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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ }

\title {\bf Applications of Interval Computations to Economic
            Input-Output Models}
\author {Max E. Jerrell}

\maketitle

\auffil{The author is with Northern Arizona University, Flagstaff, AZ,
USA.}

\begin{abstract}
Economic input-output models are empirical realizations of general
equilibrium economic models.  They are particularly useful
in determining how various industrial sectors of the economy
are interrelated and in predicting
how these sectors respond to changes in economic activity.
In the United States, state and regional models are frequently
used to help formulate local economic policy.
Wassily Leontief is generally credited
with the first useful implementation \cite{Leontief51}.
\end{abstract}

\section{Economic Input-Output Models}
Let $a_{ij}$ be the amount of the $i-$th good used to produce
one unit of the $j-$th good. Let $x_j$ be the amount of the
$j-$th good produced, and $b_j$ be the amount of final (consumer)
demand for the $j-$th good.  Demand for the $i-$th good
is $\sum a_{ij}x_j + b_i$, and in equilibrium (supply = demand)
  \[
      x_i = \sum_{i=1}^{n} a_{ij}x_j + b_i.
  \]
Equilibrium for all goods in matrix form is
  \[
      x = Ax + b,
  \]
so
  \[
      (I-A)x = Lx = b,
  \]
where $L$ is called the {\it Leontief matrix}.
If final demand is known, then the amount of the goods needed
to satisfy this demand can be found. In practice these models
are implemented using the dollar value of output
of industrial sectors rather than individual
goods - there are just too many goods.  The computer industry
is an example of an industrial sector.  Some of the output
from this industry will be used as inputs used to produce other
goods and some will be sold as final goods ($b$).

Rohn \cite{Rohn80} has derived the conditions under which
the input-output system will have economically feasible
solutions.  For our purposes, it is sufficient to note
that the Leontief matrix is most likely
an {\it M}-matrix.  Indeed, all of the production coefficients $a_{ij}$ must
be non-negative (a negative value would mean that a negative amount
of good $i$ is used to produce $j$). Hence, the off diagonal
elements of the Leontief matrix will be negative.

Further, we must have $Lu  > 0$ for some
positive vector $u$.  For our case, we can take 
$u=(1,1, \ldots, 1)'$.  Indeed, consider the product of the
first row of the Leontief matrix with $u$
(these arguments will hold for all rows).  This gives
 \begin{equation}
    1 - a_{11} - a_{12} - \cdots - a_{1n}.
 \end{equation}
The $a_{1j}$ terms represent the proportions of
good $1$ used as inputs in industrial processes.
A negative result for $(1)$ would mean that more of good $1$
is being used than
is being produced.  A zero result for $(1)$ might be possible.
This would mean that a good has no consumer demand but is used only
as an input in production processes.  Recall that the practical
implementation is in industrial rather than goods terms. In
practice this will mean that a zero result will not occur.
Even if it did, there is an easy fix: aggregate this industry
will one closely aligned that has some consumer demand.

Because $L$ is an {\it M}-matrix, the inverse and hull
solutions of the input-output model exist \cite{Neumaier90} and
 \[
    L^{-1}b = L^Hb =
    \mbox{$[\overline{L}^{-1}\down{b}, \underline{L}^{-1}\up{b}]$}
     \mbox{\qquad if $b \geq 0$}
 \]
The condition $b \geq 0$ is met because consumer demand must be
non-negative.


The inverse Leontief matrix offers some efficiencies.
It is useful to avoid solving $Lx=b$ for many different $b$.
The solution
requires that final demand, $b$, be forecast - a formidable
task in itself.
Much knowledge can be gained by considering
how economic activity will change if there is a change in
final demand. This does not require computing the level
of economic activity, only the change.  One method of examining
how a change in final demand will affect an economy is
to construct so-called {\it multipliers}.

As an example of an elementary multiplier consider the solution $x$ of
an equation 
\[
   Lx = (0,1, \ldots, 0)',
\]
which shows the effect on the economy if there is a unit
increase in final demand for good $2$.  But this solution is equal to 
 \[
   \sum_{i=1}^{n} l^{i2}
 \]
where $l^{ij}$ is an element of the Leontief inverse $L^{-1}$.
By considering the multiplier, we do not have to solve
the system $n$ times.

