In Sect. 3 we modeled individuals’ intertemporal occupational choice, i.e., providing the analysis with a solid microeconomic base. Linking individual behavior to the macrolevel, and to illustrate the role of entrepreneurs in growth, we suggest a modification of the Romer (1990) model to incorporate “pure” Schumpeterian entrepreneurs (Schumpeter 1911).Footnote 11 Hence, there are now two methods of developing new products: research laboratories in incumbent firms (inventions) and entrepreneurs (innovations). Just as in Romer’s original work we think of these products as either new types of physical capital, blueprints or “business models” that are being rented or sold to final goods producers, thus making production of final goods more effective. As, e.g., Grossman and Helpman (1991) have shown, the new varieties of capital goods can just as well be thought of as new varieties of consumer goods entering consumers’ utility function directly. The different varieties appear on markets characterized by monopolistic competition, meaning that they never become obsolete and earn an infinite stream of profits.
Table 1 Notation and definitions of variables in the theoretical model
Linking entrepreneurs to knowledge exploitation
Before the role of entrepreneurs in knowledge exploitation and growth is considered, we first briefly recapitulate the production function for researchers working in incumbent firms. Research departments within incumbent firms employ labor (L
R) as the only production factor, and research activities are influenced by the available stock of knowledge (A) and an efficiency parameter (σ
R) related to research activities.Footnote 12 The latter is a composite of a multitude of different factors. Thus, in its simplest form, the production function for research activities can be written
$$ Z_{\text{R}} \left( {L_{\text{R}} } \right) = \sigma_{\text{R}} L_{\text{R}} A, $$
(4)
where research production is positively influenced by a larger knowledge stock and higher efficiency.
In order to include the Schumpeterian entrepreneur, we first assume that entrepreneurial ability is embodied in labor, but in contrast to raw labor, it is distributed unevenly across the population. Thus, entrepreneurial activities are assumed to be characterized by decreasing returns to scale (γ < 1). The production function for entrepreneurial activities takes the following form:
$$ Z_{\text{E}} (L_{\text{E}} ) = \sigma_{\text{E}} L_{\text{E}}^{\gamma } A,\,\gamma < 1. $$
(5)
Hence, similar to R&D workers, the representative entrepreneur takes advantage of existing knowledge. On the other hand, the production technology differs (decreasing returns to scale), and they do not engage in research. Rather, they combine their entrepreneurial ability with the existing stock of knowledge to introduce new products and business models.Footnote 13 The different varieties of capital goods (x
i
) produced by entrepreneurs and researchers are employed in the final goods (Y) sector together with labor,
$$ Y = (L - L_{\text{E}} - L_{\text{R}} )^{\alpha } \int\limits_{0}^{A} {x(i)^{1 - \alpha } \text{d}i}, $$
(6)
where α (0 < α < 1) represents the scale parameter. Given that the demand for all varieties in equilibrium is symmetric, i.e., \( x_{i} = \bar{x} \) for all i ≤ A, we rewrite Eq. 6 as
$$ Y = (L - L_{\text{E}} - L_{\text{R}} )^{\alpha } A\overline{x}^{(1 - \alpha )}. $$
(7)
Assume that capital goods (K) are produced with the same technology as final goods and that it takes κ units of capital goods to produce one unit of capital (Chiang 1992). Then it can be shown that
$$ K = \kappa A\bar{x}, $$
(8)
and substituting Eq. 8 into Eq. 7 gives
$$ Y = (L - L_{\text{R}}^{{}} - L_{\text{E}} )^{\alpha } A^{\alpha } K^{1 - \alpha } \kappa^{\alpha - 1}. $$
(9)
Thus, the economy employs three factors of production, i.e., raw labor (producing finals), together with researchers and entrepreneurs that produces varieties of capital goods. Labor market equilibrium is attained when employment in R&D, entrepreneurship, and final production equals total supply:
$$ L = L_{\text{F}}^{{}} + L_{\text{E}} + L_{\text{R}}. $$
(10)
Knowledge production in an economy
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, positively influencing coming generations of research and entrepreneurial activities. Equation 11 describes the production of new knowledge, i.e., the evolution of the stock of knowledge, in relation to the amount of labor channeled into R&D (L
R) and entrepreneurial activity (L
E).
$$ \mathop A\limits^{ \bullet } = Z_{\text{R}} (L_{\text{R}} ) + Z_{\text{E}} (L_{\text{E}} ). $$
(11)
Substituting from Eqs. 4 and 5 yields
$$ \mathop A\limits^{ \bullet }/A = \sigma_{\text{R}} L_{\text{R}} + \sigma_{\text{E}} L_{\text{E}}^{\gamma }, $$
(12)
where, again, σ:s represents the knowledge efficiency in invention activities (R&D) and innovation (entrepreneurship), whereas A is the stock of available knowledge at a given point in time. The rate of technological progress is thus an increasing function in R&D, entrepreneurship, and the efficiency of these two activities.
