4. Using the Solow Growth Model

4.1. Convergence to the Balanced-Growth Path

4.1.1. The Example of Post-WWII West Germany

Economies do converge to and then remain on their balanced-growth paths. The West German economy after World War II is a case in point.

We can see such convergence in action in many places and times. For example, consider the post-World War II history of West Germany. The defeat of the Nazis left the German economy at the end of World War II in ruins. Output per worker was less than one-third of its prewar level. The economy’s capital stock had been wrecked and devastated by three years of American and British bombing and then by the ground campaigns of the last six months of the war. But in the years immediately after the war, the West German economy’s capital-output ratio rapidly grew and con verged back to its prewar value. Within 12 years the West German economy had closed half the gap back to its pre-World War II growth path. And within 30 years the West German economy had effectively closed the entire gap between where it had started at the end of World War II and its balanced-growth path.

In [9]:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

pwt91_df = pd.read_csv('https://delong.typepad.com/files/pwt91-data.csv')
In [20]:
is_Germany = pwt91_df['countrycode'] == 'DEU'
Germany_df = pwt91_df[is_Germany]
Germany_gdp_df = Germany_df[['year', 'rgdpna', 'emp']]
Germany_gdp_df['rgdpw'] = Germany_gdp_df.rgdpna/Germany_gdp_df.emp
Germany_pwg_ser = Germany_gdp_df[['year', 'rgdpw']]
Germany_pwg_ser
Germany_pwg_ser.set_index('year',  inplace=True)
Germany_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.

 

4.1.2. The Example of Post-WWII Japan

In [19]:
is_Japan = pwt91_df['countrycode'] == 'JPN'
Japan_df = pwt91_df[is_Japan]
Japan_gdp_df = Japan_df[['year', 'rgdpna', 'emp']]
Japan_gdp_df['rgdpw'] = Japan_gdp_df.rgdpna/Japan_gdp_df.emp
Japan_pwg_ser = Japan_gdp_df[['year', 'rgdpw']]
Japan_pwg_ser
Japan_pwg_ser.set_index('year',  inplace=True)
Japan_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [32]:
x = Japan_gdp_df['year']
y = np.log(Japan_gdp_df['rgdpw'])

fig, ax = plt.subplots()
ax.scatter(x, y)

plt.show()

 

4.1.3. The Post-WWII G-7

In [23]:
is_Britain = pwt91_df['countrycode'] == 'GBR'
Britain_df = pwt91_df[is_Britain]
Britain_gdp_df = Britain_df[['year', 'rgdpna', 'emp']]
Britain_gdp_df['rgdpw'] = Britain_gdp_df.rgdpna/Britain_gdp_df.emp
Britain_pwg_ser = Britain_gdp_df[['year', 'rgdpw']]
Britain_pwg_ser
Britain_pwg_ser.set_index('year',  inplace=True)
Britain_pwg_ser.plot(rgdpw)

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [26]:
is_America = pwt91_df['countrycode'] == 'USA'
America_df = pwt91_df[is_America]
America_gdp_df = America_df[['year', 'rgdpna', 'emp']]
America_gdp_df['rgdpw'] = America_gdp_df.rgdpna/America_gdp_df.emp
America_pwg_ser = America_gdp_df[['year', 'rgdpw']]
America_pwg_ser
America_pwg_ser.set_index('year',  inplace=True)
America_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [27]:
is_Italy = pwt91_df['countrycode'] == 'ITA'
Italy_df = pwt91_df[is_Italy]
Italy_gdp_df = Italy_df[['year', 'rgdpna', 'emp']]
Italy_gdp_df['rgdpw'] = Italy_gdp_df.rgdpna/Italy_gdp_df.emp
Italy_pwg_ser = Italy_gdp_df[['year', 'rgdpw']]
Italy_pwg_ser
Italy_pwg_ser.set_index('year',  inplace=True)
Italy_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [28]:
is_Canada = pwt91_df['countrycode'] == 'CAN'
Canada_df = pwt91_df[is_Canada]
Canada_gdp_df = Canada_df[['year', 'rgdpna', 'emp']]
Canada_gdp_df['rgdpw'] = Canada_gdp_df.rgdpna/Canada_gdp_df.emp
Canada_pwg_ser = Canada_gdp_df[['year', 'rgdpw']]
Canada_pwg_ser
Canada_pwg_ser.set_index('year',  inplace=True)
Canada_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [29]:
is_France = pwt91_df['countrycode'] == 'FRA'
France_df = pwt91_df[is_Canada]
France_gdp_df = France_df[['year', 'rgdpna', 'emp']]
France_gdp_df['rgdpw'] = France_gdp_df.rgdpna/France_gdp_df.emp
France_pwg_ser = France_gdp_df[['year', 'rgdpw']]
France_pwg_ser
France_pwg_ser.set_index('year',  inplace=True)
France_pwg_ser.plot()

plt.show()
/Users/delong/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:4: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
  after removing the cwd from sys.path.
In [25]:
g7_df = pd.DataFrame()
g7_df['Japan'] = Japan_pwg_ser['rgdpw']
g7_df['Germany'] = Germany_pwg_ser['rgdpw']
g7_df['America'] = America_pwg_ser['rgdpw']
g7_df['Italy'] = Italy_pwg_ser['rgdpw']
g7_df['Canada'] = Canada_pwg_ser['rgdpw']
g7_df['France'] = France_pwg_ser['rgdpw']
g7_df['Britain'] = Britain_pwg_ser['rgdpw']

print(g7_df)
g7_df.plot()