The simple output multiplier discussed above understates
the amount of activity that will occur for a unit change
in final demand \cite{Miernyk65}.  An initial increase
in final demand will increase output.
This increase in output will increase household income
which will further increase final demand.
The resulting multiplier is called a Type II multiplier.

The Type II multiplier is computed by forming a
new $A$ matrix

\[
A=
\left(
\begin{array}{lllll}
a_{11} & a_{12} & \cdots & a_{1n} & a_{1h} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
a_{h1} & a_{h2} & \cdots & a_{hn} & a_{hh}
\end{array}
\right)
\]
where the $h$ index is used to indicate households.
The elements $a_{ih}$ represents household purchases
of output from the $i-$th industry.  The elements
$a_{hi}$ represent payments by the $i-$th industry
to households (wages, rent, interest, profits).
The change in income generated by
the $i-$th industry is
\[
   \frac{l^{hi}}{l_{hi}}
\]
where $l_{hi}$ and $l^{hi}$ are elements of the
Leontief and Leontief inverse matrices, respectively.

\section{Why Interval Computations may be Useful}

The technical coefficients $l_{ij}$ of the Leontief matrix are not known
precisely.
The U.S. Department of Commerce conducts a census of U.S. businesses.
These data are used by the Bureau of Economic Analysis to construct
these coefficients for the U.S. economy.
Even if the data from the census is
not subject to error, it takes about 5 years to produce a new
table. The Department of Commerce uses these tables to estimate
Gross National Product (GNP).  The values of the technical
coefficients will change during this time period.

When the Department shifted from the
1972 census benchmark to the 1977 benchmark this raised the
1983 GNP estimate by $\$100$ billion. Errors of this magnitude can
lead policy makers to inappropriate action (e.g., creating inflation
by fighting a non-existent recession).

Things are even worse for regional analysts who rarely have the
resources to even conduct surveys of regional businesses.
Usually they are forced to infer the regional technical coefficients
by adjusting national tables and using other ad hoc procedures
\cite{Richardson72}.
I think it is reasonable to believe that the technical coefficients
are subject to some level of uncertainty.

An additional problem facing input-output analysts is that the
final demand, $b$, is also not known with precision but
must be forecast.  This creates additional uncertainty in the
Leontief system.  In practice, I believe the effects of this
uncertainty is largely ignored (although the
pioneer work \cite{Rohn80} that proposed the use of intervals for
Leontief models appeared as early as 1980).
Interval analysis
can offer an approach to dealing with these problems.
early as 1980, interval methods are still not used by economists).
Users will probably have to subjectively assign levels
of uncertainty using some ad hoc procedure.
One possibility is to let all the elements of the Leontief
matrix vary by some percentage.

\section{An Example}

Miernyk \cite{Miernyk65} uses a small, hypothetical input-output
model with six producing sectors and a household sector to illustrate
some of the mechanics of input-output analysis. We can use this
hypothetical model as a partial check on the interval results for
the Type II multiplier. Note that, in practice, the input-output
matrix produced by IMPLAN (a commercial package) for Coconino
County, Arizona contains $178$ producing sectors which would be far
to large to reproduce here.

The $A$ matrix presented below is that given by Miernyk.
Arbitrarily, I created three interval matrices symmetrical
about Miernyk's matrix and computed the interval Type II
multipliers.  The resulting interval multipliers are shown
below. The results were computed using C-XSC.
I expected the interval multiplier to
enclose Miernyk's point multipliers.  

The interval matrix with a small level of uncertainty
($A \pm 0.001A$) did {\bf not} enclose Miernyk's results.
This caused some concern.  Further investigation using
C-XSC and two other (point) matrix inversion programs
showed that Miernyk's Leontief inverse was wrong.
The following interval matrices do enclose the results 
that Miernyk should have got.  

