Universitat Autònoma de Barcelona Department of Applied Economics Annual Report Endogenous R&D investment when learning and technological distance affects absorption capacity Author: Jorge Luis Paz Panizo Director: Francesc Trillas Jané Bellaterra, June 2016
1. Introduction In the R&D literature there is a traditional negative effect of spillovers stating that higher R&D exogenous spillovers reduce R&D investment equilibrium: Firms chooses R&D taking into account that competitors' efficiency will be improved through spillovers externalities, making them tougher competitors and driving down firm's R&D spending. Furthermore, the linear specification implies that the external R&D investment reduces firms unit cost without any effort of the recipients. On the contrary, endogenous models with absorptive capacity capture the idea that to learn from a competitor s one must also be engaged in R&D investments. In our reference paper, Grunfeld (2003) shows that an extra positive learning effect is always present in endogenous absorptive capacity models. However, as he states about his model: contrary to earlier studies, absorptive capacity effects do not necessarily drive up the incentive to invest in R&D. This only happens when the market size is small or the absorptive capacity effect is weak. The main results of this paper is that the positive learning property not always hold in our endogenous model: at equilibrium, it depends on the learning/distance parameter and the market structure parameter configuration In our endogenous spillover function, higher own R&D investment improves the firm absorptive capacity up to a maximal point where the traditional learning effect dominance ends (marginal learning of own investment decreases to zero). From this point, increases in R&D investment diminishes the firm s absorption capacity from others, since the relative technological distance effect is now stronger than the learning effect. Therefore, we propose a U-inverted absorptive capacity function. The proposed general function also integrates canonical models in the R&D and spillovers literature. In particular, the Brander and Spence (BS) model without spillovers and the AJ model with exogenous absorptive capacity can be easily parameterized are particular cases. Main results of this work are: First, the endogenous spillovers effect it is not always positive, it is ruled by a relation between the learning/distance and the market structure parameters and; Second, we proposes original regions, related to our new learning/distance parameter, in which the endogenous model has a higher equal or lower R&D investment than both the AJ and BS ones. Some preliminary comparative statics and welfare results are described for pure endogenous version of the Non-Cooperative and Cooperative RJV cartelization models. As future steps we expect to find empirical evidence on the existence and effect of different degrees in technological distance; identify the theoretical aggregated value of the model, and give some initial interpretation based on the formalization of the more relevant results obtained.
2. The Model Consider a two stage game in which firms invest in R&D in the first stage and compete in the product market in the second stage. In the second period, it chooses the level of differentiated output. In the first period, given the equilibrium output, each firm choose the optimal level of R&D investment. We consider only symmetric equilibria in pure strategies and the subgame perfect equilibrium are solved using backward induction. The inverse demand function for firm is linear: (1) where is the differentiation parameter. The unit cost function is represented by (2) A firm effective R&D effort is defined as the sum of its own unit cost reduction,, and a proportion of the rivals unit costs reduction. This specification generalize the d Aspremont and Jacquemin (d Aspremont and Jacquemin, 1998) (AJ) exogenous duopoly model to an endogenous absorptive capacity model for firms: (3) In the AJ model, the absorption capacity function is constant, so the effective R&D effort is a linear function of the spillovers. Furthermore, this exogenous specification implies that the external R&D investment reduces firms unit cost without any effort of the recipients. On the contrary, endogenous models with absorptive capacity capture the idea that to learn from a competitor s one must also be engaged in R&D investments. The novelty of this paper is a spillover absorption capacity as a function of both the own and rivals R&D investments shares: (4) where is the AJ exogenous spillover parameter 1, ( ) is the relative weight the own (competitor s) R&D investment share as relative size measure have in the of the endogenous learning term, and measure the degree of exogeneity (endogeneity) of the model. 1 Parameter measures the freely exogenous learning capacity of a firm ( manna from heaven ).
