2.1. Representative consumers

The numerical value of homogeneous representative consumer groups within the model exhibits a constant growth rate denoted by $n$ and reaches a population size of $L(t)$ at time $t$ . In addition to their consumption decisions, each consumer has the option to supply one unit of labor inelastically to either of the two types of R&D sectors or to final goods production.

Data, generated as a by-product of consumer activities, can be individually sold by consumers to the two types of R&D sectors, implying that consumers possess the property rights over their own data [Cong et al. (Reference Cong, Xie and Zhang2021)]. For instance, when consumers engage in online shopping or use ride-hailing apps, their consumption patterns and behavior characteristics are recorded as data. R&D sectors can leverage this data to develop new business models, create innovative products, and enhance their ability to meet market demands.

However, when consumers sell their data to any R&D sector, their utility is impacted by concerns related to the leakage and misuse of personal information. As consumers can derive profits from selling their data, they strive to strike a balance between the financial gains and the associated privacy issues. Data subsidies can be viewed as additional bonuses beyond traditional income, enabling agents to optimize their utility based on their privacy sensitivity. Given the uncertainty surrounding how the data will be utilized, the disutility stemming from selling data to different R&D sectors varies among agents.

Each consumer’s utility maximization problem can be represented asFootnote ⁴

(1) \begin{align} \max _{c\left(t\right),D_{H}\left(t\right),D_{V}\left(t\right)} \int _{0}^{\infty }e^{-\left(\rho -n\right)\cdot t}\cdot \left[\frac{c\left(t\right)^{1-\gamma }-1}{1-\gamma }-\chi _{H}\cdot D_{H}\left(t\right)^{\sigma }-\chi _{V}\cdot D_{V}\left(t\right)^{\sigma }\right]dt; \end{align}

subject to

(2) \begin{align} \dot{a\left(t\right)}=\left(r\left(t\right)-n\right)\cdot a\left(t\right)+w\left(t\right)+p_{D,H}\left(t\right)\cdot D_{H}\left(t\right)+p_{D,V}\left(t\right)\cdot D_{V}\left(t\right)-c\left(t\right); \end{align}

and

(3) \begin{align} D_{H}\left(t\right)\leq {\Theta} _{1}(c(t)); \end{align}

(4) \begin{align} D_{V}\left(t\right)\leq {\Theta} _{2}(c(t)); \end{align}

In the equations provided, $c(t)$ represents the per capita consumption level at time $t$ , while $D_{H}(t)$ and $D_{V}(t)$ represent the quantities of data provided by a consumer to horizontal and vertical R&D sectors, respectively. The parameter $\rho$ denotes the consumer’s discount rate, and $\gamma$ , which lies within the range $(1,\infty )$ , represents the reciprocal of the elasticity of intertemporal substitution of consumption. The parameter $\sigma$ parameterizes the disutility associated with data leakage, which depends on the quantity of data provided each time. The variables $\chi _{H}$ and $\chi _{V}$ represent the degrees of negative effect on utility when consumers sell data to horizontal and vertical R&D sectors, respectively [Cong et al. Reference Cong, Wei, Xie and Zhang(2022)]. Furthermore, ${\Theta} _{1}(\cdot )$ and ${\Theta} _{2}(\cdot )$ are both increasing exogenous general functions that determine the generating process of data. The specific form of ${\Theta} _{1}(\cdot )$ and ${\Theta} _{2}(\cdot )$ reflects the constraints imposed on data generation, which can be influenced by factors such as the digital infrastructure, legal framework, and local privacy regulations. In the baseline model, it is assumed that ${\Theta} _{1}(c(t))$ and ${\Theta} _{2}(c(t))$ are sufficiently large, enabling consumers to have the autonomy to determine the quantity of data they wish to sell based on their preferences and constraints.

In the budget constraint equation (2), $a(t)$ represents the asset held by a consumer at time $t$ , $r(t)$ represents the interest rate at time $t$ , and $w(t)$ represents the wage for labor at time $t$ . The variables $p_{D,H}(t)$ and $p_{D,V}(t)$ represent the time- $t$ prices of data for horizontal and vertical R&D sectors, respectively. These prices are determined based on the nonrivalry and exclusivity of data. In order to simplify the model, it is assumed that consumers can directly sell the same data to both types of R&D sectors simultaneously, without the presence of competitive data intermediaries. This assumption aligns with previous studies such as Jones and Tonetti (Reference Jones and Tonetti2020) and Cong et al. (Reference Cong, Xie and Zhang2021). The disutility of data leakage caused by selling data is not constant and varies depending on the value of the parameter $\sigma$ . If $\sigma$ is larger than 1, the privacy cost associated with selling data will increase. Therefore, the price of data purchased by the R&D sectors is influenced by the quantity of data purchased by each sector. Constraints (3) and (4) impose limitations on the growth rate of data provision, ensuring that it remains bounded by the corresponding growth rate of consumption. This constraint reflects the idea that data, being by-products of economic activities, cannot exceed a certain proportion of consumption activities.

