Ownership as R&D incentive in business groups

Abstract

Small and medium-sized enterprises (SMEs) are supposed to be less likely to conduct formal R&D because of the lack of financial resources, weaker competencies, and the absence of scale and scope economies. These limitations may be overcome when an SME belongs to a business group. Empirical studies have shown that firms belonging to business groups have a higher propensity to engage in R&D. We demonstrate that this higher propensity depends on the ownership of controlled companies, besides the presence of coordination mechanisms. We develop a model, and we empirically test its predictions using a data set of Italian SMEs operating in the manufacturing sector. From the model we derive three main implications: (1) there is no difference in R&D propensity between standalone firms and firms at the bottom of groups; (2) head and intermediate firms have a higher R&D propensity than standalone firms and firms at the bottom of the group; (3) the intensity of R&D depends on the ownership of controlled firms and on their size. Overall, the results of the empirical analysis are in accordance with the implications of the model.

Appendix: Proof of results

First, let us consider the profit functions of firms S and B. Thanks to definitions 4 and 5 and assumptions 1, 2 and 3, we have that:

$$ r_{\text{S}} ,x_{\text{S}} \in \mathop {\arg \hbox{max} }\limits_{{r_{\text{S}} ,x_{\text{S}} }} \varPi_{\text{S}} = (a - bx_{\text{S}} )x_{\text{S}} - \left( {x_{\text{S}} + c - dr_{\text{S}} - \sum\limits_{{j \ne {\text{S}}}} {sr_{j} } } \right)x_{\text{S}} - \frac{1}{2}r_{\text{S}}^{2} $$

(1)

$$ r_{\text{B}} ,x_{\text{B}} \in \mathop {\arg \hbox{max} }\limits_{{r_{\text{B}} ,x_{\text{B}} }} \varPi_{\text{B}} = (a - bx_{\text{B}} )x_{\text{B}} - \left( {x_{\text{B}} + c - dr_{\text{B}} - \sum\limits_{{j \ne {\text{B}}}} {sr_{j} } } \right)x_{\text{B}} - \frac{1}{2}r_{\text{B}}^{2} $$

Since Π _S = Π _B, r ^*_S  = r ^*_B . Therefore firm S decides to engage in R&D spending (r ^*_S  > 0) if and only if firm B also does so. This proves Result 1.

In order to show Result 2 let us begin by considering the profit function of a firm S [see (1)]. Now let us consider the I firm. From definitions 3 and 6 and assumptions 1, 2 and 3 for firm I, the profit function is:¹⁸

$$ \varPi_{\text{I}} = (a - bx_{\text{I}} )x_{\text{I}} - \left( {x_{\text{I}} + c - dr_{\text{I}} - \sum\limits_{{j \ne {\text{I}}}} {sr_{j} } } \right)x_{\text{I}} - \frac{1}{2}r_{\text{I}}^{2} + \sum\limits_{{j \ne {\text{I}},j \ne {\text{H}}}} {\alpha_{{{\text{I}},j}} \varPi_{j} } $$

Now, note that if a firm decides to engage in R&D it is because its benefits are greater than its costs. This means that:

$$ (1 / 2) { }r_{\text{S}}^{ 2} \le dx_{\text{S}} $$

(2)

$$ (1 / 2) { }r_{\text{I}}^{ 2} \le dr_{\text{I}} + \sum_{{j \ne {\text{I}}}} \alpha_{\text{I,B}} s \, r_{\text{I}} x_{j} $$

(3)

Since ∑ _j≠I α _I,B s r _I x _j  > 0, if (2) is satisfied for some positive values of r _S , (3) is also satisfied. However the converse is not necessarily true. A similar argument holds for firm H. This proves Result 2.

In order to show Result 3, consider again the profit function for a generic firm I.

$$ \varPi_{\text{I}} = (a - bx_{\text{I}} )x_{\text{I}} - \left( {x_{\text{I}} + c - dr_{\text{I}} - \sum\limits_{{j \ne {\text{I}}}} {sr_{j} } } \right)x_{\text{I}} - \frac{1}{2}r_{\text{I}}^{2} + \sum\limits_{{j \ne {\text{I}},j \ne {\text{H}}}} {\alpha_{{{\text{I}},j}} \varPi_{j} } $$

First order conditions for an optimum are:

$$ \left\{ {\begin{array}{*{20}c} {a - 2bx_{\text{I}} - 2x_{\text{I}} - c + dr_{\text{I}} + \sum\limits_{{j \ne {\text{I}}}} {sr_{j} } = 0} \\ {dx_{\text{I}} - r_{\text{I}} = 0} \\ \end{array} } \right. $$

We thus obtain:

$$ r_{\text{I}}^{*} = \frac{{d(a - c) + d\sum\limits_{{j \ne {\text{I}}}} {sr_{j} } + 2(b + 1)\sum\limits_{{j \ne {\text{I}},j \ne {\text{H}}}} {\alpha_{{{\text{I}},j}} sx_{j} } }}{{2(b + 1) - d^{2} }} $$

Note that r _I* > 0 implies that 2(b + 1) > d ² and implies also that second order conditions for a maximum are met.¹⁹

It can be easily checked that r _I* is increasing with α _I,j and s x _j .

In the case of a Head, the profit function is similar:

$$ \varPi_{\text{H}} = (a - bx_{\text{H}} )x_{\text{H}} - \left( {x_{\text{H}} + c - dr_{\text{H}} - \sum\limits_{{j \ne {\text{H}}}} {sr_{j} } } \right)x_{\text{H}} - \frac{1}{2}r_{\text{H}}^{2} + \sum\limits_{{j \ne {\text{H}},j \ne {\text{B}}}} {\alpha_{{{\text{H}},j}} \varPi_{j} + } \sum\limits_{{j \ne {\text{H}},j \ne {\text{I}}}} {\beta_{{{\text{H}},j}} \varPi_{j} } $$

and so the optimal amount of r _H*:

$$ r_{\text{H}}^{*} = \frac{{d(a - c) + d\sum\limits_{{j \ne {\text{H}}}} {sr_{j} } + 2(b + 1)\left( {\sum\limits_{{j \ne {\text{H}},j \ne {\text{B}}}} {\alpha_{{{\text{H}},j}} sx_{j} } + \sum\limits_{{j \ne {\text{H}},j \ne {\text{I}}}} {\beta_{{{\text{H}},j}} sx_{j} } } \right)}}{{2(b + 1) - d^{2} }} $$

Also, in this case, it can be easily checked that r _H* is increasing with α _H,j , β _H,j and s x _j .