Abstract
In this paper we analyze how firms’ R&D investment decisions, firms’ profits and social welfare are affected by absorptive capacity; that is, the ability of a firm to learn from other collaborating firms. The model developed is a strategic regular network where firms have the opportunity to form pair-wise collaborative links with other firms and then compete à la Cournot. Different to the existing literature, we find that firms’ R&D efforts could increase or decrease with the degree of the network, depending on the level of absorptive capacity, the market size and the network dimension. In particular, in the case of small market size and low learning effect, the connection between firms drives up research investments. Moreover, if absorptive capacity is sufficiently low, the research collaboration between firms turns out not to be desirable from a private point of view while, in line with the existing literature, social efficiency requires a complete or intermediate level of collaborative activity. We also show that the complete network is pair-wise stable and socially optimal for an intermediate level of spillover intensity, while the empty network maximizes firms’ profits when absorptive capacity is small, yet it is not pair-wise stable.
Acknowledgments
We are grateful to Fabio Di Dio and Roy Cerqueti for their helpful suggestions. We thank Julian Wright, the editor, and two anonymous referees for their constructive comments, which helped us to improve the manuscript. Usual disclaimers apply.
Appendix A
Appendix A.1 – Result 1
Consider the condition (9). Let l′=l′(x), in order to simplify the notation. Using the implicit function theorem we know that: [16]
where, from the second order condition,
A sufficient condition to have
Regarding point i) of Result 1, first note that, for given x*, lims→∞l′=0, lims→∞l=1, lims→0l′=0 and lims→0l=0; then, for a sufficiently small value of s, the expression (20) is positive, and
Appendix A.2 – Result 2
By condition (14) we obtain:
where the expression in round bracket is the marginal effect of optimal R&D investment and:
The first term on the right hand side of expression (21) could be named as the indirect effect of connectivity on equilibrium profit. It is composed by the effect of n on x* which, according to Result 1, can be positive or negative, and by the effect of x* on π* (the marginal effect of optimal R&D investment on equilibrium profit) which can also be positive or negative, according to the network dimension N, spillover intensity s, connectivity n and R&D cost γ. The second term captures the direct effect of n on π*, which is always positive. [17]
The expression (21) is positive when, in absolute value, the indirect effect is lower than the direct one:
or when the indirect effect is positive:
Observing Table 4, it is evident that for small s, the optimal network is the empty network,
When n=0, we have
Appendix A.3 – Result 3
From equation (16) we obtain:
where
When spillover intensity is sufficiently low, we have
Figure 5 shows how the profit-maximizing degree of network (indicated as “npr”) divided by the social welfare-maximizing degree (indicated as “nw”) changes with the number of firms and with the learning effect. [18] The results differ from those of the existing literature on R&D networks with exogenous spillover: low levels of learning effect and hence of spillover rate, leading firms not to pursue reciprocal collaboration in spite of the social efficiency that requires a complete collaboration. Moreover, if the exogenous spillover rate exceeds a certain threshold, the R&D investment is sub-optimal because firms tend to form too many links ([K] and [CGP]). In the case of absorptive capacity, private and social incentives may coincide and the research investments be optimal with additionally high levels of spillovers intensity, provided that the number of firms fall into a certain range. If they do not, then firms tend to prefer a level of collaboration lower than that which social welfare would require to be maximized. On the other hand, whenever the learning effect is meaningfully strong, the opposite is true: in environments where the number of firms is sufficiently low, the profit-maximizing degree of network is excessive (like [GM]). Otherwise both the aggregate industry profits and social welfare are highest under the complete network.
Appendix B
Appendix B.1 – The Case with Manna from Heaven
In this appendix we consider a different specification of absorptive capacity, which allows firm i to realize spillover from firm j’s R&D efforts without engaging in R&D. As in Grunfeld (2003), we assume that
where α>0 is the exogenous spillover rate. Indeed, if s=0, we have the traditional spillover used in the Goyal-Moraga and Korkmaz model, where spillover is a manna from heaven.
