%Paper: ewp-lab/0004001
%From: taylorleon@aol.com
%Date: Sun, 23 Apr 2000 07:14:10 -0500


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\begin{document}

\SetTitle{Do family choices slow down economic growth?}
\SetAuthor{Leon Taylor\\
Instructor\\
Division of Finance and Economics\\
Lewis College of Business\\
Marshall University\\
Huntington, West Virginia USA}
\TitlePage{}

\begin{abstract}
Econometric simulations provide no evidence that families in West Virginia
encouraged sons to drop out of high school in order to earn income as coal
miners, at the net expense of later income that would accrue to them with
additional education. Estimates of the typical family's subjective rate of
time preference for current income over future income earned by sons are
close to zero. [\textit{JEL\ D13 D99 J24}] 
\end{abstract}

\section{Introduction}

For a quarter of a century following the end of World War II, the heart of
the region famous for its poverty, Appalachia, prospered through coal
mining. From 1948 through 1970, personal income per capita was more than a
third higher in West Virginia than in the United States. The ratio of annual
personal income per capita in West Virginia, relative to that of the United
States, correlated with the annual share of mining in gross national product
over this period (r = .76).\footnote{%
Estimates of annual personal income per capita in West Virginia, relative to
that of the United States, were obtained from Series F346 and F297
respectively of the \textit{Historical statistics of the United States}.
Estimates of the share of the mining industry in gross national product for
the United States were obtained from Series F132 and F130 respectively.}

Mining paid the uneducated well; and, over time, it seemed to pay better and
better, compared to alternative industries. From the strike year of 1946
through 1957, throughout the United States, hourly compensation in
bituminous coal mining, the type of mining that dominated in West Virginia,
was more than 60 percent greater than average hourly earnings in another
low-skill industry that provided jobs in mining regions, the railroads.%
\footnote{%
Estimates of the ratio of mining compensation to rail wages, as hourly
averages, were obtained from Series D814 and D817 respectively of the 
\textit{Historical statistics}. Series D814 was constructed by H. Gregg
Lewis.} It was more than 70 percent greater than average hourly earnings to
production workers in all manufacturing.\footnote{%
Data on manufacturing wages came from Series D802 of the \textit{Historical
statistics.}}

Rapid and broad technological change in mining may have helped account for
the relative gain in coal wages over the period. From 1946 through 1957, the
mines mechanized the cutting, loading and washing of bituminous coal, mainly
underground. In processing more coal, mines tended to mechanize all
operations at once. Over this period, the correlation between the number of
tons loaded underground mechanically, and the number cleaned mechanically,
per miner, was .98; the correlation between the number of tons loaded
mechanically, and the number cut underground mechanically, per miner, was
.77.

The increase in mechanization was especially sharp after the economic
slowdown of 1949, the year of a large strike in mining, accounting for 10.7
million idle man-days.\footnote{%
The estimate of man-days idled by strike is from Series M112 of the \textit{%
Historical statistics.}} The annual unemployment rate of the mining industry
almost tripled that year, to 8.9 percent.\footnote{%
The data on the rate of unemployment, among experienced wage and salary
workers aged 16 or older, come from Series D106 of the \textit{Historical
statistics.}} After the strike, the second in four years, mines more than
doubled the volume of mining by machine, per miner, in all three operations.
The rise in mechanization was most dramatic in the most recent of the major
technologies, the cleaning of coal. As a long-run average, the number of
short tons of bituminous coal cleaned mechanically rose 6.5 percent per year
from 1916 through 1970; per miner, the number rose 9.8 percent per year.%
\footnote{%
A regression of the natural log of the annual number of short tons cleaned
mechanically, on the year, produces the coefficient .06542 (standard error =
.002383), adjusted R$^{2}=.93.$ Visual inspection of the residuals suggests
serial correlation. A similar regression, but with tons expressed per miner
produces the coefficient .09772 (standard error = .00206502), adjusted R$%
^{2}=.98$.} From 1950 through 1957, however, the number of short tons
cleaned mechanically per miner rose by 278 percent.

