Absorptive Capacity and Innovation: When Is It Better to Cooperate?

Abstract

Cooperation can benefit and hurt firms at the same time. An important question then is: when is it better to cooperate? And, once the decision to cooperate is made, how can an appropriate partner be selected? In this paper we present a model of inter-firm cooperation driven by cognitive distance, appropriability conditions and external knowledge. Absorptive capacity of firms develops as an outcome of the interaction between absorptive R&D and cognitive distance from voluntary and involuntary knowledge spillovers. Thus, we offer a revision of the original model by Cohen and Levinthal (Econ J 99(397):569–596, 1989), accounting for recent empirical findings and explicitly modeling absorptive capacity within the framework of interactive learning. We apply that to the analysis of firms’ cooperation and R&D investment preferences. The results show that cognitive distance and appropriability conditions between a firm and its cooperation partner have an ambiguous effect on the profit generated by the firm. Thus, a firm chooses to cooperate and selects a partner conditional on the investments in absorptive capacity it is willing to make to solve the understandability/novelty trade-off.

Appendices

Appendix 1: Resolving the Investment Trade-Off (Eq. 14) to Find ρ _i

The objective is to obtain values of ρ _i that satisfy:

$$\displaystyle{ \frac{\partial \Pi _{i}} {\partial \mathit{aci}_{i}} = \frac{\partial \Pi _{i}} {\partial \mathit{rdi}_{i}}. }$$

Recall from (13) that in case of a partnership, where i needs to optimise its investment allocation conditional upon the partner’s investments,

$$\displaystyle{ \Pi = \frac{k_{i}} {1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i}}. }$$

Hence,

$$\displaystyle{ \frac{\partial \Pi _{i}} {\partial \mathit{rdi}_{i}}\ =\ \frac{\partial ( \frac{k_{i}} {1+\mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i}})} {\partial \mathit{rdi}_{i}} = \frac{\xi \mathit{rdi}_{i}^{\xi -1}(1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i}) - k_{i}\delta _{c}\mathit{ac}_{j,i}} {(1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i})^{2}}, }$$

(17)

$$\displaystyle{ \frac{\partial \Pi _{i}} {\partial \mathit{aci}_{i}}\ =\ \frac{\partial \left ( \frac{k_{i}} {1+\mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i}}\right )} {\partial \mathit{aci}_{i}} = \frac{(1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i})\left ( \frac{\partial k_{i}} {\partial \mathit{ac}_{i}}\right ) + k_{i}\delta _{c}\mathit{ac}_{j,i}} {(1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i})^{2}}, }$$

(18)

where $E^{i}(\rho _{j,t}) = \frac{\sum _{\iota =1}^{\sigma }\rho _{ j}^{t-\iota }} {\sigma } \ \Rightarrow \ \frac{\partial \mathit{ac}_{j,i}} {\partial \mathit{rdi}_{i}} = 0$ and $\mathit{rd}_{i} = \mathit{RD}_{i} -\mathit{aci}_{i}$   ⇒   $\frac{\partial \mathit{rdi}_{i}} {\mathit{aci}_{i}} = -1$. Next we set (17) equal to (18) as in Eq. (14):

$$\displaystyle{ \xi \mathit{rdi}_{i}^{\xi -1}(1 + \mathit{ac}_{ j,i}\delta _{c}\mathit{rdi}_{i}) - k_{i}\delta _{c}\mathit{ac}_{j,i}\ =\ (1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i})\left ( \frac{\partial k_{i}} {\partial \mathit{ac}_{i}}\right ) + k_{i}\delta _{c}\mathit{ac}_{j,i} }$$

and collect terms:

$$\displaystyle{ \frac{\xi \mathit{rdi}_{i}^{\xi -1}} {\left ( \frac{\partial k_{i}} {\partial \mathit{ac}_{i}}\right )}\ =\ \frac{2k_{i}\delta _{c}\mathit{ac}_{j,i}} {(1 + \mathit{ac}_{j,i}\delta _{c}\mathit{rdi}_{i})}. }$$

(19)

Recalling the expression for k _i from (11) we obtain

$$\displaystyle{ \frac{\partial k_{i}} {\partial \mathit{ac}_{i}}\ =\ \delta _{c}\mathit{rdi}_{j}\left (\frac{\partial \mathit{ac}_{i,j}} {\partial \mathit{aci}_{i}} \right ) + \mathit{ek}\left (\frac{\partial \mathit{ac}_{i,\mathit{ek}}} {\partial \mathit{aci}_{i}} \right ). }$$

