Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1.

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1 1 Growth rates Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1. We have: Ŷ = Y Y = Y t Y t 1 Y t 1 = Y t Y t 1 1 Example 1 if Y or GDP grows from 100 to 110 between 2000 and 2001, calculate Ŷ. Y t Y t = = 0.1 Y t Note that it is always possible to write: Y t = Y t 1 + Ŷ (Y t 1) which is more commonly expressed as the equivalent. Y t = Y t 1 (1 + Ŷ ) (1) This says that last year s GDP times one plus the growth rate of GDP is this year s GDP. 2 Hat calculus Rule 1. Product Rule for multiplication of two variables. Take two variables Y and Z such that X = Y Z The product rule is: ˆX = Ŷ + Ẑ In words: When X is equal to the product of Y and Z, the growth rate of X is the growth rate of Y plus the growth rate of Z. Example 2 Real and Nominal GDP. The price level is measured by the GDP deflator. This is an index which starts at 100 for some arbitrarily defined base year. The GDP deflator is a measure of the average prices of goods and services in the economy. The inflation rate is the percentage increase in the price level from one year to the next. The consumer price index (CPI) is another measure of inflation that does not take into account the whole of GDP, but only those goods purchased by consumers (leaves out investment goods for example). The CPI is the consumer price index with weights from the market basket, but the investment and government expenditure price indices use market baskets appropriate to their concept. The producer price index (PPI) is an average of the 1 Sombreros en español. 1

2 prices received by producers of goods and services at all stages of the production process. It is the wholesale price index. The GDP deflator is defined as P = Y n /Y base where Y n is nominal GDP and Y base is the same GDP but measured in base year prices. This can also be expressed as Y base = Y n /P so that the P deflates any rise in nominal GDP due to inflation. Finally, we can also write P Y base = Y n the GDP deflator times real GDP gives nominal. Now take hats ˆP + Ŷbase = ˆ Y n This says that the inflation rate, ˆP, plus the growth rate in real GDP is the growth rate in nominal GDP. Thus, if we knew that nominal GDP was growing at 5.5 percent but inflation was 3.2 percent, then real GDP would only be growing at = 2.3 percent. Example 3 The quantity equation is Mv = P Y n where M is the money supply, v is velocity. Velocity depends on how fast the money supply circulates through the economy. If the velocity is fast then a small amount of money can support a large GDP and vice-versa. Velocity is determined by the institutional arrangement of society. For example with the rise in the use of credit cards, velocity of money rises since less actual money is required for the same GDP. Applying hats to this example: ˆM + ˆv = ˆP + ˆ Y n Normally, the institutional framework of society does not change that quickly, so we can assume that ˆv = 0. Then if the money supply is growing by 10 percent, but real GDP is growing only by 4 percent, we know that inflation, by the hat equation, must be 6 percent. Rule 2. Quotient rule for when the two variable are divided The quotient rule is: X = Y/Z ˆX = Ŷ Ẑ In words: When X is equal to the quotient of Y and Z, the growth rate of X is the growth rate of Y minus the growth rate of Z. Note that this is just rearranging the product rule. 2

3 Example 4 Define GDP per capita, y, as y = Y N where N is the population of the whole country (as distinct from the labor force, L N) and Y is GDP of the whole country. To calculate the growth rate of income per capita, write: ŷ = Ŷ ˆN Rule 3. Exponent rule where z is a constant, then: Y = X z Ŷ = z ˆX Let the labor supply be related to output with the following func- Example 5 tion: This gives: 2 Can you combine the two? Y = L 0.5 Ŷ = 0.5ˆL i.e., how would we express: Y = ax b in terms of growth rates or hats? Answer: but if a is a constant, then â = 0. Ŷ = â + b ˆX Rule 4. Now there is a special case when X is e, the base of the natural logarithm system: Y = e gt Ŷ = g Example 6 Over the twentieth century the US economy has grown on average about 2.5 percent in real terms. Express this in terms of e, the base of the natural log system. Answer: Y = e 0.025t 2 Proof: calculate dots: divide by level to get percentage change: 0.5L 0.5 L/L 0.5 simplify: Ẏ = 0.5L (0.5 1) L Ẏ /Y = 0.5L 0.5 L/L 0.5 Ẏ /Y = Ŷ = 0.5 L/L = 0.5 ˆL 3