To obverse the relevant variables for transitional dynamics, we derive the system’s evolution in the form of Euler equations from the Hamilton:

(5) \begin{align} \frac{\dot{c\left(t\right)}}{c\left(t\right)}=\frac{r\left(t\right)-\rho }{\gamma }; \end{align}

(6) \begin{align} \frac{\dot{p_{D,H}\left(t\right)}}{p_{D,H}\left(t\right)}-\left(\sigma -1\right)\cdot \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}=r\left(t\right)-\rho ; \end{align}

(7) \begin{align} \frac{\dot{p_{D,V}\left(t\right)}}{p_{D,V}\left(t\right)}-\left(\sigma -1\right)\cdot \frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}=r\left(t\right)-\rho ; \end{align}

2.2. The final good producer

A representative final good producer operates within a competitive environment, characterized by a production function:

(8) \begin{align} Y\left(t\right)={L_{E}}\left(t\right)^{\alpha }\cdot \int _{0}^{N\left(t\right)}A\left(v,t\right)^{\alpha }\cdot x\left(v,t\right)^{1-\alpha }dv; \end{align}

In equation (8), $Y(t)$ represents the gross output, $L_{E}(t)$ denotes the amount of labor employed in the production of the final goods, $N(t)$ signifies the number of intermediate goods varieties utilized in the final goods production, and $A(v,t)$ represents the productivity parameter associated with the most recent version of intermediate good $v$ at time $t$ .Additionally, $x(v,t)$ refers to the total quantity of the intermediate good of variety $v$ at time $t$ , which can only be utilized in the production of final goods for a single period. The elasticity coefficient of labor in final goods production is denoted as $\alpha$ , and the rental fee for intermediate goods is denoted as $p_{x}(v,t)$ . The first-order conditions derived from the profit maximization of the final goods producer, with respect to both labor employed and the quantity of each intermediate good, are as follows:

(9) \begin{align} x\left(v,t\right)=\left[\frac{\left(1-\alpha \right)}{p_{x}\left(v,t\right)}\right]^{\frac{1}{\alpha }}\cdot A\left(v,t\right)\cdot L_{E}\left(t\right); \end{align}

(10) \begin{align} \alpha \cdot \left[L_{E}\left(t\right)\right]^{\alpha -1}\cdot \int _{0}^{N\left(t\right)}A\left(v,t\right)^{\alpha }\cdot x\left(v,t\right)^{1-\alpha }dv=w\left(t\right); \end{align}

In equation (9), the quantity of intermediate good variety $v$ at time $t$ is determined by a combination of the rental fee, productivity parameter, and the amount of labor employed. Equation (10) represents the condition where the marginal productivity of labor in the final goods production is equal to the wage rate, denoted as $w(t)$ .

2.3. Intermediate producers

The decision-making process of potential intermediate producers in conducting research is not limited, allowing them to determine whether to engage in research activities and the extent of their investment. Successful horizontal innovation grants these producers a monopoly over the intermediate product they have developed. Consequently, the market will witness the emergence of a certain number of monopoly intermediate producers. Conversely, if vertical innovation proves successful, an existing monopoly intermediate producer will be replaced. Until the next instance of vertical innovation occurs, new intermediate producers can continue to enjoy monopoly profits. It is important to note that we assume that once previous innovators exit the market, they are unable to reenter. Therefore, the latest innovator will never face competition from past innovators [Howitt and Aghion (Reference Howitt and Aghion1998)]. To address the dynamics of intermediate producers, we employ a backward induction methodology, allowing for a comprehensive analysis of the problem.

2.3.1. Final good production phase

For each monopoly intermediate producer entering the market, the profit obtained from the intermediate good of variety $v$ in a single period, denoted as $t$ is determined by the following profit-maximizing function:

(11) \begin{align} \max _{p_{x}\left(v,t\right)} \left[p_{x}\left(v,t\right)\cdot x\left(v,t\right)-\psi \cdot x\left(v,t\right)\right]; \end{align}

In equation (11), $\psi$ represents the constant marginal cost associated with this production process in the economy.

Substituting equation (9) into equation (11) and taking the derivative with respect to $p_{x}(v,t)$ yields the optimal price for each variety of intermediate goods:

(12) \begin{align} p_{x}\left(v,t\right)=\frac{{\psi} }{1-\alpha }; \end{align}

It is evident that the optimal price of intermediate goods, $p_{x}(v,t)$ , is independent of the specific intermediate good variety $v$ and the time period.

Substituting equations (12) and (9) into equation (11), we can derive the quantity of intermediate good of variety $v$ :

(13) \begin{align} x\left(v,t\right)=\frac{\left(1-\alpha \right)^{\frac{2}{\alpha }}}{\psi }\cdot A\left(v,t\right)\cdot L_{E}\left(t\right); \end{align}

By substituting equation (13) into equation (11), the maximum profit obtained from the intermediate good of variety $v$ can be expressed as

(14) \begin{align} {\pi} \left(\mathrm{v},\mathrm{t}\right)=\frac{\left(1-\alpha \right)^{\frac{2}{\alpha }-1}}{\psi ^{\frac{1}{\alpha }-1}}\cdot \alpha \cdot A\left(v,t\right)\cdot L_{E}\left(t\right); \end{align}

Furthermore, the gross output, $Y(t)$ , and the wage rate, $w(t)$ , can be expressed as

(15) \begin{align} Y\left(t\right)=\left[\frac{\left(1-\alpha \right)^{2}}{\psi }\right]^{\frac{1}{\alpha }-1}\cdot L_{E}\left(t\right)\cdot \int _{0}^{N\left(t\right)}A\left(v,t\right)dv; \end{align}

(16) \begin{align} w\left(t\right)=\alpha \cdot \left[\frac{\left(1-\alpha \right)^{2}}{\psi }\right]^{\frac{1}{\alpha }-1}\cdot \int _{0}^{N\left(t\right)}A\left(v,t\right)dv; \end{align}

In equations (15) and (16), the gross output, $Y(t)$ , is determined by the integral over all varieties of intermediate goods, denoted by $N(t)$ , and their respective productivity parameter $A(v,t)$ . The wage rate, $w(t)$ , is a function of the integral over the same range.

