Government and the public need to  plan for  slower long term US economic growth.

There are reasons to expect  the US economy to grow more slowly in the future than it has in the past. It  doesn't seem to be growing at an exponential rate,  which means its yearly % growth will go down over time. The public should rely  on conservative growth estimates because of the difficulty of forecasting the future.  Although it isn't possible  to accurately predict future growth, it is useful to try to get some idea of its possible magnitude because it influences government and personal plans.

  Unfortunately
g
overnment entities often perform estimates of future tax revenue, debt  and spending ( e.g. of social security and medicare)  based on   unrealistically optimistic  hopes for how fast our economy will grow.   Private retirement plans are often made assuming future percentage returns on investment will match past returns, which may not be the case for those who invest only in the US
. Another page on this site will address the concern that increased national debt will slow economic growth even further than discussed below.

The only way to attempt to predict the future is to find patterns in  past data and hope they continue.
It is easier to predict some aspects of the growth of  less developed countries than  to predict  US growth since they are likely to mirror the growth path other countries have already gone through. They have the potential to grow  faster than the US did as they  apply technology and methods that are already widespread in advanced economies like the US. They are largely  playing "catch up" by accumulating capital and investing it in modernization
In contrast the  US   needs to rely heavily on  new innovations to grow. It is safe to predict that new things  will   continue to be invented, but there is no way to know     what impact those unknown innovations will have on future growth rates.  It may be growth will pick up due to some new revolutionary technology, but is safest not to expect it.

People are familiar with the concept that money in a bank account earning a fixed X% interest is growing at an exponential rate.   People   usually also talk about the percentage yearly growth rate for an economy rather than focusing on  the $ growth since it is a convenient way to think about it. That way of discussing the issue can lead to the trap (even among many economists)  of implicitly thinking  economic growth must be exponential, but the data calls that into question. Although most long term forecasts acknowledge    the economy is on average growing by a smaller yearly % over time, many projections are still done implicitly assuming growth is exponential, but merely  slower exponential growth. If the growth pattern of an economy  isn't truly exponential it is to be expected that the yearly % growth rate will constantly drop over time as it has. 
Considering other potential growth patterns can yield more useful projections of future growth.

Note for the math phobic: this page won't be discussing equations, even  though  some appear in one image below. Some people  may not be interested in all the gory details below supporting the case for why slower growth is likely, however there are many graphs   which  make it easier to skim through for those that want to try.

The usual measure of the size of an economy is GDP (Gross Domestic Product), which is provided in the US   by the federal Bureau of Economic Analysis. Papers exploring economic growth often use one of three types of values   related to GDP which this page will explore to consider   growth rates.  The "nominal"  US  GDP is the actual $ size of the economy, not adjusted for inflation. "Real" GDP values are adjusted for inflation so they are are usually more useful. Some  forecasts oddly  focus on nominal GDP. The Social Security Adminstration (SSA)  appears to   even though it does provide  real GDP growth figures as well. Some of our economic growth is driven merely by an increase in population. If you have twice as many people the economy can produce more wealth, but the wealth is then obviously split up between more people. "Per Capita" real GDP values are therefore often the most useful.


The  SSA  has 3 different long range  forecast scenarios for future GDP through 2090: expected (intermediate), high and low forecasts. This page will show their expected estimates appear too optimistic. The focus is on   SSA since other  government entities like the Congressional Budget Office (CBO) and Government Accountability Office (GAOhave    "expected" forecasts  that are almost the same, but don't provide best and worst case alternatives. Some of those forecasts use models to predict the future, but  it is useful to start with a different approach  to "sanity check" their forecasts before coming back to that issue.

  Data can be  analyzed to find what sort of  equation most closely describes a trend to try to predict the future, e.g. a line, a quadratic curve, or an exponential curve 



Obviously those are just some examples of growth curves, and the real   pattern may be more complicated. However it is useful to at least get a general idea of how growth matches simple curves to   see that it isn't useful to plan for it to grow as fast as an exponential curve.

