Cong, Xie, and Zhang

Knowledge Accumulation, Privacy, and Growth in a Data Economy

Knowledge Accumulation, Privacy, and
Growth in a Data Economy††thanks: We thank Jasmine Cheng, Kay Giesecke,
Shiyang Huang, Laura Veldkamp, Ji Yan, Buyuan Yang, Liyan Yang, and the
anonymous associate editor and referees for their feedback. We also thank
conference and seminar participants at the Midwest Macroeconomics Meeting,
Australasian Finance and Banking Conference, Delhi School of Economics and the
Econometric Society Winter School, Machine Lawyering’s Conference for helpful
comments and discussions. Cong and Xie are grateful for the financial support
from the Ewing Marion Kauffman Foundation and from the National Natural
Science Foundation of China (Grant No. 71973076), respectively. The contents
of this publication are solely the responsibility of the authors.

Lin William Cong SC Johnson College of Business, Cornell University, Ithaca,
NY 14853<EMAIL_ADDRESS>Danxia Xie Institute of Economics, Tsinghua
University, Beijing, China 100084<EMAIL_ADDRESS>Longtian Zhang
School of International Trade and Economics, Central University of Finance and
Economics, Beijing, China 102206<EMAIL_ADDRESS>
We build an endogenous growth model with consumer-generated data as a new key
factor for knowledge accumulation. Consumers balance between providing data
for profit and potential privacy infringement. Intermediate good producers use
data to innovate and contribute to the final good production, which fuels
economic growth. Data are dynamically nonrival with flexible ownership while
their production is endogenous and policy-dependent. Although a decentralized
economy can grow at the same rate (but are at different levels) as the social
optimum on the Balanced Growth Path, the R&D sector underemploys labor and
overuses data—an inefficiency mitigated by subsidizing innovators instead of
direct data regulation. As a data economy emerges and matures, consumers’ data
provision endogenously declines after a transitional acceleration, allaying
long-run privacy concerns but portending initial growth traps that call for
interventions.

Big Data, Data Ownership, Endogenous Growth, Innovation, Non-rivalry, Privacy
Regulation Initially written in October 2019 and first submitted on April 16,
2020.

##  1 Introduction

Data not only help produce new products and services, but also are used in
research and development and knowledge creation, which in turn improves
production efficiency.111Data have become a key input to many technology and
financial firms (e.g., Veldkamp and Chung 2019, Arrieta-Ibarra et al. 2018,
Cong et al. 2019); Manyika et al. (2011) estimate that big data research is
believed to save over 100 billion Euros for Europe, and reduce medical care
cost of the United States by 8% or 300 billion dollars annually. Endogenous
generation, dynamic nonrivalry, and flexible ownership of data distinguish
them from labor and capital, with implications for labor market allocations
and policies. Meanwhile, the proliferation of big data applications often
comes at the expense of consumer privacy and is associated with discrimination
and misuse.222Cambridge Analytica and Facebook’s data scandal presents a
salient example (Confessore 2018). The United States have the highest ratio of
data records stolen relative to population (over 6 billion stolen records,
exceeding the population by 19 times, Sobers 2020). Data from China Judgements
Online and China Academy of Information and Communications Technology also
reveal that civil and criminal cases of data leakage have been increasing in
lock-steps with the growth of China’s digital economy. Despite the
introduction of data privacy laws ranging from the General Data Protection
Regulation (GDPR) to the California Consumer Privacy Act (CCPA) to Japan’s Act
on the Protection of Personal Information, we know little about how data
(mis)usage, digital infrastructure, and privacy regulation affect knowledge
accumulation and the growth of a dynamic economy where data are the new key
driver.

To fill in this gap, we build on Romer (1990) to develop an endogenous growth
model of the data economy. Our key innovation lies in that consumer data add
to R&D and knowledge accumulation. At the same time, data are by-products of
economic activities with potential privacy issues, which differ from other
input factors such as labor or capital because growth can endogenously feed
back to data generation. Consumers in our baseline model choose the quantity
of data to sell to intermediate firms, cognizant of potential information
leakage and misuses. Innovative intermediate firms utilize the raw data for
research that contributes to the final good production. Specifically, data are
transformed into intermediate goods (information products included)—a feature
absent in other models. Data can generate spillovers through knowledge
accumulation, which is further enhanced when they are traded over time and
used by multiple parties with little reproduction costs (both static and
dynamic nonrivalry). These effects are moderated in the model by the
disutility from potential data privacy violation.

We show that a decentralized economy grows at the same rate as the social
optimum on the Balanced Growth Path (BGP). But social welfare and consumer
surplus are strictly lower due to underemployment and data overuse in the R&D
sector. Monopolistic markups in the production of intermediate goods lead to a
crowding out of labor in R&D (Jones 1995), producers then compensate the
underemployment of labor in R&D by more aggressive utilization of data. The
crowding-out of labor crowds in data usage in R&D to a socially excessive
level. R&D labor employment and data usage can thus deviate significantly from
those in the social planner’s solution, especially in the initial phase of BGP
(albeit less severe in the long run).

Different from recent studies such as Jones and Tonetti (2020), data are
overused even when they are nonrival and owned by consumers who directly
factor in the disutility from data leakage or abuse. Direct regulation on data
usage comes at the cost of economic growth and constitutes a wealth transfer
from future generations to current generations. Rather than taxing data
overuse, our model reveals that subsidizing R&D wage or intermediate producers
are more effective at mitigating the social inefficiency. Moreover, because
data expand the innovation possibility frontier which exhibits diminishing
returns to scale, historical data usage reduces the benefit of future data
usage, potentially leading to a declining data provision per capita in the
long run.

As the economy transitions into a steady growth, data provision by consumers
can undergo accelerated growth before declining. Importantly, a low initial
growth may limit data generation even in the planner’s solution, which further
delays the transition to high growth stages in the long run—a form of growth
trap. Interventions such as foreign aid for digital infrastructure development
can help escape the trap, but only to the extent that the data generating
constraint is still binding, i.e., when economic activities translate into
limited volumes of data.

