NOTES 1: Real and Nominal Variables
Some notes on computing real growth rates in variables.
Over the years of teaching this course, I have had some recurring questions from students about computing real growth rates in variables. One question I received was:“Where does the approximation formula for calculating real growth rates come from?” (the one which says: real growth in GDP ≈ nominal growth in GDP - the inflation rate). Another question is:“Why does the approximation formula provide less accurate results when the inflation rate is high?” I will answer both of these questions in this chapter. But first, I provide a review of how to use price indices.
I will use GDP (Y) or, in some cases household wealth, as the example although we could be talking about any variable such as movie receipts, corporate profits, etc..
real Y(t) = nominal Y (t)/ P(t) (1)
(where P = some price index and (t) indexes the year, t).
This is straightforward from class. By deflating by the price index, we are converting GDP from prices in year (t) to prices in the base year (the year in which the price index is anchored). Some people (wrongly)think that the price index is unit-less. This is not true. The price index has units. In this case, the units are dollarsin year t per dollarsin base year. By definition, nominal Y(t) is measured in year t dollars (i.e., 2005 dollars). We are converting nominalY from being measured in year (t) dollars (i.e., 2005 dollars) to dollars in the base year (i.e., 1982 dollars). By doing this, we can compare the real change in GDP after the price effect has been removed. It is helpful to think of the example from class.
In many of my earlier papers, I measuredhousehold wealth and savings in 1996 dollars. Let’s see how I used price indices in those instances:
The CPI currently has a base year of 1982 (Note: the price index in the base year always equals 1.000 by definition - this would imply that real variables and nominal variables in the base year would be equivalent).
The actual CPI numbers for 1994, 1996, and 1998 are, respectively, 1.480, 1.568, and 1.630.
Suppose median nominal household (in current year dollars) wealth in 1994, 1996, and 1998is, respectively, $48,300 (in 1994 dollars), $59,200 (in 1996 dollars) and $72,400 (in 1998 dollars).
What is the value of real wealth in each of those years? This question, by itself, has infinite answers depending on the metric. Real wealth could be measured in 1982 prices, 1983 prices, 1741 prices, and 1998 prices… (you get my drift - it is like saying any length can be measured in feet, inches, yards, miles, meters). You may be thinking to yourself that it would be easy to measure real wealth in 1982 prices (that is the base year). You would be partially correct. It is also easy to measure prices in some other year's prices. We will do both and compute real wealth in two different units.
First, we will measure 1994 real wealth in 1982 prices and then I will show you how to measure 1994 real wealth in 1996 prices. When defining real variables, it is important to specify which year's prices you are using to deflate the nominal variable. For real variables, it is not asimportant which units we choose to measuring prices in - as long as we measure all prices that we wish to compare in the same units (i.e., the same prices). Again, it is helpful to think about the length example. It really doesn't matter if you measure length in feet or inches - as long as you are consistent with your measurement.
To get real wealth (measured in 1982 dollars), we use the formula above (equation (1)). (Make sure you keep track of units!)
The CPI above, for 1994, says that if you had $1 in 1982 (base year) you would need $1.48 to buy the same amount of goods in 1994. For 1.48 (1994 dollars) you would need 1.00 (1982 dollar) or, mathematically, we can express this as 1.48 (94$)/ (82$). The units on the 1994 CPI are (94$/82$) or 1994 dollars/1982 dollars.
Let's measure real wealth in 1994 (measured in 1982 dollars): Nominal wealth in 1994 is $48,300 (measured in 1994 dollars). Let’s express nominal wealth in 1994 as: 48,300 (94$), where (94$) are the units on nominal wealth measured in 1994.
What is real wealth in 1994 (measured in 1982 dollars)? Using formula (1), we get: 1994 real wealth (in 82$) = 1994 nominal wealth (in 94$)/ Price Index (94$/82$)
= 48,300 (94$)/ 1.48 (94$/82$) = 32,635.14 (82$)
<Keep track of the units - they should cancel out!!!>.
This says that real wealth in 1994 is $32,635 (measured in 1982 dollars).
We can also measure 1996 wealth and 1998 wealth in (82$) using the same procedure (try this at home).
