Toward a theory of the entrepreneurial process

Abstract

This paper models the entrepreneurial process as both creation and discovery composed of an iterative two-step process where entrepreneurs create social networks based on subjective expectations about the future effectiveness of those networks, and then choose the innovation to pursue and map a search process to discover how to bring the innovation to fruition. Critical to this process is the mix of strong ties and weak ties that make up social networks and the ability to carry forward the social capital embodied in such networks. The tendency of long-existing entrepreneurs to be less innovative can be explained using this model.

Appendix: Derivation of toward a theory of the entrepreneurial process

1.1 The search for an innovation

Given a desired innovation and a social network, the entrepreneur’s efforts focus on exploring various combinations of knowledge, actions, and resources (hereafter inputs) thought to have a reasonable chance of producing the innovation. Let N be the total number of possible inputs so that input sets can be represented by N × 1 vectors x ∈ R ⁿ (some entries in the vectors perhaps being zero).

The search for a combination of inputs x that will generate the desired outcome is assumed to take place sequentially over time. Letting A _t ∈ R ⁿ represent the region of the input space explored in time t, the entrepreneur searches over increasing larger regions. Thus:

$${\mathbf{A}}_{0} \subset {\mathbf{A}}_{ 1} \subset {\mathbf{A}}_{ 2} \subset \ldots$$

(1)

Searching is a costly process. Assume that the cost of searching is a positive function of the size of the region explored and the degree to which the entrepreneur has the ability to engage in creative cognition, and assume that the ability to engage in creative cognition is a positive function of the effectiveness, γ, of the entrepreneur’s social network. Thus, let A _t be the Lebesgue measure (that is, the size) of region A _t :

$$\it A_{t} = \, A({\mathbf{A}}_{t} ) \, = \, \smallint_{{{\mathbf{x}} \in {\mathbf{A}}}} \bf{x}{\text{ d}}{\mathbf{x}}.$$

(2)

We can thus define the cost of searching as:

$$c^{S}_{t} = \, c^{S} (A_{t} , \, \gamma ).$$

(3)

Assume that the costs of searching increase at a (weakly) increasing rate with the size of the search region:

$$\partial c^{S} / \, \partial A_{t} > 0, \, \partial^{2} c^{S} / \, \partial A_{t}^{2 } \ge 0,$$

(4)

and decrease at a decreasing rate with the effectiveness of the entrepreneur’s social network:

$$\partial c^{S} / \, \partial \gamma \, < \, 0, \, \partial^{2} c^{S} / \, \partial \gamma^{2} < \, 0.$$

(5)

The average cost of searching can then be defined as:

$$\bar{c}^{S} = c^{S} (A_{t} ,\gamma )/A_{t} .$$

(6)

Assume also that the average cost, $\bar{c}^{S} ,$ is convex with respect to A _t

$$A_{t} (\partial^{2} c^{S} / \, \partial A_{t}^{2} ) \, {-} \, 2 \, (\partial c^{S} / \, \partial A_{t} ) \, + \, 2 \, c^{S} / \, A_{t} > \, 0.$$

(7)

Because the entrepreneurial process is an uncertain one, the likelihood of finding a successful input combination in a given region A _t is objectively unknown. As a result, the entrepreneur is guided by subjective estimates of the likelihood of success. Let the entrepreneur’s subjective likelihood of success in region A _t be defined by the function Λ(A _t |γ), and note that this likelihood function is a function of the entrepreneur’s social network. Given the pattern of search regions described by Eq. (1), this subjective likelihood function will increase as the entrepreneur widens the search region. Thus:

$$\varLambda ({\mathbf{A}}_{t} | \, \gamma )\,{ \ni }\,0 \, \le \, \varLambda \left( \mathbf{A}_{t} | \, \gamma \right)\,{\text{ and }}\,\varLambda (\mathbf{A}_{t} | \, \gamma ) \, < \, \varLambda (\mathbf{A}_{t + 1} | \, \gamma ).$$

(8)