Our paper primarily contributes to the emerging literature on information and
the data economy. Relative to earlier studies on the social value, sales, and
property rights of information (e.g., Hirshleifer 1971, Admati and Pfleiderer
1990, Murphy 1996), recent studies focus on connecting digital information
with privacy issues (e.g., Akçura and Srinivasan 2005, Casadesus-Masanell and
Hervas-Drane 2015) or the nonrivalry of data and competition (e.g., Ichihashi
2020a, Easley et al. 2019). We differ by being the first to connect data usage
to knowledge accumulation and endogenous growth with privacy considerations.
Furthermore, our model complements studies microfounding data privacy concerns
(e.g. Ichihashi 2020b, Liu et al. 2020), and is broadly consistent with
empirical patterns on the correlation of data economy size with regulation,
privacy issues, and declining labor income share in the United States and
around the world (Karabarbounis and Neiman 2014, Tang 2019, Abis and Veldkamp
2020, Barkai 2020, Liao et al. 2020).

As such, our theory adds to the large literature on economic growth,
especially recent studies embedding data into growth models by allowing them
to directly enter production. For example, Jones and Tonetti (2020) highlights
the underutilization of data due to its nonrivalry and the importance of
giving data property rights to consumers; Farboodi and Veldkamp (2020)
emphasizes that data have bounded returns in long-run growth. We complement
Jones and Tonetti (2020) by allowing data to facilitate knowledge accumulation
in a semi-endogenous growth model (Jones 1995, 2016) and accounting for the
consumers’ data privacy concerns in the spirit of Stokey (1998) and Acemoglu
et al. (2012).333Unlike the growth literature focusing on firms’ property
rights, we allow consumers to own and sell data. Through a mechanism different
from Farboodi and Veldkamp (2020), we also find that data play a limited role
in the long run.

##  2 The Model

We incorporate data production, data-based innovation, knowledge accumulation,
and data privacy concerns into a macroeconomic model to characterize economic
growth. Our dynamic “data economy” consists of representative agents who are
both consumers and workers, innovative intermediate producers, and a final
good producer. Time is continuous and infinite.

###  2.1 Representative Consumers

A population of homogeneous representative consumers (or households) grow at a
constant rate $n$ and is $L(t)$ at time $t$. Besides consumption, they each
choose in each period to supply one unit of labor inelastically in either the
R&D sector (intermediate good production) or final good production.

Each consumer produces data as by-products of consumption (e.g., Veldkamp
2005) and can sell the data to intermediate good producers.444“Selling” can be
broadly interpreted: For example, not having to pay for the use of Gmail can
be viewed as compensation to consumers for allowing Google to use their data
in ways delineated in the usual terms and conditions. In Section 3.6, we allow
firms to own data as in Jones and Tonetti (2020) and derive additional
insights in addition to showing robustness of the main results. However, the
data involve personal information and risk leakage and misuse, leading to a
disutility that they consider when selling data. We follow the literature
(e.g., Jones and Tonetti 2020) to allow data to fully depreciate in every
period in the baseline model, but relax this assumption in Online Appendix 7.2
and further discuss it in Section 3.5.

Each consumer’s utility maximization problem is then:

$\max_{c(t),\varphi(t)}\int_{0}^{\infty}e^{-(\rho-n)t}\left[\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}\right]\mathrm{d}t,$
(1)

subject to

$\dot{a}(t)=(r(t)-n)a(t)+w(t)+p_{\varphi}(t)\varphi(t)-c(t),\forall
t\in[0,\infty),$ (2)

and

$\frac{\dot{\varphi}(t)}{\varphi(t)}\leq\frac{\dot{c}(t)}{c(t)}+s.$ (3)

Here, $c(t)$ is the per capita consumption level at time $t$ and $\varphi(t)$
is the quantity of data a consumer provides to potential intermediate
producers for R&D. $\rho$ is the consumers’ discount rate, the reciprocal of
$\gamma$ (also indistinguishable from the coefficient of risk aversion) is the
elasticity of intertemporal substitution (EIS) of consumption, and $\sigma$
parameterizes the disutility of data misuse or privacy violation,
$\varphi(t)^{\sigma}$, which also depends on the quantity of data provided.
$s$ represents the tightness of this constraint and depends on the digital
infrastructure, legal development, and privacy regulation policy of the
country. We normalize $s$ to zero in the baseline and discuss comparative
statics with respect to $s$ later to understand the impact of policy
intervention.

In the budget constraint (2), $a(t)$ is the asset held by a consumer at time
$t$ and $r(t)$ is its interest rate. $w(t)$ and $p_{\varphi}(t)$ are the
time-$t$ wage for labor and price of data, respectively. The relevant
variables for transitional dynamics are the growth rates of data contribution
and consumption. To see this, we can derive the system’s evolution in the form
of Euler equations from the Hamilton:

$\frac{\dot{c}(t)}{c(t)}=\frac{r(t)-\rho}{\gamma}$ (4)

and

$\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}-(\sigma-1)\frac{\dot{\varphi}(t)}{\varphi(t)}=r(t)-\rho.$
(5)

Constraint (3) therefore requires the growth rate of data provision to be
bounded by the corresponding growth rate of consumption, which is natural and
ensures analytical tractability. Moreover, (3) directly implies that
$\varphi(t)\leq\chi c(t)$ for some constant $\chi>0$, which other recent
studies feature. In other words, data are by-products of economic activities
and cannot exceed a fixed proportion of consumption activities.