2.3.2. Vertical innovation phase

This paper examines the vertical innovations that result in quality improvements. Vertical R&D sectors are capable of conducting research and development (R&D) activities by utilizing both labor, denoted as $L_{V}(t)$ , and data acquired from consumers, represented as $D_{V}(t)\cdot L(t)$ . The Poisson arrival rate of vertical innovations in any given sector is determined by $\Phi (t)=\mu \cdot \phi (t)$ , $\mu \gt 0$ , where $\mu$ is a parameter indicating the productivity of vertical R&D, and $\phi (t)$ represents the innovation success rate equation adjusted for productivity in vertical R&D within each sector [Howitt (Reference Howitt1999)]. The productivity-adjusted innovation success rate equation can be formulated as follows:

(17) \begin{align} \frac{{L_{V}}\left(t\right)^{\beta }\cdot \left[D_{V}\left(t\right)\cdot L\left(t\right)\right]^{1-\beta }}{N\left(t\right)\cdot A^{\max }\left(t\right)}; \end{align}

Equation (17) illustrates that vertical R&D sectors can enhance the success rate of innovation by utilizing a substantial amount of data during the scientific research process. In this equation, the leading-edge productivity parameter, $A^{\max}(t)$ , represents the maximum value among all productivity parameters associated with the latest version of intermediate products. The parameter $\beta$ signifies the contribution of labor in the vertical innovation process. $L_{V}(t)$ denotes the labor employed in vertical R&D sectors, while $l_{V}(t)$ represents the fraction of labor allocated to vertical R&D sectors. It is evident that as technology advances and the number of intermediate goods expands, vertical innovation becomes more intricate and challenging.Footnote ⁵

Upon the successful completion of R&D, a new intermediate producer enters the monopolistic market, replacing the previous producer. The present value of the intermediate good with the leading-edge productivity parameter in a single period, denoted as $t$ , can be calculated using the following expression:

(18) \begin{align} V\left(t\right)=\int _{t}^{\mathrm{\infty }}e^{-\int _{t}^{\tau }\left[r\left(s\right)+\Phi \left(s\right)\right]ds}\cdot \tilde{\pi }\left(\tau \right)d\tau, \end{align}

In equation (18), $\tilde{\pi }(\tau )$ represents the profit obtained from the intermediate good with the leading-edge productivity parameter in a single period, given by

(19) \begin{align} \tilde{\pi }\left(t\right)=\frac{\left(1-\alpha \right)^{\frac{2}{\alpha }-1}}{\psi ^{\frac{1}{\alpha }-1}}\cdot \alpha \cdot A^{\max }\left(t\right)\cdot L_{E}\left(t\right); \end{align}

The present value expression implies that higher investment in R&D leads to a greater success rate of innovation and a higher probability of becoming a monopoly producer of intermediate goods. However, it also implies that a higher success rate of innovation in the next phase results in a shorter cycle of replacement and a lower present value that can be obtained. This concept is rooted in the notion of creative destruction [Aghion and Howitt (Reference Aghion and Howitt1992)].

Vertical R&D sectors determine the optimal allocation of labor, $L_{R}(t)=l_{R}(t)\cdot L(t)$ , and data, $D_{V}(t)\cdot L(t)$ , to maximize the expected net profit. This decision can be formulated as followsFootnote ⁶:

(20) \begin{align} \max _{D_{V}\left(t\right),l_{V}\left(t\right)} \left\{\Phi \left(t\right)\cdot V\left(t\right)-w\left(t\right)\cdot L_{V}\left(t\right)-p_{D,V}\left(t\right)\cdot D_{V}\left(t\right)\cdot L\left(t\right)\right\}; \end{align}

The first-order condition gives rise to two free-entry conditions:

(21) \begin{align} \frac{\beta \cdot \mu \cdot {l_{V}}\left(t\right)^{\beta -1}\cdot {D_{V}}\left(t\right)^{1-\beta }\cdot \tilde{\pi }\left(t\right)\cdot \left[r\left(t\right)-n\right]}{w(t)\cdot \left[r\left(t\right)+\Phi \left(t\right)-n\right]^{2}\cdot N(t)\cdot A^{\max }\left(t\right)}=1; \end{align}

and

(22) \begin{align} \frac{\left(1-\beta \right)\cdot \mu \cdot {l_{V}}\left(t\right)^{\beta }\cdot {D_{V}}\left(t\right)^{-\beta }\cdot \tilde{\pi }\left(t\right)\cdot \left[r\left(t\right)-n\right]}{p_{D,V}\left(t\right)\cdot \left[r\left(t\right)+\Phi \left(t\right)-n\right]^{2}\cdot N\left(t\right)\cdot A^{\max }\left(t\right)}=1; \end{align}

Equations (21) and (22) represent the two free-entry conditions derived from the first-order condition, which are integral to the decision-making process of vertical R&D sectors.

