Appendix to Technology Transfer and Spillovers in International Joint Ventures

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1 Appendix to Technology Transfer and Spillovers in International Joint Ventures Thomas Müller Monika Schnitzer Published in Journal of International Economics, 2006, Volume 68, Bundestanstalt für Finanzdienstleistungsaufsicht (BaFin), International/Financial Markets Department, Georg-von- Boeselager-Str. 25, Bonn, Germany, Tel.: , Fax.: , Department of Economics,University of Munich, Akademiestr. 1/III, Munich, Germany, Tel.: , Fax.: ,

2 Proof of Result 1 MNE maximizes U MNE = qr q(1 )βs K(q). Given the assumptions on K(q) the optimal level of investment q is uniquely characterized by the following first order condition: K (q) = R 1 βs. Using the implicit function theorem we can show that dq = 1 β K (q) = (1 )β K (q) < 0 dq ( (1 )) = βs 2 K (q) = βs 2 K (q) > 0 Using these results we can show that du MNE dq dq R (1 )βs q(1 )β K (q) [R (1 )βs K (q)] q(1 )β = q(1 )β < 0 Furthermore du HC dq (1 )(R + S) + q(1 ) (1 ) K (q) (1 )[R + S K (q)] + q(1 ) 1 (1 )[R βs K (q) ( ) } {{ } (1 ) [ 1 βs + S] } {{} (+) + q(1 ) (+) + 1 βs + S] + q(1 ) Note that for S = 0, the first derivative for U HC is positive, whereas for S getting very large, the derivative gets negative. Hence, HC s payoff is maximized at S > 0. 1

3 Finally, note that d(u MNE + U HC ) = q(1 )(1 β) + dq (1 ) [ 1 βs + S] } {{} ( ) (+) For β 1, this derivative is negative for all S, whereas for β < 1 it is positive for S = 0 and it gets negative for large S. Hence welfare is maximized for S > 0. Proof of Result 2 Using the results from above we can show that du MNE = qr + dq dq dq R + qβs (1 )βs K(q) K (q) [R (1 )βs K (q)] +q(r + βs) K(q) > 0 Furthermore, du HC dq (1 )(R + S) q(r + S) + K(q) (1 ) K (q) 1 (1 )[R βs K (q) + 1 βs + S] [q(r + S) K(q)] (1 )[ 1 βs + S] [q(r + S) K(q)] (+) (+) Note that for = 1, this derivative is negative. For 0 it gets positive. Hence, HC s payoff is maximized at < 1. Finally, note that d(u MNE + U HC ) = q(β 1)S) + dq (+) (1 ) [(1 )βs + S] This expression is positive for β 1 and welfare is maximized for < 1 if β < 1. Lemma A 2

4 For any (0, 1), HC s maximization problem has a unique interior solution T () ( (1 ) β β+ S, R 1 βs). Proof of Lemma A We first show that HC s profit function is strictly concave in T. theorem, dqt (T ) dt = 1 K (q T ) < 0. Differentiating U T HC with respect to T we get dt T (T ) dt = 1 [ K (q T ) (1 )[R T K (q T ) +S] + T + qt (T ) = 1 βs by (7) (1 ) β β + ] S + T + q T (T ). d 2 UHC T = 1 [ ] K β + [(1 )β S + T ] < 0. dt 2 K [K ] 2 By the implicit function Hence, the optimal T () must be unique. Furthermore, it is never optimal to choose T R 1 βs, because this would imply qt (T, ) = 0 and U T HC = 0, while a strictly positive payoff can be obtained by choosing T < R 1 βs. Finally, it cannot be optimal to choose T = T (1 ) β β+ S. To see this note that at T = T we have q T (T, ) > 0.Thus, dt T =T = q T (T, ) > 0. Hence, if > 0, a strictly higher payoff can be obtained by choosing T > T. Proof of Lemma 1 By the implicit function theorem we can show that dq T = 1 [ 1 K β + dt ]. 3