###  2.2 The Final Good Producer

A representative final good producer operates in a competitive environment
with a production function,

$Y(t)=L_{E}(t)^{\beta}\int_{0}^{N(t)}x(v,t)^{1-\beta}\,\mathrm{d}v,$ (6)

where $L_{E}(t)$ is the labor employed and $N(t)$ is the number of varieties
of intermediate goods used in the final good production at time $t$. $x(v,t)$
is the total amount of intermediate good of variety $v$ which can only be used
in the final good production for one period. We can view price $p_{x}(v,t)$ as
the rental fee of patents. Finally, $\beta$ is the elasticity coefficient of
labor in the final good production. The final good producer’s profit
maximization over labor employed and the amount of each intermediate good
yields the first order conditions:

$x(v,t)=\left[\frac{1-\beta}{p_{x}(v,t)}\right]^{\frac{1}{\beta}}L_{E}(t),$
(7)

and

$w(t)=\beta L_{E}(t)^{\beta-1}\int_{0}^{N(t)}x(v,t)^{1-\beta}\,\mathrm{d}v.$
(8)

###  2.3 Intermediate Producers

An unlimited pool of potential intermediate producers decide whether and how
much to conduct research, the success of which gives them monopoly over the
intermediate product developed. An intermediate producer therefore enters the
market by conducting R&D and evaluating the prospective profit from success
less the costs of labor and data as inputs for R&D. We solve the intermediate
producers’ problem through backward induction.

##### Production phase.

Upon R&D success, each entrant produces a distinct variety of intermediate
goods in a monopolistic market. The present value of an intermediate good of
variety $v$ is

$V(v,t)=\int_{t}^{\infty}\exp{\left(-\int_{t}^{s}r(\tau)\,\mathrm{d}\tau\right)}\pi(v,s)\,\mathrm{d}s,$
(9)

where the profit from the intermediate good of variety $v$ in a single period
of time $t$ is

$\pi(v,t)=p_{x}(v,t)x(v,t)-\psi x(v,t).$ (10)

Here, $\psi$ is the marginal cost of this production process, which is set as
constant in this economy.

Substituting (7) into (10) and taking derivative with respect to $p_{x}(v,t)$
yields the optimal price of each variety of intermediate good:

$p_{x}(v,t)=\frac{\psi}{1-\beta},$ (11)

which is the same among different varieties and different periods. Next,
substitute (11) into (7):

$x(v,t)=\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}}L_{E}(t)\equiv
x(t),$ (12)

which implies that production quantity is independent of the variety.

Substituting (11) and (12) into (6) and (10), respectively, we get

$\pi(v,t)=\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}L_{E}(t)\equiv\pi(t)$
(13)

and

$Y(t)=\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}N(t)L_{E}(t).$
(14)

Also, we derive the wage rate as:

$w(t)=\beta\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}N(t).$
(15)

##### Entry and R&D phase.

Potential intermediate producers enter by conducting R&D using both labor
(researchers, including those working on computing, and AI to allow us to use
data more efficiently), $L_{R}(t)$, and data purchased from consumers,
$\varphi(t)L(t)$.555In some sense, data acquired by firms are their intangible
capital, akin to customer capital discussed in recent studies (e.g., Dou et
al. 2019). We assume that intermediate producers pay a “data processing cost”
before they use data for R&D:

$\theta\varphi(t)^{\phi},\quad\textit{where}\quad\theta\geq
0\quad\textit{and}\quad\phi>1.$ (16)

For simplicity, we set $\theta=0$ in the baseline model, and solve in Section
3.6 the general case in which $\theta>0$ and $\phi>1$ ensure that the cost
function is increasing and convex. We need $\theta>0$ when we allow firm
ownership of data because otherwise the solution is trivial: Firms certainly
use up all the data if there is not any data processing cost.

We also specify the evolution of the aggregate innovation possibility frontier
(number of varieties) as:

$\dot{N}(t)=\eta N(t)^{\zeta}[\varphi(t)L(t)]^{\xi}L_{R}(t)^{1-\xi}=\eta
N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t),$ (17)

where $\eta>0$ is an efficiency term of innovation, $\xi\in(0,1)$ represents
the relative contribution of data and labor in innovating process, $0<\zeta<1$
captures the spillover effect of knowledge, $\varphi(t)L(t)$ corresponds to
data provided by all the consumers in the period, $L_{R}(t)$ is labor employed
in R&D sector, and $l_{R}(t)=L_{R}(t)/L(t)$ denotes the fraction of labor
employed in the R&D sector.666We follow the convention in the literature to
maintain constant return to scale of inputs; $\zeta>0$ corresponds to the
positive external returns due to the cumulative nature of innovation (e.g.,
Chang 1995, Cong and Howell 2020), whereas $\zeta<1$ follows the innovation of
Jones (1995) over Romer (1990) to reflect that it becomes harder to come up
with new varieties of intermediate goods as the variety expands. Labor market
clearing condition requires $L_{E}(t)+L_{R}(t)\leq L(t)$.

The fact that data enter the R&D of intermediate goods distinguishes our paper
from studies such as Jones and Tonetti (2020) that have data only entering
directly into final good productions (similar to the $x(v,t)$ in our model).
While it holds for data-intensive industries (e.g., self-driving cars) that
use data directly as inputs, other industries instead use non-data
intermediate goods as inputs. Data, in addition to R&D labor, can be useful
for creating those intermediate goods, which our model captures. As Romer
(1990) aptly puts, “an intermediate-goods sector uses the designs from the
research sector together with forgone output to produce the large number of
producer durables that are available for use in final-goods production at any
time.” Our model speaks to non-data-intensive industries as well.

An intermediate producer decides how much labor $L_{R}(t)=l_{R}(t)L(t)$ and
data $\varphi(t)L(t)$ to employ and purchase to maximize the expected net
profit:

$\max_{L_{R}(t),\varphi(t)}\eta
N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t)V(t)-w(t)l_{R}(t)L(t)-p_{\varphi}(t)\varphi(t)L(t).$

The first order conditions yield two free-entry conditions:

$\eta\xi N(t)^{\zeta}\varphi(t)^{\xi-1}l_{R}(t)^{1-\xi}V(t)=p_{\varphi}(t),$
(18)

and

$\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}V(t)=w(t).$ (19)

Intermediate producers enter until the marginal benefits of adding data or
labor equal to the marginal costs.