G
rowth may not continue as steadily as any of those curves suggest, it may e.g. follow one of many potential "s-curve" shaped patterns:

During the initial stages of an 
S-curve a set of data may look like   an exponential or other growth pattern at first and there may be absolutely no way to tell it will slow in the future.
Unfortunately sometimes  different trend curves matched   to the same data can  all fit fairly well but yield very different future projections.

There are many different niches in the economy and they  grow at varied rates since they are driven by different kinds innovations and   varying speeds of diffusion of those ideas  through the economy. The hope forecasters have is  that on average those combine to produce growth which  follows a simple pattern. There are factors which tend to smooth over the differences between niches, e.g.   investment shifts between niches with comparable risk levels to encourage those with the highest rate of return. There is no guarantee   that reality follows  a straightforward pattern. Growth may merely happen to match a given equation for a short period of time but be following a different trend in the long term. For instance it might  slow down if  national debt interferes with growth or speed up after some  new innovation appears. It is useful to try to make estimates, but important to remember how inherently unpredictable the future economy is and plan conservatively.

 
Various studies have found different  types of GDP growth patterns depending on what time period  they look at and what countries. The ones mentioned next  show  that at minimum it is best not to plan on long term  exponential growth. This  study notes "Our argument is based on an empirical investigation of real GDP per capita growth in 25 OECD countries (and three country aggregates) during the post-war period using the Box-Cox transformation method. The conclusion is that per capita growth is generally (more or less) linear (and definitely not exponential)" and and another notes that even further back" "the growth in real gdp per capita is a linear one since 1870 with a break in slope between 1940 and 1950".  

Another study finds: "The S-shaped logistic pattern provides good descriptions and forecasts for both nominal and real GDP per capita in the US over the last 80 years." and this work pursues an s-curve approach: "Economic Growth and Transition: Econometric Analysis of Lim's S-curve Hypothesis" and this applies it to economic growth even in currently fast growing China which is still modernizing: "China is likely to follow an S-Curve-shaped path of slowing growth".

  Often most economic data doesn't exactly match any type of curve exactly. This shows an example of data which might be seen as coming close to matching a linear trend,  from wikipedia


The problem is it may not always be clear which type of trend to use to fit data like that, it may fit many different types of curves. The graph below of yearly values shows an example of  data generated by exact equations for linear, quadratic and exponential growth where the curves  are all fairly similar.  It illustrates a problem that shows up in some economic data:
 


If we just had the data and didn't know where it came from, the data  for each  of those lines could be matched fairly well to   any of these different  trends: linear, exponential or quadratic, regardless of how it was generated. One measure of how well an equation  fits data, R-Squared,  runs from 0 to 1, where 1 is better. R-squared is about 0.98 or higher for all of those lines for each of those trends, which   is usually considered a good fit.  Many types of economic forecasts, such as those for Social Security are made using trends on data sets that have much lower R-squared figures, e.g. their description of a real wage trend says "The R-squared value was 0.53". 

Even if there were an exact equation     responsible for a   trend, there may be errors in collecting data or other factors which cause minor fluctuations above and below the main trend like the scattered dots shown around the line earlier.   Those variations    can lead to a different type of trend best matching the data than the actual equation that generated it.  There are more advanced techniques beyond a simple R-squared measure to try to spot what trend fits best, but with a limited amount of data it can be impossible to know for sure which equation produced the data   no matter what math you use.

Even if different trends  look almost the same historically,  they may yield very different results in the future. The graph below plots the future of the lines we saw  through 2090. The line currently on top would be left behind:
 

 GDP values  exhibit this problem of    matching different trends well (as does some other economic data sometimes used to derive GDP trends, like productivity growth). Data from the federal Bureau of Economic Analysis for GDP starts in 1929, and many consider data after World War II to be more reliable and to leave out the atypical WWII and Great Depression periods. To highlight a few examples illustrating the problem: BEA data for real per capita GDP from 1929 to 2011 yields an r-squared for a linear trend of 0.97, exponential: 0.96 and quadratic: 0.99. Total real GDP from 1947 to 2011 fits a linear trend: 0.96, exponential: 0.99158 and  quadratic: 0.99505. Nominal GDP from 1947 to 2011 yields an r-squared for a linear trend: 0.86, exponential: 0.98987 and quadratic: 0.99713.  

