by Milton Harris and Artur Raviv - Faculty

Recall
A(Q) Q(1 p 2 r 2 ) D G Q,
B xy (Q) 2(1 x 2 )(Q 2y) K xy (Q) 2(G Q),
C xy (Q) 4y(1 x 2 ) 2x 2 (1 y 2 )Q 2D(Q) K xy (Q),
D(Q) 2(p r) p 2 r 2 (s Q),
E(Q) 2[2r p 2 r 2 (s Q)] 2D(Q) 4p,
G 2(p r) p 2 r 2 s,
J
2(1 pr)
,
p r
T 2p p 2 s
2(p r)
,
r 2
K xy (Q) 4x 2x 2 (2y y 2 s Q).
The following equalities are easy to check:
Appendix
B xy A A C xy , (5)
B rp B pr C pr C rp K rp K pr 2(p 2 r 2 )(Q J), (6)
A D Q G, (7)
B xy K xy 2(A D) 2(Q G), (8)
C xy D D K xy , (9)
D 4p E D, (10)
K xy 4p E C xy , (11)
Note that B xy(Q) is upward sloping in Q and negative at Q = 0, whereas A(Q) is upward sloping
in Q and A(Q) = 0 at Q = 0. Therefore B xy(Q) crosses A(Q) at most once. Let Q B(x,y) be such that
B xy(Q) = A(Q) (if it exists), i.e.,
D:\Userdata\Research\Hierarchies\temp.wpd 34 February 22, 2000 (11:18AM)