Entrepreneurship and structural economic transformation

Abstract

This paper provides an endogenous growth model to illuminate the role of entrepreneurial start-up firms in structural economic transformation. We follow the Lewis-model distinction between a traditional and modern sector and underpin this distinction with micro-foundations. We specify mature and start-up entrepreneurs and make a distinction between survivalist self-employment activities in the traditional sector and opportunity-driven entrepreneurship in the modern. The model shows how opportunity-driven entrepreneurship can drive structural transformation in both the modern and traditional sectors through innovation and the provision of intermediate inputs and services (which permits greater specialization in manufacturing) and by increasing employment and productivity.

Appendices

Appendices

1.1 Appendix 1: Proof of Proposition 1

Start-up sector: Matching of ideas and opportunities:

Separation of contracts is a positive function of the surplus rate of business ideas

$$ \vartheta = \vartheta (\delta_{N} ) = \delta_{N}^{\zeta } ,\,\;\;\frac{d\vartheta }{{d\delta_{N} }} = \zeta \delta_{N}^{\zeta - 1} > 0. $$

Start-up opportunities in modern sector O:

$$ O = \vartheta N = \delta_{N}^{\zeta } N. $$

Probability of successful start-up

$$ P_{N} = \lambda_{N} {\text{e}}^{{ - \lambda_{N} }} . $$

Contract tightness

$$ \begin{aligned} \theta_{N} = & \Updelta_{N}{/}O = \vartheta^{ - 1} \delta_{N} . \\ \theta_{N} = & \Updelta_{N}{/}O = \left( {\delta_{N}^{\zeta } } \right)^{ - 1} \delta_{N} = \delta_{N}^{1 - \zeta } . \\ \end{aligned} $$

Matching function

$$ \begin{aligned} \lambda_{N} = & \lambda_{N} (\theta_{N} ,\,h) = \theta_{N}^{{ - \varepsilon_{N} }} h^{{\upsilon_{N} }} \;\;0 < \varepsilon_{N} ,\quad \mu_{N} < 1. \\ = & \delta_{N}^{{ - \left( {1 - \zeta } \right)\,\varepsilon_{N} }} h^{{\upsilon_{N} }} . \\ \end{aligned} $$

Business opportunities and start-up matching equilibrium:

$$ \begin{aligned} O\,=\,& \vartheta N \\ \lambda_{N} (\theta_{N} ,\,h)\Updelta_{N}\,=\,& O. \\ \end{aligned} $$

$$ \begin{aligned} \vartheta^{{\varepsilon_{N} }} \delta_{N}^{{ - \varepsilon_{N} }} \alpha^{{\upsilon_{N} }} \delta_{N}\,=\,& \vartheta \\ \delta_{N}^{{\left( {1 - \zeta } \right) - \left( {1 - \zeta } \right)\,\varepsilon_{N} }} h^{{\upsilon_{N} }}\,=\,& 1. \\ \end{aligned} $$

Equilibrium surplus rate of business ideas

$$ \begin{aligned} \delta_{N} = & h^{{ - \tfrac{{\upsilon_{N} }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}}}} , \\ \frac{{d\delta_{N} }}{dh} = & - \frac{{\upsilon_{N} }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}}h^{{^{{ - \left( {\tfrac{{\upsilon_{N} }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}} + 1} \right)}} }} < 0 \\ \end{aligned} $$

Equilibrium contract separation rate

$$ \begin{aligned} \vartheta (\delta_{N} ) = & \delta_{N}^{\zeta } = h^{{ - \tfrac{{\upsilon_{N} \zeta }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}}}} , \\ \frac{d\vartheta }{dh} = & - \frac{{\upsilon_{N} \zeta }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}}h^{{ - \left( {\tfrac{{\upsilon_{N} \zeta }}{{\left( {1 - \varepsilon_{N} } \right)\,\left( {1 - \zeta } \right)}} + 1} \right)}} < 0 \\ \end{aligned} $$