plt.show()
             Japan       Germany        America         Italy        Canada  \
year                                                                          
1950   6814.862160  14368.662170   35758.421599  14929.831065  29138.617616   
1951   7275.227832  15471.740415   37287.630803  16129.735821  29648.800257   
1952   8010.266282  16605.397754   38339.674207  16685.032813  31879.525448   
1953   8375.352271  17637.602344   39577.022348  17527.068053  33275.406141   
1954   8681.110567  18446.061483   40087.192964  17999.635785  33161.425048   
1955   9002.603161  20022.304250   41734.244667  18970.207206  35801.723305   
1956   9489.309012  21134.848462   41617.372635  19684.652562  37509.164494   
1957   9968.106351  22001.515932   42286.397917  20343.311706  37989.686821   
1958  10545.443924  22807.239865   42745.338335  21223.570297  39792.297812   
1959  11439.834590  24353.544086   44670.381851  22575.248564  40738.639869   
1960  12639.396833  26123.802399   45057.660626  24334.070189  42169.464932   
1961  13942.690373  27154.798342   46176.095794  26189.640709  43254.661405   
1962  14948.543177  28373.127547   48282.060954  27878.547941  45104.689773   
1963  16124.101711  29108.869512   49687.159891  29919.852514  46410.167025   
1964  17696.615806  30871.041372   51381.664408  30846.683925  47789.194156   
1965  18404.026390  32551.404088   53452.734532  32671.783025  48820.830999   
1966  19882.410399  33395.298167   55290.835186  35214.037205  50430.070856   
1967  21668.681770  34047.790017   55528.561462  37250.319553  50864.632566   
1968  23851.662664  35752.764642   57117.060971  39739.925656  53476.822295   
1969  26501.359076  37969.225845   57537.401640  42444.416967  54592.198807   
1970  28916.499943  39544.844576   57338.088375  47555.678294  56102.849828   
1971  30023.574150  40813.336417   59222.727089  48380.425587  57212.187469   
1972  32366.276376  42554.244056   60708.076634  50285.267815  58146.454272   
1973  34185.403434  44290.629442   62141.253082  53183.698438  59075.802719   
1974  33899.792331  45030.600071   60671.819582  55339.880471  58245.925781   
1975  35043.037016  45354.240133   61319.308791  54176.789986  58729.828773   
1976  36122.415455  47489.046814   62661.158171  57447.342123  60955.246734   
1977  37247.512265  48934.270949   63387.607261  58715.764212  61709.695238   
1978  38831.791243  50019.345380   64260.379711  60383.572218  62286.046324   
1979  40543.740157  51448.571901   64598.868443  63207.828737  61845.299035   
...            ...           ...            ...           ...           ...   
1988  55241.346102  58213.083088   72720.003088  74273.048687  66316.076424   
1989  57095.111728  60120.487664   73917.213296  76260.342432  66526.733552   
1990  58899.202353  63727.091815   74560.337755  76543.418460  66355.898453   
1991  59731.431246  68543.874260   75379.253415  76312.199872  66092.091386   
1992  59585.388676  70898.793112   77879.858880  77515.211059  67330.082801   
1993  59067.418905  71192.229622   79157.387720  78971.733181  68561.874921   
1994  59588.274922  72800.332535   80661.009892  82001.204623  70259.466324   
1995  61016.766886  73688.372278   81677.563067  84534.101553  71014.146143   
1996  62844.853542  74314.939914   83527.375019  85166.053538  71458.988235   
1997  63094.351392  75775.522029   85277.546881  86569.658416  73117.077666   
1998  63162.277695  76362.212656   87702.070343  87164.860697  74240.554889   
1999  63874.642042  76700.677317   90374.191907  87539.840511  75996.246109   
2000  66155.943202  77318.533812   92843.586655  88914.772938  78145.460113   
2001  66727.670863  78928.927500   93773.213193  88469.540606  78898.369023   
2002  67643.014845  79344.868459   95754.476185  86908.545909  79422.555847   
2003  68678.010217  79617.082408   98117.491173  85650.867090  79101.336348   
2004  69768.963170  80221.989323  100708.259737  86526.109326  80159.397949   
2005  70308.013079  80809.585531  102637.462224  86822.611605  81400.198919   
2006  70754.560979  83185.935054  103699.380968  86715.648259  82167.966772   
2007  71325.427309  84518.656216  104734.315057  87007.358277  81990.606302   
2008  70591.932280  84384.362422  104744.942914  86205.521302  81550.527722   
2009  67477.782250  79533.518680  105748.671031  83048.481540  80538.325961   
2010  70393.481448  82397.358576  108867.772207  85130.305369  81525.844216   
2011  70319.107550  84049.939372  109447.170068  85459.633756  82715.066218   
2012  71562.418444  83511.951872  110021.877338  83573.110731  83335.141202   
2013  72519.069480  83432.396398  110969.595645  84689.656354  84386.927420   
2014  72329.292480  84663.670048  111937.948016  85627.217054  86417.929867   
2015  72993.414161  85854.160910  113519.103370  85893.989827  86414.816636   
2016  72985.752912  87116.236884  113520.252712  85706.024659  86891.084244   
2017  73509.422806  88025.143786  114681.633303  86162.632945  88081.952584   

            France       Britain  
year                              
1950  29138.617616  22327.596015  
1951  29648.800257  22939.267516  
1952  31879.525448  23131.660426  
1953  33275.406141  24083.326545  
1954  33161.425048  24453.541076  
1955  35801.723305  24895.522451  
1956  37509.164494  24923.302030  
1957  37989.686821  25202.888034  
1958  39792.297812  25560.085981  
1959  40738.639869  26940.408904  
1960  42169.464932  27573.780323  
1961  43254.661405  27852.213048  
1962  45104.689773  28141.077646  
1963  46410.167025  29195.109613  
1964  47789.194156  30476.980160  
1965  48820.830999  30871.462545  
1966  50430.070856  31285.418401  
1967  50864.632566  32447.402146  
1968  53476.822295  34983.661120  
1969  54592.198807  34656.254105  
1970  56102.849828  35516.268331  
1971  57212.187469  37149.954116  
1972  58146.454272  38653.648644  
1973  59075.802719  40545.807232  
1974  58245.925781  39395.331527  
1975  58729.828773  38914.431311  
1976  60955.246734  40252.939578  
1977  61709.695238  41167.643146  
1978  62286.046324  42662.183715  
1979  61845.299035  43858.328005  
...            ...           ...  
1988  66316.076424  53927.375846  
1989  66526.733552  53912.411856  
1990  66355.898453  54067.272377  
1991  66092.091386  54948.705207  
1992  67330.082801  56498.688146  
1993  68561.874921  58451.320766  
1994  70259.466324  60225.043267  
1995  71014.146143  61805.961502  
1996  71458.988235  62753.136014  
1997  73117.077666  64253.439086  
1998  74240.554889  65701.669131  
1999  75996.246109  66875.667138  
2000  78145.460113  68397.486920  
2001  78898.369023  69794.525996  
2002  79422.555847  71007.250319  
2003  79101.336348  72712.731721  
2004  80159.397949  73658.814866  
2005  81400.198919  75148.447503  
2006  82167.966772  76212.150165  
2007  81990.606302  77340.650111  
2008  81550.527722  76204.484968  
2009  80538.325961  73943.940540  
2010  81525.844216  74915.316624  
2011  82715.066218  75754.723754  
2012  83335.141202  76028.938612  
2013  84386.927420  76735.211439  
2014  86417.929867  77308.280711  
2015  86414.816636  77951.204199  
2016  86891.084244  78329.751716  
2017  88081.952584  78954.386474  

[68 rows x 7 columns]

 

4.2. Analyzing Jumps in Parameter Values

What if one or more of the parameters in the Solow growth model were to suddenly and substantially shift? What if the labor-force growth rate were to rise, or the rate of technological progress to fall?

One principal use of the Solow growth model is to analyze questions like these: how changes in the economic environment and in economic policy will affect an economy’s long-run levels and growth path of output per worker Y/L.

Let’s consider, as examples, several such shifts: an increase in the growth rate of the labor force n, a change in the economy’s saving-investment rate s, and a change in the growth rate of labor efficiency g. All of these will have effects on the balanced- growth path level of output per worker. But only one—the change in the growth rate of labor efficiency—will permanently affect the growth rate of the economy.

We will assume that the economy starts on its balanced growth path—the old balanced growth path, the pre-shift balanced growth path. Then we will have one (or more) of the parameters—the savings-investment rate s, the labor force growth rate n, the labor efficiency growth rate g—jump discontinuously, and then remain at its new level indefinitely. The jump will shift the balanced growth path. But the level of output per worker will not immediately jump. Instead, the economy's variables will then, starting from their old balanced growth path values, begin to converge to the new balanced growth path—and converge in the standard way.

Remind yourselves of the key equations for understanding the model:

The level of output per worker is:

(4.1) $ \frac{Y}{L} = \left( \frac{K}{Y} \right)^{α/(1−α)}E $

The balanced-growth path level of output per worker is:

(4.2) $ \left( \frac{Y}{L} \right)^* = \left( \frac{s}{n+g+δ} \right)^{α/(1−α)}E $

The speed of convergence of the capital-output ratio to its balanced-growth path value is:

(4.3) $ \frac{d(K/Y)}{dt} = −(1−α)(n+g+δ) \left[ \frac{K}{Y} − \frac{s}{(n+g+δ)} \right] $

 

4.2.1. A Shift in the Labor-Force Growth Rate

Real-world economies exhibit profound shifts in labor-force growth. The average woman in India today has only half the number of children that the average woman in India had only half a century ago. The U.S. labor force in the early eighteenth century grew at nearly 3 percent per year, doubling every 24 years. Today the U.S. labor force grows at 1 percent per year. Changes in the level of prosperity, changes in the freedom of migration, changes in the status of women that open up new categories of jobs to them (Supreme Court Justice Sandra Day O’Connor could not get a private-sector legal job in San Francisco when she graduated from Stanford Law School even with her amazingly high class rank), changes in the average age of marriage or the availability of birth control that change fertility—all of these have powerful effects on economies’ rates of labor-force growth.