(20)

Accounting for the difference in d _ij and d _iek in ac _i, ⋅ (9) we obtain the derivative of the absorptive capacity function with respect to distance as follows:

$$\displaystyle{ \frac{\partial \mathit{ac}_{i,\cdot }} {\partial \mathit{aci}_{i}} \ =\ \frac{4\beta _{2}\psi d_{i\cdot }\mathit{aci}_{i}^{\psi -1}} {\beta _{1}(1 + \mathit{aci}_{i}^{\psi })^{2}}\left [ \frac{2\beta _{2}d_{i\cdot }} {\beta _{1}(1 + \mathit{aci}^{\psi })} - 1\right ]. }$$

(21)

Inserting (21) into (20) accordingly:

$$\displaystyle\begin{array}{rcl} \frac{\partial k_{i}} {\partial \mathit{ac}_{i}}\ & =& \ \delta _{c}\mathit{rdi}_{j}\left ( \frac{4\beta _{2}\psi d_{\mathit{ij}}\mathit{aci}_{i}^{\psi -1}} {\beta _{1}(1 + \mathit{aci}_{i}^{\psi })^{2}}\left [ \frac{2\beta _{2}d_{\mathit{ij}}} {\beta _{1}(1 + \mathit{aci}^{\psi })} - 1\right ]\right ) \\ & +& \ \mathit{ek}\left (\frac{4\beta _{2}\psi d_{\mathit{iek}}\mathit{aci}_{i}^{\psi -1}} {\beta _{1}(1 + \mathit{aci}_{i}^{\psi })^{2}} \left [ \frac{2\beta _{2}d_{\mathit{iek}}} {\beta _{1}(1 + \mathit{aci}^{\psi })} - 1\right ]\right ).{}\end{array}$$

(22)

Note that the absorptive capacity of firm j directed on firm i is:

$$\displaystyle{ \mathit{ac}_{j,i}\ =\ \frac{\alpha \beta _{1}d_{\mathit{ij}} +\alpha \beta _{1}d_{\mathit{ij}}\mathit{aci}_{j}^{\psi } -\alpha \beta _{2}d_{\mathit{ij}}^{2}} { \frac{1} {4\alpha \beta _{2}} \left [\alpha \beta _{1}(1 + \mathit{aci}_{j}^{\psi })\right ]^{2}} \ \ \mbox{ as}\ \ d_{\mathit{ij}} = d_{\mathit{ji}}. }$$

(23)

When (22) and (23) are inserted in (19) and the latter is rearranged, we obtain

$$\displaystyle\begin{array}{rcl} \mathit{rdi}_{i}& =& \frac{32\beta _{2}^{2}} {\xi \alpha \beta _{1}^{4}\left (\beta _{1}+\beta _{1}\mathit{aci}_{j}^{\psi }-\beta _{2}d_{\mathit{ij}}\right )\left (1+\mathit{aci}_{i}^{\psi }\right )^{5}}\Biggl (\delta _{c}\mathit{rdi}_{j}d_{\mathit{ij}}\left (2\beta _{2}d_{\mathit{ij}}-\beta _{1}\left (1+\mathit{aci}_{i}^{\psi }\right )\right ) + \\ & & +\ \mathit{ek}d_{\mathit{iek}}\left (2\beta _{2}d_{\mathit{iek}} -\beta _{1}\left (1 + \mathit{aci}_{i}^{\psi }\right )\right )\Biggr )\frac{\mathit{aci}_{i}^{\psi -1}} {\mathit{rdi}_{i}^{\xi -1}} \left (\beta _{1} +\beta _{1}\mathit{aci}_{j}^{\psi } -\beta _{ 2}d_{\mathit{ij}}\right ) \cdot \\ & &\cdot \Biggl (\frac{\mathit{rdi}_{i}^{\xi }} {4\alpha \beta _{2}} \left (\alpha \beta _{1}\left (1 + \mathit{aci}_{i}^{\psi }\right )\right )^{2}\ +\ \alpha \delta _{ c}\mathit{rdi}_{j}d_{\mathit{ij}}\left (\beta _{1} +\beta _{1}\mathit{aci}_{i}^{\psi } -\beta _{ 2}d_{\mathit{ij}}\right )\ + \\ & & +\ \alpha d_{\mathit{iek}}\mathit{ek}\left (\beta _{1} +\beta _{1}\mathit{aci}_{i}^{\psi } -\beta _{ 2}d_{\mathit{iek}}\right )\Biggr ) - \frac{\beta _{1}\left (1 + \mathit{aci}_{j}^{\psi }\right )^{2}} {4\beta _{2}\delta _{c}d_{\mathit{ij}}\left (\beta _{1} +\beta _{1}\mathit{aci}_{j}^{\psi } -\beta _{2}d_{\mathit{ij}}\right )}. {}\end{array}$$