2.3.3. Horizontal innovation phase

Horizontal innovations are the result of R&D efforts aimed at creating new intermediate goods. In the horizontal R&D sector, both labor, $L_{H}(t)$ , and data purchased from consumers, $D_{H}(t)\cdot L(t)$ , are employed [Romer (Reference Romer1990); Jones (Reference Jones1995)]. The evolution of the aggregate innovation possibility frontier can be described by the following equation:

(23) \begin{align} \dot{N\left(t\right)}=\frac{\eta \cdot N\left(t\right)^{\xi }\cdot {(D_{H}}(t)\cdot L(t))^{\theta }{\cdot L_{H}}\left(t\right)^{1-\theta }}{A^{\max }\left(t\right)}; \end{align}

This equation can be simplified as

(24) \begin{align} \dot{N\left(t\right)}=\frac{\eta \cdot N\left(t\right)^{\xi }\cdot {l_{H}}\left(t\right)^{1-\theta }\cdot {D_{H}}\left(t\right)^{\theta }\cdot L\left(t\right)}{A^{\max }\left(t\right)}; \end{align}

From equation (24), we can observe that data can be transformed into digital knowledge in the process of horizontal innovation. Here, $\eta \gt 0$ is an efficiency term of innovation, $\theta \in (0,1)$ represents the contribution of data in the horizontal innovation process, and $\xi \in (1,+\mathrm{\infty })$ represents the spillover effect of digital knowledge. $L_{H}(t)$ denotes the labor employed in the horizontal R&D sectors, and $l_{H}(t)$ represents the fraction of labor employed in the horizontal R&D sectors. Notably, digital knowledge is created through data spillovers to future periods by creating new varieties [Cong et al. (Reference Cong, Xie and Zhang2021)]. The larger the leading-edge productivity parameter, the more complex the process of horizontal innovation becomes [Howitt (Reference Howitt1999)].

Each horizontal innovation gives rise to a new intermediate product whose productivity parameter is randomly drawn from the distribution of existing intermediate products.

The expected value of a horizontal innovation can be expressed as

\begin{equation*} E\left[\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}\right]\cdot V\left(t\right). \end{equation*}

In the decision-making process, horizontal R&D sectors determine the optimal allocation of labor, $L_{H}(t)=l_{H}(t)\cdot L(t)$ , and data, $D_{H}(t)\cdot L(t)$ , to maximize the expected net profit. This decision can be formulated as follows:

(25) \begin{align} \max _{l_{H}\left(t\right),D_{H}\left(t\right)} \left\{E\left[\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}\right]\cdot V\left(t\right)\cdot \dot{N\left(t\right)}-w\left(t\right)\cdot L_{H}\left(t\right)-p_{D,H}\left(t\right)\cdot D_{H}\left(t\right)\cdot L\left(t\right)\right\}; \end{align}

The first-order condition leads to two free-entry conditions:

(26) \begin{align} \eta \cdot \left(1-\theta \right)\cdot N\left(t\right)^{\xi }\cdot {l_{H}}\left(t\right)^{-\theta }\cdot {D_{H}}\left(t\right)^{\theta }\cdot V\left(t\right)\cdot E\left[\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}\right]=w\left(t\right)\cdot A^{\max }\left(t\right); \end{align}

and

(27) \begin{align} \eta \cdot \theta \cdot N\left(t\right)^{\xi }\cdot {l_{H}}\left(t\right)^{1-\theta }\cdot {D_{H}}\left(t\right)^{\theta -1}\cdot V\left(t\right)\cdot E\left[\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}\right]=p_{D,H}\left(t\right)\cdot A^{\max }\left(t\right); \end{align}

Equations (26) and (27) represent the two free-entry conditions derived from the first-order condition, which play a crucial role in the decision-making process of horizontal R&D sectors.

2.3.4. Spillovers

The growth in the leading-edge parameter, $A^{\max }(t)$ occurs due to the knowledge spillovers produced by vertical innovations. $A^{\max }(t)$ always grows at a rate proportional to the aggregate rate of vertical innovations [Caballero and Jaffe (Reference Caballero and Jaffe1993)]. The factor of proportionality, $\frac{\varphi }{N(t)}$ , $N(t)$ , by the flow of vertical innovations per sector. This relationship can be expressed as

(28) \begin{align} \frac{\dot{A^{\max }\left(t\right)}}{A^{\max }\left(t\right)}=\varphi \cdot \mu \cdot {\phi} \left(t\right); \end{align}

The distribution of productivity parameters among new intermediate goods mirrors the distribution across existing intermediate goods at time $t$ . The distribution of relative productivity parameters, $b\left(v,t\right)=\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}$ , can be described by the following expression [Howitt (Reference Howitt1999)]Footnote ⁷:

\begin{equation*} F\left(b\left(v,t\right)\leq b\right)=\left[b\left(v,t\right)\right]^{\frac{1}{\varphi }}; \end{equation*}

In the long term, it follows that:

(29) \begin{align} E\left[\frac{A\left(v,t\right)}{A^{\max }\left(t\right)}\right]=\frac{1}{1+\varphi }; \end{align}

Equation (29) represents the expected value of the ratio of the productivity parameter of a new intermediate good to the leading-edge parameter in the long term.