What effects do such changes have on output per worker Y/L—on our mea sure of material prosperity? The faster the growth rate of the labor force n, the lower will be the economy’s balanced-growth capital-output ratio s/(n + g - δ). Why? Because each new worker who joins the labor force must be equipped with enough capital to be productive and to, on average, match the productivity of his or her peers. The faster the rate of growth of the labor force, the larger the share of current investment that must go to equip new members of the labor force with the capital they need to be productive. Thus the lower will be the amount of invest ment that can be devoted to building up the average ratio of capital to output.

A sudden and permanent increase in the rate of growth of the labor force will lower the level of output per worker on the balanced-growth path. How large will the long-run change in the level of output be, relative to what would have happened had labor-force growth not increased? It is straightforward to calculate if we know the other parameter values, as is shown in

 

4.2.1.1. An Example: An Increase in the Labor Force Growth Rate: Consider an economy in which the parameter α is 1/2, the efficiency of labor growth rate g is 1.5 percent per year, the depreciation rate δ is 3.5 percent per year, and the saving rate s is 21 percent. Suppose that the labor-force growth rate suddenly and permanently increases from 1 to 2 percent per year.

Before the increase in the labor-force growth rate, in the initial steady-state, the balanced-growth equilibrium capital-output ratio was:

(4.4) $ \left( \frac{K_{in}}{Y_{in}} \right)^* = \frac{s_{in}}{(n_{in}+g_{in}+δ_{in})} = \frac{0.21}{(0.01 + 0.015 + 0.035)} = \frac{0.21}{0.06} = 3.5 $

(with subscripts "in" for "initial).

After the increase in the labor-force growth rate, in the alternative steady state, the new balanced-growth equilibrium capital-output ratio will be:

(4.5) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \frac{s_{alt}}{(n_{alt}+g_{alt}+δ_{alt})} = \frac{0.21}{(0.02 + 0.015 + 0.035)} = \frac{0.21}{0.07} = 3 $

(with subscripts "alt" for "alternative").

Before the increase in labor-force growth, the level of output per worker along the balanced-growth path was equal to:

(4.6) $ \left( \frac{Y_{t, in}}{L_{t, in}} \right)^* = \left( \frac{s_{in}}{(n_{in}+g_{in}+δ_{in})} \right)^{α/(1−α)} E_{t, in} = 3.5 E_{t, in} $

After the increase in labor-force growth, the level of output per worker along the balanced-growth path will be equal to:

(4.7) $ \left( \frac{Y_{t, alt}}{L_{t, alt}} \right)^* = \left( \frac{s_{alt}}{(n_{alt}+g_{alt}+δ_{alt})} \right)^{α/(1−α)} E_{t, alt} = 3 E_{t, alt} $

This fall in the balanced-growth path level of output per worker means that in the long run—after the economy has converged to its new balanced-growth path—one-seventh of its per worker economic prosperity has been lost because of the increase in the rate of labor-force growth.

In the short run of a year or two, however, such an increase in the labor-force growth rate has little effect on output per worker. In the months and years after labor-force growth increases, the increased rate of labor-force growth has had no time to affect the economy’s capital-output ratio. But over decades and generations, the capital-output ratio will fall as it converges to its new balanced-growth equilibrium level.

A sudden and permanent change in the rate of growth of the labor force will immediately and substantially change the level of output per worker along the economy’s balanced-growth path: It will shift the balanced-growth path for output per worker up (if labor-force growth falls) or down (if labor-force growth rises). But there is no corresponding immediate jump in the actual level of output per worker in the economy. Output per worker doesn’t immediately jump—it is just that the shift in the balanced-growth path means that the economy is no longer in its Solow growth model long-run equilibrium.

 

4.2.1.2. Empirics: The Labor-Force Growth Rate Matters: The average country with a labor-force growth rate of less than 1 percent per year has an output-per-worker level that is nearly 60 percent of the U.S. level. The average country with a labor-force growth rate of more than 3 percent per year has an output-per-worker level that is only 20 percent of the U.S. level.

To some degree poor countries have fast labor-force growth rates because they are poor: Causation runs both ways. Nevertheless, high labor-force growth rates are a powerful cause of low capital intensity and relative poverty in the world today.

 

Figure 4.2.1: The Labor Force Growth Rate Matters: Output per Worker and Labor Force Growth

Labor-force-growth-matters

 

How important is all this in the real world? Does a high rate of labor-force growth play a role in making countries relatively poor not just in economists’ models but in reality? It turns out that it is important. Of the 22 countries in the world in 2000 with output-per-worker levels at least half of the U.S. level, 18 had labor-force growth rates of less than 2 percent per year, and 12 had labor-force growth rates of less than 1 percent per year. The additional investment requirements imposed by rapid labor-force growth are a powerful reducer of capital intensity and a powerful obstacle to rapid economic growth.

It takes time, decades and generations, for the economy to converge to its new balanced-growth path equilibrium, and thus for the shift in labor-force growth to affect average prosperity and living standards. But the time needed is reason for governments that value their countries’ long-run prosperity to take steps now (or even sooner) to start assisting the demographic transition to low levels of population growth. Female education, social changes that provide women with more opportunities than being a housewife, inexpensive birth control—all these pay large long-run dividends as far as national prosperity levels are concerned.

U.S. President John F Kennedy used to tell a story of a retired French general, Marshal Lyautey, “who once asked his gardener to plant a tree. The gardener objected that the tree was slow-growing and would not reach maturity for a hun dred years. The Marshal replied, ‘In that case, there is no time to lose, plant it this afternoon.’”

 

4.2.2. The Algebra of a Higher Labor Force Growth Rate

But rather than calculating example by example, set of parameter values by set of parameter values, we can gain some insight by resorting to algebra, and consider in generality the effect on capital-output ratios and output per worker levels of an increase Δn in the labor force growth rate, following an old math convention of using "Δ" to stand for a sudden and discrete change.

Assume the economy has its Solow growth parameters, and its initial balanced-growth path capital-output ratio

(4.8) $ \left( \frac{K_{in}}{Y_{in}} \right)^* = \frac{s_{in}}{(n_{in}+g_{in}+δ_{in})} $

with "in" standing for "initial".

And now let us consider an alternative scenario, with "alt" standing for "alternative", in which things had been different for a long time:

(4.9) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \frac{s_{alt}}{(n_{alt}+g_{alt}+δ_{alt})} $

For the g and δ parameters, their initial values are their alternative values. And for the labor force growth rate:

(4.10) $ n_{alt} = n_{in} + Δn $

So we can then rewrite:

(4.11) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \frac{s_{in}}{(n_{in}+g_{in}+δ_{in})} \frac{(n_{in}+g_{in}+δ_{in})}{(n_{in} + Δn +g_{in}+δ_{in})} = \frac{s_{in}}{(n_{in}+g_{in}+δ_{in})} \left[\frac{1}{1+\frac{Δn}{(n_{in}+g_{in}+δ_{in})}} \right] $

The first term on the right hand side is just the initial capital-output ratio, and we know that 1/(1+x) is approximately 1−x for small values of x, so we can make an approximation:

(4.12) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \left( \frac{K_{in}}{Y_{in}} \right)^* \left[ 1 - \frac{Δn}{(n_{in}+g_{in}+δ_{in})} \right] $

Take the proportional change in the denominator (n+g+δ) of the expression for the balanced-growth capital-output ratio. Multiply that proportional change by the initial balanced-growth capital-output ratio. That is the differential we are looking for.