1.2 Appendix 2: Proof of Proposition 2

Traditional sector: Wages per capita income and population dynamics.

a) Wage and per capita income in the traditional sector:

$$ \begin{aligned} Y_{T} = & \bar{w}L,\,\;\;\bar{w}\,=\,a_{T} \\ y_{T} = & y_{T} (\bar{w},\,\delta_{T} )\,=\,\frac{{Y_{T} }}{Pop}\,=\,\frac{{a_{T} L}}{{L(1 + \delta_{T} )}}\,=\,\frac{{a_{T} }}{{(1 + \delta_{T} )}} \\ Ey_{T} = & a_{T} \frac{L}{{L + \delta_{T} }} + 0\frac{{\Updelta_{T} }}{{L + \Updelta_{T} }}\,=\,y_{T} \,=\,\frac{{a_{T} }}{{(1 + \delta_{T} )}} \\ \end{aligned} $$

b) Population dynamics: $\gamma_{L} \, \equiv \,\frac{{\dot{L}}}{L}\, = \,g_{L} (y_{T} )\, = \,\left( {\frac{{a_{T} }}{{(1 + \delta_{T} )}}} \right)^{\varphi }$with φ < 0

c) Properties of the population dynamics curve in the $\gamma_{L} \, - \,\delta_{T} \, -\,{\text{diagram}}$ slope:

$$ \begin{aligned} \frac{{d\gamma_{L} }}{{d\delta_{T} }} = & - \varphi \left( {a_{T} (1 + \delta_{T} )^{ - 1} } \right)^{\varphi - 1} a_{T} (1 + \delta_{T} )^{ - 2} \\ = & - \varphi a_{T}^{\varphi } (1 + \delta_{T} )^{{ - \left( {\varphi + 1} \right)}} > 0 \\ \end{aligned} $$

Location:

$$ \frac{{d\gamma_{L} }}{{da_{T} }}\,= \,\varphi a_{T}^{\varphi - 1} (1 + \delta_{T} )^{ - \varphi } < 0 $$

$$ \begin{aligned} \ln \gamma_{L} = & \varphi \ln \left( {\frac{{a_{T} }}{{(1 + \delta_{T} )}}} \right) \\ \frac{{d\gamma_{L} }}{d\varphi }\frac{1}{{\gamma_{L} }} = & \ln \left( {\frac{{a_{T} }}{{(1 + \delta_{T} )}}} \right) > 0\;\;{\text{for}}\;\;\frac{{a_{T} }}{{(1 + \delta_{T} )}} > 1 \\ \end{aligned} $$

1.3 Appendix 3: Proof of Propostion 3

a) Labour market frictions and surplus labour:

separation rate:

$$ V\,=\,\sigma L $$

matching rate:

$$ \begin{aligned} \lambda = & \lambda (\theta_{T} ,\,\gamma_{L} ) = \theta_{T}^{ - \varepsilon } \gamma_{N}^{\upsilon } ,\,\quad \quad \quad \quad \;\;\theta_{T} = \sigma^{ - 1} \delta_{T} \\ = & \lambda (\sigma ,\,\delta_{T} ,\,\gamma_{N} ) = \sigma^{\varepsilon } \delta_{T}^{ - \varepsilon } \gamma_{N}^{\upsilon } \;\;1 < \varepsilon ,\quad 0 < \upsilon < 1. \\ \end{aligned} $$

search and matching equilibrium and equilibrium surplus labor rate:

$$ \begin{aligned} \lambda \Updelta_{T} = & \sigma L \\ \lambda (\sigma ,\,\delta_{T} ,\,\gamma_{L} )\Updelta_{T} = & \sigma L \\ \sigma^{\varepsilon } \delta_{T}^{ - \varepsilon } \gamma_{N}^{\upsilon } \delta_{T} = & \sigma . \\ \end{aligned} $$

$$ \begin{aligned} \sigma^{\varepsilon } \delta_{T}^{1 - \varepsilon } \gamma_{N}^{\upsilon } = & \sigma \\ \delta_{T}^{1 - \varepsilon } = & \sigma^{1 - \varepsilon } \gamma_{N}^{ - \upsilon } \\ \delta_{T}^{1 - \varepsilon } = & \sigma^{1 - \varepsilon } \gamma_{N}^{ - \upsilon } \\ \end{aligned} $$

$$ \delta_{T} \, = \,\delta_{T} (\sigma ,\,\gamma_{N} )\, = \,\sigma \gamma_{N}^{{ - \tfrac{\upsilon }{1 - \varepsilon }}} $$

b) Properties of the surplus labour curve in the $\gamma_{N} \,-\,\delta_{T} \,-\,{\text{diagram}}$ slope:

$$ \frac{{d\delta_{T} }}{{d\gamma_{N} }}\, = \, - \frac{\upsilon }{1 - \varepsilon }\sigma \gamma_{N}^{{ - \tfrac{\upsilon }{1 - \varepsilon } - 1}} < 0 $$

Location:

$$ \frac{{d\delta_{T} }}{d\sigma }\, = \,\gamma_{N}^{{ - \tfrac{\upsilon }{1 - \varepsilon }}} > 0 $$

1.4 Appendix 4: Proof of Proposition 4

a) Deriving the start-up curve:

Combining (Eq. 20) and (Eq. 17) gives the deposit rate offered to the households

$$ r_{d} \, = \,\left( {\left( {\frac{{\left( {1 - \vartheta \left( h \right)} \right)\left( {1 - \alpha } \right)}}{\varepsilon }H^{2} \alpha^{{\frac{1 + 3\alpha }{1 - \alpha }}} c_{x}^{{ - 2\frac{\alpha }{1 - \alpha }}} A^{{\frac{1}{1 - \alpha } + 1}} } \right)\left( {1 - \vartheta \left( h \right)} \right) - c_{b} \left( {\vartheta \left( h \right)} \right)} \right)\left( {1 + \frac{1}{{B\eta_{{D,r_{d} }} }}} \right) $$

With this deposit rate, and (Eqs. 14, 23) and the Ramsey rule (Eq. 3), we can determine the optimal growth rate of the consumption path and the entrepreneurial start-up growth rate $\gamma_{N}$ of the modern sector

$$ \begin{aligned} \gamma_{N} = & \frac{1}{\Uptheta }\left[ {\left( {\left( {\begin{array}{*{20}c} {\tfrac{{\left( {1 - \vartheta \left( h \right)} \right)^{2} (1 - \alpha )}}{\varepsilon \left( A \right)}H^{2} } \\ {\left[ {\bar{c}_{x} + \tfrac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right]^{{ - 2\tfrac{\alpha }{1 - \alpha }}} } \\ {\alpha^{{\tfrac{1 + 3\alpha }{1 - \alpha }}} A^{{\tfrac{1}{1 - \alpha } + 1}} } \\ \end{array} } \right) - c_{b} \left( {\vartheta \left( h \right)} \right)} \right)\,\left( {1 + \frac{1}{B\eta }} \right) - \rho } \right], \\ = & g_{N} (\mathop {\underbrace {{\delta_{T} ,\,a_{T} }}}\limits_{\text{traditional \ sector}} ,\,\mathop {\underbrace {{H,\,h,\,\bar{c}_{x} ,\,B,\,\Uptheta ,\,\rho }}}\limits_{\text{modern \ sector}} ). \\ \end{aligned} $$

b) Properties of the start-up curve in the $\gamma_{N} \,-\,\delta_{T} \,-\,{\text{diagram}}$ slope:

$$ \frac{{d\gamma_{N} }}{{d\delta_{T} }}\, = \,\frac{{2\alpha a_{T} \left( {1 - \vartheta \left( h \right)} \right)^{2} (1 - \alpha )}}{{\Uptheta \left( {1 - \alpha } \right)\,\varepsilon }}\left( {\begin{array}{*{20}c} {H^{2} \alpha^{{\tfrac{1 + 3\alpha }{1 - \alpha }}} A^{{\tfrac{1}{1 - \alpha } + 1}} } \\ {\left[ {\bar{c}_{x} + \tfrac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right]^{{ - \tfrac{\alpha + 1}{1 - \alpha }}} } \\ \end{array} } \right)\,\left( {1 + \frac{1}{B\eta }} \right) > 0 $$

Location:

$$ \begin{aligned} \frac{{d\gamma_{N} }}{dH} = & \frac{{2\left( {1 - \vartheta \left( h \right)} \right)^{2} (1 - \alpha )}}{\Uptheta \varepsilon }H\alpha^{{\tfrac{1 + 3\alpha }{1 - \alpha }}} c_{x}^{{ - 2\tfrac{\alpha }{1 - \alpha }}} A^{{\tfrac{1}{1 - \alpha } + 1}} \left( {1 + \frac{1}{B\eta }} \right) > 0 \\ \frac{{d\gamma_{N} }}{{da_{T} }} = & - \frac{{2\alpha \left( {1 - \delta_{T} } \right)\,\left( {1 - \vartheta \left( h \right)} \right)^{2} (1 - \alpha )}}{{\Uptheta \left( {1 - \alpha } \right)\,\varepsilon }}\left( {\begin{array}{*{20}c} {H^{2} \alpha^{{\tfrac{1 + 3\alpha }{1 - \alpha }}} A^{{\tfrac{1}{1 - \alpha } + 1}} } \\ {\left[ {\bar{c}_{x} + \tfrac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right]^{{ - \tfrac{\alpha + 1}{1 - \alpha }}} } \\ \end{array} } \right)\,\left( {1 + \frac{1}{B\eta }} \right) < 0 \\ \frac{{d\gamma_{N} }}{dh} = & - 2\frac{{\left( {1 - \vartheta \left( h \right)} \right)(1 - \alpha )}}{\varepsilon }H^{2} c_{x}^{{ - 2\tfrac{\alpha }{1 - \alpha }}} \alpha^{{\tfrac{1 + 3\alpha }{1 - \alpha }}} A^{{\tfrac{1}{1 - \alpha } + 1}} \left( {1 + \frac{1}{B\eta }} \right)\,\frac{d\vartheta \left( h \right)}{dh} - \frac{{dc_{b} }}{d\vartheta }\frac{d\vartheta \left( h \right)}{dh} > 0 \\ \end{aligned} $$

1.5 Appendix 5: Proof of Proposition 5

Given the system

$$ \begin{array}{*{20}c} {0 = F_{1} \left( {\delta_{T} ,\,\gamma_{N} ,\,\sigma } \right) = \delta_{T} \left( {\gamma_{N} ,\,\sigma } \right) - \delta_{T} } \hfill & { ( {\text{surplus labour rate)}}} \hfill \\ {0 = F_{2} \left( {\delta_{T} ,\,\gamma_{N} ,\,a_{T} ,\,H, \ldots ,\,\rho } \right) = g_{N} (\delta_{T} ,\,a_{T} ,\,H, \ldots ,\,\rho ) - \gamma_{N} } \hfill & { ( {\text{start-up rate)}}} \hfill \\ {0 = F_{3} \left( {\delta_{T} ,\,\gamma_{L} ,\,a_{T} ,\,\phi } \right) = g_{L} \left( {\delta_{T} ,\,a_{T} ,\,\phi } \right) - \gamma_{L} } \hfill & { ( {\text{population dynamics)}}} \hfill \\ \end{array} $$

we want to solve for the set of implicit functions

$$ \begin{array}{*{20}c} {\tilde{\delta }_{T} = \tilde{\delta }_{T} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{surplus labour rate}} \hfill \\ {\tilde{\gamma }_{N} = \tilde{\gamma }_{N} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{start-up rate}} \hfill \\ {\tilde{\gamma }_{L} = \tilde{\gamma }_{L} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{population dynamics}} \hfill \\ \end{array} $$

a) Partial derivatives of F ₁, F ₂, F ₃ with respect to all relevant variables exist and are continuous: For F ₁ we obtain:

$$ \frac{{dF_{1} }}{{d\delta_{T} }}\,= \, - 1,\quad \frac{{dF_{1} }}{{d\gamma_{N} }} = - \frac{\upsilon }{1 - \varepsilon }\sigma \gamma_{N}^{{ - \tfrac{\upsilon }{1 - \varepsilon } - 1}} < 0 $$

$$ \frac{{dF_{1} }}{d\sigma }\, = \,\gamma_{N}^{{ - \tfrac{\upsilon }{1 - \varepsilon }}} ,\quad \frac{{dF_{1} }}{dH}\, = \cdots = \frac{{dF_{1} }}{d\rho } = 0. $$

For $F_{2}$ we obtain:

$$ \frac{{dF_{2} }}{{d\delta_{T} }}\,= \,\frac{{dg_{N} }}{{d\delta_{T} }},\quad \frac{{dF_{2} }}{{d\gamma_{N} }}\,= \, - 1 $$

$$ \frac{{dF_{2} }}{{da_{T} }} = \frac{{dg_{N} }}{{da_{T} }} < 0,\;\;\frac{{dF_{2} }}{dH} = \frac{{dg_{N} }}{dH} > 0,\;\;\frac{{dF_{2} }}{dh} = \frac{{dg_{N} }}{dh} > 0. $$

For $F_{3}$ we obtain:

$$ \frac{{dF_{3} }}{{d\delta_{T} }}\,=\,\frac{ - \varphi }{{(1 + \delta_{T} )}}\left( {\frac{{a_{T} }}{{(1 + \delta_{T} )}}} \right)^{\varphi } > 0,\quad \frac{{dF_{3} }}{{d\gamma_{L} }} = - 1 $$

$$ \frac{{dF_{3} }}{{d\alpha_{T} }}\, = \,\frac{\varphi }{{\left( {1 + \delta_{T} } \right)}}\left( {\frac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right)^{\varphi - 1} < 0,\;\;\frac{{dF_{3} }}{d\varphi } = \left( {\frac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right)^{\varphi } \ln \left( {\frac{{a_{T} }}{{\left( {1 + \delta_{T} } \right)}}} \right) > 0 $$

$$ \frac{{dF_{3} }}{dH} = \cdots = \frac{{dF_{3} }}{d\rho }\, = \,0 $$

b) Jakobian of (Eq. 32) system can be determined by:

$$ \left| {J^{ * } } \right|\, = \,\left| {\begin{array}{*{20}c} { - 1} & {\mathop {\tfrac{{dF_{1} }}{{d\gamma_{N} }}}\limits^{( - )} } & 0 \\ {\tfrac{{dF_{2} }}{{d\delta_{T} }}} & { - 1} & 0 \\ {\mathop {\tfrac{{dF_{3} }}{{d\delta_{T} }}}\limits^{( - )} } & 0 & { - 1} \\ \end{array} } \right| \ne 0 $$

Considering (a) and (b) than there exists an m-dimensional neighbourhood (m is the number of exogenous variables), in which the variables $\delta_{T} ,\,\gamma_{N} ,\,\gamma_{L}$ are functions of the form

$$ \begin{array}{*{20}c} {\tilde{\delta }_{T} = \tilde{\delta }_{T} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{surplus labour rate}} \hfill \\ {\tilde{\gamma }_{N} = \tilde{\gamma }_{N} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{start-up rate}} \hfill \\ {\tilde{\gamma }_{L} = \tilde{\gamma }_{L} (a_{T} ,\,H,\,h,\,B,\,\phi ,\,\bar{c}_{x} , \ldots ,\,A,\,\Uptheta ,\,\rho )} \hfill & {\text{population dynamics}} \hfill \\ \end{array} $$