(24)

Recall from (1) that $\mathit{rdi}_{i} =\rho _{i}\mathit{RD}_{i}\ \mbox{ and}\ \mathit{aci}_{i} = (1 -\rho _{i})\mathit{RD}_{i}$; when this is applied to Eq. (24) it takes the form:

$$\displaystyle\begin{array}{rcl} \rho _{i}\ & =& \ \frac{32\beta _{2}^{2}} {\xi \alpha \beta _{1}^{4}\mathit{RD}_{i}\left (\beta _{1} +\beta _{1}\left (\left (1 -\rho _{j}\right )\mathit{RD}_{j}\right )^{\psi } -\beta _{2}d_{\mathit{ij}}\right )\left (1 + \left (\left (1 -\rho _{i}\right )\mathit{RD}_{i}\right )^{\psi }\right )^{5}} \\ & \cdot & \ \Biggl (\delta _{c}\rho _{j}\mathit{RD}_{j}d_{\mathit{ij}}\left (2\beta _{2}d_{\mathit{ij}} -\beta _{1}\left (1 + \left (\left (1 -\rho _{i}\right )\mathit{RD}_{i}\right )^{\psi }\right )\right ) \\ & +& \ \mathit{ek}d_{\mathit{iek}}\left (2\beta _{2}d_{\mathit{iek}} -\beta _{1}\left (1 + \left (\left (1 -\rho _{i}\right )\mathit{RD}_{i}\right )^{\psi }\right )\right )\Biggr )\frac{\left (1 -\rho _{i}\right )^{\psi -1}\mathit{RD}_{i}^{\psi -\xi }} {\rho _{i}^{\xi -1}} \ \\ & \cdot & \ \left (\beta _{1} +\beta _{1}\left ((1 -\rho _{j})\mathit{RD}_{j}\right )^{\psi } -\beta _{2}d_{\mathit{ij}}\right )\Biggl (\frac{(\rho _{i}\mathit{RD}_{i})^{\xi }} {4\alpha \beta _{2}} \left (\alpha \beta _{1}\left (1 + \left ((1 -\rho _{i})\mathit{RD}_{i}\right )^{\psi }\right )\right )^{2} \\ & +& \ \alpha \delta _{c}\rho _{j}\mathit{RD}_{j}d_{\mathit{ij}}\left (\beta _{1} +\beta _{1}\left ((1 -\rho _{i})\mathit{RD}_{i}\right )^{\psi } -\beta _{2}d_{\mathit{ij}}\right ) \\ & +& \ \alpha d_{\mathit{iek}}\mathit{ek}\left (\beta _{1} +\beta _{1}\left ((1 -\rho _{j})\mathit{RD}_{j}\right )^{\psi } -\beta _{2}d_{\mathit{iek}}\right )\Biggr ) \\ & -& \ \frac{\beta _{1}\left (1 + \left ((1 -\rho _{j})\mathit{RD}_{j}\right )^{\psi }\right )^{2}} {4\beta _{2}\delta _{c}d_{\mathit{ij}}\mathit{RD}_{i}\left (\beta _{1} +\beta _{1}\left ((1 -\rho _{j})\mathit{RD}_{j}\right )^{\psi } -\beta _{2}d_{\mathit{ij}}\right )}. {}\end{array}$$

(25)

Shifting ρ _i from the left hand side to the right one, one gets F(ρ _i ) = 0.