5 Using again the implicit function theorem and taking account of the direct effect of an increase in S on q, i.e. 1 1 K β, we find that dt K dq T [K ] 2 [(1 ) β β+ S + T ] + dqt β β+ (1 ) + dqt dt = K dq T β β+ [(1 ) S + T [K ] 2 dt ] + dqt + dqt dt dt ( ) 1 K = β [(1 ) β β+ S + T ] [K ] 2 β K [(1 ) β β+ S + T [K ] 2 ] =A>0 ( ) 1 = β A < 0, with A < 1 if β > 1. Proof of Result 3 Using Lemma 1, we can show that dq T = 1 1 β[a 1]. K Differentiating U T MNE and U T HC with respect to S and re-arranging we get: du T MNE T T = q T dt qt (1 )β + dqt [R T K dq T (q)] (1 )βs } = 1 βs by (7) {{ } = q T (1 )β[a 1]. [ +S] + T + qt (1 ) + dt = 1 βs by (7) (1 )[R T K (q) [ (1 ) β β + ] S + T + q T (1 )[1 βa]. ] Note that for β > 1 we get βa > 1 and vice versa. Summarizing the effects: (i) β = 1 A = 1 dt (ii) β < 1 A > 1 dt dqt < 0, dqt < 0,, du MNE T, du MNE T, = 0, d(u T MNE +U T HC ) = 0, > 0, d(u T MNE +U T HC ) > 0 4

6 (iii) β > 1 A < 1 dt dqt < 0,, du MNE T, < 0, d(u T MNE +U T HC ) < 0 For β = 1 we have dqt = 0. Thus, it follows from equation (7) that for β = 1 we must have T (; S) = T (, 0) 1 S, where T characterizes the optimal choice of T for S = 0. Proof of Lemma 2 By the implicit function theorem, it is straightforward to show that dq T = 1 [ dt K 1 ] βs. 2 Using again the implicit function theorem and taking account of the direct effect of an increase in on q T (T, ), i.e. 1 K 1 2 βs, we can show that dt = 1 K βs 2 + [(1 ) β β+ [K ] 2 K [(1 ) β β+ [K ] 2 S + T ] β S + T ] } {{ } =B>0 q S (T ) K [(1 ) β β+ [K ] 3 S + T ] + (1 + ) 1 K }{{ } =D>0 = 1 2 βsb + D > 0, withb < 1 for β > 1. Proof of Result 4 Differentiating UHC T with respect to and re-arranging we get: T (1 )[R T K (q T ) +S] + T = 1 βs by (7) + K(q T ) q T [R T + S] = 1 [ dt K 1 ] βs 2 + qt dt β + (1 )β S + T }{{ } =K q T by (8) 5 + qt dt

7 + K(q T ) q T [R T + S] = K(q T ) q T [R T + S] + q T 1 βs. (A1) <0 >0 A marginal increase of reduces HC s share of total surplus, q T [R T ] K(q T ), and reduces the received spillover, q T S. On the other hand, a marginal increase of induces HC to increase total taxation by dt dqt and it induces MNE to change investment by. Both effects sum up to q T 1 βs, which is basically the direct effect of an increase in on the investment qt. This effect may dominate and thus HC may prefer to increase, if β is sufficiently large. Note that (A1) can be positive only if β > 1. To see this, note further that MNE will choose q T > 0 only if U T MNE > 0, i.e. q T [R T ] q T (1 )βs K(q T ) I > 0. Condition (A1) is positive if, after re-arranging, we have q T [R T ] q T βs + q T S K(q T ) < 0. Both conditions can be fulfilled simultaneously only if β > 1. Differentiating U T MNE with respect to and re-arranging we get: du T MNE = q T [R T + βs] K(q T ) q T dt + dqt [ R T K (q T ) ] } = 1 βs by (7) {{ } = q T [R T + βs] K(q T ) >0 dqt (1 )βs q T dt } {{ } >0. (A2) Thus, the impact of on MNE s payoff may be ambiguous. A marginal increase of increases MNE s share of the total net payoff, q T [R T ] K(q T ), and reduces the loss due to the spillover, q T βs. On the other hand, a marginal increase of induces HC to increase total taxes by dt, of which MNE has to pay the share in case of a successful project, which happens with probability q T. If is close enough to 0, the second effect vanishes and MNE always 6

