Question

3. Revealed Preference

Felix chooses between clothing, q1, and food, q2. His initial income is

$1,000 a month, p1=$100, and p2=$10. At his initial bundle he

consumes q1=2 and q2=80. Later, his income rises to $1,200 a

month, and the price of clothing rises to p1=$200, but the price of

food does not change (p2=$10). As a result, he reduces his

consumption of clothing to q1=1, and he increases his consumption of

food to q2=100. Using a revealed preference reasoning, can you

determine how he ranks the two bundles? Explain your answer.

Answer 1

Here we assume Good X as name for Clothing and good Y as name for food for our simple understanding. We can easily understand this problem with reference of following diagram.

Suppose rise in price of good X i.e. To prove your argument let us divide this in two stages. Namely 1 is consumer is pending his entire income on two goods X and Y. LM is his original price income line where the consumer is observed to have chosen the combination represented by R in figure. Secondly the triangle OLM is the consumer’s area of choice for the different combination of X and Y available to him, given by his price – income line LM. By choosing the only combination R, the consumer is revealed to have preferred this combination to all other in or on the triangle OLM.

As you said in the question the income of individual is increased and price of good X risen from 100 to 200 and price of good Y remains the same the new income – price line to be LS. Now he chooses a new combination is chosen say point A, which shows that the consumer prefers the less of good X and as the price is risen. In order to compensate the consumer for the loss in his real income as a result of rise in price of good X let us give him LP amount of money in terms of good Y. as a result, PQ becomes the new price – income line in which parallel to LS line passes through point R. This effect is known as compensation effect. Now the triangle OPQ becomes his area of choice. Since R was revealed preferred to any other point on the original price – income line LM, all points below R lying on the Segment RQ of PQ line will be inconsistent with consumer behavior. This is because he cannot have more good X when its price has risen.

The consumer will, therefore, rejects all combination below R and choose either combination R or any other combuination, say B in the shaded area LRP on the segment PR of the Price – income line PQ. If he chooses the combination, he will buy the same quantites of X and Y which is he before buying before the rise in the price of X. on the other hand if he chooses the combination B, he will buy less of X and more of Y than before. In the second stage, if the packet of extra money LP given to the consumer is taken back, he will be to the left of R at point Aon the price – income line LS where he will buy less of X, if the income if the income elasticity of the demand for the X is positive. Since with the rise in the price of X, its demand has fallen (when the consumer is at point A), it is proved when income elasticity is positive, price elasticity is negative.

And bundle is ranks according to the following table

Bundles of commodity and ranks preferred after price rise
			Ranks
Bundle 1 Old Price	Good X quantity 2 & Price $100	Good Y Quantity 80 & Price$ 10	2
Bundle 2 New Price	Good X quantity 1 & price$ 200	Good Y quantity 100 & price $10	1