Managerial Delegation, Product R&D and Subsidies on R&D Investment Costs

Appendix A: The Proof of Satisfaction of Second-Order Conditions

From (6a) and (6b),

From (9),

From (14a) and (14b),

Hence, if c > 3 ( 1 − s ) 2 8 , then second-order conditions are satisfied.

This completes the proof.□

Appendix B: Proof of Proposition 1

From Table 1,

1. λ m * − 1 = 1 − ( 1 − s ) 2 2 c < 0   if  c < ( 1 − s ) 2 2   > 0   if  c > ( 1 − s ) 2 2   p m * − p o * = 2 c − ( 1 − s ) 2 2 c < 0   if  c < ( 1 − s ) 2 2   > 0   if  c > ( 1 − s ) 2 2   ⋅ .

2. x m * = ( 1 − s ) 2 c > x o * = ( 1 − s ) 3 c .

This completes the proof.□

Appendix C: The Derivations of Consumers’ Surplus and Social Welfare

1. Firms are operated by managers

   c s m * = 2 ∫ 0 1 2 u 0 + q m * − p m * − t d t = u 0 − 9 4 + 1 − s c ,

   π m * = 1 − 3 ( 1 − s ) 2 8 c .

   Hence,

   w m * = c s m * + 2 π m * = u 0 + ( 1 − s ) ( 4 − 3 c − 3 c s ) 4 c − 1 4 .

2. Firms are operated by owners

   c s o * = 2 ∫ 0 1 2 u 0 + q o * − p o * − t d t = u 0 − 5 4 + ( 1 − s ) ( 1 + s ) 3 c ,

   π o * = 1 2 − ( 1 − s ) 2 18 c .

   Hence,

   w o * = c s o * + 2 π o * = u 0 + 2 ( 1 − s ) ( 1 + 2 s ) 9 c − 1 4 .

This completes the proof.□

Appendix D: Proof of Proposition 2

From Table 1,

1. π m * − π o * = 36 c − 23 ( 1 − s ) 2 72 c < 0 if  c < 23 ( 1 − s ) 2 36   > 0 if  c > 23 ( 1 − s ) 2 36   .

   23 ( 1 − s ) 2 36 − 3 1 − s 2 8 = 23 + s 1 − s 72 > 0 .

   Hence, 23 ( 1 − s ) 2 36 > 3 ( 1 − s ) 2 8 .

2. c s m * − c s o * = 2 − s 1 − s − 3 c 3 c < 0 if  c < 2 − s 1 − s 3   > 0 if  c > 2 − s 1 − s 3   .

3. w m * − w o * = 1 − s ( 27 c s − 16 s − 27 c + 28 ) 36 c > 0 if  c < 4 ( 7 − 4 s ) 27 ( 1 − s )   < 0 if  c > 4 ( 7 − 4 s ) 27 ( 1 − s )  

   4 ( 7 − 4 s ) 27 ( 1 − s ) − 2 − s 1 − s 3 = 5 − 3 s 2 + 7 s − 3 s 2 27 ( 1 − s ) > 0 .

   2 − s 1 − s 3 − 3 1 − s 2 8 = 7 + s 1 − s 24 > 0 .

   Hence, 4 ( 7 − 4 s ) 27 ( 1 − s ) > 2 − s 1 − s 3 > 3 ( 1 − s ) 2 8 .

This completes the proof.□

Appendix E: Proof of Proposition 3

1. From (19b) and Table 2,

   d q o * d s = − 2 s 3 c < 0 .

   d c s o * d s = − 2 s 3 c < 0 .

2. From Table 2,

   d w o * d s = 2 ( 1 − 4 s ) 9 c > 0 if  s < 1 4   < 0 if  s > 1 4   .

This completes the proof.□

Appendix F: Proof of Proposition 4

1. From (17b) and Table 2,

   d q m * d s = − s c < 0 .

   d c s m * d s = 2 s − 3 2 c < 0 .

2. From Table 2,

   d w m * d s = 3 c ( 1 − s ) − 2 2 c < 0 if  c < 2 3 ( 1 − s ) .   > 0 if  c > 2 3 ( 1 − s )  

   2 3 ( 1 − s ) − 3 1 − s 2 8 = 7 + 27 s − 27 s 2 + 9 s 3 24 1 − s > 0 .

   Hence, 2 3 ( 1 − s ) > 3 1 − s 2 8 .

This completes the proof.□