Nominal GDP Trends

 Lets look at various possible trends and then consider methods to decide what matches reality. The Social Security Administration (SSA) seems to focus on nominal figures even though   much of the increase is due to inflation which may be less predictable  than other elements of growth. The graph below shows  various possible trend curves that match nominal GDP   data  from 1929 on (discovered using     Eureqa software from the Cornell Creative Machines lab which searches for patterns) and  plotted  from history through the 2090 ending year of the SSA forecast, including a linear trend, quadratic trend and various logistic s-curves (the s-curves are the best fit)



Eureqa finds other s-curves  that fit the data well which are similar to those. It  would be hard to fit  more lines on that graph so a representative few were chosen to illustrate the point. Adding in even the lowest SSA projection for nominal GDP to that same graph ("Low" below) shows it is far higher than any of those trends:

The best-fit exponential trend would produce a curve that grows  faster than   any of the estimates from SSA so it was left off the graph since  it would change the scale to make it harder to see the difference between the other lines. As you'll see later it doesn't seem to match reality as well as other trends. This page is focused on considering realistic   expected or worst case possibilities rather than trying to find even more optimistic ones.

Real GDP Trends

This shows some of  Eureqa's Real GDP trends, with again a large variation.The best fits are the s-curves, the lower the curve the better the fit in this case. 



Oddly the SSA forecast with the lowest real GDP is the "high cost scenario". It is still higher than most of the S-curves.The graph below includes that as well as the expected medium forecast GDP which is   the highest line well above all the trends (their highest GDP estimate is well above even that and was left off the graph to avoid changing the scale):


Real Per capita GDP Trends

The graph below used  real per capita GDP   to find trends:

The two s-curves match better than the linear and quadratic curves. They  yield very different results since one s-curve is still in its rising phase and one flattening out. Most of the s-curves Eureqa finds are in the range of the  the lower one so they were left off. The intermediate   GDP in the social security forecast   eventually rises higher than all of them:



Even the lowest real per capita GDP from the SSA forecast (from the "high cost" scenario) is higher than the flatter s curve and the linear trend:

Comparing real GDP per capita trends to reality

The next issue is to try to get a sense of which trends might match reality, starting with   real GDP per capita. For simplicity we will make the optimistic assumption  for the moment we are looking for a steady growth pattern and not an S-curve (consideration of s-curve trends might be the topic of a future page, there are many options to consider). The first step is to decide best case, if growth is continuing, which sort of growth curve comes closest to matching reality so far.

In science predictions about reality are tested with experiments. Obviously we don't have a time machine to test predictions of future GDP today.
One
way to test whether a particular type of trend might match reality is to use part of the historical data to try to predict the rest. For example you might pretend you were back in 1970 making  a forecast using only the data up through 1970. You could use that data predict   the 2011 GDP and compare it to reality.   

You might use all the data starting with the earliest available from 1929 to make predictions, and compare it to using only  post war data to see if the depression and WWII make a difference. The overall results seems to be similar regardless of the choice of start year. The graphs below show the actual   2011 real GDP per capita compared to the forecast that would have been made in prior years using different types of trends. e.g. the values for 1992 are the prediction you would have made in 1992 year for 2011 GDP based on a linear trend, a quadratic trend and an exponential trend.




A closer look at the last part of that graph confirms that the real value tends to fall between the quadratic and linear trends:


This approach can be generalized to look for example at the  percentage  error in 20 year forecasts that would have been made in each year using data available up to that date. e.g. it compares the forecast that would have been made in 1980 for the year 2000 and calculates the percentage error compared  to the actual 2000 GDP.