Note that because the cost of searching increases with the size of the search region, the entrepreneur has an incentive given any region A _t to define the boundaries of A _t so as to maximize Λ(A _t |γ). As a result the A _t is uniquely associated with Λ(A _t |γ), and we can redefine this subjective likelihood function in terms of the size A_t of the region A _t:

$$L(A_{t} | \, \gamma )\;{ \ni }\;0 \, \le \, L(A_{t} | \, \gamma ){\text{ and }}L(A_{t} | \, \gamma ) \, < \, L(A_{t + 1} | \, \gamma ).$$

(9)

To fund the innovation process, assume that the entrepreneur uses capital markets. As a result, the entrepreneur’s access to capital is constrained by the capital market’s expected value, V ^e, of the project. That expected value can be defined as the product of the ultimate expected value V of the project were it to succeed and the (subjective) probability of success P(A _t |K), where K represents the capital market’s M-dimensional knowledge set (K ∈ R ^M):

$$V^{\text{e}} (V, \, A_{t} |{\mathbf{K}}) \, = \, V \, \cdot \, P(A_{t} |{\mathbf{K}}).$$

(10)

Thus, the entrepreneur’s resource constraint will be:

$$c^{S} (A_{t} , \, \gamma ) \, \le \, V^{\text{e}} (V, \, A_{t} |{\mathbf{K}}).$$

(11)

The solution to the entrepreneur’s problems depends on the objective of the entrepreneur. We assume that the objective of the entrepreneur is to maximize the likelihood of successfully innovating. Thus, the entrepreneur’s objective is to choose a region of size A* that will maximize the likelihood L(A _t |γ) of success in achieving the desired entrepreneurial outcome subject to the budget constraint (11). Because increasing the size, A _t , of a search region will always increase the likelihood of success (recall Eq. (9)), the resource constraint (11) will always hold as an equality:

$$c^{S} (A_{t} , \, \gamma )\, = \,V^{\text{e}} (V, \, A_{t} |{\mathbf{K}})$$

(12)

which is equivalent to the condition that average cost is equal to the average value of searching:

$$c^{S} (A_{t} , \, \gamma ) \, / \, A_{t} = \, V^{\text{e}} (V, \, A_{t} |{\mathbf{K}}) \, / \, A_{t} .$$

(13)

Figure 3 provides an illustration of this problem and its solution. Note that the outcomes noted above are ex ante. In practice, the entrepreneur engages in a sequential process of exploration. If he or she finds success before the search area reaches A*, he/she will stop searching, and profits ex post will be higher than expected. If he/she does not find success after having searched the region A*, he/she will stop searching, and profits ex post will be lower than expected, and in fact will be negative.

1.2 Entrepreneurial network creation

Social networks are created by entrepreneurs to aid in deciding which innovation to pursue and in searching for that innovation. The determination of the various search regions, A _t , and the costs associated with searching them, depends on the effectiveness, γ, of the social network. Because the process of innovation search described in the text can only take place after a social network is in place, the process of creating the social network must take place before, and independent of the later innovation search process.

Social networks are composed of bonds between individuals and/or organizations with varying types of knowledge. The effectiveness of an entrepreneur’s social network is determined by the degree of heterogeneity in the set of knowledge embodied in the social network, and by the degree of closure, that is, the degree to which the individuals/organizations, in the network are bound closely into an integral whole. We assume that the degree of heterogeneity and of closure can be represented, respectively, by the non-negative variables α and β.

The effectiveness, γ, of a given network is inherently and irremediably uncertain. It is therefore a matter of subjective conjecture by the entrepreneur. Nonetheless, we assume that entrepreneurs believe that α and β are both valuable components of any network and that there is to some degree or other a trade-off between the two, that is, that a reduction in one can to some extent be compensated by an increase in the other. Assume then that the entrepreneur’s subjective estimation of the effectiveness of a social network is a positive, strict quasi-concave function of α and β:

$$\gamma = \gamma (\alpha ,\beta ) \, { \ni }\, \frac{\partial \gamma }{\partial \alpha } > 0, \frac{\partial \gamma }{\partial \beta } > 0, {\text{and}} \; 2\frac{{\partial^{2} \gamma }}{\partial \alpha \partial \beta } \frac{\partial \gamma }{\partial \alpha } \frac{\partial \gamma }{\partial \beta } - \frac{{\partial^{2} \gamma }}{{\partial \alpha^{2} }} \left( {\frac{\partial \gamma }{\partial \beta }} \right)^{2} - \,\frac{{\partial^{2} \gamma }}{{\partial \beta^{2} }} \left( {\frac{\partial \gamma }{\partial \alpha }} \right)^{2} > 0.$$

