9 Revealed Preference Theory of Demand

Amit Kumar Basantaray

 

1. Learning Outcome

 

After completing this module the students will be able to:

 

Answer how preference is derived from actual choices.

 

Learn the assumptions of revealed preference theory of demand.

 

Learn derivation of indifference curve from revealed preference theory. Know Weak and Strong axioms of revealed preference theory

 

Learn Advantages and Disadvantages of revealed preference theory

 

2. Introduction

 

Paul Samuelson, in the year 1938, introduced the term ‘revealed preference’. The theory of revealed preference is considered a very important theory of consumer behaviour for it establishes ‘law of demand’ without the use of indifference curves. This theory does not really assume the convexity of indifference curves; rather it establishes the existence and convexity of indifference curve. This is the reason why the theory of revealed preference is considered as an improvement over Hicks’ Indifference curve analysis. Economics, in general, and consumer theory, in particular, deals with two types of preferences. One is called ‘stated preference’ and the other one is called ‘revealed preference’. In the Hicksian or Indifference curve analysis, a consumer first states his/her preference and with the information of his/her budget we go on to determine his/her demand. So in this theory, problem is to find the demand bundle given the preference and budget constraint. But what if the consumer does not state his preference? Then there is no way we can get to his equilibrium demand. In the case where preferences are not directly given, an indirect process is used to know consumer’s preferences. And that is to derive the preference from the actual demand behaviour of consumers. In real life, every person demands some commodities or other. Now the main task, in the situation where preferences are not directly observable, is to know the preferences of the consumers from their actual demand behaviour. The theory of ‘Revealed Preference does exactly this. In this theory, consumer’s references are revealed from their demand behaviour. To get to this, one of the assumptions that need to be made is that consumer’s tastes do not change substantially in the short-run. In other words, consumer’s tastes/preferences are stable in, at least, the time period in which he/she is observed for his/her demand.

  1. More about the idea of Revealed Preference:

 

Take the example of a rational consumer who wants to maximise utility given his budget/income. Figure-1 is depicting the budget line and bundles of commodities that this consumer can afford.

Along the X-axis, good X is taken and along Y-axis, good Y is taken. Line AB represents budget line of the consumer. All the commodity bundles that lie on (like point E) and below (like point F) the budget line is affordable to the consumer. But suppose he chooses bundle (X1, Y1) represented by point E over all other bundles he could have chosen. That means bundle (X1, Y1) is optimal bundle for the consumer at the given budget, which means it must be superior to other bundles on and below line AB. In this case we say that the bundle (X1, Y1) that the consumer chooses is revealed preferred to any other bundle that was available to him at that budget. Or in other words, the available bundles that were not chosen are revealed worse than the demanded bundle. For comparison purpose, we can note that bundle (X1, Y1) is revealed preferred to the bundle (X2, Y2). One more assumption can be made at this point and that is the consumer demands a unique bundle at each budget line or preferences are convex.

  1. Assumptions of the theory of Revealed Preference:

 

First, the consumer is a rational consumer as he prefers those bundles of goods that contain more quantities of commodities.

 

Second, the consumer behaves consistently with regard to his/her choices. We will discuss more about this assumption when the reader would be introduced weak and strong axioms of the theory of revealed preference.

 

Third, but most importantly, the consumer by choosing a particular bundle of good at given budget reveals his preference for that particular bundle. The chosen bundle is revealed preferred to all other bundles available at that budget. And the basket of goods so chosen maximizes his/her utility.

  1. Inference about preference from revealed preference:

 

Consider a case where prices of X and Y are P1 and P2 respectively. At this price consumer chooses (X1, Y1) over (X2, Y2) although it is affordable. In other words, he chooses (X1, Y1) when

 

P1X1 + P2Y1 ≥ P1X2 + P2Y2

 

Here the consumer is choosing the most preferred affordable bundle and we say (X1, Y1) is directly revealed preferred to (X2, Y2) bundle which further means that the consumer ranks the former bundle ahead of the latter bundle.