And by amplifying or damping that change by raising to the α/(1−α) power, we get the differential for output per worker.

 

4.2.3. A Shift in the Growth Rate of the Efficiency of Labor

4.2.3.1. Efficiency of Labor the Master Key to Long Run Growth: By far the most important impact on an economy’s balanced-growth path values of output per worker, however, is from shifts in the growth rate of the efhciency of labor g. We already know that growth in the efhciency of labor is absolutely essential for sustained growth in output per worker and that changes in g are the only things that cause permanent changes in growth rates that cumulate indehnitely.

Recall yet one more time the capital-output ratio form of the production function:

(4.13) $ \frac{Y}{L} = \left( \frac{K}{Y} \right)^{α/(1−α)} E $

Consider what this tells us. We know that a Solow growth model economy converges to a balanced-growth path. We know that the capital-output ratio K/Y is constant along the balanced-growth path. We know that the returns-to-investment parameter α is constant. And so the balanced-growth path level of output per worker Y/L grows only if, and grows only as fast as, the efficiency of labor E grows.

 

4.2.3.2. Efficiency of Labor Growth and the Capital-Output Ratio: Yet when we took a look at the math of an economy on its balanced growth path:

(4.14) $ \left( \frac{Y}{L} \right)^* = \left( \frac{s}{n+g+δ} \right)^{α/(1−α)} E $

we also see that an increase in g raises the denominator of the first term on the right hand side—and so pushes the balanced-growth capital output ratio down. That implies that the balanced-growth path level of output per worker associated with any level of the efficiency of labor down as well.

It is indeed the case that—just as in the case of an increased labor force growth rate n—an increased efficiency-of-labor growth rate g reduces the economy’s balanced-growth capital-output ratio s/(n + g - δ). Why? Because, analogously with an increase in the labor force, increases in the efficiency of labor allow each worker to do the work of more, but they need the machines and buildings to do them. The faster the rate of growth of the efficiency of la or, the larger the share of current investment that must go to keep up with the rising efficiency of old members of the labor force and supply them with the capital they need to be productive. Thus the lower will be the amount of investment that can be devoted to building up or maintaining the average ratio of capital to output.

 

4.2.4. The Algebra of Shifting the Efficiency-of-Labor Growth Rate

The arithmetic and algebra are, for the beginning and the middle, the same as they were for an increase in the rate of labor force growth:

Assume the economy has its Solow growth parameters, and its initial balanced-growth path capital-output ratio:

(4.15) $ \left( \frac{K_{in}}{Y_{in}} \right)^* = \frac{s}{(n_+g_{in}+δ_)} $

(with "in" standing for "initial"). Also consider an alternative scenario, with "alt" standing for "alternative", in which things had been different for a long time, with a higher efficiency-of-labor growth rate g+Δg since some time t=0 now far in the past:

(4.16) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \frac{s}{(n+g+Δg+δ)} $

We can rewrite this as:

(4.17) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = $ $ \frac{s}{(n+g_{in}+δ)} \frac{(n+g_{in}+δ)}{(n +g_{in}+Δg+δ)} = $ $ \frac{s}{(n+g_{in}+δ)} \left[\frac{1}{1+\frac{Δg}{(n+g_{in}+δ)}} \right] $

Once again, the first term on the right hand side is just the initial capital-output ratio, and we know that 1/1+x is approximately 1−x for small values of x, so we can make an approximation:

(4.18) $ \left( \frac{K_{alt}}{Y_{alt}} \right)^* = \left( \frac{K_{in}}{Y_{in}} \right)^* \left[ 1 - \frac{Δg}{(n+g_{in}+δ)} \right] $

Take the proportional change in the denominator of the expression for the balanced-growth capital output ratio. Multiply that proportional change by the initial balanced-growth capital-output ratio. That is the differential in the balanced-growth capital-output ratio that we are looking for.

But how do we translate that into a differential for output per worker? In the case of an increase in the labor force growth rate, it was simply by amplifying or damping the change in the balanced-growth capital-output ratio by raising it to the power (α/(1−α)) in order to get the differential for output per worker. We could do that because the efficiency-of-labor at every time t Et was the same in both the initial and the alternative scenarios.

That is not the case here.

Here, the efficiency of labor was the same in the initial and alternative scenarios back at time 0, now long ago. Since then E has been growing at its rate g in the initial scenario, and at its rate g+Δg in the alternative scenario, and so the time subscripts will be important. Thus for the alternative scenario:

(4.19) $ \left( \frac{Y_{t, alt}}{L_{t, alt}} \right)^* $ $ \left( \frac{s}{(n+g_{in} + \Delta g + \delta)} \right) ^{\alpha / (1− \alpha)} $ $ (1+(g_{in}+ \Delta g))^t E_0 $

while for the initial scenario:

(4.20) $ \left( \frac{Y_{t, ini}}{L_{t, ini}} \right)^* $ $ \left( \frac{s}{(n+g_{in} + \delta)} \right) ^{\alpha / (1− \alpha)} $ $ (1+g_{in})^t E_0 $

Now divide to get the ratio of output per worker under the alternative and initial scenarios:

(4.21) $ \left( \frac{Y_{t, alt}/L_{t, alt}}{Y_{t, ini}/L_{t, ini}} \right)^* $ $ = \left( \frac{n+g_{in}+\delta}{(n+g_{in}+\Delta g + \delta) \right)^{\alpha /(1− \alpha )} (1+ \Delta g)^t $

Thus we see that in the long run, as the second term on the right hand side compounds as t grows, balanced-growth path output per worker under the alternative becomes first larger and then immensely larger than output per worker under the initial scenario. Yes, the balanced-growth path capital-output ratio is lower. But the efficiency of labor at any time t is higher, and then vastly higher if Δgt has had a chance to mount up and thus (1+Δg)t has had a chance to compound.

Yes, a positive in the efficiency of labor growth g does reduce the economy’s balanced-growth path capital-output ratio. But these effects are overwhelmed by the more direct effect ofa larger g on output per worker. It is the economy with a high rate of efficiency of labor force growth g that becomes by far the richest over time. This is our most important conclusion. In the very longest run, the growth rate of the standard of living—of output per worker—can change if and only if the growth rate of labor efficiency changes. Other factors—a higher saving-investment rate, lower labor-force growth rate, or lower depreciation rate—can and down. But their effects are short and medium effects: They do not permanently change the growth rate of output per worker, because after the economy has converged to its balanced growth path the only determinant of the growth rate of output per worker is the growth rate of labor efficiency: both are equal to g.

Thus, if we are to increase the rate of growth of the standard of living permanently, we must pursue policies that increase the rate at which labor efficiency grows—policies that enhance technological and organizational progress, improve worker skills, and add to worker education.

 

4.2.4.1. An Example: Shifting the Growth Rate of the Efficiency of Labor: What are the effects of an increase in the rate of growth of the efficiency of labor? Let's work through an example:

Suppose we have, at some moment we will label time 0, t=0, an economy on its balanced growth path with a savings rate s of 20% per year, a labor force growth rate n or 1% per year, a depreciation rate δ of 3% per year, an efficiency-of-labor growth rate g of 1% per year, and a production function curvature parameter α of 1/2. Suppose that at that moment t=0 the labor force $ L_0 $ is 150 million, and the efficiency of labor $ E_0 $ is 35000.

