Growth and entrepreneurship

Abstract

In this paper we suggest that the spillover of knowledge may not occur automatically as typically assumed in models of endogenous growth. Rather, a mechanism is required to serve as a conduit for the spillover and commercialization of knowledge from the source creating it, to the firms actually commercializing the new ideas. In this paper, entrepreneurship is identified as one such mechanism facilitating the spillover of knowledge. Using a panel of entrepreneurship data from 18 countries, we provide empirical evidence that, in addition to measures of Research & Development and human capital, entrepreneurial activity also serves to promote economic growth.

Appendix

Entrepreneurs and researchers engage in knowledge production in order to develop a new variety of a differentiated capital good that is used in final production. Different varieties of capital goods compete in a monopolistic competition fashion, meaning that they never become obsolete and earn an infinite stream of profits. As a side effect of their efforts, researchers and entrepreneurs produce new knowledge that will be publicly available for use in future capital good development. Equation A1.1 describes the production of new knowledge, i.e., the evolution of the stock of knowledge, in relation to resources channelled into R&D (L _R ) and entrepreneurial activity (L _E ).

$$ {\frac{{\dot{A}}}{A}} = \sigma_{R} L_{R} + \sigma_{E} Z\left( {L_{E} } \right) $$

(A1.1)

Entrepreneurial activities take the following form

$$ Z\left( {L_{E} } \right) = L_{E}^{\gamma } ,\gamma < 1 $$

(A1.2)

Production of final goods (Y) takes place using labour and different varieties of capital-goods:

$$ Y = L_{m}^{\alpha } \int\limits_{0}^{A} {x(i)^{1 - \alpha } di} $$

(A1.3)

Given the symmetry of different varieties in (A1.3), demand for all varieties in equilibrium is symmetric, i.e., $x_{i} = \bar{x}$ for all i ≤ A. We therefore rewrite (A1.3) as

$$ Y = L_{m}^{\alpha } A\bar{x}^{1 - \alpha } $$

(A1.4)

Assume that capital goods are produced with the same technology as final goods and that it takes K units of capital goods to produce one unit of capital (see, for example, Chiang 1992). Then it can be shown that

$$ K = \kappa A\bar{x} $$

(A1.5)

(A1.4), and (A1.5) then gives

$$ Y = L_{m}^{\alpha } A^{\alpha } K^{1 - \alpha } \kappa^{\alpha - 1} $$

(A1.6)

Labour market equilibrium implies that employment in R&D, entrepreneurship and final production equal total labor supply.

$$ L = L_{m} + L_{R} + L_{E} $$

(A1.7)

Finally, we assume that consumer preferences can be described by constant elasticity utility

$$ U\left( C \right) = {\frac{{C^{1 - \theta } }}{1 - \theta }} $$

(A1.8)

We form the Hamiltonian for the representative consumer

$$ H_{C} = {\frac{{C^{1 - \theta } }}{1 - \theta }} + \lambda_{A} \left( {\sigma_{R} L_{R} A + \sigma_{E} L_{E}^{\gamma } A} \right) + \lambda_{K} \left( {\kappa^{\alpha - 1} A^{\alpha } K^{1 - \alpha } \left( {L - L_{R} - L_{E} } \right) - C} \right) $$

(A1.9)

Maximizing (A1.9) gives the first-order conditions

$$ \lambda_{K} = C^{ - \theta } \to {\frac{{\dot{\lambda }_{K} }}{{\lambda_{K} }}} = - \theta {\frac{{\dot{C}}}{C}} $$

(A1.10)

$$ \Updelta = {\frac{{\lambda_{A} \sigma_{R} A}}{{\lambda_{K} \alpha }}}\left( {L - L_{R} - L_{E} } \right) $$

(A1.11)

$$ \Updelta = {\frac{{\lambda_{A} \gamma \sigma_{E} L_{E}^{\gamma - 1} A}}{{\lambda_{K} \alpha }}}\left( {L - L_{R} - L_{E} } \right) $$

(A1.12)

where $\Updelta = \left( {\kappa^{\alpha - 1} A^{\alpha } K^{1 - \alpha } \left( {L - L_{R} - L_{E} } \right)} \right)$. Combining (A1.11) and (A1.12) gives

$$ L_{E} = \left( {{\frac{{\sigma_{R} }}{{\gamma \sigma_{E} }}}} \right)^{{{\frac{1}{\gamma - 1}}}} $$

(A1.13)

Thus, on a balanced growth path, where both R&D and entrepreneurship is profitable, the amount of resources engaged in entrepreneurial activities is independent of consumer preferences. As γ is less than 1, entry into entrepreneurship is increasing in σ _E and decreasing in σ _R . The maximization of (A1.9) also gives the equations of motion for the shadow prices of knowledge and capital as

$$ {\frac{{\dot{\lambda }_{K} }}{{\lambda_{K} }}} = - \left( {1 - \alpha } \right)K^{ - 1} \Updelta + \rho $$

(A1.14)

$$ {\frac{{\dot{\lambda }_{A} }}{{\lambda_{A} }}} = - \sigma_{R} L_{0} - \sigma_{E} L_{E}^{\gamma } + \sigma_{R} L_{E} + \rho $$

(A1.15)

where ρ denotes the subjective discount rate (rate of time preferences). On the balanced growth, knowledge, final production and consumption all grow at the same rate, while ${\frac{{\dot{\lambda }_{K} }}{{\lambda_{K} }}} = {\frac{{\dot{\lambda }_{A} }}{{\lambda_{A} }}}$. Combining (A1.10) and (A1.15) gives

$$ L_{R} = {\frac{1}{{\theta \sigma_{R} }}}\left( {\sigma_{R} \left( {L_{0} - L_{E} } \right) + \left( {1 - \theta } \right)\sigma_{E} L_{E}^{\gamma } - \rho } \right) $$

(A1.16)

Combining (A1.16) with (A1.13) and (A1.1) gives

$$ g = {\frac{1}{\theta }}\left( {\left( {\sigma_{R} L - \rho } \right) - \sigma_{R}^{\gamma } \gamma^{\gamma - 1} \sigma_{{_{E} }}^{\gamma - 1} + \sigma_{{_{E} }}^{{{\frac{2\gamma - 1}{\gamma }}}} \gamma^{{{\frac{\gamma - 1}{\gamma }}}} \sigma_{{_{R} }}^{{{\frac{\gamma }{\gamma - 1}}}} } \right) $$

(A1.17)

where it can be shown that the growth rate is increasing in $L,\sigma_{R} {\text{ and }}\sigma_{E} .$ but decreasing in ρ. It should be noted that (A1.17) only applies when both R&D and entrepreneurship is profitable. The given specification implies that some entrepreneurial activity will always be profitable as long as A > 0. This does not apply to R&D activities however. If R&D is not sufficiently profitable (following from A1.16), then we can combine (A1.10), (A1.12), (A1.14) and (A1.15) to derive the reduced-form growth rate. However, the resulting expression provides few new insights and is not shown here.