Firstly, observe that
Secondly,
Intuitively, a positive α reduces firm’s incentive to invest in R&D; indeed, in Table 13 we observe that the R&D effort increases in collaboration but only for very low values of the exogenous spillover rate α, even if s is small. Table 14 shows that for intermediate values of α, firms’ profit and social welfare are maximized at the maximum level of collaboration (complete network). Instead, when α is sufficiently high, firms’ profit and social welfare maximization require an intermediate level of collaborative activity. In general, being that they are exposed to “manna from heaven,” firms can learn from their collaborators even with a low R&D effort. Then, for intermediate and high levels of α, they will tend to reduce their R&D spending and form dense networks.
N | α=0.01 | α=0.3 | α=0.5 | α=1 |
---|---|---|---|---|
4 | + | – | – | – |
10 | + | – | – | – |
20 | + | – | – | – |
50 | + | – | – | – |
N | α=0.01 | α=0.3 | α=0.5 | α=1 | ||||
---|---|---|---|---|---|---|---|---|
4 | 0 | 3 | 3 | 3 | 3 | 3 | 2 | 2 |
10 | 0 | 9 | 9 | 9 | 9 | 9 | 6 | 5 |
20 | 0 | 19 | 19 | 19 | 19 | 19 | 13 | 10 |
50 | 0 | 49 | 49 | 49 | 49 | 49 | 33 | 25 |
Appendix B.2 – The Case with Involuntary Spillovers
In this appendix we assume involuntary spillovers across non-linked firms. The cost of firm i is given as follows:
where β∈[0, 1) is the parameter that reflects the level of spillovers among firms with no collaboration links. First order conditions with respect to quantity and R&D effort are, respectively:
Reasonably, we adopt an intermediate-low level of the involuntary spillover fixing β=0.3. Results 1, 2 and 3 are also confirmed in this version of the model. For the sake of brevity we only report the results in regard to the optimal degree maximizing profits and welfare (Table 15). [19] We observe that firms’ profit and social welfare maximization requires a smaller level of collaboration in respect to the case of β=0. The intuition behind this result is that when β>0 firms can also absorb knowledge from non-collaborating firms. Therefore, from a private and social point of view, it would be optimal to reduce the number of links and compensate the absorptive capacity loss with the spillovers stemming from non-collaborating firms. This phenomenon is particularly marked when s (and then spillover) is high. [20]
N | s=0.1 | s=0.5 | s=5 | s=50 | s=500 | s=5000 | … | l=1 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
… | |||||||||||||||
4 | 0 | 3 | 0 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | … | 2 | 1 |
10 | 0 | 9 | 0 | 9 | 0 | 9 | 9 | 9 | 9 | 9 | 6 | 4 | … | 5 | 3 |
20 | 0 | 19 | 0 | 19 | 0 | 19 | 19 | 19 | 19 | 19 | 14 | 8 | … | 10 | 6 |
50 | 0 | 49 | 0 | 49 | 0 | 49 | 0 | 49 | 49 | 49 | 49 | 49 | … | 24 | 15 |
Appendix B.3 – Kamien and Zang’s Approach
In this appendix, we basically develop the same model in the text by assuming a different specification of absorptive capacity in accordance with Kamien and Zang (2000). This is further analyzed in Wiethaus (2005) and Leahy and Neary (2007). This second special case assumes that firm i’s realized cost reduction is
rather than simply
The profits of firm i are given by
Applying backward induction and imposing symmetry, we derive the optimal output
and the optimal R&D expenditure level
Second order condition is:
In Table 16 we show the sign of the relationship between R&D spending and network degree. The pattern in the table confirms points i) and ii) of the Result 1 in the text. Indeed, additionally with an absorptive capacity à la Kamien and Zang, the optimal R&D spending increases with the network degree n when δ is sufficiently large (low spillover intensity). For lower levels of δ (high spillover intensity), R&D spending decreases with n first in small network, then also in large networks.