3.1. Growth rate in the decentralized economy

Proposition 1. The economic growth rates of the decentralized economy on the BGP can be expressed as follows:

(34) \begin{align} m=\frac{n}{\mu \cdot \varphi }\cdot \frac{\left(1+\xi \right)\cdot \left(\sigma -\beta \right)}{\left(1-\beta -\theta \right)\cdot \left(1-\gamma \right)+\left(1+\xi \right)\cdot \left[\left(\beta -1\right)\cdot \left(1-\gamma \right)+\sigma \right]}\gt 0; \end{align}

(35) \begin{align} \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}=\frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}=\left(\mu \cdot \varphi \cdot m-n\right)\cdot \frac{1+\xi }{\theta +\left(1-\beta \right)\cdot \xi }\lt 0; \end{align}

(36) \begin{align} \frac{\dot{N\left(t\right)}}{N\left(t\right)}=\left(\mu \cdot \varphi \cdot m-n\right)\cdot \frac{1-\theta -\beta }{\theta +\left(1-\beta \right)\cdot \xi }\gt 0; \end{align}

(37) \begin{align} l_{H}^{*}\left(t\right)=\displaystyle \frac{1}{1+\displaystyle\frac{\beta \cdot \theta }{\left(1-\beta \right)\cdot \left(1-\theta \right)}+\displaystyle\frac{\left[\mu \cdot m\cdot \left(\gamma \cdot \varphi +1\right)+\rho -n\right]\cdot \left[\theta +\left(1-\beta \right)\cdot \xi \right]}{\left(\mu \cdot \varphi \cdot m-n\right)\cdot \left(\theta +\beta -1\right)\cdot \left(1-\theta \right)\cdot \left(1-\alpha \right)}}; \end{align}

Equation (34) represents the expression for $m$ on the BGP, and it can be easily proven that $(\mu \cdot \varphi \cdot m-n)\lt 0$ is tenable. Equation (35) indicates that in the decentralized economy, the per capita data provision of both types of R&D sectors will experience a downward trend, effectively alleviating the problem of consumer privacy disclosure. However, the aggregate data provision can still grow in the long term [similar to Cong et al. (Reference Cong, Xie and Zhang2021)].

In equation (36), it is necessary to assume that $\theta +\beta -1\gt 0$ . The first partial derivatives of $m$ with respect to key variables are as follows:

(38) \begin{align} \frac{\partial m}{\partial \beta }=\frac{\left(1+\xi \right)\cdot \sigma \cdot \left(-1-2\cdot \xi +\gamma \cdot \xi \right)+\left(1-\gamma \right)\cdot \left(\theta +\xi \right)}{\left\{\left(1-\gamma \right)\cdot \left[\left(\beta -1\right)\cdot \xi -\theta \right]+\sigma \cdot \left(1+\xi \right)\right\}^{2}}\lt 0, \end{align}

(39) \begin{align} \frac{\partial m}{\partial \sigma }=\frac{\left(1+\xi \right)\cdot \left[\left(1-\gamma \right)\cdot \left(\beta \cdot \xi -\theta -\xi \right)+\beta \cdot \left(1+\xi \right)\right]}{\left\{\left(1-\gamma \right)\cdot \left[\left(\beta -1\right)\cdot \xi -\theta \right]+\sigma \cdot \left(1+\xi \right)\right\}^{2}}\gt 0; \end{align}

Most of the findings from Proposition 1 are in line with expectations. However, one notable result is the negative effect of the contribution of labor in the process of vertical innovation. This implies that a higher contribution rate of data to the success of innovation leads to a higher success rate. It demonstrates that data integration into vertical innovation processes can reduce uncertainty and improve the likelihood of success. There are two main reasons why data utilization can enhance the success rate of innovation. First, according to equation (16), wages increase with the expansion of intermediate goods varieties and improvements in productivity parameters. Consequently, labor costs continue to rise. However, equation (7) shows that the price of data is dependent on per capita data usage. If data remains unused, the cost of utilizing it would be relatively low. Therefore, from a cost perspective, incorporating data can significantly reduce innovation costs, thereby increasing the monopoly profits of intermediate producers. Second, equation (14) reveals that monopoly profits are contingent on the labor used in final goods production. Consequently, if vertical R&D sectors replace some labor with data, there will be an increase in labor allocated to final goods production, resulting in improved monopoly profits. Hence, when data utilized in the innovation process offers greater revenue incentives, the success rate of innovation is substantially enhanced.

In contrast to the conclusion drawn by Howitt and Aghion (Reference Howitt and Aghion1998), data differ fundamentally from physical capital. While physical capital incurs production costs that affect the interest rate and consequently impact the monopoly profits of intermediate producers, data, as a by-product of consumption, only entails the usage cost associated with privacy concerns and lacks production costs. Although both factors can enhance the success rate of innovation when incorporated into the innovation process, their mechanisms are distinct.

