Stock and the government expenditure variables e l a

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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 non-negative numbers. The Hawkins-Simon conditions are imposing restrictions on the Leontief input matrix that guarantee this sort of non-negativity. non-negativity. Hawkins-Simon conditions: The leading principal minors The of matrix B are all positive of ? ⇔ There exists a non-negative 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 two-industry 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 16...
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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.

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