###  2.4 Equilibrium Definition

An equilibrium in our model is an allocation in which all intermediate
producers choose
$\\{p_{x}(v,t),\varphi(t),L_{R}(t)\\}_{v\in[0,N(t)],t=0}^{\infty}$ to maximize
the discounted value of profits, the evolution of $\\{N(t)\\}_{t=0}^{\infty}$
is determined by free entry, the evolution of
$\\{r(t),w(t),p_{\varphi}(t)\\}_{t=0}^{\infty}$, is consistent with market
clearing, the evolution of $\\{L_{E}(t),x(v,t)\\}_{v\in[0,N(t)],t=0}^{\infty}$
is consistent with the final good producer’s profit maximization, and the
evolution of $\\{c(t),\varphi(t),L_{E}(t),L_{R}(t)\\}_{t=0}^{\infty}$ is
consistent with consumers’ utility maximization.

###  2.5 Distinguishing Features of Data

Although the functional forms of how data enter the production function of
intermediate goods resemble that of R&D-specific labor, the role of data in
economic growth is fundamentally different from labor or capital. It is worth
clarifying the distinguishing features of data before we proceed to solve the
model.

First, whereas population dynamics are exogenous in many growth models (such
as the possible supply of labor), data are endogenized by consumption which is
itself endogenous and depends on data usage in knowledge accumulation. Privacy
regulations such as the GDPR and CCPA can affect the endogenous production and
usage of data while population growth is more organic and hard to regulate
with immediate effects. Also, note that neither capital nor labor usage causes
disutility from privacy concerns.

Another key distinguishing feature of data our paper highlights is dynamic
nonrivalry. In Jones and Tonetti (2020), all firms need data for production
and the nonrivalry is static and cross-sectional. In our model, only potential
intermediate producer needs to use raw data in every period, although the
entrants and incumbents are benefiting from the same data without incurring
high reproduction costs. Because data are traded between intermediate
producers entering in different periods, the focus is more on data nonrivalry
over time.

This is implicit in our setup: Even though data are fully depreciated every
period, they contribute to knowledge accumulation over time in terms of
varieties of intermediate goods, which differs from the literature that only
allows labor to enter the evolution. Intuitively, a firm may innovate
incrementally by observing other firms’ previous data-based innovations. As
such, data create knowledge spillovers to future periods by creating new
varieties—a form of dynamic nonrivalry.

One more distinguishing feature of data from R&D-specific labor is that data
can be owned by firms but labor cannot. Traditional firms use long-term labor
contracts while recent on-demand labor or freelance services as seen in Uber,
TaskRabbit, Scripted, and Amazon’s Mechanical Turks require spot compensations
(The Economist 2018). Neither entails firm ownership.

We further elaborate on data’s endogenous production and usage as by-products
of economic activities, nonrivalry under knowledge accumulation and creative
destruction, and flexible ownership in Sections 3.4, 3.5, and 3.6,
respectively, as we characterize the equilibrium.

##  3 Data Economy on the Balanced Growth Path

We first solve the model along the Balanced Growth Path (BGP), which requires
that all variables are growing at the same constant rate—a steady state of
transformed variables that the literature focuses on. Constant growth then
implies $r(t)=r^{*}$ (Eq.(4)). We then identify inefficient data overuse and
underprovision of R&D labor, explore policy remedies, and discuss the
implications of data nonrivalry and ownership.

###  3.1 Growth Rate and Labor Share in a Decentralized Economy

###### Proposition 3.1

The economic growth of the decentralized economy on the balanced growth path
does not exhibit scale effect, and the BGP growth rates can be expressed as
follows

$g_{c}^{*}=g_{y}^{*}=g_{N}^{*}=g^{*}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n.$
(20)

The constraint on data provision per capita never binds under BGP, and its
growth rate is:

$g_{\varphi}^{*}=\frac{1-\zeta}{\xi}g^{*}-\frac{1}{\xi}n=\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n<0.$
(21)

Online Appendix 6.2 contains the proof. Here, $g_{\varphi}^{*}$ is negative:
As the economy grows on a BGP, the data each person contributes steadily
decrease in the long run, although the aggregate data use can still grow.

Because we recognize data as an input into the innovation possibility
frontier, the BGP growth rates are related to not only the population growth
($n$), but also consumers’ EIS ($1/\gamma$), relative contribution of data to
innovation ($\xi$), and knowledge accumulation in the form of varieties of
intermediate goods.

Specifically, note that all growth rates on BGP are ultimately driven by the
exogenous population growth rate $n$; growth rates become zeros when $n$
becomes zero.777Like Jones (1995), our model does not suffer from the scale
effect in, for example, Romer (1990), whose endogenous growth rates depend on
population level. In Online Appendix 7.1, we derive the parameter ranges for a
BGP equilibrium to exist and to be unique. We restrict our discussions within
these parameter ranges throughout the paper.

Next, when $\gamma$ converges to 1, consumers’ utility function converges to a
logarithmic form as in Jones (1995), the BGP growth rate becomes, according to
(20)

$g^{*}_{\gamma\rightarrow 1}=\frac{n}{1-\zeta}.$ (22)

Here, only knowledge accumulation and population growth rate influence the BGP
growth rate. Yet, in contrast with $g^{*}_{Jones}=n(1-\xi)/(1-\zeta)$ in Jones
(1995), our BGP growth rate is higher under the same set of parameters because
data add positively to the innovation possibility frontier.