(14)

As a result, we can represent the relationship between γ and the various values of α and β by an iso-effectiveness diagram similar to the iso-quant diagram used in the standard microeconomic theory of the firm. Figure 4 represents such a diagram with γ ₁ < γ ₂ < γ ₃.

The cost of assembling a social network is assumed to be a positive, linear function of α and β:

$$c^{N} = p_{\alpha } \alpha + p_{\beta } \beta$$

(15)

where p _α and p _β are the marginal costs of α and β. In addition, based on Burt’s (2005) characterization of the problems of echo and rigidity that arise from closure, assume that the marginal cost of α is an increasing function of the entrepreneur’s endowment, β ₀ (see below for the characterization of the entrepreneur’s endowment):

$$p_{\alpha } = p_{\alpha } (\beta_{0} )\;{ \ni }\; \frac{{\partial p_{\alpha } }}{{\partial \beta_{0} }} > 0.$$

(16)

In choosing the optimal mix of α and β, the entrepreneur cannot access funding from capital markets. Those funds are provided on the basis of the capital market’s estimation of the value of the entrepreneur’s project and the probability of success. But the nature of the project and the probability of its success are predicated on the existence of an entrepreneurial social network and therefore cannot be evaluated before the social network has been created. Therefore, the entrepreneur must rely on internal resources to fund the network creation process.

Assume that the entrepreneur has access to two sources of internal resources. The first source is a pre-existing social network that the entrepreneur has already created. That pre-existing social network essentially means that the entrepreneur will have an endowment of α = α ₀ and an endowment of β = β ₀. The second source is a general monetary endowment c ^N₀ that can be used to acquire α and β. Thus, the entrepreneur’s production possibilities frontier can be derived by setting c ^N = c ^N₀ in Eq. (15) and accounting for the endowments (see Fig. 4):

$$c^{N} = p_{\alpha } (\alpha - \alpha_{0} ) + p_{\beta } (\beta - \beta_{0} )\; { \ni }\; \alpha \ge \alpha_{0} \;{\text{and}}\; \beta \ge \beta_{0} .$$

(17)

The solution to this network creation problem is to choose that mix of α and β that achieves the greatest expected value of the network, that is, that mix such that the marginal rate of substitution is equal to the marginal rate of transformation (see Fig. 4):

$$\frac{{\frac{\partial \gamma }{\partial \alpha }}}{{\frac{\partial \gamma }{\partial \beta }}} = \frac{{p_{\alpha } }}{{P_{\beta } }}.$$

(18)

The effect on the eventual levels of α and β of different marginal cost and endowments will depend on the specific structure of the entrepreneur’s subjective iso-effectiveness of the social network map. However, for a given social network map, we can note the following observations:

• A higher marginal cost of α, p _α , will mean a steeper budget line and hence a desire for relatively more β and less α.

• A higher marginal cost of β, p _β , will mean a flatter budget line and hence a desire for relatively less β and more α.

• A higher resource endowment, c ^N₀ , will mean a budget line further to the northwest. The effect on α and β will depend on the nature of the iso-effectiveness map.

• A higher endowment of α will mean the budget line will be further to the right but with the same slope. Hence, the effect will be similar to that associated with a higher resource endowment except that the minimum amount of α will be higher.

• A higher endowment of β will mean the budget line will be vertically higher and steeper. As a result, the effect will be a combination of the effect of a higher endowment and a higher marginal cost of α, with the added restriction that the minimum amount of β will be higher.