The innovation success rate is positively influenced by the parameter representing the disutility of data leakage and misuse. This suggests that in equilibrium, consumers require a higher success rate of innovation to compensate for the increased disutility resulting from heightened privacy concerns. Furthermore, the success rate of innovation is positively associated with the natural population growth rate. This implies that with a larger population, there is an increase in both the amount of labor involved in the innovation process and the overall availability of data for innovation purposes. Conversely, the success rate of innovation is negatively correlated with the elasticity of intertemporal substitution of consumption. This indicates that if individuals prefer substituting current consumption with future consumption, the amount of data accessible to vertical R&D sectors will be diminished, leading to a lower success rate of innovation.

4.1. Growth rate in the social planner’s economy

In the analysis of the social planner’s economy on the BGP, the paper examines the relationship between key variables. The following relationships are established:

(50) \begin{align} \frac{{l_{V}}\left(t\right)^{\beta }\cdot {D_{V}}\left(t\right)^{1-\beta }\cdot L\left(t\right)}{N\left(t\right)\cdot A^{\max }\left(t\right)}\rightarrow M; \end{align}

(51) \begin{align} \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}=\frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}; \end{align}

(52) \begin{align} \frac{\dot{N\left(t\right)}}{N\left(t\right)}=\frac{1-\beta -\theta }{\xi }\cdot \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}; \end{align}

(53) \begin{align} \frac{\dot{c\left(t\right)}}{c\left(t\right)}=\frac{\sigma }{1-\gamma }\cdot \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}; \end{align}

Equation (50) indicates that on the BGP, the success rate of the vertical innovation approaches a specific value denoted as $M$ . Equations (51) and (52) demonstrate that on the BGP, the growth rate of per capita data provision is the same for both types of R&D sectors, and it relates the growth rate of per capita data provision to the growth rate of varieties of intermediate goods. Equation (53) reveals that on the BGP, the growth rate of per capita consumption is determined by the growth rate of per capita data provision.

Proposition 3. The economic growth rates of the social planner’s economy on the BGP can be expressed as follows:

(54) \begin{align} M=\frac{n}{\mu \cdot \varphi }\cdot \frac{\sigma \cdot \xi }{\sigma \cdot \xi -\left(1-\gamma \right)\cdot \left[\left(1-\beta \right)\cdot \left(\xi -1\right)+\theta \right]}\gt 0;\end{align}

(55) \begin{align} \frac{\dot{N\left(t\right)}}{N\left(t\right)}=n\cdot \frac{\left(1-\gamma \right)\cdot \left(1-\beta -\theta \right)}{\sigma \cdot \xi -\left(1-\gamma \right)\cdot \left[\left(1-\beta \right)\cdot \left(\xi -1\right)+\theta \right]}\gt 0; \end{align}

(56) \begin{align} \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}=\frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}=n\cdot \frac{\left(1-\gamma \right)\cdot \xi }{\sigma \cdot \xi -\left(1-\gamma \right)\cdot \left[\left(1-\beta \right)\cdot \left(\xi -1\right)+\theta \right]}\lt 0; \end{align}

(57) \begin{align} l_{H,s}^{*}\left(t\right)=\displaystyle \frac{\displaystyle 1-\theta }{\displaystyle \frac{\left(1+\beta \right)\cdot \theta }{\left(1-\beta \right)}+\displaystyle \frac{\left(1-\sigma -\beta \right)\cdot n\cdot \left(1-\gamma \right)\cdot \xi +\rho \cdot {\Lambda} }{n\cdot \left(1-\gamma \right)\cdot \left(1-\theta -\beta \right)}-\left(\theta +\xi \right)}, \end{align}

where ${\Lambda} =\sigma \cdot \xi -(1-\gamma )\cdot [(1-\beta )\cdot (\xi -1)+\theta ]$ .

According to equation (59), it is evident that $\frac{\partial M}{\partial \beta }\lt 0$ holds, indicating that in the social planner’s economy, a higher success rate of innovation is achieved with a greater contribution of data in the vertical innovation process. This is primarily because fewer labor resources are employed in the R&D sectors, allowing for more labor in the final goods production sector and a looser resource constraint, as indicated by equation (14). On the other hand, equation (56) demonstrates that in the social planner’s economy, the per capita data provision in both types of R&D sectors continues to decline, indicating that the issue of consumer privacy disclosure does not deteriorate over time.

4.2. Misallocation in the decentralized economy

In this section, we examine the misallocation of resources in a decentralized economy. By using standard values from the existing literature, with $n=0.02, \gamma =2.5$ , $\sigma =3, \theta =0.5, \text{and}\, \xi =0.85$ , as well as reasonable discretionary values, Figure 1 illustrates the difference between the success rate of vertical innovation in two cases, as per equations (35) and (54). The observed difference varies with the contribution of labor in the process of vertical innovation.

Figure 1. Success rate of the vertical innovation in both types of economies.

Similarly, utilizing $n=0.02, \gamma =2.5$ (standard values from existing literature), $\sigma =3, \beta =0.7, \xi =0.85$ (reasonable discretionary values), Figure 2 displays the difference between the growth rate of varieties of intermediate goods in the two cases, according to equations (36) and (55). This observed difference varies with the contribution of data in the process of horizontal innovation.

Figure 2. Growth rate of varieties of intermediate goods in both types of economies.