In general, when $\gamma>1$ (as empirical studies in the macro and behavioral
literature consistently estimate, e.g., Lucas 1969, Coen 1969, Vissing-
Jørgensen 2002), we find that BGP growth rate increases with $\xi$, the extent
that data influence innovation possibility frontier. Counter-intuitively, BGP
growth rate also increases in $\sigma$, the severity of privacy concerns, and
converges to (22). This general equilibrium effect stems from the fact that
consumers in equilibrium require a higher growth rate to compensate for the
disutility from greater privacy concerns or information leakage.888Similar
phenomenon can also be found in Stokey (1998): Higher growth rates are needed
in equilibrium when there is greater pollution because people dislike the harm
caused by pollution and require compensation. Finally, the relationship
between $\gamma$ and growth rate is negative: Consumers prefer lower growth
rate and thus less production when they are less willing to substitute current
consumption with future consumption.

Besides growth rates, we derive the following result regarding labor shares in
Online Appendix 6.3:

###### Proposition 3.2

In this decentralized economy, the share of labor employed by the R&D sector
$s_{D}$ is constant in BGP, which is determined by

$s_{D}=\frac{1}{1+\Theta_{D}},\quad\textit{where}\quad\Theta_{D}=\frac{g^{*}\gamma+\rho-n}{g^{*}(1-\xi)(1-\beta)}.$

Here, the subscript “$D$” stands for decentralized. A larger growth rate
$g^{*}$ encourages firms to employ more labor in R&D. When the population
stops growing ($g^{*}=0$), labor in the R&D sector becomes zero because
without growth, using labor in R&D only leads to distuility of consumers.

###  3.2 Growth Rate and Labor Share under the Planner’s Solution

We now derive the BGP growth rates and the shares of labor allocated in the
two sectors under socially optimal allocations, which constitute a benchmark
for comparison with those in the decentralized economy. The equilibrium in a
decentralized economy is not socially optimal because of monopolistic
competition and knowledge spillover (the First Welfare Theorem fails here).

A social planner maximizes the utility of representative consumer/household,
(1), subject to the resource constraint. The resource constraint requires that
the aggregate consumption $C(t)=c(t)L(t)$ equals the aggregate net output,
which we denote by $\widetilde{Y}(t)$. In other words,

$C(t)=\widetilde{Y}(t)\equiv
L_{E}(t)^{\beta}\int_{0}^{N(t)}x(v,t)^{1-\beta}\,\mathrm{d}v-\int_{0}^{N(t)}\psi
x(v,t)\,\mathrm{d}v.$

We first characterize the static allocation given $N(t)$ in every period. The
social planner chooses the optimal level of intermediate goods input
$[x(v,t)]_{v\in[0,N(t)]}$ at each time $t$ given the time paths of $C(t)$,
$\varphi(t)$, and $N(t)$, which is equivalent to maximizing $\widetilde{Y}(t)$
with respect to $x(v,t)$. Thus, the optimal net output is:

$\widetilde{Y}_{S}(t)=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}}\beta
N(t)L_{E}(t),$ (23)

where the subscript “$S$” indicates the social planner’s problem. Relative to
the decentralized economy (14), $\widetilde{Y}_{S}$ is larger with a markup of
$(1-\beta)^{1-\frac{1}{\beta}}/(2-\beta)$ given the same level of labor and
technology. This difference comes from the monopoly power in a decentralized
economy, which lowers the provision of intermediate goods and thus, the net
output of the final good.

Given (23), the social planner solves (dropping the subscript “$S$” to
simplify notation):

$\max_{c(t),\varphi(t)}\int_{0}^{\infty}e^{-(\rho-n)t}\left[\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}\right]\mathrm{d}t,$
(24)

subject to

$\dot{N}(t)=\eta N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t),$ (25)
$c(t)=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}}\beta
N(t)l_{E}(t),$ (26) $l_{R}(t)+l_{E}(t)=1.$ (27)

(25) is the innovation possibility frontier, (26) is the simplified resource
constraint, and (27) requires labor market clearing. In Online Appendix 6.4,
we derive the following proposition:

###### Proposition 3.3

With $\gamma>1$, BGP growth rates in the social planner’s problem are the same
as those in the decentralized economy.

Despite the lower net aggregate output relative to the planner’s solution, the
decentralized data economy grows at the same rate. The growth rates are
determined by the final goods production and the usage of data. The former is
the same under the decentralized and the planner’s solutions. The latter
exhibits a gap due to the markup created by the data price mechanism in the
decentralized economy (as opposed to the planner’s directly setting data
usage), which causes the decentralized equilibrium to be less socially
efficient. But, the gap is a constant and does not manifest in growth rates.

Reminiscent of Jones (1995), growth rates alone cannot fully characterize the
performance of an economy. We thus also examine the labor share in R&D and
derive in Online Appendix 6.5:

###### Proposition 3.4

In the social planner’s problem, the share of labor allocated in the R&D
sector $s_{S}$ is constant in BGP, which is determined by

$s_{S}=\frac{1}{1+\Theta_{S}},\quad\textit{where}\quad\Theta_{S}=\frac{(\sigma-\xi)n+\xi\rho}{\xi(1-\xi)g^{*}_{S}}-\frac{(\sigma-\xi)(1-\zeta)}{\xi(1-\xi)}.$
(28)

Without monopolistic production, the result differs from that in the
decentralized economy. To ensure $s_{S}\in[0,1]$, i.e., $\Theta_{S}\geq 0$, we
also note that the BGP growth rate under the planner’s solution cannot be too
high:

$0<g^{*}_{S}\leq\frac{1}{1-\zeta}\left(n+\frac{\xi}{\sigma-\xi}\rho\right).$
(29)

This upper limit consists of three components: knowledge accumulation,
population growth, and a data-related term. Intuitively, a social planner does
not want the growth of the economy to be too fast since higher growth rates
require more data usage, which can create excessive data leakage or privacy
violations in expectation. When $\gamma>1$, $g_{S}^{*}$ is given by (20) and
(29) always holds. But when $\gamma<1$, it is possible that the upper limit
binds and the planner’s solution features slower growth.