Such technology may have proven a mixed blessing for the miner.
Mechanization in cleaning coal, and probably in cutting coal at the face of
the underground mine, seem to have enabled the average miner to produce
more. Over the period from 1946 through 1957, the elasticity of the number
of short tons per miner was .51 with respect to the number of tons cleaned
mechanically per miner; and .58 with respect to the number of tons cut
mechanically per underground miner. The estimation tried to hold constant
the number of miners and the number of tons loaded mechanically per
underground miner (Appendix A). The adoption of each of the three
technologies, per miner, correlated highly with each of the others, so the
reader should interpret these statistical results with care. The basic point
is this:\ As the mines mechanized across operations, productivity rose,
especially after the slowdown of 1949. After dropping by more than half from
1947 through 1949, annual productivity more than doubled from 1950 through
1955, to nearly 2,100 short tons per miner.

Moreover, as mines quickly mechanized in the coal boom of the 1950s, their
compensation of miners grew more rapidly than did wages on the rail tracks
or in the factories. Real mine wages tracked closely with the spreading use
of machines to wash coal. Hourly compensation in bituminous mining, adjusted
by the consumer price index, correlated positively with the annual number of
tons mechanically washed per miner (r = .95).\footnote{%
The CPI used to compute real coal wages came from series E135 of the \textit{%
Historical statistics}. The measure of the number of coal tons cleaned
mechanically per miner was calculated from Series M104 (for tons cleaned)
and from Series M107 (for all mine workers on active days).} By 1957,
compensation in mining was more than 90 percent greater than manufacturing
wages.

But mechanization may also have destroyed jobs. From 1929 through 1970, the
elasticity of the number of miners on an active day, with respect to the
annual number of tons cleaned mechanically, was about -.19, controlling for
real compensation and coal demand. While that estimate is too imprecise to
merit much confidence ($t=-1.3$; Appendix A), one may note the reduction in
the mine workforce during the period of mechanization after 1949. The
average number of bituminous coal miners on an active day dropped by 45
percent from 1950 to 1957, from 416,000 to 229,000. This was much steeper
than the 8 percent drop in the demand for U.S. bituminous coal over this
period, to 412 million short tons in 1957.\footnote{%
I calculated annual demand as the annual production of bituminous coal mines
minus the stocks at the end of the year. Data sources were Series M93 (for
output) and M102 (for stocks) from the \textit{Historical statistics.}}

The phenomenon of higher wages and fewer jobs may have been visible to
communities throughout Appalachia, which provided most of the coal to the
United States in the 1950s. The loss of jobs may have been distributed
somewhat evenly across mining areas and thus may have been evident to many
of them:\ The number of mines open dropped only 10 percent over this period,
to 8,539 mines in 1957.\footnote{%
Data on the number of bituminous coal mines in operation is from Series M103
of the \textit{Historical statistics.}} A tradeoff between jobs and wages
may itself have been evident. From 1929 through 1970, the elasticity of the
number of miners on an active day, with respect to a measure of real
compensation per hour, was -.65 ($t=-3.8$; Appendix A). At the mean
compensation of \$3.45 in 1967 dollars, an increase of \$1 was associated
with a decrease of 60,000 fulltime jobs.

The local spectre of better but fewer jobs might have induced the parents of
a teenage boy in West Virginia to encourage him to secure a good job while
he could; to work in the mines rather than complete a high school education.
One might expect that the early entry of young men into the mining labor
force would lower wages and spur total output from the mines. The loss of
human capital, however, might have reduced the gain in long-run personal
income per capita.

The goals of this paper are to estimate the trade-off, if any, that West
Virginian families made between mining wages and education; and to estimate
the reduction in long-run income due to any subsequent loss of education.
These estimates may roughly suggest the subjective rate of time preference
applied in some family decisions.

The analysis focuses on family decisions made in the period from 1948
through 1970. Men who entered mines rather than schools by 1970 would likely
have felt much of the subsequent impact of that decision on their long-run
incomes by 1995. Data for the period exist.

\section{Model}

The family maximizes the discounted stream of income from the teen male by
picking the number of years that the teen would spend in school rather than
work in a mine. This number of school years is $s$. The analysis assumes
that the teen would either study fulltime or mine fulltime; it excludes
part-time jobs. The longer that the teen remains in school, then, the lower
his earned income over the school-age years. The expectation of this earned
income is $V_{e}$, which depends negatively on the choice of $s$. The
expectation also depends positively on the real marginal value of the
product of mine labor that prevails when the expectation is formed. That
value is $P(t_{0})$, where $t_{0}$ is the time that the expectation is
formed.

In addition to $V_{e}$, the teen is expected to earn income $Y_{e}$ over the
rest of his anticipated career of length $T-s-t_{0}$. His income may
increase with his level of education as well as with the stock of
technology. Thus $Y_{e}$ depends positively on $s$ and on time variable $t$.