We have two extreme cases. When absorption capacity depends positively on its own firm R&D investment relative size (share), which implies that increases R&D always improve absorptive capacity and that relative technological distance from rivals does not negatively affect the spillover rate. When, only the relative distance with respect to others negatively affect spillovers, so increase in own R&D decreases the firm s absorptive capacity. At the beginning, higher own R&D investment improves the firm absorptive capacity up to a maximal point where the traditional learning effect dominance ends (marginal learning of own investment decreases to zero). From this point, increases in R&D investment diminishes the firm s absorption capacity from others, since the relative technological distance effect is now stronger than the learning effect. Therefore, we propose a U-inverted absorptive capacity function. As we will show, at equilibrium the maximal absorptive capacity will depend only on a relationship between and. The proposed general function integrates canonical models in the R&D and spillovers literature. In particular, the Brander and Spence (BS) model without spillovers and the AJ model with exogenous absorptive capacity can be easily parameterized as a particular cases: When and, we are in a R&D model without spillovers (BS) When and, we are in a model with linear exogenous R&D absorption capacity (AJ) When, we are in a mixed exogenous/endogenous model. When, we are in a model with pure endogenous absorption capacity Second stage: Cournot competition In the product market, firms act as Cournot competitors maximizing their individual profits conditional on the first stage R&D levels. Firm profits in the second stage is then given by (5) where is a quadratic investment cost function. First-order necessary conditions for the second stage, given the technology acquired in the first stage, are solved as usual to obtain the Cournot-Nash equilibrium: 2 (6) 2 See Appendix 1.
We consider two regimes of R&D investment: non-cooperative R&D competition and cooperative RJV cartelization. In the non-cooperative R&D competition firms invest in R&D unilaterally in order to maximize their own profits. In the RJV cartelization, firms coordinate their R&D decisions to maximize joint profits sharing the cost and result of their R&D investments. 2.1. First stage: Non-cooperative R&D In this section, we assume that firms behave non-cooperatively in the first stage. We define the net (of R&D investment costs) profits as following: (7) where is a quadratic investment cost function. In the first stage, each firm simultaneously chooses its R&D investment in order to maximize (5) subject to (6). Therefore SPNE is determined by simultaneously solving the following first order conditions: In a symmetric equilibrium, we obtain 3. (8) In the R&D literature there is a traditional negative effect of exogenous spillovers stating that higher linear R&D spillovers reduce R&D investment equilibrium: Firms chooses R&D taking into account that competitors' efficiency will be improved through spillovers externalities, making them tougher competitors and driving down firm's R&D spending. Grunfeld (2003) shows that there is a positive learning effect in the endogenous absorptive capacity models: If we separate out the negative traditional effect of spillovers, by replacing the endogenous spillover rates in the AJ model F.O.C., the endogenous models still obtain higher R&D investment in equilibrium than the modified AJ exogenous one. This property does not always hold in the present model, as the following proposition states. 3 Let (see Appendix 2).
Proposition 1. If the exogenous spillover rate in the AJ exogenous game is the same as the endogenous spillover rate generated by the game with absorptive capacity effects, the equilibrium R&D investment in the absorptive capacity game will be higher, equal or lower than in the AJ game if and only if is higher, equal or lower than. Proof. See Appendix 2. The intuition of this result comes directly from the marginal effect of own R&D investment in the absorption capacity function: at the symmetric equilibrium, the balance between the (conventional) positive marginal contribution of own R&D and the negative contribution through the distance channel is summarized by and, respectively 4. Given the symmetry assumption of equilibrium, as increases the (negative) marginal effect of our own R&D on the aggregate R&D investment tends to zero, and the (traditional) positive learning effect of the endogenous models prevails. Figure 1. Equilibirium R&D investment and the learning/distance effect 5 As proved in Proposition 1 the endogenous effect is ruled by a relation between the learning/distance and the market structure parameters. As observed in Figure 1, R&D investment in our pure 4 As in Grunfeld, the proof rests on the sign of. In our case, so the learning/distance effect depends, at the symmetric equilibrium, on the sign of this exogenous relationship. Note that, at equilibrium, learning/distance effect is always positive for (see Appendix 2). 5 Parameter values are:.