Endogenous growth with knowledge-exploiting entrepreneurs
Assuming that demand is governed by consumer preferences characterized by constant intertemporal elasticity of substitution (1/θ), the maximization problem can be expressed in following way:
$$ \mathop {\max }\limits_{{C,L_{\text{E}} ,L_{\text{R}} }} \int_{0}^{\infty } {{\frac{{C^{1 - \theta } }}{1 - \theta }}} \text{e}^{ - \rho t} {\text{d}}t $$
(13)
subject to the laws of motion for knowledge and capital
$$ \mathop A\limits^{ \bullet } = \sigma_{\text{R}} L_{\text{R}} A + \sigma_{\text{E}} L_{\text{E}}^{\gamma } A, $$
(14a)
$$ \mathop K\limits^{ \bullet } = Y - C = \left( {L - L_{E} - L_{R} } \right)^{\alpha } A^{\alpha } K^{1 - \alpha } \kappa^{\alpha - 1} - C. $$
(14b)
The current value Hamiltonian for the representative consumer is then
$$ H_{\text{C}} = {\frac{{C^{1 - \theta } }}{1 - \theta }} + \lambda_{\text{A}} \left( {\sigma_{\text{R}} L_{\text{R}} A + \sigma_{\text{E}} L_{\text{E}}^{\gamma } A} \right) + \lambda_{\text{K}} \left( {\kappa^{\alpha - 1} A^{\alpha } K^{1 - \alpha } \left( {L - L_{\text{R}} - L_{\text{E}} } \right) - C} \right). $$
(15)
The first-order conditions for a maximum, letting \( \Updelta \equiv \left( {L - L_{\text{E}} - L_{\text{R}} } \right)^{\alpha } A^{\alpha } K^{1 - \alpha } \kappa^{\alpha - 1} \), are as follows:
$$ {\frac{{\partial H_{\text{C}} }}{\partial C}} = C^{ - \theta } - \lambda_{\text{K}} = 0, $$
$$ \lambda_{\text{K}} = C^{ - \theta } \to {\frac{{\dot{\lambda }_{\text{K}} }}{{\lambda_{\text{K}} }}} = - \theta {\frac{{\dot{C}}}{C}}, $$
(16)
$$ {\frac{{\partial H_{\text{C}} }}{{\partial L_{\text{E}} }}} = \lambda_{\text{A}} \gamma \sigma_{\text{E}} L_{\text{E}}^{\gamma - 1} A - \lambda_{\text{K}} \alpha \left( {L - L_{\text{E}} - L_{\text{R}} } \right)^{ - 1} \Updelta = 0, $$
(17)
$$ {\frac{{\partial H_{\text{C}} }}{{\partial L_{\text{R}} }}} = \lambda_{\text{A}} \sigma_{\text{R}} A - \lambda_{\text{K}} \alpha \left( {L - L_{\text{E}} - L_{\text{R}} } \right)^{ - 1} \Updelta = 0. $$
(18)
Combining Eqs. 17 and 18 gives
$$ L_{\text{E}} = \left( {{\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}} \right)^{{{\frac{1}{\gamma - 1}}}}. $$
(19)
Thus, on a balanced growth path, where both R&D and entrepreneurship are profitable, the amount of resources engaged in entrepreneurial activities is independent of consumer preferences (ρ). As γ is less than 1, entry into entrepreneurship is increasing in σ
E and decreasing in σ
R.