The family seeks to solve

\[
\max_{\{s\}}\int_{s+t_{0}}^{T}e^{-rt}Y_{e}(s,t)\ dt+V_{e}(s,P) 
\]
where $r$ is the subjective rate of time preference and where $s=[0,T-t_{0}]$%
.

The first-order condition is

\[
\int_{s+t_{0}}^{T}e^{-rt}\frac{\partial Y_{e}}{\partial s}\
dt-e^{-r(s+t_{0})}Y_{e}(s,s)\leq -\frac{\partial V_{e}}{\partial s} 
\]
if $s<T-t_{0}$, as the analysis will assume. (The teen male is presumed not
to become a lifetime professional student.) Given fulltime study, $%
Y_{e}(s,s)=0$ and

\[
\int_{s+t_{0}}^{T}e^{-rt}\frac{\partial Y_{e}}{\partial s}\ dt\leq -\frac{%
\partial V_{e}}{\partial s}. 
\]

The family chooses $s$ to balance, at the margin, the long-run costs of
immediate work, in foregone income, against the immediate gain of income.

For simplicity, the analysis assumes that the family uses a constant
estimate of annual mining income $W_{e}(P)$ for the short-run expectation $%
V_{e}$:

\[
V_{e}=A(P)-W_{e}(P)s. 
\]

Annual mining income is estimated as

\[
W_{e}[P(t_{0})]=c_{0}+c_{1}P(t_{0})+c_{2}t_{0} 
\]
on annual data for 1946 through 1970. The estimate of $W_{e}$ provides $%
-\partial V/\partial s$ for specifying the first-order condition.

The family projects income to be received in year $t$ as

\[
Y_{e}(t)=d_{0}+d_{1}s+d_{2}t. 
\]
The estimate of $d_{1}$provides $\partial Y_{e}/\partial s$ for the
first-order condition.

The first-order condition yields a nonlinear estimate of the subjective rate
of time preference:

\begin{equation}
\frac{r}{e^{-r(s+t_{0})}-e^{-rT}}=\frac{d_{1}}{W_{e}}.  \label{simulate this}
\end{equation}

\section{Estimation}

\subsection{Specifying the expectation of the hourly wage}

\subsubsection{The marginal value of labor}

I assume that the family is well-informed about local mine conditions and
that, as a long-run average, it is usually correct in forecasting the wage
for the coming two years. I thus specify the family's expectation for the
hourly wage at time $t_{0}$, $w_{e}(t_{0}),$ by regressing a three-year
average of the real mining wage, for year $t_{0}$ and for the following two
years, on the current and the recent marginal value of product for mining
labor, expressed in hours: $p(t_{0})$ and $p(t_{0}-1)$. The expectation of
the hourly wage combines with an expectation of the number of hours worked
per year, $h_{e}(t_{0})$, to yield an expectation of annual income, $%
W_{e}(t_{0})$. I assume that the wage expectation is independent of the
hours expectation, so that $W_{e}(t_{0})=w_{e}(t_{0})h_{e}(t_{0})$.

I obtained estimates of $p(t_{0})$ by regressing the total real annual value
of bituminous coal (\textit{TVP}) on the annual average number of bituminous
miners on an active day (\textit{AllMen}) as well as on the square of that
number; on a measure of physical capital, the annual number of short tons of
bituminous coal cleaned mechanically (\textit{Clean)}; and on a time trend
to capture technological change (\textit{Year)}.\footnote{%
To estimate the total real annual value of coal, I multiplied the average
value of a short ton of coal, f.o.b. at the mine, by the level of output. I\
deflated the value with the CPI. Data sources, all from the \textit{%
Historical statistics of the United States}, were Series M96 (value); Series
M93 (output); and Series E135 (CPI).} The equation estimated was

\begin{equation}
TVP=a_{0}+a_{1}Year+a_{2}AllMen+a_{3}AllMen^{2}+a_{4}Clean.  \label{TVP}
\end{equation}

The marginal value of labor product was estimated as

\begin{equation}
p(t_{0})=a_{2}+2a_{3}AllMen(t_{0}).  \label{MVP}
\end{equation}

Descriptive statistics for the variables follow.

