Suppose the typical consumer only purchases food and clothing, and her utility can be expressed as U = F ? C. Currently, food costs $5 per unit and clothing costs $2 per unit. Her income is $70
If the price of food increases to $6, compare the resulting Laspeyres price index with a true cost of living index.
Maximizing utility subject to the initial constraint (5F + 2C = 70 ) yields C/F = 5/2 or F = 7 and C = 17.5. The Laspeyres price index calculates the ratio of the income necessary to achieve the original bundle relative to the original income. In this case, [(6 ∗ 7 ) + (2 ∗ 17.5 )]/70 = 1.10. The true cost of living index calculates the ratio of the income necessary to achieve the original level of utility relative to the original income. Utility is held constant when C ∗ F = 17.5 ∗ 7 = 122.5. The consumer is on the new budget line when C/F = 3. Combining yields F = 6.39 and C = 19.17. At the new prices, this requires an income of 76.68 and a resulting cost of living index of 76.68/70 = 1.095.
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