Similar patterns appear for the percentage errors in e.g. 5 and 10 year forecasts. They show a  quadratic trend is the most accurate overall,   with reality usually being somewhere in between linear and quadratic trends. Below are linear&quadratic  trends using data from 1947 onward for  per capita GDP compared to SSA forecasts. The CBO  forecast  is  included this time to show it is almost the same as  the SSA medium forecast (it stops a few years early or it would cover up the SSA forecast). The trends are lower than all but the worst case SSA forecast (its lowest real GDP is in the "high cost" scenario):


Even though reality seems to fall between   quadratic and linear trends, it is only the government's lowest growth case forecast that falls between them, which indicates it (or the quadratic trend) is a better choice for an expected case forecast Since government should plan conservatively, even the linear trend might be worth considering as   a conservative expected forecast choice (though no growth would be the safest to plan on).

Earlier it was pointed out that it can be hard to determine for sure which trend type matches best. One possibility to check on is whether  the choice of start year accidentally worked best for quadratic trends.  The issue is whether for  instance if we used data from 1955 thru 2009, would it have fit an exponential curve better instead of a quadratic curve.

One way to check that is to consider every possible StartYear to EndYear combination  to see which type of trend fits best. One way to visualize the results of that  is to use a spreadsheet grid where the Y axis is the start year and the X axis is the end year, running from 1929 to 2011   For example the grid cell in the lower right corner would consider the fit for 1929->2011. A grid cell at coordinates (X=1975,Y=1952) would consider the best fit for data running from 1952 thru 1975. Each grid cell     is then  given a different color corresponding to which trend matches best. In the images below a spreadsheet display was shrunk in size to make it easier to see.

The first image compares a quadratic trend to an exponential trend. The cells are black if an exponential trend matches best
, which isn't usually the case so it seems reasonable to view a quadratic trend as a better approximate growth curve:


Again, the lower left corner is 1929->1929, the upper right corner is 2011->2011.
The vertical and horizontal lines are an odd artifact that appeared when the sheet was viewed at a smaller size. They aren't the cells which are the tiny colored rectangles.

Other polynomial trends like cubic and quartic sometimes match the data even better and yield similar long term projections to the quadratic trend. A plot of the type above above comparing whether those  polynomials fit better than an exponential trend shows only a few black squares. The pattern is more mixed for a linear trend in comparison to an exponential trend, but even that often fits better than exponential. In the case below  the cells are red if a linear trend fits best and black if exponential fits best:


Real total GDP forecasts

Population growth tends to be a simple trend so similar patterns appear using forecasts for total real GDP (rather than per capita) Here are the projections of 2011 GDP that would have been made in the past compared to the reality:


 
Error rates for for what past predictions would have been:


. Below is a graph of projections provided by the CBO and SSA, compared to quadratic and linear trends. Once again the SSA expected medium GDP and the CBO GDP are   higher than the quadratic trend   and the linear GDP trend is below all of them.



A spreadsheet grid comparing the fit for every combination of start and end years shows a similar  result to the real per capita GDP example. It  indicates a quadratic fit is closest and seems a reasonable choice for an "expected" GDP forecast (though more even conservative would be safer).The quadratic trend also matche nominal GDP better, and similar graphs   show that. The impact of inflation leads exponential trends to match nominal GDP a little better than they matched real GDP,   but they are still too optimistic. To avoid belaboring the point and this  page getting even longer, the graphs won't be included here now (spreadsheets might be added with the data for those interested in actually seeing the details).

Data available on the web from a non-government source compared using a basic statistics program
show total real GDP from 1900 onward  also  fits quadratic growth  better than linear or exponential (even if you don't include the slower growth of the last few years).   A database of world economic statistics since 1950  compared in that program show the vast majority of countries also best match  a  quadratic growth trend.

