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Chapter 3: Problem 17

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.)

Short Answer

Expert verified
The ideal and Laspeyres cost-of-living indices for Meredith for 2000 are 83.33 and 250 respectively.

Step by step solution

01

Calculate the consumption in 1990.

Given that the prices of food and clothing are $1 per unit each in 1990, and Meredith's income is $1200. Also, considering that Meredith had equal spending preferences, the consumption of food (F) and clothing (C) would be equal to \(F_1 = C_1 = \frac{1200}{2*1} = 600\)
02

Calculate the expenditure in 2000 for achieving same utility.

In 2000, the prices of food and clothing increased to $2 and $3 per unit respectively. To achieve the same level of utility as in 1990, we need to solve for food and cloths in equation \(2F = 3C\). Since Meredith is assumed to spend equal amount on food and clothing, we have \(2F + 3F = 1200\). This gives us \(F = 200\) and \(C = 133.33\) respectively. The total expenditure in 2000 is thus \(2F + 3C = 2*200 + 3*133.33 = 1000\)
03

Calculate the ideal cost-of-living index.

The Ideal Cost of Living Index shows the cost to achieve the same level of utility in a different price regime. It equals the ratio of the expenditure to achieve the same level of utility in the current period to the expenditure in the base period. Thus, Ideal Cost of Living Index for 2000 is \(\frac{1000}{1200} * 100 = 83.33\)
04

Calculate Laspeyres cost-of-living index.

Laspeyres cost-of-living index for 2000 is calculated by using the quantities for 1990 and the prices for 2000, as \(\frac{2*600 + 3*600}{1*600 + 1*600} * 100 = 250\)

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