We get real wealth 1996 (82$) = $37,755
We get real wealth 1998 (82$) = $44,417
Now, we can also express real wealth in 1994 in 96$. Let's see how we can do this:
real wealth 1994 (96$) = nominal wealth 1994 (94$) * [(96$/82$)]/(94$/82$)]
We convert first from 94$ to 82$ and then from 82$ to 96$. Again, think of the length example. If I am given a measure in yards and wanted to get a measure in inches, I can convert yards to feet and then feet to inches. For our example of writing 1994 wealth in 1996 dollars, we get (48,300 * 156.8 /148.0) = 51,171 (96$). Notice that I made the conversion in one step (just like I could do if I was going to go directly from yards to inches). <Keeping track of the units - we see that all we are left with is 96$>. The way we do this is first convert wealth from 94$ to 82$. After that, we convert 82$ to 96$:
[1 (82$) = 1.568 (96$)].
So, let’s recap: We have expressed 1994 wealth in 3 ways (in 82$, 94$ and 96$):
real wealth 1994 (82$) $32,635
real wealth 1994 (94$) $48,300
real wealth 1994 (96$) $51,171
The value of real wealth in 1994 depends on what year we are measuring wealth in (i.e., inches, feet, meters). As long as we pick a measure and are consistent, it doesn't matter what year we report our real variables in (but, we need to express what year we are measuring prices in to the relevant audience).
We can do the same for 1996 and 1998 wealth.
<you should attempt this on your own >
Ok - back to the question at hand. How do we compute growth in a real variable? I will outline three ways - all of them are essentially equal (although, some are approximations).
- Growth in real Y (t, t+1) = [Real Y (t+1) - Real Y (t)] / Real Y (t) (2)
This formula ALWAYS WORKS! This is how we compute growth in any variable (think the inflation formula). Compute real GDP in any two years, where the real variable is measured in the same year prices. Then take the growth rate in the GDP between the two years.
- The second way is a derivation of the first. Let’s substitute in the definition for real Y into the above formula; remember, real = nominal/price index.
Growth in real Y(t, t+1) = [Real Y (t+1) - Real Y (t)]/Real Y (t)
Growth in real Y (t,t+1) = [(Nominal Y(t+1)/ P(t+1)) - Nominal Y(t) / P(t)] / (Nominal Y(t)/P(t))
We can rewrite this expression by multiplying the numerator and the denominator by P(t)/Nominal Y(t):
Growth in real Y(t, t+1) =
{ [Nominal Y(t+1)/Nominal Y(t)] / [P(t+1)/P(t)] } - 1
Define g as the growth rate in nominal Y between t and t+1 (i.e., % increase in nominal Y). Define as the inflation rate (i.e., growth rate in prices between t and t+1)
Doing so, we can rewrite the growth rate in real Y as:
[(1+g)/(1+)] - 1
<How did I do that?
We remember: = (P(t+1) - P(t))/ P(t) = (P(t+1)/P(t)) - 1. Re-writing, we get that 1+ = P(t+1)/P(t). This is what we used to get the above equation>.
So, growth rate in realY = [(1+g)/(1+)] – 1(3)
3.The third way is an approximation method based off of equation (3).
Let's do one more step to equation (3) - rewrite (3) as:
(1 + g(r)) = [(1+g)/(1+)]
To get this, just define g(r) as the real growth rate in Y (and add one to each side of the equation).
Now - this doesn't look like the approximation formula I gave you in class. Where does the approximation formula come from?
Let’s multiply both sides by (1+π) such that:
(1 + g) = (1+g(r)) * (1+)
Then, expand the right side of the equation such that:
1 + g = 1+ g(r) + π + g(r) *
Subtracting 1 from each side yields:
g = g(r) + π + g(r) * π
If the inflation rate () is low (3% per year) such that g(r) * π is really low (compared to either g(r) or π), we can ignore the interaction term. In that case,
g(r) ≈ g – π (4)
This is the approximation formula I gave you in class. This approximation formula is a good approximation when g(r) * π is so small that it can be assumed to be zero. This occurs when the inflation rate is really low (or when the growth in real GDP is real low).
To restate: When is this approximation formula correct???? When the inflation rate is close to zero. The larger the inflation rate (like in the example from the notes), the less likely the approximation formula works. In this case, it would be best to compute the real growth in Y using method (1) from above!
On tests and quizzes, I will usually tell you which method to use (so learn all three methods). But, in general, all are valid when the inflation rate is low - with the first two methods being 100% correct all the time.
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