Question

Suppose that there are two goods, and two time periods – a base period, and a current time period. An individual’s utility is increasing in each good, and the individual has to choose the consumption bundle that maximises utility subject to the budget constraint. Is it possible for the following three to hold simultaneously: (a) the Laspeyres quantity index is greater than one thereby implying that one cannot use WARP to conclude that the individual is better off in the ‘base’ period, (b) the Paasche quantity index is less than one thereby implying that one cannot use WARP to conclude that the individual is better off in the ‘current’ period, and (c) the individual’s utility is higher in one particular time period than in the other time period? If your answer is yes, present a complete example where (a)−(c) above hold simultaneously. If your answer is no, clearly reason why (a) − (c) above cannot hold simultaneously.

Answer 1

Yes, it is possible for the Laspeyres and Paasche quantity indexes to have certain values while the individual's utility is higher in one time period than the other.

Step-by-step explanation:

Yes, it is possible for the Laspeyres quantity index to be greater than one, the Paasche quantity index to be less than one, and the individual's utility to be higher in one time period compared to the other. These conditions can occur when there are changes in relative prices between the two periods. Here's an example:

1. Base period: Price of Good 1 = $2, Price of Good 2 = $4, Quantity of Good 1 = 2, Quantity of Good 2 = 3.
2. Current period: Price of Good 1 = $4, Price of Good 2 = $2, Quantity of Good 1 = 3, Quantity of Good 2 = 2.

In this example, the Laspeyres quantity index is 1.5, indicating that the individual requires more income in the current period to achieve the same level of utility as in the base period. The Paasche quantity index is 0.67, suggesting that the individual would need less income in the base period to achieve the same level of utility as in the current period. Additionally, if the individual's preferences are such that they derive more utility from Good 1 compared to Good 2, and the higher price of Good 1 in the current period does not offset the increase in quantity, the individual can experience higher utility in the base period.