###  3.3 Misallocation and Data Overuse in a Decentralized Economy

Endogenous labor allocations between production and R&D sectors influence
other variables in equilibrium. In particular, with $n$, $\beta$, $\gamma$ and
$\rho$ taking on standard values from the existing literature, the labor
allocation in the R&D sector in the social planner’s problem is always larger
than that in the decentralized economy. Figure 1 plots this difference as we
vary the influence of data in innovation possibility frontier ($\xi$),
knowledge accumulation through innovation possibility frontier ($\zeta$), and
the severity of privacy concerns in the consumers’ utility function
($\sigma$). The larger $\sigma$ is, the more the failure to internalize
privacy concerns creates misallocation; the overuse of data and
underallocation of labor in R&D are especially severe when data are important
for innovation (large $\xi$) and knowledge accumulation is slow (small
$\zeta$).

(a) $\sigma=1.5$

(b) $\sigma=2.5$

(c) $\xi=0.5$

(d) $\xi=0.6$

Figure 1: Difference in labor employed in the R&D sector between the two cases

Notes: The figure shows the difference of labor allocated in R&D sector
between decentralized economy and the social planner’s problem. Light color
represents larger differences (greater misallocation) and dark represents
smaller ones. Other parameters are set as $n=0.02$, $\beta=2/3$, $\gamma=2.5$,
$\rho=0.03$, which are standard values used in existing literature.
$\xi\in[0,1]$, $\sigma>1$ ensures the convexity of disutility term in
consumer’s utility function and $\zeta\in[0,1]$ ensures the existence of BGP.

Our model thus reveals a new source of inefficiency in the decentralized
economy, even though the growth rates are the same as those in the planner’s
solution: undersupply of labor and overuse of data in the R&D sector. This
finding contrasts and complements recent studies such as Jones and Tonetti
(2020), which predicts an underutilization of data due to its nonrival nature.
In our setting, data are overused even when owned by consumers who directly
factor in the disutility from data leakage and/or abuse.

Similar to Jones (1995), the underallocation of labor in the R&D sector comes
from monopolistic markups in the production of intermediate goods. The final
good producer employs more labor to compensate for the lower production and
usage of intermediate goods, which in turn crowds out labor employed in R&D.
The crowding out of labor in R&D is exacerbated because intermediate producers
are less reliant on labor, once data enter the evolution of the innovation
possibility frontier (unlike Jones 1995, in which labor is the only input for
the innovation possibility frontier). To maintain the same growth rate as in
the planner’s solution, intermediate producers have to compensate the
underemployment of labor in R&D by more aggressive utilization of data. In
other words, this crowding-out effect of labor crowds in data usage in the R&D
sector to a socially excessive level. Given the parameter set we have, data
usage is four to five times higher in the decentralized economy than in the
social planner’s problem.

###  3.4 Data Generation and Regulatory Policies

Different from R&D-specific labor, the production of data is endogenized by
consumption, which itself is also endogenous. More data supplied add to
innovations on variety, which in turn spurs consumption (e.g., in BGP), which
then further relaxes the data generating constraint. Such a feedback is absent
when it comes to labor or capital in that consumption is typically decoupled
from the exogenous evolution of population size or the growth in potential
capital to be allocated.

As such, privacy regulations reducing $s$ in (3) may affect the production and
thus the usage of data.999Both the GDPR and the CCPA give individuals certain
rights over the collection and usage of their personal information, but they
differ in multiple aspects (e.g. Hospelhorn 2020). For example, California’s
being a much larger economy implies more severe penalties than that of the
GDPR, which maps to a tighter constraint (3). However, reducing $s$ would
reduce growth rate and would not lead to welfare improvement without
sacrificing growth, if it improves welfare at all. A full analytical
characterization is not tractable, not to mention that such an intervention
entails intergenerational transfers in practice which can be controversial. We
therefore restrict our discussion in this section to policy interventions that
preserve growth rates in the planner’s BGP solution.

We find that levying a tax on the usage of data alters the transitional
dynamics but is ineffective in bringing the equilibrium allocations in
decentralized economy closer to the social planner’s solution because as
discussed in Online Appendix 8.1.1, it does not solve the underemployment of
labor in R&D but only slows down the economy before it eventually returns to
the original BGP path. However, subsidizing labor wage rate in the R&D sector
or subsidizing intermediate producers in terms of profit proves to be
effective. Appendices 8.1.2 and 8.1.3 derive these optimal subsidies.

The intuition is that labor wage is pinned down by both the R&D and production
sectors, and a subsidy directly affecting the wage level alters the labor
share. Specifically, because labor allocations are derived from equalizing
wages in the R&D sector and production sector, a government can apply taxes or
subsidies to adjust the prices of factors to alleviate the overuse of data in
intermediate producers’ R&D. For example, a subsidy of rate $0<\tau(t)<1$ to
the R&D sector for employing labor would modify (19) into

$\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}V(t)=\tau(t)w(t),$

while (15) remains unchanged. Then, one can derive the desirable $\tau(t)$ by
equating the labor share to that in the social planner’s problem.101010In
Online Appendix 8.1.2, we show that this subsidy rate should be a constant.
Because only the changing rate of variables may influence growth rates (as
seen in Online Appendix 6.2), applying a fixed subsidy rate does not introduce
further distortions into the model. To avoid overuses of data and potential
privacy violations, intermediate producers should be incentivized to employ
more labor for innovation with subsidies on wages in the R&D sector, which
would lower data usage.

In contrast, data price is pinned down by intermediate producers and
consumers. The consumers’ joint decision on consumption and data provision
implies that they care about the growth rate of data provision, not the level,
as seen in (5). In a sense, a direct tax on data purchase is decoupled from
the data provision and because of that, would not alter the equilibrium labor
share.111111A non-constant tax rate would not work either because it alters
the BGP growth rate in the decentralized economy which is originally the same
as that in the social planner’s solution. Consequently, the underemployment in
the R&D and data overuse persist.