endogenous model (, dotted line) is not always above the modified AJ model (continuous line). In particular, Figure 1 shows that for a given value of the learning/distance parameter, the learning/distance effect is positive if. For low, the negative learning/distance effect (associated to the difference between the endogenous model dotted line and modified continuous AJ model line) adds to the traditional negative spillover effect (associated to the difference between BS line and AJ line), resulting in a lower R&D investment for the pure endogenous model with respect to the exogenous AJ model. For, traditional positive learning effect dominates distance effects, which causes that pure endogenous model R&D investment tends to exceed the AJ one. 6 It is worth noting that the present analysis is not strictly comparable with Grunfeld s one, based the latter on his market size variable:. Furthermore, the model extension for firms and the inclusion of product differentiation in the model add some issues overlooked in Grunfeld work (e.g. the competitive effects. At the same time, however, those adds complexity to both the effects and mechanisms underlying our results, so we are still developing analytical results in order to formalize some of the simulations findings. 7 We shed some light on the interaction between the market structure and the parameter of learning/distance. Figure 2 shows the set of points for which our endogenous R&D investment in the non-cooperative game is higher than the R&D for the AJ model. We find that there exist for which a low correspondent value of does not necessarily imply a lower R&D equilibrium in the exogenous model. In his work, Grunfeld states that as the learning parameter increases, the (critical) market size that equates R&D in both models is reduced (and also the R&D is reduced). For example, if we set (i.e. large average market size) and, we observe in Figure 2 a higher R&D in the exogenous model with respect to the our endogenous one, as in Grunfeld s result. However, there are alternatives market size per firm (or, in an alternative interpretation, alternatives degrees of competition) for which this result is reversed (e.g. for ). In general, there exists a new parameter region in which the endogenous model has a higher R&D investment than the AJ one. 6 In fact, for higher, learning/distance effect dominates both distance and spillover effects. See Figure 3 for an equivalent figure with 7 This section contains a first interpretation of results. It should be noted that at the moment, the analysis is based on the pure endogenous version of the model in order to isolate effects with respect to the exogenous models and the analogous Grunfeld analysis.
Figure 2. Equilibrium R&D investment thresholds with respect to AJ model 8 It is worth noting that the set of critical values for and are also affected by other parameters 9. In particular, Figure 2 shows an intuitive results: a higher implies a larger area in which the endogenous model predominates because of the negative traditional effect. Therefore, for any fixed, it is required a lower value of in order to balance the negative effect of the linear AJ spillovers model. Figure 3. Learning/distance effect and the BS model without spillover effects 10 8 Parameter values are:. 9 In this first monograph, we assume a pure endogenous model in order to extract some main results with the AJ model as a reference. Thus, is assumed that 10 Parameter values are:.
Finally, as Grunfeld points out: earlier theoretical literature on spillovers where it is claimed that spillovers have an unambiguous negative incentive effect on R&D investment. Figure 3 shows that our pure endogenous model could achieve more R&D investment than in the BS model. In contrast with Grunfeld counterexample, obtained for a sufficiently small market size, our absorptive capacity game generates higher R&D investment than a game with no spillovers if the learning/distance parameter is high and the market per firm size (low ) is large. 2.2. First stage: Cooperative RJV cartelization In the case of cooperation firms coordinate their first stage R&D decisions so as to maximize the aggregated industry profits. In a cooperative Research Joint Venture (RJV) cost minimization is restored and, for any level of production, R&D external effect are internalized optimally in the cooperative first stage. Therefore, in cooperative cartelization model there is no under or over investment. Firms maximize joint profits: and the first order conditions are given by In a symmetric equilibrium : 11 In presence of R&D without spillovers effects, first stage R&D is pre-committed under strategic considerations. This imply that at equilibrium, and given total output, total costs is not at the minimum level. In particular, as BS states, there is an over-investment situation. In the case of models with exogenous spillovers (AJ model), over/under-investment is conditioned by the incentives strength to reduce R&D: there exists a critical value for the spillover rate that determines the direction of the inefficiency. Consequently, we 11 The expressions for and are shown in the Appendix 3.
