The Revealed Preference Theory of Demand PDF

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The Revealed Preference Theory of Demand! In both the Marshallian cardinal utility theory of demand and Hicks-Allen indifference curve theory of demand introspective method has been applied to explain the consumer’s behaviour. In other words, both these theories provide psychological explanation of consumer’s demand; they derive laws about consumer’s demand from how he would react psychologically to certain hypothetical  changes in price and income. But the Revealed Preference Theory which has been put forward  by Prof. Samuelson seeks to explain consumer’s demand from his actual behaviour in the market in various price-income situations. Thus, in sharp contrast to psychological or introspective explanation Prof. Samuelson’s revealed preference theory provides behaviouristic explanation of consumer’s demand. Besides, revealed preference theory is based upon the concept of ordinal utility. Preference Hypothesis and Strong Ordering: Prof. Samuelson’s revealed preference theory has Preference Hypothesis as a basis of his theory of demand. According to this hypothesis when a consumer is observed to choose a combination A out of various alternative combinations open to him, then he ‘reveals’, his preference for A over all other alternative combinations which he could have purchased. In other words, when a consumer chooses a combination A, it means he considers all other alternative combinations which he could have purchased to be inferior to A.

From the hypothesis of ‘choice reveals preference’ we can obtain definite information about the preferences of a consumer from observing his behaviour in the market  and about his preference scale. Let us graphically explain the preference hypothesis. Given the prices of two commodities X and Y and the income of the consumer, budget line PL is drawn in Fig. 12.1. The budget line PL, represents a given price-income situation.

Given the price-income situation as represented by PL, the consumer can buy or choose any combination lying within or on the triangle OPL. In other words, all combinations lying on the line PL such as A, B,C and lying below the line PL such as D, E, F and G are alternative combinations open to him, from among which he has to choose any combination. If our consumer chooses combination  A out of all those open to him in the given price-income situation, it means he reveals his preference for A over all other combinations such as B, C, D, E and F which are rejected by him. As is evident from Fig. 12.1, in his observed chosen  combination A, the consumer is buying OM quantity of commodity X and ON quantity of commodity Y. Strong Form of Preference Hypothesis: It should be carefully noted that Prof. Samuelson’s revealed preference theory is based upon the strong form of preference hypothesis.Strong ordering implies that there is definite ordering of various combinations in consumer’s scale of preferences and

therefore the choice of a combination by a consumer reveals his definite preference for that over all other alternatives open to him. Thus, under strong ordering, relation of indifference is ruled out. When in Fig. 12.1 a consumer chooses a combination A out of various alternative combinations open to him, it means he has a definite preference for A over all others, the possibility of the chosen combination A being indifferent to any other possible combination is ruled out by strong ordering hypothesis.

Consistency Postulate or Weak Axiom of Revealed Preference (WARP): The revealed preference theory rests upon another basic assumption which has been called the ‘consistency postulate’. In fact, the consistency postulate is implied in strong ordering preference hypothesis. The consistency postulate can be stated thus: “no two observations of choice behaviour are made which provide conflicting evidence to the individual’s preference.” In other words, consistency  postulate asserts that if an individual chooses A rather than B in one particular instance, then he cannot choose B rather than A in any other instance. Thus, consistency postulate requires that if once A is revealed to be preferred to B by an individual, then B cannot be revealed to be preferred to A by him at any other time when A and B are present in both the cases. If a person chooses a combination A rather than combination B which he could purchase with the given budget constraint, then it cannot happen that he would choose (i.e. prefer) B over A in some other situation in which he could have bought A if he so wished.

This is called revealed preference axiom. We illustrate, revealed preference axiom in Figure 12.2. Suppose with the given prices of two goods X and Y and given his money income to spend on two goods, PL is the budget line facing a consumer. In this budgetary situation PL, the consumer chooses A when he could have purchased B (note that combination B would have even cost him less than A). Thus, his choice of A over B means he prefers the combination  A to the combination B of the two goods. Now suppose that price of good X falls, and with some income adjustment, budget line changes to P’L’. Budget line P’L’ is flatter than PL reflecting relatively lower price of X as compared to the budget line PL. With this new budget line PLc if the consumer chooses combination B when he can purchase combination A (as A lies below the budget line PLc in Figure 12.2), then the consumer will be inconsistent in his preferences that is, he will be violating the axiom of revealed preference.

Such inconsistent consumer’s behaviour is ruled out in revealed preference theory based on strong ordering. This axiom of revealed preference according to which consumer’s choices are consistent is also called ‘Weak Axiom of revealed Preference or simply WARP. To sum, up according to the axiom of revealed preference if combination A is directly revealed preferred to another combination  B, then in any other situation, the

combination B cannot be revealed to be preferred to A by the consumer when combination A is also affordable. Now consider Figure 12.3 where to start with a consumer is facing budget line PL where he chooses combination A of two goods X and Y. Thus, consumer prefers combination A to all other combinations within and on the triangle OPL. Now suppose that budget constraint changes to PLC and consumer purchases combination B on it as combination B lies outside the original budget line PL it was not affordable when combination A was chosen. Therefore, choice of combination B with the budget line PLC is consistent with his earlier choice A with the budget constraint PL and is in accordance with the axiom of revealed preference.

