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  Cost-of-Living Adjustments

Cost-of-Living Adjustments:

General Price inflation refers to a rise in all prices, or the general price level. Many government support programs are indexed to the rate of inflation and in wage negotiations labor unions often make wage demand such that they are at least compensated for the rate of inflation; in some countries wages are also directly indexed to the price level. If all prices and wages increases by the same percentage rate, the real income and purchasing power of a household, which derives all their source of income from working, is unchanged. However, it is uncommon that prices of all goods rise to the same extent, relative prices changes are often mixed up with general price level increases. It is of great interest in general to calculate the “true cost-of- living” and especially how it has changed over time. Microeconomic theory suggests that the consumer is better off, in period 2, if she can afford to buy the consumption basket (bundle of goods) that she chose to consume in period 1, i.e., before the price and wages changed.

The government’s statistical agency (bureau) is in charge of the task of calculating the official inflation rate, or the Consumer Price Index (CPI). The statistical agency defines a standard, or average, consumption basket and records each month the actual cost of buying this bundle. The starting year is called a base year and the index value is usually set to 100 at that time. If the cost of the standard basket has increased by 1.5% during the course of a year, the CPI in the beginning of period 1 is 101.5. Now, if the price of a particular good has increased by say 3%, its real price increase is approximately 1.5%.

If we consider an example with only two goods, (food, F) and clothing (C), with first period prices p1F and p1C, the cost of the bundle (C1,F1) in year 1 is:

Y1 = p1F . F1 + p1C .C1,

while the cost of the same bundle in year 2 is:

Y2 = p2F . F1 + p2C . C1

Y1 and Y2 are the money income necessary to buy the “standard” bundle in year 1 and 2, respectively. The ratio of these incomes Y2/Y1 shows how much the cost of the bundle has increased. E.g., if Y1 = 100kr and Y2 = 101.50kr, the ratio is 1.015, hence the rate of inflation is 1.5%. We can also write the ratio as,

812_cost of living1.jpg

By multiplying and dividing the first term in the numerator by p1F and the second term by p1C, we can rewrite the expression as,

1427_cost of living2.jpg

where θF = p1F F1/Y1 and θC = p1C C1/Y1 are the budget shares of food and clothing during the base year (year 1). Hence, the CPI is a weighted average of the price increases for each good, with budget shares as weights.

An important point is that if a group (retirees for example) is compensated for inflation in the sense that their pensions are adjusted upwards by the inflation rate (or multiplied by the ratio Y2/Y1), they will actually be overcompensated compared to a compensation which aims to keep their utility constant.

Example:

Assume that the prices of food and clothing both are equal to 1, initially. Our consumer has income Y1 = 100kr, and if she has a Cobb-Douglas utility function; U(C,F) = C0.5F0.5, she buys an equal amount of the two goods (given that the relative price ratio is equal to 1), i.e., she buys 50 units of both food and clothing. This means that the budget shares are: θF = θC = 0.5. If the price of food increases by 3%, to 1.03 but the price of clothing stays unchanged, the CPI increases by 1.5%. If the money income is increased to Y2 = 101.5, she will now buy F2 = (0.5x101.5)/1.03 = 49.27 units of food and C2 = (0.5x101.5)/1 = 50.75 units of clothing. Note that her initial utility was: U1 = 500.5 . 500.5 = 50, but here final utility is: U2 = 49.270.5 . 50.750.5 = 50.005. Hence, her utility has actually increased a little bit.

The reason that our consumer is better off is that the two prices did not increase to the same extent. That means that the relative price ratio changed (food being relative more expensive than clothing). The consumer will therefore substitute away from food and buy more clothing. The effect is that to some extent the consumer is able to shield herself from the effects of the price increase of food, and if she gets compensation such that she can buy the same bundle as she did last year, she will be overcompensated.

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