Graphics Programs Reference
In-Depth Information
Linear Economic Models
MATLAB's linear algebra capabilities make it a good vehicle for studying
linear economic models
, sometimes called
Leontief models
(after their
primary developer, Nobel Prize-winning economist Wassily Leontief) or
input-output models
. We will give a few examples. The simplest such model
is the
linear exchange model
or
closed Leontief model
of an economy. This
model supposes that an economy is divided into, say,
n
sectors, suchas
agriculture, manufacturing, service, consumers, etc. Eachsector receives
input from the various sectors (including itself) and produces an output,
which is divided among the various sectors. (For example, agriculture
produces food for home consumption and for export, but also seeds and new
livestock, which are reinvested in the agricultural sector, as well as
chemicals that may be used by the manufacturing sector, and so on.) The
meaning of a
closed
model is that total production is equal to total
consumption. The economy is in equilibrium when each sector of the
economy (at least) breaks even. For this to happen, the prices of the various
outputs have to be adjusted by market forces. Let
a
ij
denote the fraction of
the output of the
j
thsector consumed by the
i
th sector. Then the
a
ij
are the
entries of a square matrix, called the
exchange matrix A
, eachof whose
columns sums to 1. Let
p
i
be the price of the output of the
i
thsector of the
economy. Since eachsector is to at least break even,
p
i
cannot be smaller
than the value of the inputs consumed by the
i
thsector, or in other words,
p
i
≥
a
ij
p
j
.
j
But summing over
i
and using the fact that
i
a
ij
=
1, we see that both sides
must be equal. In matrix language, that means that (
I
−
A
)
p
=
0
,
where
p
is
the column vector of prices. Thus
p
is an eigenvector of
A
for the eigenvalue
1, and the theory of stochastic matrices implies (assuming that
A
is
irreducible, meaning that there is no proper subset
E
of the sectors of the
economy suchthat outputs from
E
all stay inside
E
) that
p
is uniquely
determined up to a scalar factor. In other words,
a closed irreducible linear
economy has an essentially unique equilibrium state
. For example, if we
have
A = [.3, .1, .05, .2; .1, .2, .3, .3; .3, .5, .2, .3; .3,
.2, .45, .2]
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