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The R&D Productivity Puzzle: Innovation Networks with Heterogeneous Firms

M. Sadra Heydari, Zafer Kanik, Santiago Montoya-Blandón

Abstract

We introduce heterogeneous R&D productivities into an endogenous R&D network formation model, generalizing the framework of Goyal and Moraga-González (2001). Heterogeneous productivities endogenously create asymmetric gains from collaboration: less productive firms benefit disproportionately from links, while more productive firms exert greater R&D effort and incur higher costs. When productivity gaps are sufficiently large, more productive firms experience lower profits from collaborating with less productive partners. As a result, the complete network -- stable under homogeneity -- becomes unstable, and the positive assortative (PA) network, in which firms cluster by R&D productivity, emerges as pairwise stable. Using simulations, we show that the clustered structure delivers higher welfare than the complete network; nevertheless, welfare under this formation follows an inverted U-shape as the fraction of high-productivity firms increases, reflecting crowding-out effects at high fractions. Altogether, we uncover an R&D productivity puzzle: economies with higher average R&D productivity may exhibit lower welfare through (i) the formation of alternative stable networks, or (ii) a crowding-out effect of high-productivity firms. Our findings highlight that productivity is a structural force reshaping the organization of innovation. Productivity-enhancing policies must therefore account for their effects on endogenous R&D alliances and effort, as they may reverse the intended welfare gains.

The R&D Productivity Puzzle: Innovation Networks with Heterogeneous Firms

Abstract

We introduce heterogeneous R&D productivities into an endogenous R&D network formation model, generalizing the framework of Goyal and Moraga-González (2001). Heterogeneous productivities endogenously create asymmetric gains from collaboration: less productive firms benefit disproportionately from links, while more productive firms exert greater R&D effort and incur higher costs. When productivity gaps are sufficiently large, more productive firms experience lower profits from collaborating with less productive partners. As a result, the complete network -- stable under homogeneity -- becomes unstable, and the positive assortative (PA) network, in which firms cluster by R&D productivity, emerges as pairwise stable. Using simulations, we show that the clustered structure delivers higher welfare than the complete network; nevertheless, welfare under this formation follows an inverted U-shape as the fraction of high-productivity firms increases, reflecting crowding-out effects at high fractions. Altogether, we uncover an R&D productivity puzzle: economies with higher average R&D productivity may exhibit lower welfare through (i) the formation of alternative stable networks, or (ii) a crowding-out effect of high-productivity firms. Our findings highlight that productivity is a structural force reshaping the organization of innovation. Productivity-enhancing policies must therefore account for their effects on endogenous R&D alliances and effort, as they may reverse the intended welfare gains.
Paper Structure (14 sections, 7 theorems, 47 equations, 8 figures)

This paper contains 14 sections, 7 theorems, 47 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Equilibrium Characterization
  4. Market Competition
  5. Equilibrium R&D Efforts
  6. R&D Network Formation
  7. Simulation preview: Link sustainability for random networks
  8. Simulation exercises for stability and welfare
  9. Welfare Comparison
  10. Crowding-out Effect
  11. Network Structure vs. Density Effect
  12. Conclusion
  13. Proofs
  14. Additional Simulation Results

Key Result

Proposition 1

There exists a unique optimal effort profile ${\mathbf{e}}^{*}(\mathbf{G}) \coloneqq \{{e}^{*}_j(\mathbf{G})\}_{j=1}^{n}$ with ${e}^{*}_i(\mathbf{G}) > 0$ for all $i \in \mathcal{N}$ that solves equation eq:optimal_effort_matrix if the cost parameter $\phi$ satisfies:

Figures (8)

  • Figure 1: Percentage changes in profits of two arbitrary firms after forming an R&D collaboration link between them in random networks with varying connection density and firm productivity distributions
  • Figure 2: Pairwise stability domains for $n=4$ firms with two high- and two low-productivity firms for different productivity gap levels
  • Figure 3: Welfare, profit, and effort comparison of pairwise stable structures in $n=6$ setting, with $\rho=1/2$
  • Figure 4: Crowding-out effect of high-productivity firms on welfare
  • Figure 5: Welfare over Network Link Density for Random vs. PA/Complete Structures
  • ...and 3 more figures

Theorems & Definitions (15)

  • Proposition 1: Cost parameter bound
  • Definition 1: Symmetric position
  • Proposition 2: Effort and profit ratios under symmetric positions
  • Corollary 1
  • Proposition 3: Thresholds for complete network stability
  • proof : Proof of Proposition \ref{['prop:equilibrium_existance']}
  • proof : Proof of Proposition \ref{['prop:summetric_ratio']}
  • proof : Proof of Corollary \ref{['corrolary_profit']}
  • proof : Proof of Proposition \ref{['prop:FC_stable']}
  • Lemma 1
  • ...and 5 more