Remembering that for firm i performing R&D activity without a partner δ _c  = 0, it is straightforward to show that for this firm (25) takes a simpler form as follows:

$$\displaystyle\begin{array}{rcl} F^{a}(\rho _{ i})\ & =& \mathit{ek}\frac{4\beta _{2}\psi d_{\mathit{iek}}((1 -\rho _{i})\mathit{RD}_{i})^{\psi -1}} {\beta _{1}(1 + ((1 -\rho _{i})\mathit{RD}_{i})^{\psi })^{2}} \left ( \frac{2\beta _{2}d_{\mathit{iek}}} {\beta _{1}(1 + ((1 -\rho _{i})\mathit{RD}_{i})^{\psi })} - 1\right ) - \\ &-& \ \xi \left (\rho _{i}\mathit{RD}_{i}\right )^{\xi -1} = 0. {}\end{array}$$

(26)

Appendix 2: Finding Optimal Solution for F(ρ _i ) and F ^a(ρ _i ) Using Heuristics

Thanks to the recent advances in computing technology, new nature-inspired optimization methods (called heuristics) tackling complex combinatorial optimization problems and detecting global optima of various objective functions have become available (Gilli and Winker 2009). Differential Evolution (DE), proposed by Storn and Price (1997), is a population based optimization technique for continuous objective functions. In short, starting with an initial population of solutions, DE updates this population by linear combination and crossover of four different solutions into one, and selects the fittest ones among the original and the updated population. This continues until some stopping criterion is met. Algorithm 1 provides a pseudocode of the DE implementation.

Algorithm 1 Pseudocode for Differential Evolution

  1:    Initialize parameters p, F and $\Omega$

  2:    Randomly initialize $P_{i}^{(1)} \in \Omega$, i = 1, ⋯ , p

  3:    while the stopping criterion is not met do

  4:        P ⁽⁰⁾ = P ⁽¹⁾

  5:        for i = 1 to p do

  6:            Generate r ₁,r ₂,r ₃ ∈ 1, ⋯ ,p, r ₁ ≠ r ₂ ≠ r ₃ ≠ i

  7:            Compute $P_{i}^{(\upsilon )}$ = $P_{r_{1}}^{(0)}$ + F × $(P_{r_{2}}^{(0)}$ - $P_{r_{3}}^{(0)})$

  8:            if $P_{i}^{(\upsilon )} \in \Omega$ then $P_{i}^{(n)} = P_{i}^{(\upsilon )}$ else repair $P_{i}^{(\upsilon )}$

  9:            if $F(P_{i}^{(n)}) < F(P_{i}^{(0)})$ then $P_{i}^{(1)} = P_{i}^{(n)}$ else $P_{i}^{(1)} = P_{i}^{(0)}$

10:        end for

11:    end while

In contrast to other DE applications to optimization problems (as described in, for example, Blueschke et al. 2013), our solution is represented by a single value within [0, 1] according to (1). Therefore, DE starts with a population of size p of random values drawn from [0, 1] ($\Omega$) (2:). For the same reason, current DE implementation has no need in the crossover operator (otherwise, one would have to compare $F(P_{i}^{(0)})$ with itself and potentially waste computational time). Tuning our DE code we set p = 30, F = 0. 8 and as a stopping criterion we choose a combination of two conditions: either a maximum number of generations is reached (which is set to be equal 50²⁹) or the global optimum is identified ($F(P_{i}^{(1)}) = 0$). To make sure that our candidate solutions constructed by linear combination (7:) satisfy our constraint on ρ _i , we explicitly check it in (8:)—and if it is not met we ‘repair’ it by adding/deducting one unit—before comparing its fitness with the current solutions in (9:).

As an illustration of the DE convergence for the tuning parameters stated consider Fig. 4 below. On the left plot one can see F(ρ _i ) simulated for different ρ _i  ∈ [0, 1], while on the right plot the cumulative density function of F(ρ _i ) for 100 restarts and different number of maximum generations g (10, 30 and 50) is given. Obviously, with g = 50 DE converges to zero (or a very close approximation of it) in almost 100 % of restarts. To ensure a good solution, therefore, we take g = 30 and restart DE three times. Using Matlab 7.11 on Pentium IV 3.3 GHz a single DE restart with thirty generations requires about 0.02 s.

Fig. 4

F(ρ _i ) for different ρ _i and empirical distribution of F(ρ _i ) for different g