It is straightforward to calculate the economy at that time 0. Because the economy is on its balanced growth path, its capital-output ratio K/Y is equal to the balanced-growth path capital-output ratio (K/Y)*:

       K0Y0=(KY)∗=sn+g+δ=0.20.01+0.01+0.03=0.20.05=4


And with an efficiency of labor value E0=70000, output per worker at time zero is:

       Y0L0= (K0Y0)(α1−α)(E0)= (4)(0.51−0.5)(70000)=   (4)1(35000)=140000

Since the economy is on its balanced growth path, the rate of growth of output per worker is equal to the rate of growth of efficiency per worker. Since the efficiency of labor is growing at 1% per year, we can calculate what output per worker would be at any future time t should the parameters describing the economy remain the same:

       (YtLt)ini=(Y0L0)egt=(140000)e(0.01)t

where the subscript "ini" tells us that this value belongs to an economy that retains its initial parameter values into the future. Thus 69 years into the future, at t=69:

       (Y69L69)ini=(140000)e(0.01)t=(140000)(1.9937)=279120

Now let us consider either a jump in g—a permanent, discontinuous change—or an alternative scenario in which output per worker is the same in year 0 but in which the efficiency of labor growth rate g is a higher rate. Suppose galt=gini+Δg, with the subscript "alt" reminding us that this parameter or variable belongs to the alternative scenario just as "ini" reminds us of the initial scenario or set of values. How do we forecast the growth of the economy in an alternative scenario—in this case, in an alternative scenario in which Δg=0.02

The first thing to do is to calculate the balanced growth path steady-state capital-output ratio in this alternative scenario. Thus we calculate:

       (KY)∗alt=   sn+g+Δg+δ= 0.20.01+0.03+0.03=

The steady-state balanced growth path capital-output ratio is much lower in the alternative scenario than it was in the initial scenario: 2.857 rather than 4. The capital-output ratio, of course, does not drop instantly to its new steady-state value. It takes time for the transition to occur.

While the transition is occurring, the efficiency of labor in the alternative scenario is growing at not 1% but 3% per year. We can thus calculate the alternative scenario balanced growth path value of output per worker as:

       (YtLt)∗alt=   [sn+g+Δg+δ(α1−α)]

e(g+Δg)t=

[(2.857)1]E0e(0.03)t

And in the 69th year this will be:

       (Y69L69)∗alt=

(2.857)1e(0.03)69=

792443

How good would this balanced growth path value be as an estimate of the actual behavior of the economy? We know that a Solow growth model economy closes a fraction (1−α)(n+g+δ) of the gap between its current position and its steady-state balanced growth path capital-output ratio each period. For our parameter values (1−α)(n+g+δ)=0.035 ( 1 − α ) ( n + g + δ

)

0.035 . That gives us about 20 years as the period needed to converge halfway to the balanced growth path. 69 years is thus about 3.5 such halvings of the gap—meaning that the economy will close 9/10 of the way. Thus assuming the economy is on its alternative scenario balanced growth path in year 69 is not a bad assumption.

But if we want to calculate the estimate exactly?

       (K69Y69)alt=

(sn+g+Δg+δ)+ [K0Y0−sn+g+Δg+δ] e−(1−α)(n+g+Δd+δ)(69)

       (K69Y69)alt=

(0.200.01+0.01+0.02+0.03)+ [4−0.200.01+0.01+0.02+0.03] e−(0.5)(0.01+0.01+0.02+0.03)(69)

(K69Y69)alt= 2.857+ [4−2.857] e−(0.035)(69)= 2.857+(1.143) e−(0.035)(69)= 2.857+(1.143)(0.089)=2.959

And with a year-69 capital-output ratio in the alternative scenario of 2.959, output per worker is then:

       (Y69L69)alt=

(K69Y69)α1−α [E69alt]= (2.959)

[E0e(g+Δg)(69)]= (2.959) [(2.959)e(0.03)(69)]=

       (Y69L69)alt=(2.959)(35000)(7.925)=820752

An actual alternative scenario output per worker value of 820752 in year 69; a balanced growth path alternative scenario value of 792443; and an initial parameter values scenario value of 279120; all from a year-0 value of 140000 for output per worker.

The takeaways are three:

For these parameter values, 69 years are definitely long enough for you to make the assumption that the economy has converged to its Solow model balanced growth path. One year no. Ten years no. Sixty-nine years, yes.

Shifts in the growth rate g of the efficiency of labor do, over time, deliver enormous differentials in output per worker across scenarios.

The higher efficiency of labor economy is, in a sense, a less capital intensive economy: only 2.959 years' worth of current production is committed to and tied up in the economy's capital stock in the alternative scenario, while 4 years' worth was tied up in the initial scenario. But the reduction in output per worker generated by a lower capital-output ratio is absolutely swamped by the faster growth of the efficiency of labor, and thus the much greater value of the efficiency of labor in the alternative scenario comes the 69th year.

Desire not an algebraic but a visual, graphical depiction of the difference between the initial and the alternative scenarios for the alternative scenario with a 2% per year faster growth rate of labor efficiency?

Continue below to Box 4.4.4

Box 4.4.4: A Faster Efficiency of Labor Growth Rate: Year by Year

2018 02 06 DeLong and Olney Macro 3rd Ch 4 4 Using the Solow Growth Model

The figure panel above shows the initial and alternative balanced growth paths for an economy on the initial growth path until year zero, and then experiencing a two percentage point per year permanent jump in its efficiency of labor growth rate. The blue curves are the initial scenario balanced growth path values; the orange curves are the alternative scenario values; the green curves are the track of the variables in the economy as it converges from the initial to the alternative balanced growth path.

The figure panel above has definite values for the parameters. There is a Δg=0.02 Δ

g

0.02 —a two percentage point permanent jump in the yearly proportional growth rate of the efficiency of labor g. There are no changes to the labor force growth rate n, the savings rate s, or any other values of parameters or initial conditions. They are:

savings rate s is 20% labor force growth rate n is 1% per year depreciation rate δ δ is 3% per year capital share production function parameter α α is 0.5 initial efficiency of labor E0 E 0 is 35000 the initial scenario rate of growth of the efficiency of labor g is 1% per year. But you do not have to restrict yourself to this one figure panel and these parameter values. The code cell below contains oru function for calculating and graphing the levels of Solow growth model variables in simulations—the simulated track of the economy, plus both the initial and the alternative scenario balanced growth paths, with initial and alternative scenarios distinguished by values of Δn Δ n , the change in the labor force growth rate, Δg Δ g , the change in the labor efficiency growth rate, and Δs Δ s , the change in the savings investment share.

After running the code cell below, create a new code cell and in it call the function:

sgm_bgp_100yr_run(L0, E0, n=0.01, g=0.01, s=0.20, alpha=0.5, delta=0.03, Delta_s=0, Delta_g=0, Delta_n=0, T = 100) The first two arguments are required, and are the initial time zero values for the labor force and the efficiency of labor. The other values are optional, and if omitted will be set to their default values in the function definition.

As you simulate different possibilities particular attention to how differences across simulations in the capital share parameter α α and the depreciation rate δ δ cause a Solow growth model economy to react differently to shifts in labor force growth, labor efficiency growth, and the savings-investment share caused by changes or differences in economic policy and the economic environment.