\begin{tabular}{|l|l|l|l|l|l|}
\hline
\textit{Statistics} & \textit{Year} & \textit{AllMen} & \textit{AllMen}$^{2}$
& \textit{Clean} & \textit{TVP} \\ \hline
\textbf{Mean} & 1943 & 390504 & 1.82626E+11 & 149923 & 2461220 \\ \hline
\textbf{Median} & 1943 & 419182 & 1.75714E+11 & 142187 & 2332768 \\ \hline
\textbf{Standard deviation} & 16.02 & 175188 & 1.35342E+11 & 117898 & 689794
\\ \hline
\textbf{Coefficient of variation} & 0.00824 & 0.449 & 0.741 & 0.786 & 0.28
\\ \hline
\textbf{Minimum} & 1916 & 124532 & 15508219024 & 13629 & 991981 \\ \hline
\textbf{Maximum} & 1970 & 704793 & 4.96733E+11 & 349402 & 4149230 \\ \hline
\textbf{Observations} & 55 & 55 & 55 & 55 & 55 \\ \hline
\end{tabular}

Regression results follow for estimating (\ref{TVP}) for 1916 through 1970.

\begin{tabular}{|l|l|}
\hline
\textit{Statistics} &  \\ \hline
Observations & 55 \\ \hline
R$^{2}$ & 0.724 \\ \hline
Adjusted R$^{2}$ & 0.702 \\ \hline
SEE & 376797 \\ \hline
\end{tabular}

The specified model was

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Variable} & \textit{Coefficient} & \textit{Standard error} & \textit{%
T-statistic} & \textit{Elasticity} \\ \hline
Intercept & 756270 & 332316 & 0.228 &  \\ \hline
\textit{Year} & -5995.40 & 17151.8 & -0.35 & -4.7 \\ \hline
\textit{Allmen} & 13.6716 & 1.89990 & 7.2 & 2.2 \\ \hline
\textit{Allmen}$^{2}$ & -6.04696E-06 & 2.0882E-06 & -2.9 & -0.45 \\ \hline
\textit{Clean} & 15.4287 & 1.99272 & 7.74 & 0.94 \\ \hline
\end{tabular}

Time may have had a relatively large impact on mining value (the elasticity
is -4.7), although the estimate of the coefficient is too imprecise to
permit one to be sure of the direction of the impact. It seems more probable
that mining value increased through mechanization (\textit{Clean) }than
through innovation (\textit{Year}).

Annual estimates for the marginal value of product, expressed in hourly
terms, were generated by applying regression results to (\ref{MVP}) and then
dividing the year-long marginal value of labor product by the estimated
number of hours worked. Since the early years of the Great Depression,
hourly compensation to miners has gained upon the marginal value of mining
product, which began dropping steadily in 1956. That was on the eve of a
six-year decline in coal demand as well as of a fall by a fourth in the real
average value of coal over the period until 1970.\footnote{%
Appendix A describes how I calculated the demand for coal. To obtain the
real average value of a short ton of bituminous coal, f.o.b. at the mine, I
used the Consumer Price Index to adjust Series M96 of the \textit{Historical
Statistics of the United States.}}

\subsubsection{Hourly wage expectations}

Let $w(t_{0})$ represent the actual hourly wage at time $t_{0}$. I specify
the wage expectation as

\[
w_{e}(t_{0})=c_{0}+c_{1}p(t_{0})+c_{2}p(t_{0}-1)+c_{3}t_{0} 
\]
where the parameters $c_{i}$ come from the regression equation

\[
\frac{w(t_{0})+w(t_{0}+1)+w(t_{0}+2)}{3}%
=c_{0}+c_{1}p(t_{0})+c_{2}p(t_{0}-1)+c_{3}t_{0}. 
\]

The series $w(t_{0})$, which includes wage supplements, extends a series by
H. Gregg Lewis to the time period from 1958 through 1970. The extension
divides the annual supplement by an estimate of annual hours. This estimate
of the hourly supplement is then added to hourly earnings, with an
adjustment to match the Lewis series. \ Appendix A\ has details.

The dataset for the regression extends from 1924 through 1968. This permits
the formation of the kind of long-held expectations that seem likely in
mining communities that change slowly. The variables, described below, are
each reasonably symmetric in distribution.