can use minimization costs F.O.C. in order to find the critical rate of spillover for the over or underinvestment analysis: In the non-cooperative endogenous model, the first order condition for the symmetric equilibrium can be written as: This imply that the over/under investment results arises from the sign of the right hand side expression, which in general is ambiguous in our model. For example, if the second order condition holds, we have that firms will underinvest if. Grunfeld s point out that in the endogenous models the critical rate of spillovers ( ) at which equilibrium R&D is the same in the RJV game as in the non-cooperative game, is higher when we take into consideration the absorptive capacity effect of R&D compared with the case with exogenous R&D spillover rates. This result does not hold in our model as explained in the following statement 12 : Let be the critical spillover rate equating equilibrium R&D for cooperative RJV cartel and endogenous noncooperative games. can be either higher equal or lower than the critical rate for the AJ exogenous model (0.5) Figure 4 shed some light on this issue. We have four regions defined by two critical locus of points associated to both the equivalence of the RJV and Non-cooperative R&D investment and the critical spillover rate for the AJ exogenous model (0.5). As an example: underinvestment in R&D is consistent with spillover rates either higher or lower than the critical spillover rate of the exogenous AJ model (see points and in Figure 4, respectively). Note that for extreme low (high) values for sigma the model predict over(under)-investment independently of the number of firms. 13 12 Grunfeld duopoly model predicts a higher spillover rate than the exogenous AJ model. 13 This last results does not always hold in a product differentiation model.
Figure 4. Cooperation and over/under-investment in the endogenous model 14 3. Welfare analysis (Preliminary results) The welfare function in a symmetric R&D equilibrium is of the form: where, in our model. Simulations in Figure 5 show how the learning/distance parameter affect welfare in our absorptive capacity model. Our first results are the following ones: Non-cooperative solution tends to first best (the unconstrained first stage solution, given optimal Cournot responses) as σ increases. Cooperative solution is welfare superior only for low. 14 Parameter values are:.
Figure 5. Welfare as a function of learning/distance parameter: cooperative, non-cooperative and 1º best 15 The second result is sensitive to the product differentiation assumption. If we increases product differentiation (decreases ) the region in which cooperation is preferred in terms of total welfare expands. Simulations for the basic model we are dealing with shows that when (monopoly power), noncooperative R&D model is always inferior in terms of welfare than the cooperative model. Figure 6. Welfare dominance regions between cooperative and non-cooperative models 16 15 Parameter values are:. 16 Parameter values are:.
Finally, the optimal level of learning distance parameter for the non-cooperative case as a function of the market size per firm is depicted in Figure 7. Grunfeld s states in his Result 4 that highest welfare in a small market is reached when the absorptive capacity effect of R&D ( ) is large, while welfare is highest in a large market when the absorptive capacity effect of R&D is small. We find the opposite result in our model: optimal sigma is negatively affected by and (see Figure 7). At the moment, we do not have a clear explanation of this apparent contradiction, however, it is clear that when N is included in the model, some mechanism seem to be strong enough to change results in such a qualitative way. 17 Figure 7 Optimal learning/distance parameter as a function of and 4. Discussion and future work As future steps we expect to: Find empirical evidence on the existence and effects of different degrees in technological distance, Identify the theoretical aggregated value of the model and its possible extensions, and Give some initial interpretation based on the formalization of the relevant results obtained in simulations. Main results: Endogenous effect is ruled by a relation between the learning/distance and the market structure parameters. 17 In this model, we guess influence at least in three ways: competitive pressure, spillover magnitude and distance effect.
With respect to the previous literature, there exists regions related to our new learning/distance parameter in which the endogenous model has a higher or lower R&D investment than both the AJ and BS ones.
References d Aspremont, C., Jacquemin, A., 1988. Cooperative and noncooperative R&D in duopoly with spillovers. American Economic Review 78 (5), 1133 1137. Brander, J.A., Spencer, B.J., 1983. Strategic commitment with R&D. The symmetric case. The Bell Journal of Economics 14 (1), 225 235. Grunfeld, L.A.,2003. Meet me halfway but don t rush: absorptive capacity and strategic R&D investment revisited. International Journal of Industrial Organization 21, 1091-1109.
Appendix Appendix 1. Second stage equilibrium output derivation: Second stage F.O.C for firm is: Right hand side expression is constant, thus: Replacing in (X.1) Appendix 2. First stage non-cooperative equilibrium In a symmetric equilibrium, first order condition was stated as following:
1 2 =0 Let Proof of Proposition 1 For we are in the AJ model with endogenous spillover rate. Given that is increasing in, we will have that the sign of the learning effect is the same as, or as of expression. Appendix 3. Cooperative R&D Objective function: Given :
We obtain the expression for : Also we have that: