Optimal Project Rejection and New Firm Start-ups

Transcription

1 Optimal Project ejection and New Firm Start-ups Bruno Cassiman IESE Business School and University of Leuven Masako Ueda University of Wisconsin - Madison and CEP This Version: August 2005

2 5 Appendix In this appendix, we provide the proof of Lemma 1, Propositions 2-7, and the analysis of the firm behavior when it does not have any commercialization capacity. 5.1 Proof of Lemma 1 Wedistinguishsixcaseseachofwhichgivesadierent combination of the fall-back options for the firm and the scientist and the ecient destiny of innovations. Note that if ( ) 1 2 the ecient destiny of the innovation is internal commercialization; if ( ) 3 4 the ecient destiny of the innovation is external commercialization; and if ( ) 5 6 the ecient destiny of the innovation is shelving. First, suppose that ( ) 1. Thenifthefirm does not commercialize the innovation internally, the scientist s payo is equal to and the firm s incremental payo is because the scientist does want to start up a new firm. Thus, the total payo in this case is equal to +. If the firm internally commercialize the innovation, the total payo is + + Since the ecient destiny of the innovation is internal commercialization in this case, following the Nash bargaining solution, the surplus created (++ )(+) is shared according to the relative bargaining power of the firm and the scientist and their respective payos are 1 = + ( ) and 1 = +(1)( ). Thisimpliesatransferof 1 = 1 Second, suppose that ( ) 2. Thenifthefirm does not commercialize the innovation internally, both the scientist and the firm s payo are equal to zero because the scientist does not want to start up a new firm Thus, the total payo inthiscaseisequaltozero. Ifthefirm commercializes the innovation internally, the total surplus is + + Since the ecient destiny of the innovation is internal commercialization in this case the second statement of the lemma follows. Third, suppose that ( ) 3. Thenifthefirm does not commercialize the innovation internally, the scientist s payo is equal to and the firm s payo is because the scientist does want to start up a new firm.thisisexactlytheecient destiny of the innovation, and therefore no negotiation occurs. Thus, the third statement of the lemma follows. 32

3 Fourth, suppose ( ) 4. Thenifthefirm does not commercialize the innovation internally, both the scientist and the firm s payo are equal to zero because the scientist does not want to start up a new firm. Thus, the total payo in this case is equal to zero. If the scientist commercializes the innovation externally, the total payo is + Since the ecient destiny of the innovation is external commercialization in this case, the fourth statement of the lemma follows. Fifth, suppose that ( ) 5. Thenifthefirm does not commercialize the innovation internally, the scientist s payo is equal to and the firm s payo is because the scientist does want to start up a new firm. Thus, the total payo in this case is equal to +. If the firm shelves the innovation, the total surplus is zero. Since the ecient destiny of the innovation is shelving, the fifth statement in the lemma follows. Sixth, suppose that ( ) 6. Thenifthefirm does not commercialize the innovation internally, both the scientist and the firm s payo are equal to zero because the scientist does not want to start up a new firm. Thus, the total payo in this case is equal to zero. This is exactly the ecient destiny of the innovation, and therefore no negotiation occurs. Thus the sixth and final statement in the lemma follows. 5.2 Analysis when there is no internal commercialization capacity Next, we derive how innovation is selected when there is no commercialization capacity. Therefore, the innovation will be either shelved or externally commercialized. Shelved innovations will be worth zero and an externally commercialized innovation worth +. Thus, it should be shelved if + 0 and externally commercialized otherwise. The payos tothefirm when it does not have any commercialization capacity depends on only two variables: and +. If is non-negative, the scientist can threaten the firm with the possibility of external commercialization. If + is non-negative, the ecient destiny of the innovation is external commercialization, and shelving otherwise. Thus, the firm s payo is summarized as follows: Lemma If + 0 and 0 then the innovation is externally commercialized, and = and 33

4 =. 3. If + 0 and 0 then the innovation is externally commercialized, and = ( + ) and =(1 )( + ). 4. If + 0 and 0 then the innovation is shelved, and = ( + ) and = (1 )( + ). 5. If + 0 and 0 then the innovation is shelved, and =0and =0. Proof. Similar to that of Lemma 1. Note that, since is time-independent and each scientist has a constant arrival rate of innovations, the firm s problem is also time-independent. As a consequence, the firm faces the stationary arrival rate of innovations, Thus, where the firm does not have any commercialization capacity, the firm s expected payo is = 1 max ( )+ 0 Due to the characteristics of the function, ( + ) Proof of ( + ) ( + ) (3) 0 and 0 if and only if ( + ) 0 and =0otherwise. 0 Because 0 0 is essential for proving many propositions, we will display it in the following lemma. Lemma 4 is strictly decreasing 34

5 Proof. Since 0 = ( ) + ( + )! ( + ) = ( ) ( ) 1 2 = = ++ =0 +0 and =0 and +0 ( + ) ++ =0 and =0 +0 and 0 =0 and +0 ( + + ) ( + ) The inequality is strict because is unbounded from above. 5.4 Proof of Proposition 2 To prove that 1 for =0. We start with proving that 1 andthenshow that if, for =0 1. The proof is done by contradiction. 1 Suppose that 1 1 then 1 ()+ 1 ( )+ (+) The transition from the first line to the second used the fact that 1 does not take the maximum if 1 6= 1 The transition from the second line to the third used the fact that is decreasing. However, = ( )+ (+) And this is a contradiction. Thus, 1 and 1 0. Next suppose that 2 1. Then 2 0 and therefore 2 1. This is a contradiction, because the first order condition (2) and the fact that () is decreasing implies 35