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Characteristic} & \textit{w} & \textit{p(t}$_{0})$ & \textit{p(t}$%
_{0}-1)$ & \textit{t}$_{0}$ \\ \hline
\textbf{Description} & real pay; 3-yr forward avg & MVP labor & MVP lag & 
Current year \\ \hline
\textbf{Mean} & 3.20233 & 5.53725 & 5.44669 & 1946 \\ \hline
\textbf{Median} & 2.76225 & 5.02764 & 4.91596 & 1946 \\ \hline
\textbf{Standard deviation} & 1.63950 & 1.60541 & 1.66137 & 13.13 \\ \hline
\textbf{Coefficient of variation} & 0.512 & 0.29 & 0.31 & 0.007 \\ \hline
\textbf{Minimum} & 1.26533 & 3.07386 & 2.46754 & 1924 \\ \hline
\textbf{Maximum} & 6.20828 & 8.86375 & 8.86375 & 1968 \\ \hline
\textbf{Observations} & 45 & 45 & 45 & 45 \\ \hline
\end{tabular}

Basic statistics for the regression, which follow, suggest that the model
explains most of the fluctuation in the wage average.

\begin{tabular}{|l|l|}
\hline
\textit{Characteristic} &  \\ \hline
\textbf{Dependent variable} & \textit{w} \\ \hline
\textbf{R}$^{2}$ & 0.959 \\ \hline
\textbf{Adjusted R}$^{2}$ & 0.956 \\ \hline
\textbf{SEE} & 0.343506 \\ \hline
\end{tabular}

Estimates of the model suggest that much of the explanatory power stems from
the intercept and the time variable which varies little. This may simplify
the family's forecast. As expected, the short-run pay average rises with the
current marginal value of mining and with the increase in that value over
the previous year. Neither effect, however, is precisely estimated or
powerful.

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Variable} & \textbf{Coefficient} & \textbf{Standard error} & \textbf{%
T-statistic} & \textbf{Elasticity} \\ \hline
Intercept & -226.061 & 12.6214 & -17.91 &  \\ \hline
\textit{p(t}$_{0})$ & 0.066001 & 0.098622 & 0.669 & 0.114 \\ \hline
\textit{p(t}$_{0}-1)$ & -0.017534 & 0.099373 & -0.176 & -0.03 \\ \hline
\textit{Year} & 0.117674 & 0.006605 & 17.815 & 71.51 \\ \hline
\end{tabular}

\subsubsection{ Expectation of workhours}

The family's expectation of the annual number of work hours hinges on its
forecast of the annual number of workdays, $d_{e}(t_{0})$, which is modeled
here. The actual number of workdays is $d(t_{0})$. For the family, two vivid
factors may shape the annual number of workdays: The current annual number
of strike days per miner, $strike(t_{0})$; and the passage of time $t_{0}$,
which reflects continuing trends in demand and supply.

The expectation of workdays is specified as

\[
d_{e}(t_{0})=g_{0}+g_{1}strike(t_{0})+g_{2}t_{0} 
\]
where the parameters $g_{i}$ are obtained from the regression

\[
\frac{d(t_{0})+d(t_{0}+1)+d(t_{0}+2)}{3}=g_{0}+g_{1}strike(t_{0})+g_{2}t_{0} 
\]
on data extending from 1923 through 1968.

The variables in the regression are described below. For \textit{strike},
the mean is nearly triple the median. A few large strikes may
disproportionately increase the number of workdays in the ensuing years.

\begin{tabular}{|l|l|l|l|}
\hline
\textit{Characteristic} & \textit{d} & \textit{t}$_{0}$ & \textit{strike} \\ 
\hline
\textbf{Description} & active days; 3-yr forward avg & year & strike days
per miner \\ \hline
\textbf{Mean} & 202.07 & 1945.5 & 7.41321 \\ \hline
\textbf{Median} & 198.83 & 1945.5 & 2.66833 \\ \hline
\textbf{Standard deviation} & 23.976 & 13.423 & 10.3102 \\ \hline
\textbf{Coefficient of variation} & 0.119 & 0.0069 & 1.391 \\ \hline
\textbf{Minimum} & 157.67 & 1923 & 0.30136 \\ \hline
\textbf{Maximum} & 267.67 & 1968 & 49.1885 \\ \hline
\textbf{Observations} & 46 & 46 & 46 \\ \hline
\end{tabular}

Basic statistics for the regression follow. The number of workdays is harder
to explain than real compensation, perhaps because it relates less
systematically to time. Compared to compensation, the number of workdays may
be more responsive to the current demand for coal; and it may be less
subject to contracts that last for several years.