Our findings have important policy implications because the current debate has
centered on privacy regulations, which decisively affect the data economy’s
growth. We demonstrate how R&D labor relates to the increasing use of data and
how labor market policies can effectively reduce excessive data overuse to
improve the social welfare.

###  3.5 Historical Data and Dynamic Data Nonrivalry

One unique and important property of data we highlight thus far is dynamic
nonrivalry. Beyond its implicit manifestation through the evolution of
intermediate good varieties in the baseline specification, we now extend our
discussion of dynamic nonrivalry by explicitly modeling the trading of
historical data and the associated creative destruction. Specifically, instead
of one potential intermediate producer using data for research (and entry)
with full depreciation, we now assume:

  * •

Data generated at time $t$ can also be used and traded in the following $M>0$
periods. Historical data depreciate at a rate of $\delta$.

  * •

Potential intermediate producers can purchase not only data from current
consumers, but also historical data from existing intermediate producers
entering in the past $M$ periods. The new and historical data bundles are
perfect substitutes.

  * •

The owners of historical data determine the proportion of data sold to
newcomers, fully recognizing potential creative destruction.

  * •

If historical data are reused by new intermediate producers, disutility of
contributing consumers is also cumulative according to the usage.

In a sense, data become a stock variable whereas labor is a flow variable in
the intermediate good production function. Online Appendix 8.2 contains
detailed derivation and discussion, which are summarized here:

###### Proposition 3.5

Dynamic data nonrivalry under creative destruction as specified does not
change the BGP growth rates or labor share in the decentralized economy and
social planner’s problem. However, incumbent intermediate producers are always
willing to sell more than the social optimal level of historical data to
entrants in the decentralized economy, as long as the negative effect of
creative destruction does not come to an extreme level.

The implicit dynamic nonrivalry in our baseline model still exists: Data are
transformed into new varieties of intermediate goods, and the existing level
of varieties can affect the R&D process in future. But, the proposition
demonstrates that the key findings are robust to explicitly modeling data
nonrivalry and the direct use of historical data. Therefore, the differences
between our findings and Jones and Tonetti (2020)’s are driven by the dynamic
nature of data usage we model, not by the simple consideration of nonrivalry.
Moreover, the proposition has implications for regulating data resale among
intermediate producers, an interesting topic for future studies.

###  3.6 Data Ownership

So far, we have allowed consumers to own data and the firms (intermediate
producers) to acquire data by paying the consumers. We now consider cases with
positive data processing cost $\theta>0$ under both consumer ownership and
firm ownership (detail derivation in Online Appendix 8.3). $\theta>0$ would
not affect much of the equilibrium outcomes when consumers own data as long as
the knowledge spillover effect is moderate, but consumers’ EIS are reasonably
moderate and the data processing cost is sufficiently convex (which holds true
due to e.g., curse of dimensionality). The main new insight from the analysis
is the positive BGP growth rate of data provision when firms own data. In
other words, data usage under BGP could be increasing when firms own data,
which is different from the decreasing trend when consumers own data.

Intuitively, because firms no longer need to pay consumers for data usage,
they no longer bear the disutility of potential privacy violation. Although
they still pay a mechanical “data processing cost”, it does not push the BGP
growth rate of data provision to negative as we have seen in the case in which
consumers own data. That said, under standard parameter values, e.g.,
$\xi=0.5$, $\zeta=0.85$, and $\phi>1$, the BGP growth rate of other variables,
$g^{*}$, is still larger than $g^{*}_{\varphi}$. This is consistent with our
baseline model: Data usage becomes trivial in BGP compared with consumption as
time passes.

Moreover, Sections 3.5 and 3.6 demonstrate that our finding on data overuse is
robust to nonrivalry and ownership variation. This result differs from and
complements Jones and Tonetti (2020) which emphasizes the underutilization of
data due to its cross-sectional nonrivalry and the importance of consumer data
ownership. Because data add to innovation over time and have persist benefits
in our model (instead of only entering contemporary production as in Jones and
Tonetti (2020)), firms are more willing to utilize and purchase data if
consumers own data.

##  4 Transitional Dynamics: A Numerical Analysis

The transitional dynamics of the data economy onto a BGP also reveal
interesting patterns, as we demonstrate numerically with the baseline model
setup. Even without regulations restricting the use of data, a decline in data
usage could occur as the economy grows, which mitigates the concern of data
privacy. At the same time, for data economies with minimal initial growth, the
process of data generation dictates that they may be trapped in low-growth
regimes for a long time without any intervention to improve digital
infrastructure or relax privacy regulation.

###  4.1 Methodology and Calibration

Similar to Jones (2016), we focus on the planner’s solution in this numerical
exercise mainly for tractability. This would be the situation if a benevolent
government implements policies, balancing privacy protection and growth, such
as the subsidy scheme we mentioned earlier. Instead of a formal calibration
designed to replicate any country’s data, this analysis is best viewed as an
illustration of the basic transitional dynamics that are possible in our
theoretical framework.

We derive in Online Appendix 7.3 a system of differential equations that
describe the dynamics of the planner’s economy. They consist of a constraint
on data generation (3) and three state-like variables whose interpretations
and steady state values are shown in Table 1. Other variables can be derived
from them. We solve the system of differential equations using “reverse
shooting” (Judd 1998)121212We start from the steady state and run the system
backward according to the three differential equations. We first find the
values of the three state-like variables that minimize the distance between
their growth rates and zero, since they all converge to nonzero constants in
the steady state. For other parameters, we choose the standard values in the
existing literature or plausible values if they are not discussed in extant
studies ($\xi$, $\sigma$, and $\zeta$). We set $\xi=0.5$ to indicate that data
and labor contribute equally in creating new varieties; we discuss various
choices of $\sigma$ in Online Appendix 7.4; we set $\zeta$ to be a value less
than one following Jones (1995). Table 2 provides a summary of
parametrization.

