at different prices without using IC and its restrictive assumptions. Moreover, this theory is also capable of establishing the existence of IC and its convexity. Because of its success, it is also known as the “third root of the logical theory of demand”.

Assumptions of Revealed Preference Theory

Rationality: A consumer is always rational, i.e. he/she always prefers more goods and services to derive maximum utility. Thus he always buys the commodity which gives him maximum utility first, and then he buys the least utility-giving commodity at the end.

Transitivity and Consistence of Choice: The consumer’s preferences are always transitive, i.e., if a consumer prefers good X over good Y and the same consumer prefers good Y over good Z, then according to this assumption of transitivity, he must prefer good X over good Z also.

If, X>Y

If, Y>Z

Therefore, X>Z.

Whereas as per the consistency of choice, if a consumer prefers good X to good Y in one period, then he must not prefer good Y to good X in another period or must not treat both goods as equal. Symbolically,

If, X>Y in one period

Then, Y>X or Y≠X in another period.

Price Inducements: Given the consumer’s choice for a basket of goods, a consumer can induce to buy a different basket of goods which provides him sufficient price incentives.

Axioms of Revealed Preference Theory

The basic axiom of the revealed preference theory is that “if a consumer chooses one basket of goods, given his budgetary constraints and the alternative baskets of goods of the same price, then he reveals his preference.

For instance, if there are two baskets, A and B, comprising of two goods, X and Y, and both are equally expensive to the consumer, then if a consumer chooses basket A over B, then the consumer is said to reveal his preference for basket A.

He may do so because either he would have a liking for that basket of goods or it is relatively less expensive than the other. But if the consumer chooses one basket over the other because it is cheaper than the other, then the consumer is not said to have revealed his preference; he is only said to have revealed his preference for a basket over the other when the price of both the baskets are same, and he chooses one basket over the other because he likes that basket over the other. Then only the consumer reveals his preference for one basket over the other.

This can be shown in the following diagram 1:

Here in the diagram, the budget line of the consumer is MN, where he can choose various baskets of goods (X and Y) given his income and the prices of X and Y. Now if the consumer chooses any basket of goods, for instance, if he chooses basket A, which comprises OX of X and OP of Y, then he is said to have revealed his preference for basket A over any other basket which lies on the same budget line.

So here, the consumer has revealed his preference for A over B. Any basket which lies below the budget line, like basket C, comprises cheaper X and Y, and the consumer will not reveal his preference for it. Any basket lying above the budget line would be too expensive for the consumer to buy. Therefore, he will also not reveal his preference for that basket (like basket D).

Decomposition of Substitution and Income Effect and Derivation of Demand Curve

The price effect and its decomposition into substitution and income effect can also be shown by the revealed preference theory apart from the indifference curve theory. For this, lets us assume the budget line as M1N1 on which a consumer chooses to bundle A, comprising AX1 of Y and OX1 of X. Since all the bundles on this budget line are equally expensive to the consumer but the consumer has revealed his preference for A over all other bundles lying on this budget line.

Now if the price of X fall,s then the budget line will pivot to M1N3, and the consumer will shift to point C. This movement of the consumer from point A to C is known as the price effect. This price effect can now be split into substitution and income effect in the following Figure 2:

This can be done by drawing a budget line M2N2 through point A. Note that we are doing the separation based on the Slutskian method. Since the new budget line M2N2 passes through point A so, it means that the combination of X and Y is still available to the consumer. Now the consumer response to the change in the price of X could be taken as the bundle which he chooses on this M2N2 budget line.

As it has been seen in the above diagram that the consumer will not choose any bundle lying between M2 A, as all these bundles are inferior to him. He would only choose either A or any bundle lying on segment A N2 and precisely between points A and H.

Now if he chose basket A, then the substitution effect would be zero, and if he chose basket B then the substitution effect would be X1X2, and the income effect would be X2X3. Since here, the substitution effect is positive. Thus, it implies that when the price of X falls, the demand for X increases. Hence the demand curve could be derived from this.

Derivation of Indifference Curve

As has been explained above that the revealed preference theory is capable of deriving the proofs for the existence of the indifference curve and its convexity; it does so using consumer behaviour, i.e. a consumer’s choice for various goods at various prices. This derivation of the IC is shown in the diagram 3.

For this, let us assume that the consumer reveals his preference for basket A over the other on the budget line MN. Moreover, all the bundles lying below the budget line are inferior to him and thus not preferred by him as they all are cheaper than A. This is represented as the triangle MON which is marked as an inferior zone.

Let us now consider the area above the budget line. This area is divided into three segments, namely, JAM, JAK and KAN. The area JAK is the preferred one because any point on JA represents a higher quantity of Y with the same quantity of X.

Similarly, any point on AK is preferred as it shows a higher quantity of X, with the same quantity of Y; and the area above AK and to the right of AJ represents a basket comprising more of both the goods X and Y. Therefore, any point on the line AJ, AK and between them is preferred over point A. Hence in the diagram, this area is marked as the preferred zone.

The areas JAM and KAN are the ignorance zone as any point in these areas represents more of one good and less of the other good as compared to point A. And the consumer’s preferences are very difficult to be known in these areas.

Thus it is clear from the above discussion that the consumer’s indifference curve will pass through points A, JAM and KAN to retain its convexity.

The course of the indifference curve in the ignorance zones can be found by ranking consumers’ choices in these areas. For this, let us assume a budget line MN which shifted to PT when the price of X falls and the price of Y increases. Now the consumer will either choose B or any bundle on the BT segment of this new budget line, but since because of the assumption of consistency, the consumer cannot choose any point on BP as they all are inferior to him.

Now if he chooses B, then any other point on or below PT is inferior to B. Thus, any bundle in the area NBT is revealed as inferior to B. Hence, the triangle NBT which is a part of the ignorance zone, KAN, is clipped off because the consumer’s ranking of this area is now known.

This procedure can be repeated for as many points as we wish to repeat in order to find out the best point and thus the ignorance area can be reduced bit by bit. For instance points C and D. Moreover the same procedure can be done for the upper ignorance zone, JAM, to find points in relation to A in the following Figure 4.

Now if we join all these points, D, A, B, and C, then we get the offer curve FF’ in the following Figure 5:

According to Samuelson the position of the offer curve would be the probable position of the Indifference curve. He has given the following points in support of this argument.

• The IC cannot be a straight line like MN because when the consumer chooses point A, then it reflects that all other points on MN are inferior to A, and hence the consumer cannot be indifferent to point A and to all other points lying on MN.
• All the points below the budget line MN are revealed as inferior to A; therefore, the IC cannot intersect the budget line, nor it could be concave (as shown in the diagram as CC’).
• Since all the points on or above the budget line MN are revealed superior to A, IC cannot pass through the preferred zone JAK. Therefore, the position of the IC would be somewhere between the ignorance zone, which passes through A and is shown as FF’ in the above diagram 5.

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