The utility that Meredith receives by consuming food \(F\) and clothing \(C\) is given by \(U(F, C)=F C .\) Suppose that Meredith's income in 1990 is \(\$ 1200\) and that the prices of food and clothing are \(\$ 1\) per unit for each. By 2000 however, the price of food has increased to \(\$ 2\) and the price of clothing to \(\$ 3 .\) Let 100 represent the cost of living index for \(1990 .\) Calculate the ideal and the Laspeyres cost-of-living index for Meredith for 2000 (Hint: Meredith will spend equal amounts on food and clothing with these preferences.)

Understanding the Laspeyres index is crucial when it comes to examining how the cost of living changes over time. Named after the German economist Etienne Laspeyres, it computes the change in the general price level or cost of a fixed basket of goods and services between two periods.

The Laspeyres index focuses on the consumption patterns in the base period. It can be expressed mathematically as: \[ \text{Laspeyres Index} = \left(\frac{\sum (p_t \cdot q_0)}{\sum (p_0 \cdot q_0)}\right) \times 100 \]where \( p_t \) represents the price of goods in the current period, \( p_0 \) the price of goods in the base period, and \( q_0 \) the quantity of goods consumed in the base period.

By using the prior consumption pattern as the benchmark, the Laspeyres index tends to overstate the increased cost of living when prices rise, as consumers often change their consumption habits in response to price changes. This is because it doesn't account for the substitution effect—where consumers opt for cheaper alternatives when prices increase.

For example, if Meredith had to adjust her spending in the year 2000 because of the rise in the price of food and clothing, the Laspeyres index would not reflect these changes in spending behavior. It would simply indicate how much more expensive it would be to purchase the same amount of goods from 1990 in the year 2000.

The Paasche index, named after the German economist Hermann Paasche, is another method to measure the cost of living. In contrast to the Laspeyres index, the Paasche index uses the current period as the base for quantity weights, reflecting what consumers are actually buying in that period.

The formula for the Paasche index is given by: \[ \text{Paasche Index} = \left(\frac{\sum (p_t \cdot q_t)}{\sum (p_0 \cdot q_t)}\right) \times 100 \]where \( p_t \) is the price in the current period, \( p_0 \) in the base period, and \( q_t \) the quantity of goods consumed in the current period.

By taking into account the possible substitution effects due to the change in prices, the Paasche index provides a measure of cost of living that adjusts for changes in purchasing habits. It can often result in a lower figure than the Laspeyres index because it reflects the consumer's ability to substitute cheaper goods for more expensive ones.

However, in the provided exercise, since the price changes for food and clothing from 1990 to 2000 are uniform and Meredith doesn't change her consumption habits (due to the exercise's assumption of fixed preferences), the Paasche index equals the Laspeyres index. In real-world scenarios, such uniformity is rare, and the two indexes usually give different readings, with the Paasche index typically reflecting consumer adaptation to price changes.

The Consumer Price Index (CPI) is a widely recognized measure that tracks changes in the cost of living over time. It serves as an economic indicator that reflects the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services.

CPI is calculated by taking: \[ \text{CPI} = \left(\frac{\text{Current period cost of basket}}{\text{Base period cost of basket}}\right) \times 100 \]Essentially, the CPI is a statistical estimate that is constructed using the prices of a sample of representative items whose prices are collected periodically. It is used for several important purposes, such as adjusting people's income (like pension benefits) to maintain their purchasing power, informing monetary policy, and as a deflator of other economic series such as retail sales.

In practice, the CPI usually resembles a Laspeyres index because quantity weights do not change frequently. However, adjustments are made periodically to better reflect the current consumption patterns. Thus, CPI is a practical tool for individuals to understand how price changes affect their living costs and for policymakers to make informed economic decisions.

In our textbook problem with Meredith's expenditures, the CPI concept could be applied to understand how her ability to consume food and clothing would be affected by the changes in prices—from 1990 to 2000—on her fixed budget.