\begin{tabular}{|l|l|}
\hline
\textit{Characteristics} &  \\ \hline
\textbf{Dependent variable} & \textit{d} \\ \hline
\textbf{R}$^{2}$ & .115 \\ \hline
\textbf{Adjusted R}$^{2}$ & .074 \\ \hline
\textbf{SEE} & 23.067 \\ \hline
\end{tabular}

The regression model follows. On average, the miner replaced each strike day
with a workday over the three-year period. The strike may thus be viewed as
a reallocation of worktime rather than as an effective signaling device for
raising the employer's estimate of the reservation wage. Another regression
suggests that, in this time period of 1924 through 1968, the number of
strike days per miner related negatively, though weakly, to the three-year
forward average of real hourly compensation (elasticity = -.02; \textit{t-}%
statistic = -1.69). That regression controls for the positive and large
impact of the year on compensation.

The model below suggests a possible reason for reallocating worktime. The
workyear itself tended to increase over time, as did real wages and capital
intensity, while the size of the work force diminished.

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Variable} & \textbf{Coefficient} & \textbf{Standard error} & \textbf{%
T-statistic} & \textbf{Elasticity} \\ \hline
Intercept & -942.4 & 504.4 & -1.868 &  \\ \hline
\textit{t}$_{0}$ & 0.5869 & 0.2591 & 2.265 & 5.65 \\ \hline
\textit{Strike} & 0.34564 & 0.3373 & 1.025 & 0.01 \\ \hline
\end{tabular}

\subsubsection{Expected annual mining income}

The two regression models described provide estimates for the family's
expectations of hourly compensation and annual number of active days over a
three-year horizon. Multiplying these estimates by the current number of
work hours per active day provides an estimate of the family's expectation
of annual mining income.\footnote{%
Appendix A explains how the number of work hours per active day were
computed. It also provides estimates for daily hours and for expected annual
income from mining.} This expectation of real income rises steadily by about
\$234 a year, from \$1,100 in 1924 to \$11,300 in 1968.

\subsubsection{Expected long-range income}

The family's expectation of annual income beyond the high school-age years
is $Y_{e}(t)$ for year $t$. The family does not presume that the teen will
necessarily stay in mining. Instead, as it ponders whether the teen should
remain in school, it considers the income that he may anticipate on average
if he were to complete $s$ years of education. In particular, it considers
the increase that it may expect in his income for year $t$ if he were to
complete another year of education.

Notation follows. The real income of a median worker in year $t$ is $Y(t)$.
The level of education attained by that worker is $s(t)$. The worker's
number of years of experience in year $t$ is $z(t)$.

The structure of the family's expectation of future income is obtained by
estimating the equation

\begin{equation}
Y(t)=d_{0}+d_{1}s(t)+d_{2}z(t)  \label{structure}
\end{equation}
for annual observations from 1939 through 1970. Its calculations of future
income are then generated with the equation

\[
Y_{e}(t)=d_{0}+d_{1}s+d_{2}(h+t-t_{0}) 
\]
where $s$ is the family's choice of education for the teen and $h$ is the
number of years that he would complete in the mines while of school age. The
parameters $d_{i}$ are obtained from (\ref{structure}).

Estimates of (\ref{structure}) follow. \textit{Income }is annual earnings
per fulltime employee, adjusted with the CPI\ (\$1967), drawing upon Series
D722 of the \textit{Historical statistics of the United States.} \textit{%
Experience} is the estimate of the number of years of job experience for the
mean male worker in the given year. \textit{School }is the estimate of the
median number of years of formal education completed by the male worker of
given age.\footnote{\textit{Experience} is calculated by estimating the mean
age of a male worker in the given year; subtracting an estimate of the
median number of years of schooling for a male worker of that age; and
subtracting the age at which the worker is presumed to have started school,
6. This provided estimates of the potential number of years that the worker
might have worked.
\par
To gain estimates of actual experience, I adjusted the estimates of
potential experience for joblessness. I multiplied the yearly unemployment
rate by an estimate of the share of the year spent in unemployment, for each
year that the worker might have worked. I then summed the products. Finally,
I multiplied the estimate of the number of years of potential experience by
1 minus the sum of unemployment products.
\par
\textit{School }data draw upon Series H619, H628, H635 and H642 of the 
\textit{Historical statistics. }The series data are decennial; estimates for
intervening years are linear interpolations. To estimate \textit{School }for
years before 1940, I regressed schooling estimates for 1940 through 1970 for
a male worker of given age on the rate of unemployment (coefficient -.023)
and on a time variable (coefficient .1171). The negative coefficient on the
unemployment rate may reflect that it was less common before 1970 for
unemployed workers to return to school than it was after 1970.
\par
For most years before 1940, I used regressions to estimate most data for the
model of long-run income. Appendix A has details.} In the descriptive
statistics, the small coefficient of variation for \textit{School }suggests
that, from 1939 to 1970, the average level of general human capital may have
risen slowly --- probably more slowly than the marginal level. Few
experienced workers returned to school.

