## 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 converged 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 :
# bring in libraries needed for python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# load data from the penn world table's estimates
# of growth among the g-7 economies
#
# (these estimates have problems; but all estimates
# have problems, and a lot of work has gone into
# making these)


In :
# pull america out of the loaded data table

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.set_index('year',  inplace=True)

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: 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/user_guide/indexing.html#returning-a-view-versus-a-copy


In :
# pull germany out of the loaded data table, and
# plot growth

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.set_index('year',  inplace=True)
np.log(Germany_pwg_ser).plot()
plt.show()

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:7: 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/user_guide/indexing.html#returning-a-view-versus-a-copy
import sys In :
# compare german real national income per worker
# with american

Germany_ratio_ser = Germany_pwg_ser/America_pwg_ser
Germany_ratio_ser.plot()
plt.show() The two figures above show, respectively, the natural logarithm of absolute real national income per worker for the German economy and real national income per worker relative to the U.S. value, both since 1950. By 1980 the German economy had converged: its period of rapid recovery growth was over, and national income per capita then grew at the same rate as that in the U.S., which had not suffered wartime destruction pushing it off and below its steady-state balanced-growth path. Then in 1990, at least according to this set of estimates, the absorption of the formerly communist East German state into the Bundesrepublik was an enormous benefit: the expanded division of labor and return of the market economy allowed productivity in the German east to more than double almost overnight. Thereafter the German economy has lost some ground relative to the U.S. as the U.S.'s leading information technology hardware and software sectors have been much stronger leading sectors than Germany's precision machinery and manufacturing sectors.

By comparison, the United States shows no analogous period of rapid growth catching up to a steady-state balanced-growth path. (There is, however, a marked boom in the 1960s, and then a return to early trends in the late 1970s and 1980s, followed by a return to previous normal growth in the 1990s and then a fall-off in growth after 2007.)

In :
# plot american growth since 1950

np.log(America_pwg_ser).plot()
plt.show() #### 4.1.2. The Example of Post-WWII Japan ¶

The same story holds in an even stronger form for the other defeated fascist power that surrendered unconditionally to the U.S. at the end of World War II.

In 1950, largely as a result of Curtis LeMay's B-29s, Japan is only half as productive as Germany, and only one-fifth as productive as the United States. Once again, it converges rapidly. After 1990 Japan no longer grows faster than and catches up to the United States. Indeed, like Germany it thereafter loses ground as its world class manufacturing sectors are also less powerful leading sectors than the United States's information technology hardware and software complexes.

In :
# pull japan out of the loaded data table, and
# plot growth

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.set_index('year',  inplace=True)
np.log(Japan_pwg_ser).plot()
plt.show()

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:7: 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/user_guide/indexing.html#returning-a-view-versus-a-copy
import sys In :
# compare japan to america

Japan_ratio_ser = Japan_pwg_ser/America_pwg_ser
Japan_ratio_ser.plot()
plt.show() #### 4.1.3. The Post-WWII G-7 ¶

The same story holds for the other members of the G-7 group of large advanced industrial economies as well.

In :
# pull britain out of the loaded data table

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_ratio_ser = Britain_pwg_ser/America_pwg_ser
Britain_pwg_ser.set_index('year',  inplace=True)

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: 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/user_guide/indexing.html#returning-a-view-versus-a-copy


In :
# pull italy out of the loaded data table

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_ratio_ser = Italy_pwg_ser/America_pwg_ser
Italy_pwg_ser.set_index('year',  inplace=True)

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: 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/user_guide/indexing.html#returning-a-view-versus-a-copy


In :
# pull canada out of the loaded data table


/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: 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/user_guide/indexing.html#returning-a-view-versus-a-copy In :
# pull italy out of the loaded data table

is_France = pwt91_df['countrycode'] == 'FRA'
France_df = pwt91_df[is_France]
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_ratio_ser = France_pwg_ser/America_pwg_ser
France_pwg_ser.set_index('year',  inplace=True)

/Users/delong1/opt/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:6: 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/user_guide/indexing.html#returning-a-view-versus-a-copy


In :
# plot g-7 natural logarithm levels of national
# income per worker

g7_df = pd.DataFrame()
g7_df['Japan'] = np.log(Japan_pwg_ser['rgdpw'])
g7_df['Germany'] = np.log(Germany_pwg_ser['rgdpw'])
g7_df['America'] = np.log(America_pwg_ser['rgdpw'])
g7_df['Italy'] = np.log(Italy_pwg_ser['rgdpw'])
g7_df['France'] = np.log(France_pwg_ser['rgdpw'])
g7_df['Britain'] = np.log(Britain_pwg_ser['rgdpw'])
g7_df.plot()
plt.show() In :
# calculate and plot g-7 levels of national income
# per worker as a proportion of american

g7_ratio_df = pd.DataFrame()
g7_ratio_df['Japan'] = Japan_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df['Germany'] = Germany_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df['America'] = America_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df['Italy'] = Italy_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df['France'] = France_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df['Britain'] = Britain_pwg_ser['rgdpw']/America_pwg_ser['rgdpw']
g7_ratio_df.plot()
plt.show() The idea—derived from the Solow model—that economies pushed off and below their steady-state balanced-growth paths by the destruction and chaos of war thereafter experience a period of supergrowth that ebbs as they approach their steady-state balanced-growth paths from below story holds for the other members of the G-7 group of large advanced industrial economies as well. In increasing order of the magnitude of their shortfall vis-a-vis the U.S. and the speed of recovery supergrowth, we have: France, Italy, Germany, and Japan. The three economies that escaped wartime chaos and destruction—the U.S., Britain, and Canada—do not exhibit supergrowth until catchup to their steady-state balanced-growth paths.

