Question

The utility that Meredith receives by consuming food F and clothing C is given by U(F,C) = FC. 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

Meredith’s ideal cost-of-living index and Laspeyres cost-of-living index is 250.

Step-by-step solution

Computing the income in the year 2000

Meredith’s budget constraint in 1990:

F + C = 1200

From the condition of utility maximization

$$\begin{gathered} \frac{M{U}_F}{P_F}=\frac{M{U}_C}{P_C} \\ C=F \\ \sin cepricesperunitis\$1each \end{gathered}$$

Using the relation in the budget equation

F + C= 1200

C=F=600

The utility for the consumption bundle (600,600) will be:

U(F, C)=FC=360000

Now, Meredith’s budget constraint in 1990:

2F + 3C = I₂₀₀₀

from the utility maximization condition

C = F = $\frac{{\mathrm{I}}_{2000}}{5}$

Meredith’s utility doesn’t change and is the same for the year 2000 (as in 1990). From the above relation, the utility function will give Meredith’s income in 2000 as:

U(F, C)=FC=360000

$$\begin{gathered} \frac{{\mathrm{I}}_{2000}^2}{25}=360000 \\ {\mathrm{I}}_{2000}=\$3000 \end{gathered}$$

The ideal cost of living adjustment is, therefore, $1800.

Computing the ideal cost-of-living index

The cost-of-living index represents the cost of attaining a given level of utility at current prices relative to the cost to achieve the same utility at base prices.

The ideal cost-of-living index is:

$\frac{3000}{25}=2.5$

If 100 represents the cost-of-living index for 1990, the ideal cost-of-living index in 2000 will be 250. A value of 250 represents the 150% increase in cost-of-living.

Computing the Laspeyres cost-of-living index

The amount of money at current-year prices that an individual requires to purchase the bundle of goods and services chosen in the base year is divided by the cost of buying the same bundle at base-year prices.

The base year (1990) consumption bundle is (600,600).

The amount of money at current year prices will be:

600 x 2 + 600 x 3 = 3000

The cost of the same bundle at base prices is $1200.

Laspeyres cost-of-living index will be:

$$\begin{gathered} \frac{\sum _{}{P}_{2000}{Q}_{1990}}{\sum _{}{P}_{1990}{Q}_{1990}}\times 100 \\ =\frac{3000}{1200}\times 100 \\ =250 \end{gathered}$$

The Laspeyres cost-of-living index is 250.