Transitivity Assumption of Revealed Preference Theory: The axiom of revealed preference described above provides us a consistency condition that must be satisfied by a rational consumer who makes an optimum choice. Apart from the axiom of revealed preference, revealed preference theory also assumes that revealed preferences are transitive. 

According to this, if an optimizing consumer prefers combination A to combination B of the goods and prefers combination B to combination C of the goods, then he will also prefer combination A to combination C of the goods. To put briefly, assumption of transitivity of preferences requires that if A > B and B > C, then A > C.In this way we say that combination A is indirectly revealed preferred to combination C. Thus, if a combination A is either directly or indirectly revealed to be preferred to another combination we say that combination A is revealed to be preferred to the other combination, Consider Figure 12.4 where with budget constraint PL, the consumer chooses A and therefore reveals his preference for A over combination B which he could have purchased as combination B is affordable in budget constraint PL.Now suppose budget constraint facing the consumer changes to P’L’, he chooses B when he could have purchased C. Thus, the consumer prefers B to C. From the transitivity assumption it follows that the consumer will prefer combination A to combination C.

Thus, combination A is indirectly revealed to be preferred to combination C. We therefore conclude that the consumer prefers A either directly or indirectly to all those combination of the two goods lying in the shaded region in Figure 12.4.

It is thus evident from above that concept of revealed preference is a very significant and powerful tool which provides a lot of

information about preferences of a consumer who behaves in an optimising and consistent manner. By merely booking at the consumer’s choices in different priceincome situations we can get a lot of information about the underlying consumer’s preferences. Deriving Demand Theorem from Revealed Preference Hypothesis: Revealed preference hypothesis can be utilised to establish the demand theorem. Prof. Samuelson has derived the Marshallian law of demand from his revealed preference hypothesis. Marshallian law of demand, as is well known, states that a rise in the price of a good must, if income and other prices are held constant, results in the reduction of amount demanded of the good and vice versa. Samuelson proceeds to establish relationship between price and demand by assuming that income elasticity of demand is positive. From positive income elasticity, he deduces the Marshallian inverse price-demand relationship. He states the demand theorem what he calls the Fundamental Theorem of Consumption Theory as under: “Any good (simple or composite) that is known always to increase in demand when money income alone rises must definitely shrink in demand when its price alone rises”. It is clear from the above statement of Fundamental Theorem of Consumption that positive income elasticity of demand has been made a necessary qualification to the inverse price-demand relationship.

The geometrical proof of the Fundamental Theorem is illustrated in Fig. 12.5. Let us suppose that the consumer spends his entire income on two goods X and Y.Now the budget line AB represents the price-income situation confronting the consumer, within or on the triangle OAB are available to the consumer, from which he can buy any combination. Suppose that the consumer is observed to choose the combination Q. This means that Q is revealed to be preferred to all other combinations that lie in or on the triangle OAB. Now, suppose that price of good X rises, price of remaining unchanged. The budget line shifts to the new position AC.We now want to know what is the effect  of this rise in price of good X on its quantity demanded, assuming that demand varies directly with income (i.e., income elasticity of demand is positive). It is evident from Fig.12.5 that combination  Q is not available to the consumer in priceincome situation AC. Let us compensate the consumer for the higher price of X by granting him extra money so that he can buy the same combination Q even at the higher price of X. The amount of money which is required to be granted to the consumer so that he could buy the original combination  Q at the higher price of X has been called C  ost-difference by Prof. J. R. Hicks. In Fig. 12.5, a line DE parallel to AC has been drawn so that it passes through Q. DE represents the higher price of X and the money income after it has been increased by cost difference.

Now, the question is which combination will be chosen by the consumer in price-income situation DE. The original combination Q is available in price-income situation DE. It is evident from Figure 12.5 that he will not choose any combination lying below Q on the line DE. This is because if he chooses any combination below Q on the line DE, his choice would be inconsistent. All combinations below Q on DE, that is, all combinations on QE could have been bought by the consumer  but had been rejected by him in price-income situation AB in favour of Q. (All points on QE were contained in the original choice triangle OAB.). It follows, therefore, that in the price-income situation DE the consumer will either choose the original combination  Q or any other combination on QD segment of DE or within shaded area QAD. It should be noted that choice of any other combination on QD or within the shaded area QAD in preference to Q by the consumer will not be inconsistent since combinations lying above Q on QD or within shaded region QAD were not available in price-income situation AB. In price-income situation DE if the consumer chooses the original combination Q, it means he will be buying the same amount of goods X and Y as before, and if he chooses any combination above Q on QD or within the shaded area QAD, it means that he will be buying less amount of commodity X and greater amount of Y than before. Thus, even after sufficient extra income has been granted to the consumer to compensate him for the rise in price of good X, he purchases either the same or the smaller quantity of X at the higher price. Now, if the extra money granted to him is withdrawn, he will definitely buy the smaller amount of X at the higher price, if the demand for good X is known always to fall with the decrease in income (that is, if income elasticity of demand fox X is positive).