N | δ=0.02 | δ=0.04 | δ=0.06 | δ=0.08 | δ=0.1 | δ∈[0.2; 1) |
---|---|---|---|---|---|---|
4 | – | – | – | – | – | + |
10 | – | – | – | – | + | + |
20 | – | – | + | + | + | + |
30 | – | + | + | + | + | + |
50 | + | + | + | + | + | + |
70 | + | + | + | + | + | + |
100 | + | + | + | + | + | + |
1000 | + | + | + | + | + | + |
On the other hand, point iii) of Result 1 is not confirmed given that, according to equation (34), the market dimension a–c does not affect the sign of the derivative ∂x*/∂n. In Table 17, we calculate the optimal level of firms’ connections from a private and social point of view as a function of network dimension N and the firms-specific approach δ. Numerical simulation confirms points i), ii) and iii) of Result 2 in the text. Moreover, with absorptive capacity à la Kamien-Zang, firms do not prefer to collaborate if learning effect is very low (high δ) while, for sufficiently high levels of absorptive capacity (low δ), firms’ profit is maximized in complete network (n=N–1) or at an intermediate level of collaborative activity (0<n<N–1). Moreover, the patterns in Table 17 also confirm Result 3 concerning the relationship between network degree and social welfare. Indeed, complete networks (
N | δ=0 | δ=0.005 | δ=0.1 | δ=0.2 | δ=0.3 | δ=0.4 | δ∈[0.5; 1) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 3 |
10 | 6 | 5 | 7 | 5 | 9 | 9 | 9 | 9 | 0 | 9 | 0 | 9 | 0 | 9 |
20 | 13 | 10 | 15 | 11 | 19 | 19 | 0 | 19 | 0 | 19 | 0 | 19 | 0 | 19 |
30 | 20 | 15 | 21 | 18 | 0 | 29 | 0 | 29 | 0 | 29 | 0 | 29 | 0 | 29 |
50 | 33 | 25 | 43 | 34 | 0 | 49 | 0 | 49 | 0 | 49 | 0 | 49 | 0 | 49 |
70 | 47 | 35 | 69 | 56 | 0 | 69 | 0 | 69 | 0 | 69 | 0 | 69 | 0 | 69 |
100 | 66 | 50 | 99 | 99 | 0 | 99 | 0 | 99 | 0 | 99 | 0 | 99 | 0 | 99 |
1000 | 666 | 500 | 0 | 999 | 0 | 999 | 0 | 999 | 0 | 999 | 0 | 999 | 0 | 999 |
Appendix C
Firms’ Profits in the Empty Network ge
Let ge be the empty network, obtained by assuming n=0. Condition (9) becomes:
by which we derive the optimal firm’s R&D investment and quantity:
As usual, firm’s profit is:
Deviating Profits in Network ge+gij
Assume some firm i forms a link with some firm j. The resulting network is ge+gij. In this network there are two types of firms. Firms i and j with n=1 links and all the other N–2 firms with n=0 links. By standard computation, and assuming symmetry, first order conditions are:
for deviating firms, and:
for the N–2 non-deviating firms, where
By numerical simulation we solve equations (40) and (41) and obtain the optimal R&D investments
Firms’ Profits in the Complete Network gc
We derive R&D investments, quantities and profits in the complete network by setting n=N–1 in the expression (9):
where
From equation (45) we obtain the optimal R&D investment x*(gc) which substituted into equation (14) gives firms’ profits under the complete collaborative agreement;
Deviating Profits in the Network gc–gij
Consider that some firm i severs its link with some firm j. The resulting network is gc–gij. In this network there are two types of firms. Firms i and j (deviating firms) which have N–2 links and the rest of the firms which have N–1 links. Let
for deviating firms and
for non deviating firms, where
We obtain the optimal R&D investments
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