NOTES 1: Real and Nominal Variables
Some notes on computing real growth rates in variables.
I had a few different questions on computing real growth rates in variables. The
questions were different in specifics, but were very similar in spirit. One question I
received was where does the approximation formula I gave in class for calculating real
growth rates come from (the one which says that real growth in GDP is approximately
equal to the nominal growth in GDP - the inflation rate). The other question is why is
it that the approximation formula provides less accurate results when the inflation rate is
high (like the example in class and on the practice quiz). I will attempt to answer both of
these now. But first, I want to give 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 real variable.
real Y(t) = nominal Y (t)/ P(t)
(where P = some price index and (t) is some year, t).
This is straightforward from class. By deflating by the price index, we are converting
GDP from prices in year (t) to some prices in the base year (the year in which the price
index is anchored). Some people want to think that the price index is unit-less. This is
not true!!! The price index has units. In this case, the units are dollars(in year t) per
dollars(in base year). By definition, nominal Y(t) is measured in year t dollars (i.e.,
1999 dollars). We are converting nominal Y from being measured in year (t) dollars (i.e.,
1999 dollars) to dollars in the base year (i.e., 1982 dollars). By doing so, we can compare
the real change in GDP after the price effect has been removed. (Think of the example
from class).
As I told most of you in class, in my papers I measure my variables in 1996 dollars. I
often study household wealth and savings. Lets see how I use the price indexes.
The CPI currently has a base year of 1982 (price index in a base year always equals
1.000).
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, 1998
is, respectively, $48,300 (in 1994 dollars), $59,200 (in 1996 dollars) and $72,400 (in
1998 dollars).
What is the single value of real wealth in each of those years? This question, by itself,
has infinite answers. Real wealth could be measured in 1982 prices, 1983 prices, 1741
prices, and 1998 prices… (you get my drift - it is just like saying any length can be
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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). In part, you
are correct. But, it is also easy to measure prices in some other year's prices. We will do
this two ways. 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
really, really important to specify what year's prices you are using to deflate the nominal
variable. For real variables, it is not so much important as what units we are measuring
prices - as long as we are measuring all goods in the same units (i.e., the same prices).
Just think about the length example. It really doesn't matter if you measure 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. (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
$1.48 to buy the same amount of goods in 1994. That says that for 1.48 (1994 dollars)
you would need 1.00 (1982 dollar) or, mathematically, we can express this as 1.48
(94$)/(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
$48,300 (measured in 1994 dollars). Let’s write nominal wealth in 1994 as: 48,300
(94$), where (94$) is the units on nominal wealth measured in 1994. What is real wealth
in 1994 (measured in 1982 dollars)? Using the formula above, we get 1994 real wealth
(82$) = 1994 nominal wealth (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$):
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real wealth 1994 (82$)
real wealth 1994 (94$)
real wealth 1994 (96$)
$32,635
$48,300
$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 the growth in a real variable? I
will outline three ways - all of them are essentially equal (although, some are
approximations).
1.
Growth in real Y (t, t+1) = [Real Y (t+1) - Real Y (t)] / Real Y (t)
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.
2.
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))
Rewriting you get
P(t)/Nominal Y(t)):
(multiplying the numerator and the denominator by
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 (% increase in
nominal Y).
Define  as the inflation rate (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
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<<How did I do that: We remember that  = (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, the growth rate in real Y = [(1+g)/(1+)] - 1
Let's do one more step - rewrite the above equation 1 more time:
The growth rate in real Y = ((1+g) - (1+)) / (1+)
algebra>>
<this was just some
Now - this doesn't look like the approximation formula I gave you in class - but, it
is close to what I gave in class (this above formula is an exact formula for the real
growth rate in Y!!!
If the inflation rate () is low (3% per year), 1+ is essentially 1. So, the
denominator of the above equation is essentially 1 (it is very close to one - this is
where the approximation part comes in….). If the denominator is 1, then we can
write the above equation as:
Growth in real Y = (1 + g) - (1 + ) = g -  = [growth rate in nominal Y inflation rate]
This is exactly what I gave you in class as our approximation formula.
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 what method to use (so learn both).
But, in general, both are valid when the inflation rate is low - with the first
method being 100% correct all the time.
As a test, re-compute the real growth in Erikland from the example I used in class
using the exact method. Notice that the results differ by a lot. The reason is the
inflation rate is so high!!!
******
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