In other words, when the price of good X rises and no extra money is granted to the consumer so that he faces price-income situation AC, he will purchase less amount of good X than at Q. Thus assuming a positive income-elasticity of demand, the inverse price-demand relationship is established  so far as rise in price is concerned. That the inverse price-demand relationship holds good in case of a fall in price also is demonstrated in Fig. 12.6. Let us suppose that AB represents  original price income situation and further that other consumer reveals his preference for Q over all other combinations in or on the triangle OAB. Now, suppose that price of good X falls so that the price line shifts to the right to the position AC. Let us take away some amount of money from the consumer so that he is left with just sufficient amount of money which enables him to purchase the original combination  Q at the lower price of good X. Thus, in Figure. 12.6, a line DE is drawn parallel to AC so that it passes through Q. Price line DE represents lower price of X as given by AC and the money income after it has been reduced by the cost difference.

It is obvious that in price-income situation DE, the consumer cannot choose any combination above Q on QD, since all such combinations were available to him in the original price-income situation AS and were rejected by him in favour of Q. The consumer will, therefore, choose either Q or any other combination on QE or from within the shaded region QEB. In price-income situation DE, his choice of Q means that he buys the same quantity of

goods X and Y as in original price-income situation AB, and his choice of any other combination on QE or from within the shaded region QEB means that he buys a larger amount of good X and a smaller amount of good y than in the original price-income situation AB. Thus, even after consumer’s income has been reduced, he buys either the same quantity of X or more at the lower price. And if we give him back the amount of money taken away from him so that he confronts again price-income situation AC he will definitely buy more of X at the lower price, provided that his demand for X rises with the rise in income (i.e., his income elasticity of demand for good X is positive). The two demonstrations given above together prove the fundamental theorem of consumption theory, according to which any good whose demand varies directly with income must definitely shrink in demand when its price rises and expands in demand when its price falls. It may be noted that Samuelson’s theory involves two implicit assumptions which have not been explicitly stated. 1. In the first place the consumer is always shown to choose a combination on the price line. In other words, he is never shown to choose a combination from within the triangle. This is based upon the assumption that a consumer always prefers a larger collection of goods to a smaller one. 2.Secondly, another implicit assumption involved in Samuelson’s theory is that the consumer is shown to choose only one combination of goods in every price-income situation. With these two implicit assumptions the inverse price-demand relationship is deduced by Samuelson by making explicit assumptions of consistency of choice and a positive income elasticity of demand.

Breaking up of Price Effect into Substitution Effect and Income Effect: Having now explained the derivation of law of demand from revealed preference approach we are now in a position to show how in the revealed preference approach price effect can be broken up into substitution and income effects. We will explain this by considering the case of fall in price of a commodity. Now consider Figure 12.7 where, to begin with, price-income situation faced by a consumer is given by the budget the AB. With price-income situation represented by the budget line, AB suppose the consumer  chooses combination Q and buys OM quantity of commodity X. Now, suppose price of commodity X falls and as a result budget line shifts to the new position AC. Now, income of the consumer is reduced so much that the new budget line DE passes through the original chosen combination Q. That is, income is reduced equal to the cost difference so that gain in real income caused by the fall in price of commodity X is cancelled out. As seen above, with the new budget line DE, to be consistent in his behaviour the consumer can either choose the original combination Q or any combination lying on the segment QE of the budget line DE. If he chooses again the original combination Q, the Slutsky substitution effect well be zero.

However, suppose that the consumer actually chooses combination S on the segment QE of the new budget line DE. Now, choice of the combination S shows that there will be substitution  effect due to which the consumer will buy MN more of good X. Note that substituation effect is negative in the sense that the relative fall in price has led to the increase in quantity demanded of X, that is, change in quantity demanded is in opposite direction to the change in price. It should be noted that choice of combination S on segment QE in preference to combination Q of the budget line DE is not inconsistent because combinations on QE segment and within the shaded area were not available before when combination Q was earlier chosen in price- income situation AB.Thus, with the new budget line DE after consumer’s income has been adjusted to cancel out the gain in real income resulting from a relative fall in price of X, the consumer chooses either Q (when substitution effect is, zero or a combination such as S on segment QE when substitution  effect leads to the increase in quantity demanded of good X by MN. This is generally known as Slutsky theorem which states that if income effect is ignored substitution effect will lead to the increase in quantity demanded of the good whose price has fallen and therefore the Marshallian law of demand describing inverse relationship between quantity demanded and price of a good will hold good, that is, due to substitution effect alone demand curve slopes downward. Now, if the consumer chooses the combination S on the line segment QE of budget line DE it means that he buys MN more due to the substitution effect. Thus he prefers combination S to combination  Q. In other words, his choice of S instead of Q reveals that he will be better off at S as compared to Q.

Now, if money income withdrawn from him is restored to him so that he is faced with the budget line AC’. If income effect is positive, he will choose a combination, say R) on the budget line AC’ to the right of point S indicating that as a result of income effect he buys NH more of the commodity X.Thus quantity demanded of commodity X increases by MN as a result of substitution effect and by NH as a result of income effect. This proves the law of demand stating inverse relationship between price and quantity demanded. On budge...