8 prefers to increase. However, if is sufficiently large, the second effect may dominate. The effect of a change of on total surplus is given by d(u T MNE + U T HC) = q T 1 βs qt dt + qt (β 1)S = q T 1 βs[1 B] q T D + q T (β 1)S. (A3) } {{} < 0 for >0 < 0 for β < 1 β < 1 By proof of Lemma 1 and Result 3 we know that for β = 1, T (, S) = T (, 0) 1 S and thus q T (; S) = q (, 0). Hence, equations (A1), (A2), and (A3), and therefore the effects of a decrease in are the same for β = 1 and for S = 0. Summarizing the effects: (i) β = 1 dt (ii) β < 1 dt dqt > 0, dqt > 0, < 0, du T MNE < 0, du T MNE < 0, d(u T MNE +U T HC ) < 0. < 0, d(u T MNE +U T HC ) < 0. (iii) β > 1 dt dqt > 0, du T MNE d(u T MNE +U T HC ) < 0. We prove by example that there indeed exist cases with the properties described in the Result. Consider the following cost function: K(q) = 1 1 q q. For = 0.98, R = 40, and S = 3 the following results are obtained for different values of β: du T MNE d(u T MNE +U T HC ) q T T U T MNE U T HC β = β = β = β = Thus, for large values of, there exist cases where MNE s payoff increases as decreases. This result can be obtained independently of the efficiency of a spillover β. For β > 1 there exist cases where HC s payoff and the efficiency of the project increase as increases. This is the 7

9 case for β = 3 in the example. For = 1 and R = 40 we get: du T MNE d(u T MNE +U T HC ) q T T U T MNE U T HC = Hence, in some cases HC benefits and the efficiency of the project is maximized if ownership is not shared. The intuition for this is easy to see for the extreme example of β being so large that the optimal q is chosen equal to zero for any < 1. In this case, HC s payoff is maximized by choosing = 1. This induces MNE to choose a positive q and hence allows HC to enjoy a positive payoff through taxation. Lemma B For any (0, 1), HC s payoff is maximized at M (), with (a) M () (0, ), if R > 1 (b) M () ( 1 βs R, ), if R 1 (c) M = 0, if R 1 βs otherwise. Proof of Lemma B βs and d2 UHC M dm 2 M=M < 0, or βs, d2 UHC M dm 2 M=M < 0, and UHC(M M ) > 0, or The optimal level of investment q is characterized by the following first order condition: K (q M ) = R + M 1 βs. (A4) By the implicit function theorem, dqm (M) dm = 1 K (q M ) > 0. Differentiating U M HC with respect to M we get duhc M dm M dm = (1 ) R + M K (q M ) +S + qm (1 ) C (M) = 1 βs by (9) β + (1 )β S + q M (M)(1 ) C (M). 1 K (q M ) 8

10 d 2 UHC M = 1 [ ] K β + (1 )β S (1 ) C (M). dm 2 K [K ] 2 Hence, HC s payoff is maximized at M (), if d2 U M HC dm 2 M=M < 0 and if moreover U M HC(M ) > 0. Given the assumptions on C(M) there must exist an upper bound for M. If < 1 and R > 1 βs, it is never optimal to choose M = 0. To see this note that in this case q M > 0 and thus du M HC dm = M 1 K (q M ) β + (1 )β S + q M (M)(1 ) > 0. Hence, if (0, 1), a strictly higher payoff can be obtained by choosing M > 0. If < 1 and R 1 βs, it follows from equation (9) that q = 0 for all M 1 βs R. Hence, HC chooses M ( 1 βs R, ) if U M HC(M ) > 0 and M = 0 otherwise. Proof of Lemma 3 By the implicit function theorem we can show that dq M = 1 [ 1 K β dm ]. Using again the implicit function theorem and taking account of the direct effect of an increase in S on q M (M, ), i.e. 1 1 K βs, we find that dm = K (1 ) β β+ dq M [K ] 2 K dq M [K ] 2 dm S + dqm β β+ (1 ) dm (1 ) β β+ S + dqm (1 ) C dm = 1 K β (1 ) β β+ S + [K ] 2 β K (1 ) β β+ > 0. S (1 ) + K [K ] 2 C =E>0 + dqm (1 ) The last inequality follows from the fact that the denominator has to be positive by Lemma B if HC s payoff is maximized at M (). 9