\[ A =
\left( \begin{array}{lllllll}
   0.16 & 0.26 & 0.03 & 0.05 & 0.13 & 0.13 & 0.19 \\
   0.08 & 0.07 & 0.18 & 0.03 & 0.08 & 0.18 & 0.24 \\
   0.11 & 0.04 & 0.21 & 0.03 & 0.13 & 0.07 & 0.07 \\
   0.17 & 0.02 & 0.05 & 0.21 & 0.16 & 0.09 & 0.06 \\
   0.06 & 0.00 & 0.03 & 0.36 & 0.08 & 0.04 & 0.12 \\
   0.03 & 0.11 & 0.18 & 0.15 & 0.05 & 0.13 & 0.11 \\
   0.25 & 0.32 & 0.18 & 0.13 & 0.18 & 0.20 & 0.01
\end{array}
\right)
\]

\begin{center} $A \pm 0.001A$  \end{center}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Industry & Multiplier & Diameter \\
1        & $[4.81,4.88]$ & 0.07  \\
2        & $[3.79,3.85]$ & 0.05  \\
3        & $[6.34,6.44]$ & 0.09  \\
4        & $[9.20,9.35]$ & 0.15  \\
5        & $[6.04,6.13]$ & 0.09  \\
6        & $[5.71,5.80]$ & 0.08  \\ \hline
\end{tabular}
\end{center}


\begin{center} $A \pm 0.01A$  \end{center}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Industry & Multiplier & Diameter \\
1        & $[4.51,5.22]$ & 0.71  \\
2        & $[3.57,4.10]$ & 0.52  \\
3        & $[5.94,6.89]$ & 0.95  \\
4        & $[8.58,10.05]$ & 1.47  \\
5        & $[5.65,6.56]$ & 0.91  \\
6        & $[5.35,6.20]$ & 0.85  \\ \hline
\end{tabular}
\end{center}

\begin{center} $A \pm 0.1A$  \end{center}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Industry & Multiplier & Diameter \\
1        & $[2.62,12.84]$ & 10.21  \\
2        & $[2.15,9.69]$ & 7.54  \\
3        & $[3.39,17.18]$ & 13.79  \\
4        & $[4.71,26.17]$ & 21.46  \\
5        & $[3.21,16.49]$ & 13.28  \\
6        & $[3.09,15.34]$ & 12.26  \\ \hline
\end{tabular}
\end{center}

 
\section{Other Approaches}

West \cite{West86} has derived a formal expression for the
probability density of the multipliers assuming that the technical
coefficients are independent and that they can be characterized
by probability distributions with small variances.
This is certainly a significant
improvement over assuming the coefficients are point values and
that the multiplier results are known with certainty.
If probability distributions are assumed, then West's method
will generate confidence intervals.

Yet significant practical problems remain.  We do not know what the
probability distributions of the coefficients are in practice.
Since the national tables are at least 5 years old before publishing,
it is reasonable to believe that the distribution has changed.
The regional analyst usually has no sample information at all.
The regional analyst, most likely, will have no basis for
assigning confidence intervals, but will have to assign
subjective levels of uncertainty.  My own feeling is that
national analysts should be most suspicious about the possibility of
assigning confidence intervals, given the delay in obtaining data.

\section{Associated Research}


Recent economic research has shown that most of the elements
of the Leontief matrix are not ``important''.  That is, they
could be given zero value and this would have little affect
on the resulting multipliers \cite{Hewings84}.
Extensions of this research will be useful because it
will identify those elements in the Leontief matrix
where resources should be directed to try to determine
the level of uncertainty in the parameters.




\begin{thebibliography}{50}


\bibitem{Hewings84} G. J. D. Hewings, ``The Role of Prior
                    Information in Updating Regional
                    Input-Output Models,'' {\it Socio-Economic
                    Planning Sciences}, 1984, Vol. 18, 1984, pp. 319--336.

\bibitem{Leontief51}  W. Leontief, 
                    {\bf The Structure of the American Economy,  
                            1919--1939}, 2nd ed.,
                     Oxford University Press,
                     New York, 1951.

\bibitem{Miernyk65} William H. Miernyk, 
                    {\bf Input-Output Analysis},
                    Random House, NY, 1965.

\bibitem{Neumaier90}
     Arnold Neumaier, 
     {\bf Interval Methods for Systems of Equations},
     Cambridge University Press, Cambridge, 1990.

\bibitem{Richardson72}
     Harry W. Richardson, 
     {\bf Input-Output and Regional Economics}
     Halsted Press, NY, 1972.

\bibitem{Rohn80}
     Jiri Rohn,
     ``Input-Output Model with Interval Data,''
     {\it Econometrica}, 1980, Vol. 48, No 3,
      pp. 767--769.

\bibitem{West86}
     Guy R. West, ``A Stochastic Analysis of an Input-Output
     Model'', {\it Econometrica}, 1986, Vol. 54, No. 2,
     pp. 363--374.


\end{thebibliography}

\end{document}