As depicted in Figures 1 and 2, both the success rate of vertical innovation and the growth rate of varieties of intermediate goods in the social planner’s economy surpass those in the decentralized economy. Intuitively, due to the inherent competition and lack of coordination between the two types of research and development (R&D) sectors, each sector independently maximizes its own profits in the decentralized economy, resulting in negative externalities between them. Conversely, in the social planner’s economy, the R&D intensity of both sectors is determined by the maximization of total utility, effectively mitigating the negative externalities.

To further investigate the misallocation within the decentralized economy, this paper examines the differences in the ratio of labor employed in R&D sectors and the growth rate of data provision. By employing standard values ( $n=0.02, \gamma =2.5,\rho =0.03, \alpha =\frac{2}{3},\beta =0.7, \theta =0.5, \xi =0.85,\mu =0.1, \text{and}\, \varphi =10$ ) from the existing literature, Figure 3 presents the difference between the ratios of labor employed in horizontal R&D sectors in the two cases, as per equations (37) and (57). The observed difference varies with the disutility effect of data. Notably, since the ratio of labor employed in vertical R&D sectors is the same proportion in both economies on the BGP, our analysis focuses solely on the ratio of labor employed in horizontal R&D sectors.

Figure 3. Ratio of labor employed in horizontal R&D sectors in both types of economies.

With $n=0.02, \gamma =2.5,\rho =0.03,\alpha =0.67$ taking on standard values from the existing literature, as well as $\sigma =3, \theta =0.5, \xi =0.85,\mu =0.1,\varphi =10$ as reasonable discretionary values, Figure 4 is presented to analyze the difference in the growth rate of data provision employed in R&D sectors between the two cases, as per equations (35) and (56). The observed difference varies with the contribution of labor in the vertical innovation.

Figure 4. Growth ratio of data provision in R&D sectors in both types of economies.

Notably, Figure 3 illustrates that on the BGP, the ratio of labor employed in horizontal R&D sectors in the decentralized economy is consistently lower than that in the social planner’s economy, regardless of the disutility effect of data. This outcome arises due to the high prices set by monopoly intermediate producers for their intermediate goods, which prompts final goods producers to substitute labor for some intermediate goods. Consequently, there is an undersupply of labor employed in R&D sectors, aligning with the findings of Jones (Reference Jones1995).

Furthermore, Figure 4 demonstrates that on the BGP, the growth ratio of data provision in R&D sectors is consistently lower in the decentralized economy compared to the social planner’s economy, regardless of the contribution of labor in the process of vertical innovation. This discrepancy primarily arises because, in the decentralized economy, R&D sectors need to purchase data using financial resources instead of receiving direct allocations for innovation. Moreover, according to equations (6) and (7), the price of data is positively influenced by data provision and per capita consumption, indicating that the price of data introduces distortions in data provision. Consequently, on the BGP, both types of R&D sectors employ less labor and data in the decentralized economy. There are two underlying reasons for this, which imply that the R&D intensity in the decentralized economy is lower compared to the planned economy.

7.1. Methodology and calibration

To facilitate tractability, this paper employs a numerical method based on the social planner’s economy, following a similar approach to Jones (Reference Jones2016). Instead of undertaking formal calibration to replicate the circumstances of a specific country, this analysis aims to demonstrate the basic transitional dynamics achievable within the theoretical framework.Footnote ¹¹ By deriving the first-order conditions with respect to variables such as $c(t), D_{H}(t), D_{V}(t), l_{H}(t), l_{V}(t), A^{\max }(t), N(t)$ from equation (49) in Section 4, the dynamical systems can be expressed as a set of differential equations.Footnote ¹² Since $l_{H}(t), l_{V}(t), A^{\max }(t), N(t)$ are all state variables, this paper sets their initial values to a sufficiently small $\epsilon$ to drive economic growth.Footnote ¹³

Table 1 provides a summary of the parametrization choices made in this study. Based on estimates from Antras (Reference Antras2004), the elasticity of substitution falls within the range of approximately 0.6–0.8. Therefore, we set the contribution of labor in final goods production α to 2/3. Considering estimates from Vissing-Jørgensen (Reference Vissing-Jørgensen2002), which suggest that the elasticity of intertemporal substitution (EIS) ranges from 0.3 to 0.4, we select the reciprocal of EIS γ as 2.5. Drawing from estimates in Krupka and Stephens (Reference Krupka and Stephens2013), the subjective discount factor ρ is set at 0.025. Population growth is estimated at around 1% per year according to Jones and Tonetti (Reference Jones and Tonetti2020), and the growth rate of R&D labor in developed economies is approximately 4%. Hence, we choose the value n to be 0.02. Considering estimates from Caballero and Jaffe (Reference Caballero and Jaffe1993), the productivity of vertical innovation μ is chosen as 10 and the marginal impact of vertical innovation on the stock of public knowledge ϕ as 0.1.

Table 1. Parameters for studying transitional dynamics

Because they could not be accurately estimated, the following parameter values are selected from Jones (Reference Jones1995) and Cong et al. (Reference Cong, Xie and Zhang2021): the contribution of data in horizontal innovation θ as 0.5, the spillover effect of knowledge in horizontal innovation ξ as 0.85, the efficiency term in horizontal innovation η as 1, and the severity of consumers’ privacy concern σ as 1.8. To meet the requirement θ + β − 1 > 0, the contribution of labor in vertical innovation β is set at 0.7.