11 Proof of Result 5 From Lemma 3 it follows dq M = 1 K 1 β[e 1] where E is defined above in the Proof of Lemma 3. Note that E > 1 if β > (1 ) + K C and vice versa. It is straightforward to see that E > 1 is compatible with both β > 1 and β < 1, and vice versa for E < 1. Differentiating U M MNE and U M HC with respect to S and re-arranging we get: du M MNE M [R + M K (q M )] dqm (1 )βs } = 1 βs by (A4) {{ } + q M dm qm (1 )β = q M (1 )β[e 1] < 0. du M HC M (1 ) R + M K (q M ) +S + qm (1 ) dm βs by (A4) = 1 + q M (1 ) C (M) dm = 1 K [ 1 β dm ] (1 ) β β + S + q M (1 ) dm + q M (1 ) C (M) dm = 1 (1 ) 2 ( ) β β + βs + q M (1 ) < 0. K The effect of a change in S on total surplus is given by d(u M MNE + U M HC) = q M (1 )β[e 1] 1 (1 ) 2 K + q M (1 ) < 0. β β + βs 10

12 Proof of Lemma 4 By the implicit function theorem we can show that dq M = 1 [ dm K + 1 ] βs. 2 Using again the implicit function theorem and taking account of the direct effect of an increase in on q M (M, ), i.e. 1 K 1 2 βs, we can show that dm = 1 K [(1 ) β β+ S] [K βs ] 2 β 2 K [(1 ) β β+ S] (1 ) + K [K ] 2 C q M (M) K [(1 ) β β+ < 0. S] (1 ) 1 + C [K ] 3 K The last inequality follows from the fact that the denominator has to be positive by Lemma B if HC s payoff is maximized at M (). Proof of Result 6 Differentiating U M HC and U M MNE with respect to and re-arranging we get: du M HC M = 1 K (1 ) R + M K (q M ) +S + qm (1 ) dm = 1 βs by (9) C (M) dm + K(qM ) q M [R + M + S] [ dm + 1 ] βs (1 ) β β + S 2 + dm [ q M (1 ) C (M) ] + K(q M ) q M [R + M + S] = K(q M ) q M [R + M + S] + 1 β + (1 )β } {{ } <0 } K {{} >0 βs 2. (A5) 3 Thus, the impact of on HC s payoff is, independently of the efficiency of the spillover β, ambiguous. A marginal increase of reduces HC s share of total surplus, K(q M ) q M [R + M ] and reduces the received spillover q M S. On the other hand, a marginal increase of induces HC 11

13 to reduce its investment in infrastructure by dm and it induces MNE to change investment by dq M. Both effects sum up to the second expression in (A5). This effect may dominate depending on the exact nature of investment costs. du M MNE = q M [R + M + βs] K(q M ) + q M dm + dqm [R + M K (q M )] dqm (1 )βs } = 1 βs by (A4) {{ } = q M [R + M + βs] K(q M ) >0 + q M dm } {{ } <0. The impact of on MNE s payoff may also be ambiguous. A marginal increase of increases MNE s share of total net payoff, q M [R + M ] K(q M ) and reduces the loss due to the spillover by q M βs. On the other hand, a marginal increase of induces HC to reduce its investment in infrastructure by dm, of which MNE enjoys the share in case of a successful project, which happens with probability q M. If is close to 0, the second effect vanishes and MNE always prefers to increase. However, if is sufficiently large, the second effect may dominate. The effect of a change in on total surplus is given by d(u M MNE + U M HC) = q M (β 1)S + 1 β + (1 )β βs 2 K < 0 for }{{ 3 } >0 β < 1 + q M dm } {{ } <0. Summarizing the effects: (i) S = 0 dm dqm < 0, < 0, du M MNE du M HC < 0, d(u M MNE +U M HC ) < 0. (ii) S > 0 dm dqm < 0, du M MNE du M HC d(u M MNE +U M HC ) < 0. We prove by example that there indeed exist cases with the properties described in the Result. Consider the following cost functions: K(q) = 1 3 q3 and C(M) = M 2. For = 0.98, R = 0.1, and S = 20 the following results are obtained for different values of β: 12

14 du M MNE du M HC d(u M MNE +U M HC ) q M M U M MNE U M HC β = β = β = Thus, for large values of, there exist cases where MNE s payoff increases as decreases. Moreover, there exist cases where HC s payoff and the efficiency of the project increase as increases. As the example highlights this can be the case even for an efficient spillover. For β = 0.9 sharing of ownership with = 0.98 results in no investment by both parties. However, for = 0.99, R = 0.1, and S = 20 we get: du M MNE du M HC d(u M MNE +U M HC ) q M M U M MNE U M HC β =