4 Note that like all applications of calculus, these rules only apply for small changes; for large changes they are only approximations. The rule of thumb: don t apply hats to growth rates over 10 percent. Rule 5. There is no rule for addition. The best we can say is that when we have a sum X = Y + Z then if both Y and Z are growing at the some common rate, then X will also be growing at that same rate. Otherwise we have X = Y + Z X Y + Z = X X X X = Y Y Y X + Z Z Z X ˆX = wŷ + (1 w)ẑ where w = Y/X and so (1 w) = Z/X. This is not that useful since as Y and Z grow, w changes. In the special case of Ŷ = Ẑ, we have X growing at the same rate, but that is about it. 3 Constant versus variable growth rates Thus the path for any variable can be represented two ways: (1) as a function of time as expressed in the table above; (2) as a starting point with a growth rate. If as in table 1, the growth rate is variable, then we must write: Y t = Y t 1 (1 + Ŷi) where the subscript on Ŷ indicates that it can change from period to period. If we impose a constant growth rate, we have equation 1 above. If the growth rate of a variable X is g then we can write: X 1 = (1 + g)x 0 that is the level of X in period 1 is just whatever it was in period 0 plus gx 0, the growth from period 0 to period 1. The same can be said for period 2: X 2 = (1 + g)x 1 Now we can combine these two equations to write X 2 = (1 + g)(1 + g)x 0 X 2 = (1 + g) 2 X 0 and now generalizing X t = (1 + g) t X 0 (2) 4

5 Table 1: Macroeconomic data Year CPI Percent Change Inflation Rate Source: Made up data To get an average rate of growth, g, we can write and taking the tth root X t X 0 = (1 + g) t ( Xt ) 1/t = 1 + g and now solving for g X 0 g = ( Xt X 0 ) 1/t 1 (3) This gives the average rate of growth g for t growth periods knowing the beginning level and the ending level. Example 7 Consider the path in table 1 for the consumer price index, what is the average inflation rate? Answer: Using equation 3 or 4 percent. g = ( ) 1/ =

6 4 Doubling Times The average growth rate equation gives rise to a convenient expression for doubling time, often called the rule of 72. Here is how it works. The basic equation for a growth process at a constant rate is given by equation 2 above X t = (1 + g) t X 0 Doubling time means how much time would it take for X 0 to double? for X t /X 0 = 2 For a given growth rate, we need only solve That is Taking the natural log of both sides 2 = (1 + g) t ln 2 = = t ln(1 + g) the problem here is the term ln(1 + g). Is there any way to simplify this or if not find some approximation to it? Fortunately there is. We have 3 so for a first approximation, we have ln(1 + g) g (4) = gt Where does 72 come from and why is this not simply called the rule of 69? When the doubling proceeds quickly, instantaneously as in bacteria or nuclear chemistry, then it is the rule of 69. (See The Story of e). In economics change typically takes place more slowly, so we use the rule of 72 (sometimes 70) instead of the instantaneous 69. Example 8 The growth rate of the economy is 3.5%. GDP to double? Answer: From the rule of 72, write How long will it take for 72/3.5 = t or years. As seen this is an approximation and will be a better approximation for smaller growth rates. 4 3 The proof of why this works depends on a power series expansion ln(1 + g) = g 1 2 g g3 1 4 g g The approximation comes from taking the first term only! 4 To express the growth rate as a decimal, the rule becomes the rule of

7 Example 9 Why can we not take the yearly rates of growth of real GDP and average them? Answer: Because of compounding. The growth in year t depends on the level in the previous year. But the level is the sum of the previous level plus the growth in the previous period. So this year s level is the sum of the growth of the previous level plus the previous growth. This makes growth geometric and not arithmetic. Here is what we mean. Let s say that GDP is 100 in the base year. If it grows by 6 percent the first year and 12 percent the second year, the average growth rate is 9 percent per year. This gives but in reality: [100 ( )]( ) = [100 ( )]( ) = a bit larger. So in general we must use equation 3 instead. 7