Research and innovation in higher education: empirical evidence from research and patenting in Brazil

Abstract

This study presents a hierarchical differential game between universities and scholars in order to examine innovation and research in higher education. In this stylized setup, scholars maximize the impact of their research, and universities maximize their market value. Innovations play a key role among the incentives given by the university to boost scholars’ productivity, as measured by academic publications and citations, which translates into scholars’ professional success. The scholars’ academic productivity increases university reputation and market value. Using Brazilian data, seemingly unrelated regression estimations suggest that the number of published papers grows with external funding and the percentage of faculty holding doctorate degrees, while the number of citations is associated with the presence of graduate programs and higher teaching quality. Market evaluation is, however, negatively affected by innovation, suggesting a lack of focus on patenting and technology transfer in Brazil.

Mathematical appendix

The Hamiltonian, first order condition (FOC) and adjoint equation for the scholar are,

$$H_{s} = aIP - \frac{b}{2}P^{2} + \delta C + \lambda [\rho P + mS - \sigma C + \varOmega ],$$

(17)

$$\frac{{{\text{d}}H_{\text{s}} }}{{{\text{d}}P}} = aI - bP + \lambda \rho = 0 \Rightarrow P = \frac{aI + \lambda \rho }{b},$$

(18)

and

$$\dot{\lambda}- r\lambda = - \frac{{{\text{d}}H_{s} }}{{{\text{d}}C}} \Rightarrow \dot{\lambda} = \lambda (r + \sigma ) - \delta ,$$

(19)

where λ is the shadow price of citations for the scholar. The university takes (7), (5) and the best-reply function of the scholar, given by (9) and (10). Its Hamiltonian, FOC and adjoint equations are,

$$H_{U} = XI - 0.5I^{2} + v\left( {\frac{aI + \lambda \rho }{b}} \right) + n\left( {1 - \frac{aI + \lambda \rho }{b}} \right) + \mu_{1} \left[ {\rho \left( {\frac{aI + \lambda \rho }{b}} \right) + mS - \sigma C + \varOmega } \right] + \mu_{2} \left[ {\lambda (r + \sigma ) - \delta } \right],$$

(20)

$$\frac{{{\text{d}}H_{P} }}{{{\text{d}}I}} = X - I + b^{ - 1} (v - n)a + \mu_{1} b^{ - 1} \rho a = 0 \Rightarrow I = X + b^{ - 1} (v - n + \mu_{1} \rho )a,$$

(21)

$$\dot{\mu}_{1} - \theta \mu_{1} = - \frac{{{\text{d}}H_{P} }}{{{\text{d}}C}} \Rightarrow \dot{\mu}_{1} = (\theta + \sigma )\mu_{1} ,$$

(22)

and

$$\dot{\mu}_{2} - \theta \mu_{2} = - \frac{{{\text{d}}H_{U} }}{{{\text{d}}\lambda }} \Rightarrow \dot{\mu}_{2} = \theta \mu_{2} - [b^{ - 1} (v - n + \mu_{1} \rho )\rho + \mu_{2} (r + \sigma )].$$

(23)

Differentiating (21) with respect to time and inserting (22) yields the differential equation for the university’s innovations appearing as (8) in the main body of the text. Using (8), one can derive a differential equation for publications, which is obtained by differentiating (18) with respect to time and inserting (8) and (19).