 

Consider another case of some other prices for X and Y at which the consumer had preferred (X2, Y2) over (X3, Y3). Since in our earlier example, (X1, Y1) was preferred over (X2, Y2) the consumer would prefer (X1, Y1) to (X3, Y3) by transitivity assumption. That is if (X1, Y1) is directly preferred to (X2, Y2) and (X2, Y2) is directly preferred to (X3, Y3), then (X1, Y1) would be indirectly preferred to (X3, Y3).

 

Revealed preference includes both direct and indirect revealed preference. Suppose there are two bundles X and Y. if X is either directly or directly revealed preferred to Y we say that X is revealed preferred to Y.

 

 

 

The inference of preference of a consumer from his/her choice is discussed through figure-2. Consider the initial budget line CD. At this budget line the consumer has the option of choosing any bundle that lie on or below the line CD. Suppose he prefers a bundle represented by (X2, Y2). Now there happens a change in prices and the budget line shifts from CD to AB. Here again the consumer has the option of choosing any bundle that lies on or below the line AB. Suppose, he chooses bundle (X1, Y1). Given these revealed preferences of consumer at different budget lines, we can conclude that the bundle (X1, Y1) is preferred to all other bundles in the shaded area either directly or indirectly. The inference that we can draw from this revealed preference is that the indifference curve for the consumer should not encroach upon the shaded area or in other words, the indifference curve should lie above the shaded area. On these lines, we can observe different demand bundles of the consumer at different budget lines and we can narrow in on the area through which his/her indifference would pass.

  1. Derivation of Indifference Curve from Revealed Preference:

The reader must be well aware of the fact that under ordinal utility analysis, a lot of information like consumer’s taste and preferences, ranking of those preferences are needed from the consumer to draw indifference curve which represents various combinations of two goods (say X and Y) resulting in a particular level of utility. The theory of revealed preference does not ask consumers to give the information about his tastes and preferences. The theory enables us to draw indifference curve of the consumer by his behaviour or choice for different bundles of commodities at various market prices provided certain conditions are fulfilled and these conditions are: his choice is consistent, his tastes do not change frequently and the consumer prefers more goods to less. We now begin the process of derivation of indifference curve for a consumer from his revealed preference.

Along the X-axis and Y-axis of figure-3, we take good-X and good-Y respectively. Let the initial budget line for the consumer is Aa which represents various combinations of good-X and good-Y that the consumer can purchase with his given income. With this budget line, let us further assume that the consumer chooses bundle-E which is on the line Aa. If this is the case, then all the other combinations of two goods X and Y on the line AA and below it denote inferior bundles to ‘E’. Now to know the bundles that are superior to ‘E’, draw two perpendiculars through ‘E’ like ‘EF’ and ‘EH’. All the bundles that are on these two lines and in the area created by them towards the right of ‘E’ are preferred to bundle-E as they contain more quantities of at least one commodity. For example, on the perpendicular EF, all the points except E represent more quantities of good-Y, quantity of good-X remaining constant. Similarly, on the perpendicular EH, all the points except E represent more quantities of good-X, quantity of good-Y remaining constant. And in the area towards the right of E and between two perpendiculars EF and EH, all the points give more quantities of two goods. Bundles of goods in the area represented by FEA and HEa are still not ordered or we cannot not conclusively say whether these are having superior or inferior bundles to E. To do that, we need to observe the consumer more in different price situations. Let us assume that price of X fall so that the new budget line ‘Bb’ passes below ‘E’. Now the consumer would choose either ‘U’ or a point to the right og ‘U’. If he chooses any point on the segment ‘BU’ or below it, then his choice would be inconsistent as these bundles were revealed as inferior bundles when budget line was Aa. Suppose, the consumer chooses point ‘U’. Using the transitivity assumption, we can conclude following.

 

E was preferred to U when budget line was Aa.

 

U is preferred to area Uab when the new budget line is Bb.

 

Therefore, E would be preferred to area Uab when budget line is Aa.