Maximization of Eq. 15 also gives the equations of motion for the shadow prices of capital (K) and knowledge (A) as
$$ {\frac{{\partial H_{\text{C}} }}{\partial A}} = \lambda_{\text{A}} \left( {\sigma_{\text{R}} L_{\text{R}} + \sigma_{\text{E}} L_{\text{E}}^{\gamma } } \right) + \lambda_{\text{K}} \alpha A^{ - 1} \Updelta = \rho \lambda_{\text{A}} - \mathop {\lambda_{\text{A}} }\limits^{ \bullet }, $$
$$ {\frac{{\mathop {\lambda_{\text{K}} }\limits^{ \bullet } }}{{\lambda_{\text{K}} }}} = \rho - \left( {1 - \alpha } \right)K^{ - 1} \Updelta, $$
(20)
\( {\frac{{\partial H_{\text{C}} }}{\partial K}} = \lambda_{\text{K}} \left( {1 - \alpha } \right)K^{ - 1} \Updelta = \rho \lambda_{\text{K}} - \mathop {\lambda_{\text{K}} }\limits^{ \bullet }, \)
$$ {\frac{{\mathop {\lambda_{\text{A}} }\limits^{ \bullet } }}{{\lambda_{\text{A}} }}} = \rho + \sigma_{\text{R}} L_{\text{E}} - \sigma_{\text{R}} L - \sigma_{\text{E}} L_{\text{E}}^{\gamma }, $$
(21)
$$ {\frac{{\partial H_{\text{C}} }}{{\partial \lambda_{\text{A}} }}} = \mathop A\limits^{ \bullet }, $$
(22)
$$ {\frac{{\partial H_{\text{C}} }}{{\partial \lambda_{\text{K}} }}} = \mathop K\limits^{ \bullet }. $$
(23)
A balanced growth path, i.e., where \( {\frac{{\mathop Y\limits^{ \bullet } }}{Y}} = {\frac{{\mathop C\limits^{ \bullet } }}{C}} = {\frac{{\mathop K\limits^{ \bullet } }}{K}} = {\frac{{\mathop A\limits^{ \bullet } }}{A}} \), requires that \( {\frac{{\dot{\lambda }_{\text{K}} }}{{\lambda_{\text{K}} }}} = {\frac{{\dot{\lambda }_{\text{A}} }}{{\lambda_{\text{A}} }}} \). From (16) and the law of motion for knowledge (14a),
$$ {\frac{{\mathop {\lambda_{\text{K}} }\limits^{ \bullet } }}{{\lambda_{\text{K}} }}} = - \theta {\frac{{\mathop C\limits^{ \bullet } }}{C}} = - \theta {\frac{{\mathop A\limits^{ \bullet } }}{A}} = - \theta \left( {\sigma_{\text{R}} L_{\text{R}} + \sigma_{\text{E}} L_{\text{E}}^{\gamma } } \right). $$
(24)
Equating Eqs. 20 and 21, using Eq. 24, yields the following expression:
$$ - \theta \left( {\sigma_{\text{R}} L_{\text{R}} + \sigma_{\text{E}} L_{\text{E}}^{\gamma } } \right) = \rho + \sigma_{\text{R}} L_{\text{E}} - \sigma_{\text{R}} L - \sigma_{\text{E}} L_{\text{E}}^{\gamma }. $$
(25)
Solving for employment in the research sector gives
$$ L_{\text{R}} = {\frac{1}{{\theta \sigma_{\text{R}} }}}\left( {\sigma_{\text{R}} \left( {L - L_{\text{E}} } \right) + \left( {1 - \theta } \right)\sigma_{\text{E}} L_{\text{E}}^{\gamma } - \rho } \right). $$
(26)
Inserting the expressions for equilibrium employment in the entrepreneurial (19) and research sectors (26) into the law of motion for knowledge, the steady-state growth rate (g) can be derived as
$$ g = {\frac{{\mathop A\limits^{ \bullet } }}{A}} = \sigma_{\text{R}} L_{\text{R}} + \sigma_{\text{E}} L_{\text{E}}^{\gamma }, $$
$$ g = \sigma_{\text{R}} \left( {{\frac{1}{{\theta \sigma_{\text{R}} }}}\left( {\sigma_{\text{R}} \left( {L - L_{\text{E}} } \right) + \left( {1 - \theta } \right)\sigma_{\text{E}} L_{\text{E}}^{\gamma } - \rho } \right)} \right) + \sigma_{\text{E}} L_{\text{E}}^{\gamma }, $$
$$ g = \sigma_{\text{R}} \left( {{\frac{1}{{\theta \sigma_{\text{R}} }}}\left( {\sigma_{\text{R}} \left( {L - \left( {{\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}} \right)^{{1/\left( {\gamma - 1} \right)}} } \right) + \left( {1 - \theta } \right)\sigma_{\text{E}} \left( {{\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}} \right)^{{\gamma /\left( {\gamma - 1} \right)}} - \rho } \right)} \right) + \sigma_{\text{E}} \left( {{\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}} \right)^{{\gamma /\left( {\gamma - 1} \right)}}, $$
$$ g = {\frac{1}{\theta }}\left( {\sigma_{\text{R}} L - \rho + \left( {1 - \gamma } \right)\gamma^{{\gamma /\left( {1 - \gamma } \right)}} \left( {{\frac{{\sigma_{\text{E}} }}{{\sigma_{\text{R}}^{\gamma } }}}} \right)^{{1/\left( {1 - \gamma } \right)}} } \right). $$
(27)
Note that some entrepreneurial activity (Eq. 19) will always be profitable, i.e., \( L_{\text{E}} > 0, \)as long as the stock of knowledge exceeds zero (\( A > 0 \)), which does not, however, always apply to R&D activities (Eq. 26).Footnote 14 The model shares a number of characteristics with previous models, e.g., growth is decreasing in the discount factor (\( \rho \)) and increasing in a larger labor force.
