Through revealed preference theory, Prof. Samuelson derives the Marshallian law of demand, i.e. inverse relationship between price and demand of a good, when other things remain the same.
While deriving law of demand, Prof. Samuelson assumes positive income elasticity of demand. The geometrical explanation of the law of demand is given.
It is assumed that a consumer spends his whole income on two goods X and Y. Suppose his budget line (Price Line), in a given price-income situation, is AK.
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Our consumer can buy any combination of goods lying on or below the AK price line. Supposing the consumer chooses the combination M out of all possible combinations lying on or below price line AK. It means he revealed his choice for combination M and rejected all other available combinations.
Now suppose the price of goods X increases, other thing being constant. As a result of increase in the price of goods X, price line shifts to the left on X-axis and becomes AL.
Now the question arises what will be the effect of increase in price of X on its demand, assuming elasticity of demand positive.
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The previously chosen combination is not available in the new price-income situation. As a result of rise in price of goods X, the real income of the consumer decreases.
Now we increase the money income of the consumer equal to ‘cost difference’ so that he can purchase the previously chosen combination of goods at increased price of goods X.
As a result of increase in money income equal to cost difference the price line becomes PQ. Price line PQ passes through point M and is parallel to price line AL. Price line AL represents the increased price of goods X and the adjusted money income.
Originally chosen combination M is available in price-income situation given by price-line PS. Now the question arises which combination will be chosen by the consumer in price income situation given by price-line PQ.
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The consumer will certainly not choose any combination lying below M on the Portion MQ. The reason is that if chooses any combination lying on MQ portion then he will be inconsistent in his behaviour.
It is, therefore clear that in the price-income situation given by price line PQ the consumer will cither choose combination M or any other combination on MP portion of price line PQ. In doing so the consumer will not be inconsistent in his behaviour.
If the consumer chooses any combination above M on M portion, then he will be purchasing less quantity of goods X and more quantity of Y than before.
Now if the increased money is taken back from the consumer he will certainly purchase the smaller quantity of the goods X at its higher price subject to the condition that income elasticity of demand is positive.
This inverse relationship between price and demand will also remain when price of goods X falls. It is also clear that according to ‘Revealed Preference Theory’ of consumer behaviour that the consumer chooses only one combination of goods in every price-income situation.