6 that = = Note that the transition from the firstlinetothesecondlineusedthat 2 is the unique optimum that satisfies the first order condition (2) if = 2. Hence, Similarly, by induction, the proposition follows. 5.5 Proof of Proposition 3 1 and Given +1 let us define, a correspondence 1 : such that the the pair, { 1 ( )} satisfies the Bellman equation (1). Similarly, let us define a correspondence 2 : such that the the pair, { 2 ( )} satisfies the first order condition (2). Our goal is to show that these two correspondences look like in Figure 4. V j φ 2 * V j * V j + 1 φ 1 o * N j N j Figure 3: Existence and Uniqueness [Uniqueness] To prove that the equilibrium is unique if it exists, below we demonstrate that (1) 2 is strictly decreasing, (2) 1 is a function, and (3) 0 1 =0when 1 and 2 crosses (by construction). These three facts establish the uniqueness. 36

7 First, totally dierentiating the equation (2) gives: 0 2 = 00 0 = Second, we prove by contradiction that 1 is a function that is well-defined for the entire domain 0. Suppose 1 is not a function then there exist 1 and 2, 1 2 such that ( 1 ) = ( )+ ( 1 +1 ) and ( 2 ) = ( )+ ( 2 +1 ). Subtracting the second equation from the first equation gives ( 1 2 )= ( ( 1 +1 ) ( 2 +1 )) This is contradiction; the left hand side is positive and the right hand side is negative because 0 0. This establishes that 1 is a function. Third, totally dierentiating 1 gives 0 1 ( )= 0 + ( 1 ( ) +1 ) 0, (4) which is equal to zero when 1 crosses 2. This is because at an intersection the numerator of 0 1 must be zero. This concludes the proof of the uniqueness. [Existence: Necessity Part] We now prove the necessity part by proving the contraposition. Suppose that ( ) 0. Then, 1(0) = 0 and equation (2) implies 2 (0) 0. As 2 is strictly decreasing, this implies that there is no combination such that 0. [Existence: Suciency Part] We now prove that there exist 1 0 and 1 0 if ( ) 0. Note that 1 goes through the origin with a positive slope because 0 (0) = 0 by definition and ( ) 0. Since 0 1 = [ 0 + ( 1 ( ) +1 )] and 0 + ( 2 ( ) +1 )= 0, then 0 1 ( ) 0 if 1 ( ) 2 ( ) for = 0 1 because 0 0. Thus, 1 is monotonically increasing at least until 1 crosses 2. Thus 1 and 2 cross and the equilibrium exists at 1 0. At equilibrium, 1 0 because 1 (0) = (0) 0 2 (0) 0 and To prove the existence for =0 2, notethat 1 accordingtoproposition2. Because is decreasing, this implies that 1 ( ) 0. Therefore, by induction, we can use the same procedure to prove the existence of 1 1 for =0 2. Q.E.D 37

8 5.6 Proof of Proposition 4 Note that from the equation (3), it is immediate that We are now going to prove 0, 0, 0, and 0. 0, and 0 for =01. Todo so, we use an inductive argument. In particular, we show that for =0 1 if +1 0 then 0 if +1 0 then 0 and if +1 0 then 0. ³ i Let = 01. Then we can rewrite the equation (2) at the optimum as h = +1. Substituting this equation into the equation (1) for and rearranging the terms give =0 (5) Dierentiating the left hand side of the equation by gives = = 0 0 We used the fact 0 = when moving from the first line to the second line. Thus, = 0 = = Using this fact, we are now ready to prove the comparative statics results. Since +1 µ = µ 0 = µ 0 µ given 0 0 and +1 > 0 where the transition to the last inequality was due to 0 =. Thus, is increasing in. = µ = = given 0 0 and

9 Thus, Since and +1 is decreasing in. Finally, = 0 0 Z = 1 0 = + Z 2! and therefore +1!! + Z Z ( + ) 4 is increasing in.!! +1 5 ( + ) 0 Next, we are going to prove the results on. Note that from the equation (3), it is immediate that 0, 0, and =0. We now prove the results on =01 Combining the equations (2) and (1) gives: = + 0 =0 1 (6) The right hand side of the equation is increasing because is strictly convex. Thus, and comove. Together with the comparative statics results of this proves the comparative statics results of = Proof of Proposition 5 We now move to prove that 1 =1 by contradiction. Suppose 1 ³ ³. Then 1. By taking the dierence of the equation (1) gives 1 = = This is a contradiction because 0 as proved above. Thus, 1 =1. To prove 1 =1 note again that Since 1 1 =1. and comove as given in equation (6). 39

10 5.8 Proof of Proposition 6 To show that it suces to demonstrate that (1 ( ( + ))) (1 [ ( + )]) (1 []) since the denominators of and are both positive. This inequality should hold because the left-hand side is average of conditional on and the right-hand side is average of conditional on. Similarly, to show that it suces to demonstrate that (1 ( ( + ))) (1 [ ( + )]) (1 []) (1 []) This inequality should also hold because the left-hand side is average of conditional on and the right-hand side is average of conditional on. 5.9 Proof of Proposition 7 The density of conditional on external commercialization under = 1 and = 2 is 1 () = (1 []) 1 (1 []) 1 and 2 () = (1 []) 2 (1 []) 2 respectively. Hence the likelihood ratio is: 1 () 2 () = 1 (1 []) 2 2 (1 []) 1 Since the second component of the right hand side is constant, µ 1 () 2 = () µ That is, the density function, 1 () and 2 () also satisfy the monotone likelihood ratio property. Therefore,