In the forthcoming figures, we examine the potential impact on the economy when the quantity of data available to either of the R&D sectors is limited.Footnote ¹⁴ To incorporate this constraint, we reformulate equations (3) and (4) as equations (76) and (77), respectively. To enforce the constraint on data provision for each R&D sector, we set the values of $s_{1}\text{ or } s_{2} \text{ as }0.01$ . This ensures that the provision of data used by the respective sectors adheres to the constraint. Conversely, if we do not intend to impose a constraint on the data provision used by the R&D sectors, we set the values of $s_{1}\text{ and } s_{2} \text{ as }100$ , rendering the constraint (3) or (4) ineffective.

(76) \begin{align} \frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}\leq \frac{\dot{c\left(t\right)}}{c\left(t\right)}+s_{1}; \end{align}

(77) \begin{align} \frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}\leq \frac{\dot{c\left(t\right)}}{c\left(t\right)}+s_{2}; \end{align}

Equation (76) states that the growth rate of $\frac{\dot{D_{H}\left(t\right)}}{D_{H}\left(t\right)}$ should not exceed the growth rate of $\frac{\dot{c\left(t\right)}}{c\left(t\right)}+s_{1}$ , while equation (77) imposes a similar constraint for $\frac{\dot{D_{V}\left(t\right)}}{D_{V}\left(t\right)}$ compared to $\frac{\dot{c\left(t\right)}}{c\left(t\right)}+s_{2}$ .

Figure 7. Growth rates of consumption and intermediate goods varieties. (a) Without any constraint. (b) With horizontal innovation constraint. (c) With vertical innovation constraint.

7.2. Results and discussions

In Panel (a) of Figure 7, the paper reveals that in the absence of constraints, different types of innovations play distinct roles during different periods of economic growth. In the early stage, vertical innovation assumes greater significance due to the limited availability of intermediate goods varieties. However, as the number of intermediate goods varieties gradually increases, horizontal innovation becomes more crucial in the later stage. In comparison to the findings of Cong et al. (Reference Cong, Xie and Zhang2021), by placing greater emphasis on vertical innovation, this economy achieves a higher economic growth rate along the transitional path in the long run.

Panel (b) of Figure 7, as per equation (6), illustrates the performance of the two types of innovations during different stages of economic growth when a constraint is imposed on horizontal innovation. The constraint implies insufficient data usage in the horizontal R&D sectors. It is evident that there is more pronounced fluctuation during the early period of economic development. Moreover, an economy constrained by horizontal innovation requires additional time to reach the BGP. Intuitively, when the data usage in horizontal R&D sectors is inadequate, the rate at which labor moves away from final goods production slows down. This can be observed in both Panels (a) and (b) of Figure 8, similar to the findings of Cong et al. (Reference Cong, Xie and Zhang2021). The insufficient data provision and labor in horizontal R&D sectors result in a lack of driving force for horizontal innovation. Consequently, vertical innovation assumes a more prominent role in the later stage. However, as per capita consumption increases and the constraint on data usage becomes more significant, the speed of intermediate goods variety expansion accelerates. The impact of vertical innovation is subject to a certain spillover effect, leading to greater fluctuations.

Figure 8. Rate of labor employed in horizontal R&D sector. (a) Without any constraint. (b) With horizontal innovation constraint.

In Panel (c) of Figure 7, it is observed that the economy with a constraint on vertical innovation reaches the BGP in approximately the same period as the economy without any constraints. Intuitively, due to the spillover effect of the expansion of intermediate product varieties, vertical innovation has minimal influence on the timing of reaching the BGP. Although the economy performs smoothly without significant fluctuations in this scenario, the growth rate of per capita consumption along the transitional path is lower compared to Panel (a) of Figure 7.

For further analysis, Figure 9 is examined. Both images demonstrate that the growth rate of the leading-edge productivity parameter initially increases to a maximum and then stabilizes. This suggests that when there are relatively few varieties of intermediate goods, vertical innovation is more easily realized. Additionally, Figure 10 shows an increasing number of laborers swiftly transitioning from final goods production to vertical R&D sectors.

Figure 9. Growth rate of leading-edge productivity parameter. (a) Without any constraint. (b) With vertical innovation constraint.

Figure 10. Rate of labor employed in vertical R&D sector. (a) Without any constraint. (b) With vertical innovation constraint.

However, as the scale of intermediate goods gradually expands, vertical innovation becomes more complex and plays a less significant role. Consequently, the growth rate of the leading-edge productivity parameter gradually slows down, with some laborers in vertical R&D sectors transitioning to other sectors. Furthermore, an economy with limited data usage requires additional time to reach the maximum value of the growth rate of the leading-edge productivity parameter, influenced by the stronger spillover effect. This is why such an economy experiences smoother progress along the traditional path with minimal fluctuation.

In summary, when the data provision in horizontal R&D sectors is limited, the economy exhibits more severe fluctuations along the transitional path, resulting in increased uncertainty during the development process and requiring additional time to reach the BGP. If the data usage in vertical R&D sectors is constrained, the economy experiences a lower growth rate without significant fluctuations, indicating that the welfare level of the local population is relatively lower in the early stages of development.