 

4.2.5. Shifts in the Saving Rate s

4.2.5.1 The Most Common Policy and Environment Shock: Shifts in labor force growth rates do happen: changes in immigration policy, the coming of cheap and easy contraception (or, earlier, widespread female literacy), or increased prosperity and expected prosperity that trigger "baby booms" can all have powerful and persistent effects on labor force growth down the pike. Shifts in the growth of labor efficiency growth happen as well: economic policy disasters and triumphs, countless forecasted "new economies" and "secular stagnations", and the huge economic shocks that were the first and second Industrial Revolutions—the latter inaugurating that global era of previously unimagined increasing prosperity we call modern economic growth—push an economy's labor efficiency growth rate g up or down and keep it there.

Nevertheless, the most frequent sources of shifts in the parameters of the Solow growth model are shifts in the economy’s saving-investment rate. The rise of politicians eager to promise goodies—whether new spending programs or tax cuts — to voters induces large government budget deficits, which can be a persistent drag on an economy’s saving rate and its rate of capital accumulation. Foreigners become alternately overoptimistic and overpessimistic about the value of investing in our country, and so either foreign saving adds to or foreign capital flight reduces our own saving- investment rate. Changes in households’ fears of future economic disaster, in households’ access to credit, or in any of numerous other factors change the share of household income that is saved and invested. Changes in government tax policy may push after-tax returns up enough to call forth additional savings, or down enough to make savings seem next to pointless. Plus rational or irrational changes in optimism or pessimism—what John Maynard Keynes labelled the "animal spirits" of individual entrepreneurs, individual financiers, or bureaucratic committees in firms or banks or funds all can and do push an economy's savings-investment rate up and down.

 

4.2.5.2 Analyzing a Shift in the Saving Rate s: What effects do changes in saving rates have on the balanced-growth path levels of Y/L?

The higher the share of national product devoted to saving and gross investment—the higher is s—the higher will be the economy’s balanced-growth capital-output ratio s/(n + g + δ). Why? Because more investment increases the amount of new capital that can be devoted to building up the average ratio of cap ital to output. Double the share of national product spent on gross investment, and you will find that you have doubled the economy’s capital intensity, or its average ratio of capital to output.

As before, the equilibrium will be that point at which the economy’s savings effort and its investment requirements are in balance so that the capital stock and output grow at the same rate, and so the capital-output ratio is constant. The savings effort of society is simply sY, the amount of total output devoted to saving and investment. The investment requirements are the amount of new capital needed to replace depreciated and worn-out machines and buildings, plus the amount needed to equip new workers who increase the labor force, plus the amount needed to keep the stock of tools and machines at the disposal of more efficient workers increasing at the same rate as the efficiency of their labor.

(4.4.12) sY=(n+g+δ)K

And so an increase in the savings rate s will, holding output Y constant, call forth a proportional increase in the capital stock at which savings effort and investment requirements are in balance: increase the saving-investment rate, and you double the balanced-growth path capital-output ratio:

(4.4.4) KY∗in=sn+g+δ

(4.4.13) KY∗alt=s+Δsn+g+δ

(4.4.14) KY∗alt−KY∗in=Δsn+g+δ

with, once again, balanced growth path output per worker amplified or damped by the dependence of output per worker on the capital-output ratio:

(4.4.2) (YL)∗=(sn+g+δ)(α1−α)(E)

Table 4.4.1: Effects of Increases In Parameters on the Solow Growth Model

Table

 

4.2.6. An Increase in the Saving-Investment Rate: An Example

To see how an increase in the economy’s saving rate s changes the balanced-growth path for output per worker, consider an economy in which the parameter α α is 2/3, the rate of labor-force growth n is 1 percent per year, the rate of labor efficiency growth g is 1.5 percent per year, and the depreciation rate δ δ is 3.5 percent per year.

Suppose that the saving rate s, which was 18 percent, suddenly and permanently jumped to 24 percent of output.

Before the increase in the saving rate, when s was 18 percent, the balanced-growth equilibrium capital-output ratio was:

       (KY)∗ini=

sn+g+δ= 0.180.01+0.015+0.035=3

After the increase in the saving rate, the new balanced-growth equilibrium capital- output ratio will be:

       (KY)∗alt=

s+Δsn+g+δ= 0.240.01+0.015+0.035=4

Before the increase in saving, the balanced-growth path for output per worker was:

       (YL)∗ini=

(KY)∗(α1−α)ini(E)= (3)(2/31−2/3) (E)= 32(E)=9E

After the increase in saving, the balanced-growth path for output per worker would be:

       (YL)∗alt=

(KY)∗(α1−α)alt(E)= (4)(2/31−2/3) (E)= 42(E)=16E

Divide the second equation by the first. We see that balanced-growth path output per worker after the jump in the saving rate is higher by a factor of 16/9, or fully 78 percent higher.

Just after the increase in saving has taken place, the economy is still on its old, balanced-growth path. But as decades and generations pass the economy converges to its new balanced-growth path, where output per worker is not 9 but 16 times the efficiency of labor. The jump in capital intensity makes an enormous differ ence for the economy’s relative prosperity.

Note that this example has been constructed to make the effects of capital inten sity on relative prosperity large: The high value for the diminishing-returns-to- investment parameter a means that differences in capital intensity have large and powerful effects on output-per-worker levels.

But even here, the shift in saving and investment does not permanently raise the economy’s growth rate. After the economy has settled onto its new balanced-growth path, the growth rate of output per worker returns to the same 1.5 percent per year that is g, the growth rate of the effciency of labor.

The figure immediately below charts the effects of such a permanent jump in the savings-investment share s with such a high production function α parameter:

2018 02 06 DeLong and Olney Macro 3rd Ch 4 4 Using the Solow Growth Model

And the code cell below allows you to run your own simulations:

In [12]:

sgm_bgp_100yr_run(L0=150000000, E0=35000, Delta_s=0.06, s=0.18, n=0.01, g=0.015, delta=0.035, alpha=2/3)

0.01 is the labor force growth rate 0.015 is the efficiency of labor growth rate 0.035 is the depreciation rate 0.24 is the savings rate 0.6666666666666666 is the decreasing-returns-to-scale parameter

Box 4.4.6: Estimating the Effects of Policy Changes: An Example

In late 2017 and early 2018 the Trump administration and the Republican congressional caucuses pushed through a combined tax cut and a relaxation of spending caps to the tune of increasing the federal government budget deficit by about 1.4% of GDP. These policy changes were intended to be permanent.

Not the consensus but the center-of-gravity analysis by informed opinion in the economics profession of the effects on long-run growth of such a permanent change in fiscal policy would have made the following points:

The U.S. economy at the start of 2018 was roughly at full employment, or at least the Federal Reserve believed that it was at full employment and was taking active steps to keep spending from rising faster than their estimate of the trend growth of the economy, so a long-run Solow growth model analysis would be appropriate.

The economy's savings-investment effort rate, s, has two parts: private and government saving: s=sp+sg

s

s p + s g .

The private savings rate sp s p is very hard to move by changes in economic policy. Policy changes that raise rates of return on capital—interest and profit rates—both make it more profitable to save and invest more but also make us richer in the future, and so diminish the need to save and invest more. These two roughly offset.

Therefore, when the economy is at full employment, changes in overall savings are driven by changes in the government contribution: Δs=Δsg Δ

s

Δ s g .

And an increase in the deficit is a reduction in the government savings rate.

The standard center-of-gravity analysis would thus start by assuming that the economy was on its balanced growth path, and investigate the consequences of a reduction in s by 1.4% points in order to get an estimate of the effect of this policy shift if it were to be a permanent change.