Alternate forecast approaches

Sometimes forecasts are based directly on trends on an aggregate figure like GDP. Other GDP forecasts like the ones from SSA   derive their totals from the trends of various factors that combine to produce that aggregate value. For example   a projection might combine forecasts of growth in the number of workers and of their productivity  to derive a GDP forecast "bottom up".

 It is useful to do both top down and bottom up forecasts and compare the results to try to get a better idea of which trends might match reality. Even the different GDP trends considered on this page could be compared by combining   projections of population and inflation growth to see how well they match. A future page might take on that task.

Unfortunately   If you build up a forecast from the bottom up the uncertainty and errors in the data  (and  of the resulting trends) for every   factor compounds the potential uncertainty in the end result. Some of the trends in the SSA forecast have far worse fits than the GDP trends by themselves do. Although it seems unlikely, it  is possible in theory for  a  forecast based on individual factors that don't match  trends  well    to   be closer to reality than a forecast based on an aggregate trend, but there isn't reason  to  be confident it would be  in this case.
It is   important to "sanity check" the trends for each bit of data that is combined  into an aggregate trend. This site examined some of the SSA's data trends and found them questionable and will address that on a new page soon. All the SSA trends need to be skeptically analyzed by outsiders to see if values are appropriately selected rather than optimistically selected.

Some organizations  use a "model" which attempts to look at the interaction of different aspects our economy rather than a trend to forecast future   GDP. Unfortunately if you don't know enough to model the details of a system well enough then a flawed partial model can be "too wrong to be useful" or at least not provide a reliable indicator of the future even if it gives the appearance of being thorough. It is important to do a "sanity check" on such forecasts to  see if they make sense, and a sanity check on any trends that are used as inputs to the model. The CBO's long term forecast for future real GDP seems to match an exponential curve a bit better than linear or quadratic.. unlike historic real GDP. It isn't too much different than the SSA's intermediate forecast which as this page indicates appears  too  optimistic. If CBO will give its model to the public  that might be examined in the future.  (the model doesn't appear to be on its site at first glance, but a better search will be tried to be sure  and if it isn't there perhaps they will email it).

Conclusion

It appears best to assume GDP growth isn't exponential, and that government forecasts are   too optimistic and should likely consider limiting their growth rates to at best a quadratic trend. The issue of whether growth may slow further and match an   S-curve or some other pattern, or what specifically a more realistic estimate of GDP would be, is difficult to resolve and will be left for the future. Since an S-curve can't be ruled out, it would safer    for the government to make plans based on  the possibility growth will slow or stop.

Appendix (Technical notes & Spread the word)

Technical notes:

For convenience   analysis on this page was done using traditional least squares regression, just as the SSA does for its trend analysis. However if a more accurate actual  forecast were being created it  would be more appropriate  to consider other forms of regression with  different weighting factors like "least squares percentage regression". A standard least squares regression method determines how   well data fits a  curve   based on the absolute difference between the curve and the data. The problem is that what matters more often is the percentage error. An error of $1 trilllion   in a data point when the GDP was $1 trillion is of more of a concern  than an error of $2 trillion when the GDP is $16 trillion. Other alternative approaches might weight the fit of   recent data more heavily since it may be more representative of the current trend, and newer data may be more accurate than old data.

Those that sometimes use spreadsheets to analyze data rather than statistical packages  should be aware that Excel is fine for simple calculations but has a history of numerical bugs in things like its statistics routines and trend analysis routines (e.g. see here, here, here and here just for a start).  For example the spreadsheet that compares  fits to produce different colored cells  was imported into the most recent version of Mac Excel available in Jan. 2013 and it choked on the calculations, errors appeared and some cells would change values when the same formula was re-entered a second time. Open Office (used by default for this site) and Libre Office both handled it with no problems.

 It should be straightforward for any curious analysts to replicate   results on this page for outside confirmation if desired since they just use data from the BEA in simple spreadsheets. The spreadsheets used are internal scratch "work in progress" files, but some   might be cleaned up a little and posted in the future for the curious.

Spread the Word: