Induced innovation and its impact on productivity growth in China: a latent variable approach

Abstract

This paper proposes a latent variable approach to examine the induced innovation hypothesis originating from Hicks (The theory of wages, Macmillan, London, 1932). It allows a flexible decomposition of total factor productivity into Type I (exogenous) and Type II (induced) technical changes using a latent variable d as an argument in a variable cost function. By making extensive use of duality theory, we are able to estimate the Hicksian input demand functions, which are explicit in d but may lack a closed-form representation in terms of observable variables such as input prices and output level. The “unobservability” of d is solved by using a numerical inversion estimation method. We apply this methodology with an appropriate estimator to investigate the impact of induced innovation on China’s total factor productivity growth. Our findings indicate that the induced innovation hypothesis is strongly supported by the data and that induced innovation has a positive and profound effect on China’s productivity growth.

Appendix 1 Derivation of Eq. (30)

Define the proportionate rate of growth of a variable v as $\dot{v}$ and totally differentiating the variable cost function $C\left[ {{\mathbf{w}}, \, {\mathbf{z}}, \, y{\text{, t, D}}^{c} \left( {{\mathbf{w}}, \, {\mathbf{z}}{, }y{\text{, t}}, \, c^{T} } \right)} \right]$ with respect to time, we obtain:

$$\frac{d\log \left( C \right)}{{dt}} = \sum\limits_{i}^{{}} {\frac{\partial \log \left( C \right)}{{\partial \log \left( {w_{i} } \right)}}} \dot{w}_{i} - \sum\limits_{j} {S_{{Z_{j} }} \dot{z}_{j} } + E_{Cy} \dot{y} + \frac{\partial \log \left( C \right)}{{\partial t}} + \frac{\partial \log \left( C \right)}{{\partial \log \left( d \right)}}\frac{{d\log \left( {D^{c} } \right)}}{dt},$$

(33)

where $S_{{Z_{i} }} = - \frac{\partial \log \left( C \right)}{{\partial \log \left( {z_{i} } \right)}}$, and $E_{Cy}$ and $\frac{{d\log \left( {D^{c} } \right)}}{dt}$ are defined in (21) and (31), respectively. Likewise, by taking the total derivative of the definition of the total variable cost ($c = \sum\limits_{i^{\prime}} {w_{i^{\prime}}^{{}} x_{i^{\prime}} }$) with respect to t, dividing through by c and then rearranging terms, it yields:

$$\frac{d\log \left( C \right)}{{dt}} = \sum\limits_{i^{\prime} = 1}^{{}} {S_{i^{\prime} } } \dot{w}_{i^{\prime} } + \sum\limits_{i^{\prime} = 1}^{{}} {S_{i^{\prime} } } \dot{x}_{i^{\prime} } .$$

(34)

Equating Eqs. (33) and (34) after some algebraic manipulations gives:

$$E_{Cy} \dot{y} - \sum\limits_{i^{\prime} = 1}^{{}} {S_{i^{\prime} } } \dot{x}_{i^{\prime} } - \sum\limits_{j^{\prime} } {S_{{Z_{j^{\prime} } }} \dot{z}_{j^{\prime} } } = - \frac{\partial \log \left( C \right)}{{\partial t}} - \frac{\partial \log \left( C \right)}{{\partial \log \left( d \right)}}\frac{{d\log \left( {D^{c} } \right)}}{dt}.$$

(35)

Let $C^{*} = \sum\limits_{i^{\prime} = 1}^{4} {\tfrac{\partial C}{{\partial w_{i^{\prime}} }}w_{i^{\prime}} } - \sum\limits_{j^{\prime} = 1}^{2} {\tfrac{\partial C}{{\partial z_{j^{\prime}} }}z_{j^{\prime}} }$ denote the total cost function of variable and fixed inputs (including public capital stock). Then, the total cost share functions of the variable input i and fixed input j are defined as $S_{i}^{*} = \frac{{w_{i} X_{i} \left( {{\mathbf{w}}, \, {\mathbf{z}}, \, d,{\text{ y}}} \right)}}{{C^{*} }}$, and $S_{{Z_{j} }}^{*} = \frac{{W_{{Z_{j} }} \left( {{\mathbf{w}}, \, {\mathbf{z}}, \, d,{\text{ y}}} \right) \cdot z_{j} }}{{C^{*} }}$, respectively. Dividing (35) by $E_{Cy}$, we obtain:

$\dot{y} - \sum\limits_{i^{\prime} = 1}^{{}} {\frac{{S_{i^{\prime} } }}{{E_{Cy} }}} \dot{x}_{i^{\prime} } - \sum\limits_{j} {\frac{{S_{{z_{j} }} }}{{E_{Cy} }}\dot{z}_{j} } = - \frac{\partial \log \left( C \right)}{{\partial t}}\frac{1}{{E_{Cy} }} - \frac{\partial \log \left( C \right)}{{\partial \log \left( d \right)}}\frac{{d\log \left( {D^{c} } \right)}}{dt}\frac{1}{{E_{Cy} }}.$

It could be further rewritten as:

$$\begin{aligned} \dot{y} - \sum\limits_{{i^{\prime} }}^{{}} {S_{{i^{\prime} }}^{*} \dot{x}_{{i^{\prime} }} - \sum\limits_{{j^{\prime} }} {S_{{Z_{{j^{\prime} }} }}^{*} } \dot{z}_{{j^{\prime} }} } & = - \frac{{\partial \log \left( C \right)}}{{\partial t}}\frac{1}{{E_{{Cy}} }} - \frac{{\partial \log \left( C \right)}}{{\partial \log \left( d \right)}}\frac{{d\log \left( {D^{c} } \right)}}{{dt}}\frac{1}{{E_{{Cy}} }} \\ & \quad + \sum\limits_{{i^{\prime} = 1}}^{{}} {\left( {\frac{{S_{{i^{\prime} }} }}{{E_{{Cy}} }} - S_{{i^{\prime} }}^{*} } \right)\dot{x}_{{i^{\prime} }} } + \sum\limits_{{j^{\prime} }} {\left( {\frac{{S_{{z_{{j'}} }} }}{{E_{{Cy}} }} - S_{{Z_{{j^{\prime} }} }}^{*} } \right)\dot{z}_{{j^{\prime} }} } \\ & = RTP - \frac{{E_{{Cd}} }}{{E_{{Cy}} }}\frac{{d\log \left( {D^{c} } \right)}}{{dt}} + \sum\limits_{{i^{\prime} }}^{{}} {S_{{i^{\prime} }}^{*} } \dot{x}_{{i^{\prime} }} \left( {\frac{{C^{*} }}{{C \cdot E_{{Cy}}^{{}} }} - 1} \right) \\ &\quad + \sum\limits_{{j^{\prime} }} {C_{{Z_{{j^{\prime} }} }}^{{}} } = \mathop {TFP_{t} }\limits^{ \cdot } , \\ \end{aligned}$$

which is the expression in Eq. (30).