 

In this way, we have managed to reduce the earlier unranked area HEa by area Uab. Like this if we observe the behaviour of the consumer at different price situations, we can rank the remaining area of HEa. Similarly, we can also rank the other part of the area FEA (in figure-3) which was not ranked. The ranking of this area is explained through figure-4.

In figure-4, initial situation is just same as figure-3 where initial budget line was Aa, E bundle was chosen and area HEa and FEA were not ranked. Then later in figure-3, we showed how area HEa can be narrowed down. Now in this figure-4, we want to rank the area represented by FEA. Assume that price of good-X increases and therefore new budget line Cc passes through point E. the consumer has the choice of either choosing E or choose a point such as V which lies towards the left of E and on line Cc. Through rationality assumption we can show that area MVN would be preferred to point V. Now using transitivity assumption we show that as V is preferred to E, MVN is preferred to V, MVN is preferred to E. So area MVN is superior to E and hence the indifference curve which passes through E must not cross area MVN. Repeating this procedure would eventually narrow down the earlier unranked zone would give us the exact path through which indifference curve would pass. With this we have successfully managed to derive the indifference curve (see figure-5) from the revealed preference of the consumer which was again from the actual behaviour of the consumer in various market situations

 

We have mentioned before that revealed preference theory does not assume the convexity of indifference curve but establishes its convex nature. It is quite evident from figure-5 which depicts various shapes of indifference curve out of which I1 is the derived indifference curve from consumer’s revealed preference. All other indifference curves (like I2, I3, and Aa) are deliberately drawn to show why these shapes would not be valid. First take the case of straight line indifference curve Aa is not valid as all other points except E on Aa are shown inferior in the previous section. Similarly, in the case of I2 and I3, some of parts of each of these curves are crossing either the inferior zone (in case of I2) or both inferior and superior zone (in case of I3). Therefore, the only possible shape of indifference curve is convex to the origin represented by I1.

  1. The Weak Axiom of Revealed Preference:

One of the key assumptions of this theory is that the consumer always chooses best amongst the affordable goods. In the circumstances where this assumption is violated, there is no meaning of the derivation of indifference curve though the revealed preference. We know that when a consumer chooses the best bundle affordable, he/she is maximising his/her objective. How would we check whether or not the consumer is not maximizing? Consider the situation in figure-6.

 

 

In the above diagram, assume that the consumer chooses bundle (X1, Y1) when the budget line is AB. When budget line changes its position to CD, the consumer chooses another bundle (X2, Y2). Clearly this kind of choice making does not represent a maximizing consumer. Why? Because the consumer chooses (X1, Y1) when (X2, Y2) was available giving one preference. At the same time, the same consumer chooses (X2, Y2) when (X1, Y1) was available giving exactly an opposite preference. This type of choice is not consistent for which we cannot derive any inference on the preference of this consumer. This is precisely the reason for which the revealed preference must satisfy the weak axiom. What is the weak axiom of revealed preference then?

 

If one bundle, say (X1, Y1), is directly revealed preferred to another bundle, say (X2, Y2), and these two bundles are not the same, then it cannot happen that (X2, Y2) is directly revealed preferred to (X1, Y1). In other words, if a particular bundle (say bundle-1) is chosen over another bundle (say bundle-2) then if the consumer buys bundle-2, bundle-1 must not be available. The example in figure-3 violates this weak axiom. That is why on the basis of the preferences that violates weak axiom of revealed preference, indifference curves cannot be drawn.

  1. The Strong Axiom of Revealed Preference:

Whereas weak axiom is based upon the concept of direct revealed preference, the strong axiom is based upon the added concept of indirect revealed preference. As we already know, indirect revealed preference follows transitivity principle that is if bundle-1 is preferred to bundle-2 and bundle-2 is preferred to bundle-3, then bundle-1 is indirectly revealed preferred to bundle-3. For consumer’s choice to be maximizing we should not get a case where indirect revealed preference is violated or strong axiom of revealed preference is violated. What is the strong axiom of revealed preference then?

