Unformatted text preview: is an input
1 2 3 ... n coefficient that says
“industry 2 must supplya23
Suppose industry 2
1 a11 a12 a13 ... a1n Suppose
dollars’ worth
a is producing
2 21 a22 a23 ... a2 n of its products for
something that is
3 a31 a32 a33 ... a3n industry 3 to produce
used by industry 3:
used $1’s worth of its
... ..... ..... .... .... .... product. n an1 an 2 an 3 ... ann Each column j in the input coefficient matrix contains the dollar
amounts needed to be produced by the other industries in order for $1
dollar’s worth of industry j’s product to be produced. 11 Adding Final Demand
Let d = ( d1 , d 2 ,..., d n ) be a vector of final demands in the economy. That is,
Let
each industry is producing n inputs for the other industries (including
itself), plus it satisfies the consumer demand for its products.
itself
We can formalize this assumption in the following manner: x1 = a11 x1 +a12 x2 +... +a1n xn +d1
Total amount of good 1
Total
produced in the economy
produced Amount of good 1
Amount
needed to produce x1
of itself.
of Amount of good 1
Amount
needed to produce
$1 worth of good 2
$1 Consumer demand for
Consumer
product 1
product Amount of good 1 needed
to produce x2 dollars’ worth of good 2. 12 Leontief Model in Matrix Form
Writing down the input and consumer demand equations for each good in
Writing
the economy, we end up with a system of n linear equations: x1 = a11 x1 + a12 x2 + ... + a1n xn + d1 x = a x + a x + ... + a x + d ⇔x = Ax +d , A =[aij ], i =1..n, j =1..n
2
21 1
22 2
2n n
2
′ d =( d1 , d 2 ,..., d n )
... xn = an1 x1 + an 2 x2 + ... + ann xn + d n Rearranging a bit we end up with the following system: ( I − A) x = d
The solution thus will look as follows: x = ( I − A) d
* −1 13 HawkinsSimon Conditions
It is natural to require that the solution values for the n outputs produced in
It
the economy be nonnegative numbers. The HawkinsSimon conditions are
imposing restrictions on the Leontief input matrix that guarantee this sort of
nonnegativity.
nonnegativity.
HawkinsSimon conditions:
The leading principal minors
The
of matrix B are all positive
of ?
⇔ There exists a nonnegative
There
solution to Bx = d 14 Leading Principal Minors
The concept of leading principal minors is easily illustrated graphically.
Consider a square matrix B: b11
B = b21 b31 b12
b22
b32 b13 b23 b33 The first order principal minor is just one element
The first B1 ≡ b11 The second order principal minor is formed as follows:
The second B2 ≡ b11 b12 b21 b22 Analogously, we can compute the third order principal minor:
Analogously,
the
b11 b12 b13 B3 = b21
b31 b22
b32 b23
b33
15 Economic Meaning of HawkinsSimon Conditions
Consider the twoindustry case:
1 −a11
I −A = −a21 −a12 1 −a22 The positive first order principal minor means that you shouldn’t be
The
producing $1s worth of the first commodity with more than one dollar’s
worth of the first commodity.
worth B1 > 0 ⇔ a11 < 1
The positive second order principal minor is a bit more intricate: B2 > 0 ⇔(1 −a11 )(1 −a22 ) −a12 a21 > 0 a11 +a12 a21 <1
The condition says that the direct and indirect use of commodity 1 used
The
indirect
for the production of $1’s worth of itself should be less than $1.
for
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This note was uploaded on 09/14/2013 for the course STAT 1001 taught by Professor Kim during the Fall '11 term at Yonsei University.
 Fall '11
 Kim

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