\begin{tabular}{|l|l|l|l|}
\hline
\textit{Characteristic} & \textit{Income} & \textit{Experience} & \textit{%
School} \\ \hline
\textbf{Description} & annual real earnings(\$) & years of work & years of
schooling \\ \hline
\textbf{Mean} & 4781 & 10.86 & 11 \\ \hline
\textbf{Median} & 4679 & 7.883 & 11.2 \\ \hline
\textbf{Standard deviation} & 1047 & 4.737 & 1.14 \\ \hline
\textbf{Coefficient of variation} & 0.219 & 0.436 & 0.105 \\ \hline
\textbf{Minimum} & 3038 & 5.784 & 8.8 \\ \hline
\textbf{Maximum} & 6504 & 17.57 & 12.4 \\ \hline
\textbf{Observations} & 32 & 32 & 32 \\ \hline
\end{tabular}

Basic statistics for the regression suggest that expectations of income
based on even a simple model may have explanatory power.

\begin{tabular}{|l|l|}
\hline
\textit{Characteristics} &  \\ \hline
\textbf{Dependent variable} & \textit{Income} \\ \hline
\textbf{R}$^{2}$ & 0.946 \\ \hline
\textbf{Adjusted R}$^{2}$ & 0.942 \\ \hline
\textbf{SEE} & 251.21 \\ \hline
\end{tabular}

The model (below) suggests that the earnings generated by another year of
schooling may substitute for those generated by roughly another six years of
fulltime experience, net of the labor income itself. The relatively high
returns to education suggested here may owe in part to the fact that workers
accumulated education slowly.

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Variable} & \textbf{Coefficient} & \textbf{Standard error} & \textbf{%
T-statistic} & \textbf{Elasticity} \\ \hline
Intercept & -2325.7 & 689.09 & -3.37 &  \\ \hline
\textit{Experience} & 85.99 & 18.92 & 4.55 & 0.195 \\ \hline
\textit{School} & 564. & 78.3 & 7.2 & 1.29 \\ \hline
\end{tabular}

\subsection{Estimates of the subjective rate of time preference}

Microcomputer simulations of (\ref{simulate this}) estimated the value of $r$
that locally minimized the absolute value of the difference between the two
sides of the equation.\footnote{%
While, in principle, the minima are local, the extremely large and growing
differences that occurred in runs at larger values for \textit{r} suggest
that the minima were global as well.
\par
Readers may obtain a copy of the simulation program, written in Visual Basic.%
} I estimated the subjective rate of time preference for each year from 1940
through 1968. For the entire time period, I assumed that the family
anticipated that the teen would work until age 65.

Throughout the time period, the estimated value of $r$ was low, ranging
between $5*10^{-4}$ and $6*10^{-4}$, with estimation errors of one or two
orders of magnitude larger than the estimates of $r$. For practical
purposes, the annual rate of subjective time preference applied by mining
families to education decisions appeared to be zero.

\section{Conclusions}

Human capital accumulated slowly in the U.S. labor force over the mid-20th
century. The average level of education in a male worker, about 40 years
old, rose 41 percent from 1940 to 1970. The reasons do not seem to rest with
the family's impatience to consume, however. Simulations examined a typical
family that could have taken a teen out of the schools and put him to work
in the mines for current income. They suggest that, for such decisions, the
family essentially viewed income received in the future the same as current
income. Families did not appear to discount heavily the higher income that
the teenager would receive, in the future, from a more substantial education.

\section{Appendix A}

\subsection{Determinants of mine employment, 1929-1970}

I regressed the average number of miners on active days, \textit{AllMen}, on
a measure of annual coal demand, \textit{Demand, }which subtracted
end-of-year stocks from annual output; on a measure of mechanization, the
annual number of short tons cleaned mechanically, \textit{Clean; }and on a
measure of real hourly compensation,\textit{\ RealPay}. Data for estimating 
\textit{Demand} came from Series M93 (for output) and Series M102 (for
stocks) of the \textit{Historical statistics. }Data for \textit{Clean }came
from Series M104 of the \textit{Historical statistics}, with linearly
interpolated values for 1922 and 1924-26.