Set up the Solow growth model, with the Labor force growth rate n = 1.0% per year, the labor efficiency growth rate g = 1.5% per year, the depreciation rate δ δ = 3% per year, the production function diminishing returns to investment parameter α α = 1/3, and the initial efficiency of labor E0 E 0 = 65000. That produces an initial state of the economy's balanced growth path of:

       (KY)∗ini=4

( K Y ) i n i

4

       (YL)∗ini=(KY)∗ini(α1−α)E0=4(1/2)(65000)=130000

( Y L ) i n i

( K Y ) i n i ∗ ( α 1 − α ) E

0

4 ( 1 / 2 ) ( 65000

)

130000

Along the alternative balanced growth path, the same variables are:

       (KY)∗alt=(0.22−0.0140.01+0.015+0.03)(1/31−1/3)=3.745

( K Y ) a l t

( 0.22 − 0.014 0.01 + 0.015 + 0.03 ) ( 1 / 3 1 − 1 / 3

)

3.745

       (YL)∗alt=(KY)∗alt(α1−α)E0=(0.22−0.0140.01+0.015+0.03)(1/31−1/3)(65000)=3.745(1/2)(65000)=130000

( Y L ) a l t

( K Y ) a l t ∗ ( α 1 − α ) E

0

( 0.22 − 0.014 0.01 + 0.015 + 0.03 ) ( 1 / 3 1 − 1 / 3 ) ( 65000

)

3.745 ( 1 / 2 ) ( 65000

)

130000

That is, the alternative balanced growth path has an output per worker level 3.3 percent below the initial path The policy is expensive for the economy in the long run.

How fast does this growth retardation make itself felt? We know that the velocity of convergence vc v c in the Solow growth model is:

vc=−(1−α)(n+g+δ) v

c

− ( 1 − α ) ( n + g + δ )

In this case:

vc=−(1−α)(n+g+δ)=−(1−1/3)(0.01+0.015+0.035)=−0.0329 v

c

− ( 1 − α ) ( n + g + δ

)

− ( 1 − 1 / 3 ) ( 0.01 + 0.015 + 0.035

)

− 0.0329

The economy closes about 1/30 of the gap between its initial and its alternative balanced growth path every year. The first-year effect is thus about (-0.033)(0.33) = -0.001: a drop in the growth rate of 0.1% point, and a drop in the level of 0.1% point after one year. After 10 years, the economy will have closed about 28 percent of the 3.3 percentage point gap—a total effect on the level of real GDP ten years out of 0.9%: nine-tenths of a percentage point.

 

Box 4.4.7: Speed of Convergence and Estimating the Effects of Policy Changes: An Alternative

It is worth noting an alternative calculation of the likely effects of the Trump administration's economic policies, carried out by four Stanford economists and five others. The most important thing to know to understand and evaluate this calculation is that all nine of these economists are strong Republicans. They wrote http://delong.typepad.com/2017-11-26-nine-unprofessional-republican-economists.pdf, in a piece that was notionally a letter to U.S. Treasury Secretary Steven Mnuchin but that was in actuality primarily intended to be published in the Wall Street Journal to influence the debate, that Trump administration fiscal policy—the tax cut—would:

increase... the capital stock... raise the level of GDP in the long run by just over 4%. If achieved over a decade, the associated increase in the annual rate of GDP growth would be about 0.4% per year.... [In] the House and Senate bills... the increase in capital accumulation would be less, and the gain in the long-run level of GDP would be just over 3%, or 0.3% per year for a decade...

The four Stanford University economists are:

Michael J. Boskin, Tully M. Friedman Professor of Economics, Stanford University; Chairman of the Council of Economic Advisers under President George H.W. Bush

John Cogan, Leonard and Shirley Ely Senior Fellow, Hoover Institution, Stanford University; Deputy Director of the Office of Management and Budget under President Ronald Reagan

George P. Shultz, Thomas W. and Susan B. Ford Distinguished Fellow, Hoover Institution, Stanford University; Secretary of State under President Ronald Reagan; Secretary of the Treasury under President Richard Nixon

John. B. Taylor, Mary and Robert Raymond Professor of Economics, Stanford University; Undersecretary of the Treasury for International Affairs under President George W. Bush

The five others are:

Robert J. Barro, Paul M. Warburg Professor of Economics, Harvard University

Douglas Holtz-Eakin, President, American Action Forum, former director of the Congressional Budget Office

Glenn Hubbard, Dean and Russell L. Carson Professor of Finance and Economics (Graduate School of Business) and Professor of Economics (Arts and Sciences), Columbia University; Chairman of the Council of Economic Advisers under President George W. Bush

Lawrence B. Lindsey, President and Chief Executive Officer, The Lindsey Group; Director of the National Economic Council under President George W. Bush

Harvey S. Rosen, John L. Weinberg Professor of Economics and Business Policy, Princeton University; Chairman of the Council of Economic Advisers under President George W. Bush

Their conclusions—"the gain in the long-run level of GDP would be just over 3%, or 0.3% per year for a decade..."—look in their effects on levels of output per worker like the calculation in box 4.4.6, with one crucial difference: the sign is reversed. In 4.4.6, the first order effect of the policy changes was to reduce national savings and investment and thus make America a less capital intensive and poorer economy. And this calculation, the first order effect is to raise national savings and investment and us make America a more capital intensive and richer economy. Moreover, the effect on the growth rate is not only of the wrong sign, but three times the magnitude: instead of a slowdown in annual growth of 0.1% point, there is a speedup of

Why the difference?

Why does not the increased government deficit and thus government anti-saving reduce the national savings investment rate s? The authors do not say.

Where is the analysis stating that increased after tax rates of return on savings and investment have offsetting substitution and income effects, with the substitution effect raising saving and the income effect lowering it? That analysis, also, is absent.

What, then, is present? This:

Fundamental tax reform... [is] a set of tax changes that reduces tax distortions on productive activities (for example, business investment and work) and broadens the tax base to reduce tax differences among similarly situated businesses and individuals. Fundamental tax reform should also advance the objectives of fairness and simplification.... The proposals emerging from the House, Senate, and President Trump’s administration, fall squarely within this tradition.... There is some uncertainty about just how much additional investment is induced by reductions in the cost of capital, but... many economists believe that a 10% reduction in the cost of capital would lead to a 10% increase in the amount of investment. Simultaneously reducing the corporate tax rate to 20% and moving to immediate expensing of equipment and intangible investment would reduce the user cost by an average of 15%, which would increase the demand for capital by 15%.... Such an increase in the capital stock would raise the level of GDP... just over 3%, or 0.3% per year for a decade...

That's all she writes. And note: "many" economists—not "most economists", not "nearly all economists", not "the center of gravity of informed economic opinion".

And the claims about "the proposals emerging from the House, Senate, and President Trump’s administration" being "within this tradition" of "broaden[ing] the tax base to reduce tax differences among similarly situated businesses and individuals... advanc[ing] the objectives of fairness and simplification..." are simply false.

Even more alarming than the reversal-of-sign of the effect, is the estimate of the growth rate: a jump of + 0.3 percentage points per year. It comes from the nine economists' observation that:

increase... [would] raise the level of GDP in the long run by just over 4%. If achieved over a decade, the associated increase in the annual rate of GDP growth would be about 0.4% per year.... [In] the House and Senate bills... the increase in capital accumulation would be less, and the gain in the long-run level of GDP would be just over 3%, or 0.3% per year for a decade...

But the nine economists know just as well as you do that only 28 percent of the total gain accrues in the first decades, not all of it.

