Hawkins-Simon condition PDF

Preview

Summary

Lecture notes...

Description

The Existence of Nonnegative Solutions (Page 116) In the previous numerical example, the Leontief matrix I − A happens to be nonsingular, so solution values of output variables do exist.  Moreover, the solution values all turn out to be nonnegative, as economic sense would dictate.  Such desired results, however, cannot be expected to emerge automatically; they come about only when the Leontief matrix possesses certain properties.  These properties are described in the so-called Hawkins-Simon condition. To explain this condition, we need to introduce the mathematical concept of principal minors of a matrix, because the algebraic signs of principal minors will provide important clues in guiding our analytical conclusions. We already know that, given a square matrix, say, B, with determinant |B|,  a minor is a subdeterminant obtained by deleting the ith row and jth column of |B|,  where i and j are not necessarily equal.  If we now impose the restriction that i = j , then the resulting minor is known as a principal minor.  For example, given a 3 × 3 matrix B, we can write its determinant generally as || |

|

The simultaneous deletion of the ith row and the ith column (i = 3, 2, 1, successively) results in the following three 2 × 2 principal minors:  By deleting 3rd row and 3rd column |

||

|

 By deleting 2nd row and 2nd column |

| |

|

 By deleting 1st row and 1st column |

| |

|

In view of their 2 × 2 dimensions, these are referred to as second-order principal minors. 1

We can also generate first-order principal minors (1 × 1) by deleting any two rows and the same-numbered columns from |B|. They are | | | | | | & Finally, to complete the picture, we can consider |B| itself as the third-order principal minor of |B|.

While certain economic applications require checking the algebraic signs of all the principal minors of a matrix B, quite often our conclusion depends only on the sign pattern of a particular subset of the principal minors referred to variously as the leading principal minors, naturally ordered principal minors, or successive principal minors. In the 3 × 3 case, this subset consists only of the first members of (5.25) through (5.27): || | |

|| |

|| |

| |

Here, the single subscript m in the symbol | |, unlike in the subscript usage in the context of Cramer’s rule, is employed to indicate that the leading principal minor is of dimension . An easy way to derive the leading principal minors is to section off the determinant |B| with the successive broken lines as shown:

2

Taking the top element in the principal diagonal of |B| by itself alone gives us |B1|; taking the first two elements in the principal diagonal, b11 and b22 , along with their accompanying off-diagonal elements yields |B2|; and so forth. Given a higher-dimension determinant, say, n × n, there will of course be a larger number of principal minors, but the pattern of their construction is the same.  A -order principal minor is always obtained by deleting any rows and the same-numbered columns from |B|.  And its leading principal minors |Bm | (with m = 1, 2, . . . , n ) are always formed by taking the first m principal-diagonal elements in |B| along with their accompanying off-diagonal elements. With this background, we are ready to state the following important theorem due to Hawkins and Simon:  Given (a) an n × n matrix B, with (i = j ) (i.e., with all off-diagonal elements nonpositive), and  (b) an n × 1 vector d ≥ 0 (all elements nonnegative),  there exists an n × 1 vector such that , if and only if |Bm | > 0 (m = 1, 2, . . . , n) i.e., if and only if the leading principal minors of B are all positive. || | | || |

|

|| |

|

The relevance of this theorem to input-output analysis becomes clear when  we let B represent the Leontief matrix I − A (where for i = j are indeed all nonpositive),  and d, the final-demand vector (where all the elements are indeed nonnegative).  Then is equivalent to ( ) , and the existence of guarantees nonnegative solution output levels.  The necessary-and-sufficient condition for this,  known as the Hawkins-Simon condition, o is that all the principal minors of the Leontief matrix I − A be positive.

3

The proof of this theorem is too lengthy to be presented here, but it should be worthwhile to explore its economic meaning, which is relatively easy to see in the simple two industry case (n = 2). Economic Meaning of the Hawkins-Simon Condition For the two-industry case, the Leontief matrix is [

[

[

]

]

]

] [

The first part of the Hawkins-Simon condition, |B1| > 0, requires that || | | | |  | | | | Economically, this requires the amount of the first commodity used in the production of a dollar’s worth of the first commodity to be less than one dollar. The other part of the condition, |B2| > 0, requires that || | || || ( ( ( (

| |

( )( ) ( )( ) ( )( ) ( )( ) )( ) ( )( ) ) ( ) ( )( ) ) ( ) ( )( ) ) ( )( ) ( ) ( ) ( ) ( )

Further, since (

|

) is positive, the previous inequality implies that

4

Economically,  measures the direct use of the first commodity as input in the production of the first commodity itself, and  measures the indirect use—it gives the amount of the first commodity needed in producing the specific quantity of the second commodity that goes into the production of a dollar’s worth of the first commodity.  Thus the last inequality mandates that the amount of the first commodity used as direct and indirect inputs in producing a dollar’s worth of the commodity itself, must be less than one dollar.  Thus, what the Hawkins-Simon condition does is to specify certain practicability and viability restrictions for the production process. o If and only if the production process is economically practicable and viable, can it yield meaningful, nonnegative solution output levels. Page 119

5...