Apart from these properties, the specification of the model implies that the impact on growth of the distribution of labor between R&D and entrepreneurial activities can be derived. Similarly, the optimal distribution of labor between final goods production and knowledge production (R&D and entrepreneurs) can also be inferred from the model. The following propositions follow from the model:
Proposition 1
Given that an economy has an optimal distribution of workers between the final goods sector and the knowledge producing sectors (R&D and entrepreneurs), optimal steady-state growth implies that a marginal redistribution between entrepreneurship and R&D workers has no effect on growth (assuming that the efficiency parameters is constant and that the knowledge stock exceeds zero).
Proof
In steady state \( {\frac{{\mathop A\limits^{ \bullet } }}{A}} = {\frac{{\mathop K\limits^{ \bullet } }}{K}} \Rightarrow {\frac{{\mathop Y\limits^{ \bullet } }}{Y}} = {\frac{{\mathop A\limits^{ \bullet } }}{A}} \), i.e., an optimal distribution of labor between final goods and knowledge production implies that a marginal increase in either sector is exactly counterbalanced by a decrease in the other sector. This is, however, not equivalent to an optimal distribution of labor in the knowledge producing sector (\( L_{A} \)) between R&D workers and entrepreneurs, \( L_{\text{A}} = L - L_{\text{F}} = L_{\text{R}} + L_{\text{E}} \). Using this relationship and Eqs. 27 and 10, growth can be rewritten as\( g = {\frac{{\mathop A\limits^{ \bullet } }}{A}} = \sigma_{\text{R}} (L_{\text{A}} - L_{\text{E}} ) + \sigma_{\text{E}} L_{\text{E}}^{\gamma } \). Differentiating with respect to \( L_{\text{E}} \) yields \( {\frac{\partial g}{{\partial L_{\text{E}} }}} = - \sigma_{\text{R}} + \gamma_{{}} \sigma_{\text{E}} L_{\text{E}}^{\gamma - 1} = 0 \), which is equivalent to Eq. 19, \( L_{\text{E}}^{*} = ({\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}})^{1/(\gamma - 1)} \), where \( L_{\text{E}}^{*} \) represents an optimal allocation between R&D workers and entrepreneurs. Consequently, \( {\frac{\partial g}{{\partial L_{\text{E}} }}} > 0 \Rightarrow L_{\text{E}} < \left({\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}\right)^{1/(\gamma - 1)} = L_{\text{E}}^{*} \), i.e., social optimum is not attained. Given that there is an optimal allocation of labor between the final goods sector and the knowledge sector (\( dL_{\text{A}} = 0 \)), there is also a direct mapping to R&D workers (\( dL_{\text{R}} = - dL_{\text{E}} \)); too few entrepreneurs is mirrored by too many R&D workers.
A second, and rather obvious, implication of the model concerns the efficiency of an economy in accumulating and exploiting knowledge, which should influence the rate of growth. Moreover, if the efficiency in converting knowledge to commercial use is influenced asymmetrically over time due to events that are exogenous to firms (national and international institutional change), there may be time-inconsistent effects of R&D and entrepreneurship on growth over time.
Proposition 2
Growth is increasing in higher efficiency of research (\( \sigma_{\text{R}} \)) and of entrepreneurship (\( \sigma_{\text{E}} \)).
Proof
First, differentiating the growth Eq. 27 with respect to the efficiency parameter of research,
$$ {\frac{\text{d}g}{{\text{d}\sigma_{\text{R}} }}} = {\frac{1}{\theta }}\left[ {L - \left({\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}\right)^{{{\frac{1}{\gamma - 1}}}} } \right] = {\frac{1}{\theta }}\left[ {L - L_{\text{E}}^{{}} } \right], $$
which is unambiguously nonnegative.
Second, differentiating the growth Eq. 27 with respect to the efficiency parameter of entrepreneurial activities,
$$ {\frac{\text{d}g}{{\text{d}\sigma_{\text{E}} }}} = {\frac{1}{\theta }}\left[ {\left({\frac{{\sigma_{\text{R}} }}{{\gamma \sigma_{\text{E}} }}}\right)^{{{\frac{\gamma }{\gamma - 1}}}} } \right] = {\frac{1}{\theta }}\left[ {L_{\text{E}}^{\gamma } } \right], $$
which is unambiguously nonnegative.
From these two propositions the following testable hypotheses emerge: If countries have attained an optimal growth path there will be no growth effect of a (i) marginal redistribution of labor between sectors (the final goods sectors and the knowledge producing sectors), (ii) marginal redistribution of labor within the knowledge producing sectors (R&D and entrepreneurial activities), (iii) marginal redistribution of knowledge workers between time periods, while growth should be positively influenced by (iv) altering variables that influence how efficiently an economy works.