Knowledge spillover entrepreneurship in an endogenous growth model

Early entrepreneurship scholars like Kirzner (1973), von Hayek (1937) and Baumol (1991) all followed Schumpeter and considered innovation to be a profit-driven, rent-seeking economic activity while keeping opportunities exogenous. The entrepreneurship literature, however, largely turned away from mainstream equilibrium macro modeling, and the two strands of literature got separated. This article is an attempt to bring them back together.

Harrod neutrality implies pure labor augmenting technical change. Kennedy (1964) failed in the sense that he replaced the assumption of the Harrod neutral technical change in production by the assumption of a particular type of efficiency improvements in innovation. See Thirtle and Ruttan (1987) for an excellent overview of this literature from Schumpeter up to the modern growth literature.

This externality was both intra- and inter-temporal in the sense that the knowledge increased the productivity of all firms equally and increased the productivity of all future firms who would build upon the accumulated knowledge.

The bulk of expenditure on R&D is corporate, and large corporations and established firms make up for the bulk of corporate R&D. Private industry accounted for 64 % of industrial R&D in OECD countries according to Science and Technology Indicators in 2004 (Inklaar et al. 2006).

See for example Barro and Sala-I-Martin (2004), Aghion and Durlauf (2005), Jones (2006), Acemoglu (2008) and Aghion and Howitt (2011) for surveys.

To our knowledge there are no models in the literature that disconnect the decision to enter from the decision to generate knowledge. Earlier papers have made attempts at introducing entrepreneurship in growth models but never made the separation in functions we suggest here. Acs et al. (2005, 2009) and Michelacci (2003) are some of the few that recognize the importance of entrepreneurs but also model innovation as a co-production where entrepreneurs are complementary to R&D. In Acs and Sanders (2012), we have presented an earlier version of the present model in which we introduce both knowledge creation and commercialization as separate economic activities.

In the extant KSTE (Acs et al. 2009 p. 17), the commercialization efficiency hypothesis predicts “the more efficiently incumbents exploit knowledge flows, the smaller the effect of new knowledge on entrepreneurship,” suggesting entrepreneurs can only pick up knowledge that incumbents fail to develop. Plummer and Acs (2012) argue and show, however, that in fact entrepreneurs compete for the best ideas, implying all locally produced knowledge is in principle “in play.” We follow the latter in assuming all ideas for commercially viable new products spill over to the entrepreneurs.

The distinction and importance of intra-temporal knowledge spillovers in the knowledge spillover theory of entrepreneurship have been pointed out by Acs (2012).

And obviously money flows in the opposite direction where total income for consumers equals the sum of wage and capital income to close the model.

Jones (2006) offers several alternatives to this specification that would not suffer from this problem, but as the issue has no bearing on our purpose, we chose to stick to the Romer specification.

Note that in fact our model would essentially reduce to the Romer (1990) model for δ = 0 and γ = 0. The only difference would then be that Romer’s flow of human capital is replaced by a stock of accumulated R&D knowledge in final goods production.

We feel it is intuitively plausible that both R&D and entrepreneurship can substitute one's own experience and accumulated knowledge for intermediate product variety and outside opportunities, respectively. The constant returns to both sources of knowledge retain the basic assumption in Romer (1990) that the returns to knowledge accumulation are constant at the aggregate level. Not imposing that assumption would eliminate the steady state and cause growth in growth as the rate of new knowledge creation would then be positively related to the size of the aggregate knowledge stock.

Computing the growth rates for (9) and (14), it can immediately be verified that in any steady-state equilibrium the wage will therefore grow at the rate: \( \frac{{\dot{w}_{E} }}{{w_{E} }} = \frac{{\dot{X}}}{X} - \gamma \left( {\frac{{\dot{A}}}{A} - \frac{{\dot{n}}}{n}} \right) = \frac{{\dot{X}}}{X} + \frac{{\dot{A}}}{A} - \frac{{\dot{n}}}{n}. \)

The assumption of a stable equilibrium interest rate is consistent with a steady-state equilibrium later on but convenient to also make here. The interest rate cannot have a positive or negative growth rate as it would imply bond prices going to 0 or infinity, which is not consistent with rational expectations. It is a very common assumption in the literature. See, for example, Campbell and Mankiw (1989)

As final output is homogeneous and we normalized its price to 1, sales equal production.

Summing over all final goods producers, j then yields the result that total expenditure on intermediates in the economy is (1 – α − β)X.

Time arguments have been included in the transversality condition as the limit is taken for time to infinity.

It can be shown that the right hand side of (B11) is actually positive in A _j when the optimal amounts of labor and intermediates have been employed. In that case, output in (4) substituting for labor and intermediates by (8) and (47) equals: \( X_{j} = A_{j} \left( {\frac{\beta }{{w_{P} }}} \right)^{{\frac{\beta }{\alpha }}} \left( {\frac{1 - \alpha - \beta }{{\bar{\chi }}}} \right)^{{\frac{1 - \alpha - \beta }{\alpha }}} n^{{\frac{\alpha + \beta }{\alpha }}} \) where \( \bar{\chi } \) represents the average price for intermediates. Plugging this expression in the threshold wage in (47) and solving for the wage yields an expression that is positive and concave in A _j .

Taken literally, this result may strike one as unrealistic, and it yields the undesirable result that initial levels of production knowledge have to be exactly equal. At this point, however, it is worth noting that, for example, uncertainty in the R&D process and fixed costs have been assumed away. In real life, the uncertainty in R&D outcomes would create a bandwidth, not a precise level for the threshold wage, and fixed costs would cause firms to actually exit when employment levels fall below a critical level. Then the prediction is that a group of technology leaders will be able to survive in the market, where they must “run to stand still,” and a shakeout will cause firms with less than efficient production processes to exit in the transition to the steady state. Such processes are well known in the empirical literature on industrial dynamics. They are not present in our model as they complicate but do not change the key results.

Such that \( X(t) = X(T){\text{e}}^{{\dot{X}/Xt}} \) and \( n(t) = n(T){\text{e}}^{{\dot{n}/nt}} \)