Linear Programming and Economic Analysis by Robert Dorfman & Paul A. Samuelson & Robert M. Solow

Author:Robert Dorfman & Paul A. Samuelson & Robert M. Solow
Language: eng
Format: epub
ISBN: 9780486142111
Publisher: Dover Publications
Published: 2012-10-11T04:00:00+00:00

Now we multiply the first inequality by a21 and the second by 1 − a11. Both of these are positive numbers (the latter by the Hawkins-Simon conditions), so the inequalities remain valid. This gives us

Now we add term by term to get

[(1 − a11)(1 − a22) − a21a12]X2 ≥ a21(ΔS1 + C1) + (1 − a11)(ΔS2 + C2)

Naturally it is legitimate to add inequalities this way. Finally the coefficient (1 − a11)(1 − a22) − a21a12 on the left will be recognized as the determinant

and the Hawkins-Simon condition tells us that this too is a positive number. Hence,

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