by Milton Harris and Artur Raviv - Faculty

5 Comparative Statics
For Q Optimal Design
[0,2r)
[2r,2p)
[2p,G)
[G,2r+r 2 s)
[2r+r 2 s,2p+p 2 s)
[2p+p 2 s,")
Table 2: Optimal Design as a Function of Q and F
In this table,
A(Q) = Q(1 p2r2 ) = D(Q) G+Q
Bxy(Q) = 2(1 x2 )(Q 2y) = Kxy(Q) 2(G Q)
Cxy(Q) = 4y(1 x2 )+2x2 (1 y2 )Q = 2D(Q) Kxy(Q) D(Q) = 2(p+r)+p2r2 (s Q)
Centralized Flat if 4F > A(Q)
Matrix otherwise
Centralized Flat if 4F > max{A(Q),B pr(Q)}
Matrix if 4F < min{A(Q),C pr(Q)}
Centralized P-hierarchy otherwise
Centralized Flat if 4F > max{A(Q),B pr(Q),B rp(Q)}
Matrix if 4F < min{A(Q),C pr(Q),C rp(Q)}
Centralized P-hierarchy if C pr(Q) < 4F < B pr(Q) & Q < J
Centralized R-hierarchy otherwise
Decentralized Flat if 4F > max{D(Q),K pr(Q),K rp(Q)}
Matrix if 4F < min{D(Q),C pr(Q),C rp(Q)}
Centralized P-hierarchy if C pr(Q) < 4F < K pr(Q) & Q < J
Centralized R-hierarchy otherwise
Decentralized Flat if 4F > max{D(Q),4p,K rp(Q)}
Matrix if 4F < min{D(Q),E(Q),C rp(Q)}
Decentralized P-hierarchy if E(Q) < 4F < 4p & Q > T
Centralized R-hierarchy otherwise
Decentralized Flat if 4F > max{D(Q),4p}
Matrix if 4F < min{D(Q),E(Q)}
Decentralized P-hierarchy otherwise
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)
In this section, we analyze how the optimal design varies with the opportunity cost of the CEO,
Q, and the fixed cost of the middle managers, F. Although Table 2 defines the optimal design for each
pair (Q,F), it is difficult to see from the table the effect of changing Q or F on the optimal design.
D:\Userdata\Research\Hierarchies\temp.wpd 22 February 22, 2000 (11:18AM)