There is a lot more going on in the post-WWII history of the G-7 economies than just catchup to their steady-state balanced-growth paths after the destruction of World War II: Why do the other economies lose ground vis-a-vis the U.S. after 1990? Why does the U.S. exhibit a small speedup, slowdown, speedup, and then renewed slowdown again? What is it with Britain's steady-state balanced-growth path having so much lower productivity than the other Europeans? Why is Japan the most different from its G-7 partners? And what is it with Italy's attaining U.S. worker productivity levels in 1980, and then its post-2000 relative collapse? (The post-2000 collapse in Italian growth is real; the estimate that it was as productive as the U.S. from 1980-2000 is a data construction error.)

### 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) $\frac{Y}{L} = \left( \frac{K}{Y} \right)^{\theta}E$

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

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

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

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

where, you recall $\theta = \alpha/(1-\alpha)$ and $\alpha = \theta/(1+\theta)$

#### 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 the example below.

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 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 hundred 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) $\frac{Y}{L} = \left( \frac{K}{Y} \right)^{\theta} E$

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) $\left( \frac{Y}{L} \right)^* = \left( \frac{s}{n+g+δ} \right)^{\theta} E$

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 $\theta = (α/(1−α))$ in order to get the differential for output per worker. We could do that because the efficiency-of-labor at every time t $E_t$ 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) $\left( \frac{Y_{t, alt}}{L_{t, alt}} \right)^*$ $\left( \frac{s}{(n+g_{in} + \Delta g + \delta)} \right) ^{\theta)}(1+(g_{in}+ \Delta g))^t E_0$

while for the initial scenario:

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

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

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

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 and thus a $\theta = 1$. 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)*:

(4.22) $\frac{K_0}{Y_0} = \left( \frac{K}{Y} \right)^* = \frac{s}{n+g+\delta} = \frac{0.2}{0.01 + 0.01 + 0.03} = 4$

And with an efficiency of labor value $E_0=70000$, output per worker at time zero is:

(4.23) $\frac{Y_0}{L_0} = \left( \frac{K_0}{Y_0} \right)^{\theta} = 4^1 (35000) = 140000$

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:

(4.24) $\left(\frac{Y_t}{L_t}\right)_ini = (140000)e^{0.01t}$

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:

(4.25) $\left(\frac{Y_69}{L_69}\right)_ini = (140000)e^{(0.01)(69)} = (140000)(1.9937) = 279120$

Now let us consider 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 $g_{alt} = g_{ini} + Δg$, 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:

(4.26) $\left( \frac{K}{Y} \right)_{alt}^* = \frac{s}{n + g_{ini} + Δg + δ} = \frac{0.20}{0.01 + 0.01 + 0.02 + 0. 03} = \frac{0.20}{0.07} = 2.857$

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:

(4.27) $\left(\frac{Y_t}{L_t}\right)_{alt}^* = \left( \frac{K}{Y} \right)_{alt}^{*\theta} E_0 e^{(0.01+0.02)t}$

And in the 69th year this will be:

(4.28) $\left(\frac{Y_69}{L_69}\right)_{alt}^* = (2.857)(35000) e^{(0.03)(69)} = 792443$

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. 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? 820752.

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.

#### 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.

(4.29) $sY = (n+g+δ)K$

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:

(4.30) $\frac{K}{Y}_{ini}^* = \frac{s_{ini}}{n+g+δ}$

(4.31) $\frac{K}{Y}_{alt}^* = \frac{s_{ini}+Δs}{n+g+δ}$ KY∗alt=s+Δsn+g+δ

(4.32) $\frac{K}{Y}{alt}^* - \frac{K}{Y}{ini}^* = \frac{Δs}{n+g+δ} with, once again, balanced growth path output per worker amplified or damped by the dependence of output per worker on the capital-output ratio: (4.33)$ \frac{Y}{L}^* = \frac{K}{Y}^* E $4.2.5.3 Analyzing a Shift in the Saving-Investment Rate: An Example: To see how an increase in the economy’s saving rate s changes the balanced-growth path for output per worker, consider an economy in which the parameter$ \theta = 2 $(and$ α = 2/3 $}, the rate of labor-force growth n is 1 percent per year, the rate of labor efficiency growth g is 1.5 percent per year, and the depreciation rate δ is 3.5 percent per year. Suppose that the saving rate s, which had been 18 percent, suddenly and permanently jumped to 24 percent of output. Before the increase in the saving rate, when s was 18 percent, the balanced-growth equilibrium capital-output ratio was: (4.34)$ \frac{K}{Y}_{ini}^* = \frac{s_{ini}}{n+g+δ} = \frac{0.18}{0.06} = 3 $After the increase in the saving rate, the new balanced-growth equilibrium capital- output ratio will be: (4.35)$ \frac{K}{Y}_{alt}^* = \frac{s_{ini + {\Delta}s}{n+g+δ} = \frac{0.24}{0.06} = 4 $Before the increase in saving, the balanced-growth path for output per worker was: We see, with a value of$ \theta = 2 $, that balanced-growth path output per worker after the jump in the saving rate is higher by a factor of$ (4/3)^2 = 16/9 $, or fully 78 percent higher. Just after the increase in saving has taken place, the economy is still on its old, balanced-growth path. But as decades and generations pass the economy converges to its new balanced-growth path, where output per worker is not 9 but 16 times the efficiency of labor. The jump in capital intensity makes an enormous differ ence for the economy’s relative prosperity. Note that this example has been constructed to make the effects of capital intensity on relative prosperity large: The high value for$ \theta \$ 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.

4.2.5.4 Analyzing a Shift in the Incentives to Save and Invest: An Empirical 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.

## Lecture Notes: The Solow Growth Model ¶ 