\textit{RealPay} extends a series constructed by H. Gregg Lewis for the
average hourly compensation of bituminous coal miners, which includes wage
supplements. For 1929 through 1957, the last year in Lewis' series, \textit{%
RealPay} adjusts Lewis' estimates with the Consumer Price Index. Data came
from Series D814 (Lewis series) and Series E135 (CPI) of the \textit{%
Historical statistics.}

For 1958 through 1970, I extended the Lewis series in the following manner.
First, I estimated the hourly wage supplement. To obtain this estimate, I
divided the average annual supplement to wages and salaries of fulltime
employees in mining (Series D896) by an estimate of annual hours. The
estimate of annual hours was the product of two numbers:\ the average weekly
hours in bituminous mining for production workers (Series D812); and an
estimate of the number of weeks worked in a year by a bituminous miner. For
the estimate of the number of weeks worked, I\ divided the average annual
earnings of fulltime employees in bituminous mining (Series D743) by the
product of two numbers: the average weekly hours in bituminous mining for
production workers; and the average hourly earnings for those workers
(Series D813).

Next, I added the estimate of the hourly wage supplement to the average
hourly earning for production workers in bituminous mining (Series D813).

This estimate of compensation to the miner was 22 percent less than Lewis'
estimate in 1957. To determine the adjustment to make to my estimates of
compensation for 1958 through 1970, I regressed, on the year, the percentage
differential between the two compensation estimates, for 1929 through 1957.
The model indicated that the differential was a positive linear function of
time ($R^{2}=.93;$ $t=12.5$ for the time coefficient of .01786). I used the
model to estimate the differential to apply to my compensation estimates for
1958 through 1970.

To bring those estimates in line with the Lewis estimate of 1957, I
subtracted -.117 from each of my estimates for 1958 through 1970. This
ensured that my estimate for 1958 equalled what the differential model would
have predicted for that year, taking as given the Lewis estimate for 1957.

Finally, I adjusted the estimates for 1958 through 1970 for inflation by
deflating them with the CPI.

Descriptive statistics follow:

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Statistic} & \textit{RealPay} & \textit{Demand} & \textit{Clean} & 
\textit{AllMen} \\ \hline
Mean & 3.45333 & 415287 & 189561 & 321245 \\ \hline
Median & 3.30961 & 409416 & 189790 & 388224 \\ \hline
Standard deviation & 1.65814 & 76346.6 & 107199 & 137689 \\ \hline
Coefficient of variation & 0.480 & 0.184 & 0.566 & 0.429 \\ \hline
Minimum & 1.21134 & 280044 & 30278 & 124532 \\ \hline
Maximum & 6.42451 & 578463 & 349402 & 502993 \\ \hline
Observations & 42 & 42 & 42 & 42 \\ \hline
\end{tabular}

In \textit{AllMen}, the 21.8 percent disparity between the mean and the
median, relative to the mean, may indicate the impact of strikes on
active-day employment.

The regression estimated the model

\[
AllMen=a_{0}+a_{1}\ \func{Re}alPay+a_{2}\ Demand+a_{3}\ Clean. 
\]

Specifications from the regression follow:

\begin{tabular}{|l|l|}
\hline
\textit{Statistics} &  \\ \hline
\textit{Observations} & 42 \\ \hline
\textit{R}$^{2}$ & 0.936 \\ \hline
\textit{Adjusted R}$^{2}$ & .931 \\ \hline
\textit{SEE} & 36082.0 \\ \hline
\end{tabular}

The estimated model follows:

\begin{tabular}{|l|l|l|l|l|}
\hline
\textit{Variable} & \textit{Coefficient} & \textit{Standard error} & \textit{%
T-statistic} & \textit{Elasticity} \\ \hline
Intercept & 452045 & 40579.2 & 11.1 &  \\ \hline
\textit{RealPay} & -60031.7 & 15824.9 & -3.79 & -0.645 \\ \hline
\textit{Demand} & 0.331489 & 0.0856168 & 3.87 & 0.429 \\ \hline
\textit{Clean} & -0.322612 & 0.248668 & -1.3 & -0.190 \\ \hline
\end{tabular}

\end{document}