Constant Growth Rate, Point to Point Models.
Suppose you observe the level y 0 for a variable y at the beginning of a time interval and
observe the level yt at the end of the time interval.
Further, suppose you want to know the answer to
the following question: At what constant rate would the variable y have to grow in order for y 0 to grow
to the level yt over the time interval?
Geometric Point to Point Model
If time is discrete, meaning it is broken up into time periods, then the geometric model
(1)
y t  y 0 [1  r ]t
gives the relationship between the initial level y 0 for a variable, the ending level yt , the number of time
periods t , and the constant growth rate r . Solving for the growth rate, one obtains
1
t
(2)
y 
r   t   1.
 y0 
Given and variable y that grows over time, an estimate for the average growth rate between any two time
periods can be obtained using condition (2).
For example, nominal GDP for the U.S. was 269.2 in 1948 and 13,194.7 in 2006. Here,
t  2006  1948  58 , meaning there are 58 time periods. Substituting these values into (2), we obtain
r  0.0694  6.94%.
By inserting the growth rate obtained from (2) into condition (1), along with the initial value y 0 ,
you can create a model data series. For example, using the r  0.0694 and y 0  269.2 , condition (1)
becomes
y t  269.2[1.0694]t .
This
equation
describes
a
model
path,
where
y t  269.2[1.0694]  269.2 when t  0 , y t  269.2[1.0694]  287.9 when t  1 and so on until
0
1
y t  269.2[1.0694]58  13,194.7 when t  58 . Notice the model begins and ends with the same
number as the actual data series. This is why the model can be described as a point to point model.
Exponential Point to Point Model
If time is continuous, meaning it is not broken up into time periods but rather unfolds from
moment to moment, then the exponential model
(3)
y t  y 0 e rt
gives the relationship between the initial level y 0 for a variable, the ending level yt , the amount of time
t , and the constant growth rate r . Solving for the growth rate, one obtains
1
(4)
r
ln  yt   ln  y0 
.
t
Given and variable y that grows over time, an estimate for the average growth rate between any two time
periods can be obtained using condition (4).
Let’s again use the fact that nominal GDP for the U.S. was 269.2 in 1948 and 13,194.7 in 2006,
but now apply the exponential model. Here, it is again true that t  2006  1948  58 . However, now
the 58 does not represent the number of time periods. Rather, it represents the amount of time that has
transpired between the two points in time where the two GDP values were observed. Substituting these
values into (3), we obtain r  0.0671  6.71%.
By inserting the growth rate obtained from (4) into condition (3), along with the initial value y 0 ,
you can create a model data series. For example, using r  0.0694 and y 0  269.2 , condition (3)
becomes y t  269.2e 0.0671t . This equation describes a model path, where y t  269.2e 0.06710   269.2
when t  0 , y t  269.2e 0.06711  287.9 when t  1 and so on until y t  269.2e 0.067158  13,194.7
when t  58 . Notice, as with the geometric model, the exponential model begins and ends with the same
number as the actual data series. Again, this is why the model can be described as a point to point model.
The geometric model and exponential model each trace out the same path when they are plotted.
For the GDP data presented here, we associate t  0 with 1948 and we associate t  58 with 2006. The
Figure below illustrates that each of these point to point models results in a model data series where the
model begins and ends where the actual data begins and ends. While the growth rate along the actual
path varies, the growth rate along the model path is constant. In this case, the discrete year to year
average rate is 6.94%, while the continuous average constant growth rate is 6.71%. (Can you explain
why the continuous growth rate is lower but yields the same path?)
14,000
U.S. Gross Domestic Product
1948-2006
12,000
10,000
8,000
6,000
4,000
2,000
0
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Actual
Model
2