 

If a bundle (X1, Y1) is either directly or indirectly revealed preferred to another bundle (X2, Y2) and these two bundles are different from each other, there must not be a case where the bundle (X2, Y2) is either directly or indirectly revealed preferred to (X1, Y1).

 

Clearly, this is putting a stricter restriction as it is taking into account both direct revealed preference and indirect revealed preference. Simply it states that consumer’s preferences must be transitive. This condition ensures that the consumer always chooses the best bundle that he/she can afford. We can also say that strong axiom of revealed preference is sufficient condition for maximizing behaviour of the consumer. Thus, both weak and strong axiom of revealed preference put all the necessary and sufficient conditions on the choices of the consumer so that one can use these choices to draw inference about preferences.

 

9.  Advantages of the theory of Revealed Preference:

The theory of revealed preference has certain advantages over Indifference curve analysis and Marshallian cardinal utility analysis. It is the first theory to apply behaviouristic method to demand theory or in this theory the theory of demand is explained through the actual behaviour of the consumer. Other two theories provide introspective and psychological explanations to consumer behaviour. That is both Indifference curve and Marshallian cardinal utility analysis derive demand theorem on the basis of the ranking of different commodity bundles given by the consumer. Since behaviouristic method is more scientific than introspective and psychological method, revealed preference theory is an improvement over both ordinal and cardinal utility analysis. Another point in which this theory has an advantage over earlier two theories is that the theory of revealed preference is not build upon utility maximization assumption but on the consistency assumption about consumer behaviour. This consistency assumption is a key to derive the demand theorem. The theory of revealed preference also abandons the assumption of continuity of indifference curve because of the fact that in real world situations there is this presence of discontinuity. Actual observed choice of the consumer may deal with some discontinuity of the actually available price-income situation.

 

10.  Disadvantages of the theory of Revealed Preference:

According to Samuelson, preference can be revealed from a single instance of choice by the consumer and hence there is no possibility of indifference in consumer’s behaviour. However, it very difficult to discard the possibility of indifference between commodity bundles if there is sufficiently large number of observations. This point is further elaborated by Prof. Tapas Majumdar when he links the theory of revealed preference to welfare economics. He says “it may be remembered that in all forms of welfare theory indeed in any integral view of human activity, we may have to assume that the individual can always compare his ends. If this axiom is not granted, the whole of welfare economics falls to the ground. And if this axiom is granted, then the idea of remaining on the same level of welfare while sacrificing something of one commodity for something else of another will emerge automatically”. The view that the actual selection of bundle of goods is from a very few bundle of goods on which the consumer is indifferent, if accepted, would break down the entire hypothesis of revealed preference. This point was made by Armstrong and it has an effect of breaking down Samuelson’s proof that demand for a good decreases with the increase in its price. Another limitation of the theory is that it fails to distinguish between substitution effect and income effect because of absence of indifference relation. Samuelson has rejected the substitution effect that allows movement along the same indifference curve there is no place for the presence different bundles of goods with same level of satisfaction. The revealed preference theory would be unable to explain demand theorem when income elasticity of demand is negative and it is smaller than the substitution effect. Again this inability to enunciate demand theorem arises when income elasticity of demand is negative and is greater than substitution effect. Samuelson’s theory also cannot solve the Giffen’s Paradox. Last criticism of the theory is that when consumer employs the strategies of game theory, it very difficult to derive preference from his choices.

  1. Summary:

 

The theory concludes that Consumer’s preferences are revealed from their demand behaviour. Revealed preference includes both direct and indirect revealed preference. The validity of this whole exercise depends upon fulfilment both weak axiom and strong axiom of revealed preference. Weak axiom means If one bundle, say (X1, Y1), is directly revealed preferred to another bundle, say (X2, Y2), and these two bundles are not the same, then it cannot happen that (X2, Y2) is directly revealed preferred to (X1, Y1). Strong axiom tells us that If a bundle (X1, Y1) is either directly or indirectly revealed preferred to another bundle (X2, Y2) and these two bundles are different from each other, there must not be a case where the bundle (X2, Y2) is either directly or indirectly revealed preferred to (X1, Y1).

 

REFERENCE

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