When challenged by former U.S. Treasury Secretary Lawrence Summers and former Council of Economic Advisers Chair Jason Furman https://www.washingtonpost.com/news/wonk/wp/2017/11/28/lawrence-summers-dear-colleagues-please-explain-your-letter-to-steven-mnuchin/?utm_term=.9d690352f4b3:

Since you are explicitly talking about 10-year growth rates in your letter, would it not be better to... show that the effect in the 10th year is less than one-third of the long-run effect, translating into an annual growth rate of less than 0.1 percentage point?...

The nine economists denied that they had made claims about the speed of adjustment to the post policy change blaanced growth path and so offered a prediction that real GDP growth would be boosted by not 0.1% (or -0.1%) but rather 0.3% points per year over the next decade https://www.washingtonpost.com/news/wonk/wp/2017/11/29/economists-respond-to-summers-furman-over-mnuchin-letter/?utm_term=.8d4d8991717a:

First point you raised: Our letter addresses the impact of corporate tax reform on GDP; we did not offer claims about the speed of adjustment to a long-run result...

We believe that Stanford (and Harvard, and Columbia, and Princeton, and the American Action Forum, and the Lindsey Group) have a serious problem here: As Berkeley medieval history professor Ernst Kantorowicz wrote http://www.lib.berkeley.edu/uchistory/archives_exhibits/loyaltyoath/symposium/kantorowicz.html back in the 1940s, shortly before being fired for refusing to take a loyalty oath demanded by the Regents of the University of California, academic freedom is a grave and serious thing:

Professions... entitled to wear a gown: the judge, the priest, the scholar. This garment stands for its bearer's maturity of mind, his independence of judgment, and his direct responsibility to his conscience and to his God.... They should be the very last to allow themselves to act under duress and yield to pressure. It is... shameful and undignified... an affront and a violation of both human sovereignty and professional dignity... to bully... under... economic coercion... compell[ing] either giv[ing] up... tenure or... his freedom of judgment, his human dignity and his responsible sovereignty as a scholar...

Those possessing academic freedom are given great latitude so that they can speak what they, after great and considered research and reflection, believe sincerely to be the truth. But this freedom to be responsible solely to one's conscience and God requires that one be responsible to one's conscience and God. But what if bearers of academic freedom fear not God nor their own consciences? What then?

One possibility is to inquire and point out that something has gone wrong, as Summers and Furman did, politely, with:

Since you are explicitly talking about 10-year growth rates in your letter, would it not be better to... show that the effect in the 10th year is less than one-third of the long-run effect, translating into an annual growth rate of less than 0.1 percentage point?...

inviting the response: "yes, it would have been better; we have made an error; we will correct it".

But that is not the reply Summers and Furman got.

A second possibility is to teach young people the basics of macroeconomics. I hope everybody who read the nine economists letter who had ever taken a macroeconomics course read that "the gain in the long-run level of GDP would be just over 3%, or 0.3% per year for a decade..." and immediately thought: "that is not how the effects of an increase in the economy's capital intensity from a higher savings-investement effort work—these authors, prestigious as their academic appointments may be, are not doing economic analysis but rather playing political Three-Card Monte". I hope everybody who reads this textbook remembers enough of it that they are able to do the work of reading with a jaundiced eye that is clearly needed here.

There was, I should say, a further oblique reply by one of the four Stanford economists, Michael Boskin https://www.project-syndicate.org/commentary/republican-tax-plan-growth-effects-by-michael-boskin-2017-12. In it he made points:

"Robert Barro..." published a deeper elaboration of the tax plan’s growth effects..." (which saiclaimedd that the long-run balanced growth path boost to the level of output per worker would be not 3 percent but 7 percent).

"The current tax bill could... have been better.... But such a bill would not pass Congress."

"The question is whether a viable final bill will be better than the status quo."

"Barro and I have clearly come to a different conclusion.... While I certainly respect Summers and Furman’s right to their views, I am not about to cede my professional judgment to others, in or out of government."

"There are legitimate differences of opinion on how much and how quickly the tax plan will affect investment decisions."

"Summers... and DeLong... have made the strongest case I know that equipment investment can have a large impact... much larger than in the conventional models."

"I believe that the current reform may well have deviated further from the ideal had we not offered our analysis and advice.... Many factors other than economists’ textbook policy proposals affect the final product."

"The actual tax provisions people and businesses will be required to use have yet to be written, and will be determined partly by technical interpretations and regulations in the coming months."

Point (6) seems to me to be a red herring, at least as far as the policy change's effects on the growth rate are concerned. We—DeLong and Summers—believe that α α is higher than the 1/3 assumed in the center-of-gravity of informed economic opinion analyses. A higher value of α α both stretches out the time it takes for the economy to converge and magnifies the ultimate differential, and these effects roughly cancel out, leaving the near term growth rate effect unchanged. And a higher α α magnifies both the boost from higher savings and the drag from those savings being diverted to finance larger government deficits.

The overwhelming impression I get from Boskin's piece is one of extraordinary cognitive dissonance. If I sincerely believed that a policy change was likely to boost America's productivity and wealth by 7 percent, I would not be apologizing for it. I would be crowing from the rooftops. I would not be agreeing that "the current tax bill could... have been better". I would not be saying that the bar is the very low "better than the status quo". I would not be defending my participation in the process on the grounds that the bill would have been worse if I had washed my hands of it. I would not be saying that we need to work hard now to improve it because "the actual tax provisions people and businesses will be required to use have yet to be written". I would not be saying that there are legitimate differences of opinion and that I respect the judgemtns of those who think differently.

I thus read Boskin's piece as, in large part, and perhaps not completely of his intention, a sotto voce argument that:

We nine economists said in public what we needed to say so that we could get into the room where the decisions were really being made.

We nine economists made the bill better than it would have been otherwise.

We nine economists will continue to make the implementation of the bill better.

4.4.4.3 Saving and Investment: Prices and Quantities

The same consequences as a low saving rate—a lower balanced-growth capital- output ratio — would follow from a country that makes the purchase of capital goods expensive. An abnormally high price of capital goods can translate a reasonably high saving effort into a remarkably low outcome in terms of actual gross additions to the real capital stock. The late economist Carlos Diaz-Alejandro placed the blame for much of Argentina’s poor growth performance since World War II on trade policies that restricted imports and artificially boosted the price of capital goods. Economist Charles Jones reached the same conclusion for India. And economists Peter Klenow and Chang-Tai Hsieh argued that the world structure of prices that makes capital goods relatively expensive in poor countries plays a major role in blocking development.

4.4.4.3 How Important This Is in the Real World

How important is all this in the real world? Does a high rate of saving and investment play a role in making countries relatively rich not just in economists’ models but in reality? It turns out that it is important indeed. Of the 22 countries in the world with output-per-worker levels at least half of the U.S. level, 19 have investment that is more than 20 percent of output. The high capital-output ratios generated by high investment efforts are a very powerful source of relative prosperity in the world today.

Figure 4.4.2: Savings-Investment Shares of Output and Relative Prosperity

The average country with an investment share of output of more than 25 percent has an output-per-worker level that is more than 70 percent of the U.S. level.

The average country with an investment share of output of less than 15 percent has an output-per-worker level that is less than 15 percent of the U.S. level.

This is not entirely due to a one-way relationship from a high investment effort to a high balanced-growth capital-output ratio: Countries are poor not just because they invest little; to some degree they invest little because they are poor. But much of the relationship is due to investment's effect on prosperity. High saving and investment rates are a very powerful cause of relative wealth in the world today.

Where is the United States on this graph? For these data it has an investment rate of 21 percent of GDP and an output- per-worker level equal (not surprisingly) to 100 percent of the U.S. level.