Table 1: State-like variables for studying transitional dynamics

Variable | Meaning | Steady state value
---|---|---
$g_{N}(t)$ | Growth rate of variety of intermediate goods | (20)
$g_{\mu}(t)$ | Growth rate of shadow price corresponding to technology change | $g_{\mu}^{*}=\frac{\sigma(1-\zeta)-\xi}{\xi}g_{N}^{*}-\frac{\sigma}{\xi}n$
$l_{E}(t)$ | Ratio of labor employed in production sector | (28)

Table 2: Parameters for studying transitional dynamics

Variable | Meaning | Value | Source
---|---|---|---
$\beta$ | Contribution of labor in final good production function | $2/3$ | Standard
$\gamma$ | Reciprocal of Elasticity of Intertemporal Substitution | 2.5 | Standard
$\rho$ | Subjective discount factor | 0.03 | Standard
$\xi$ | Contribution of data in innovation possibility frontier | 0.5 | Discretionary
$\zeta$ | Knowledge accumulation through innovation possibility frontier | 0.85 | Discretionary
$\sigma$ | Severity of consumers’ privacy concern | 1.5 | Discretionary
$n$ | Population growth rate | 0.02 | Standard
$\eta$ | Efficiency term in innovation possibility frontier | 1 | Standard

###  4.2 Results and Discussions

We can simulate the paths of the three state-like variables before reaching
equilibrium for different values of $\sigma$, which stands for the extent of
disutility caused by privacy leakage. We can then derive the paths of other
variables like $c(t)$ and $\varphi(t)$. Figure 2 provides an illustration of
the transitional dynamics for $\sigma=1.5$ with and without the data provision
constraint here. Other cases exhibit similar patterns, which we discuss
further in Online Appendix 7.4.

(a) Growth rates of consumption and intermediate goods varieties (without
constraint)

(b) Labor allocation and growth rate of data provision (without constraint)

(c) Growth rates of consumption and intermediate goods varieties (with
constraint)

(d) Labor allocation and growth rate of data provision (with constraint)

Figure 2: Key variables along transitional path when $\sigma=1.5$

Notes: These figures show the transitional dynamics of the social planner’s
problem when $\sigma=1.5$, with and without the constraint of data provision.
The economies undergo long but relatively steady states before finally
reaching BGP at the end points.

Several robust patterns emerge. No matter where an economy starts before BGP,
the growth rate of consumption and variety both move to the steady state
levels, and a transition can happen relatively quickly once the economy
reaches nontrivial growth (sufficiently away from zero), as (a) and (c) show.
Moreover, as shown in (b) and (d), the growth rate of data provision decreases
from positive to negative in BGP. For an economy starting from low growth, the
provision of data increases rapidly for the accumulation of varieties of
intermediate goods, which contributes to the final good production and growth
in consumption. Along this transitional path, labor moves from production
sector to R&D sector, which reflects how labor is used to compensate for the
decreasing provision of data.

Notice that in Panel (a), different from the growth rate of variety, the
growth rate of consumption undergoes some periods of negative growth before
increasing to the positive growth rate without constraints on data provision.
The temporary pain is common in the growth literature (e.g. Brock and Taylor
2010) and indicates that at the beginning of the adjustment to high BGP
growth, labor moves out from the production sector, which causes the decrease
of output and consumption. But data are unique because economic activities
constrain their supply, as is clear from (d) in which data provision is
binding from time 0 to 400. The temporary pain is absent because moving labor
away from production is not as costly as before, due to declines in data
provision (which lead to slower growth rates).

The transitional dynamics is also broadly consistent with the empirical
patterns in Abis and Veldkamp (2020) that the labor income share in knowledge
work is decreasing. Formally, labor income share is decreasing in $l_{R}(t)$
in our model:

$\text{Labor Income
Share}=\frac{w(t)L(t)}{p_{\varphi}(t)\varphi(t)L(t)}=\frac{\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}V(t)L(t)}{\eta\xi
N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}V(t)L(t)}\frac{1}{l_{R}(t)}=\frac{1-\xi}{\xi}\frac{1}{l_{R}(t)}.$

With similar extent of labor allocation in intermediate and final goods in
production sectors under the decentralized economy, our numerical simulations
reveal this decreasing trend of labor income share as $l_{R}(t)$ increases in
transitional states.

###  4.3 Implications for Growth Trap and Privacy Concerns

Comparing the cases with and without the constraint of data provision, we find
that constraint (3) binds for an economy with low initial growth rate. The per
capita data contribution is also much lower than the case without the
constraint. Importantly, for an economy starting with a growth rate close to
zero, it takes almost 200 additional time periods (years) for it to reach BGP.
Even after reaching BGP growth, the economy’s output could be significantly
lower due to the delay of accelerated growth. The intuition is
straightforward: Initial low growth limits data generation, which negatively
feeds back to growth because data constitute an input factor for innovations
in intermediate good variety.

An intervention to boost the initial growth rate is crucial for escaping such
a growth trap. Meanwhile, we should notice that the social planner problem is
at a national level. Thus, intervention to relax the data provision constraint
(3) here is more at an international level, e.g., actions taken by the World
Bank or IMF or fellow countries in European Union to improve digital
infrastructure and data collection/storage efficiency or share expertise,
which relax (3). Also, if the central planner is originally too budget/cash
constrained to improve data infrastructure, foreign subsidies might help relax
the binding constraint temporarily.

Global waves of technological innovations such as the Internet and AI take
place when countries are having different population growth and different
environments for technology transfer and spillover. This can lead to vastly
different transitional dynamics. The tightness of data constraint $s$ does not
affect the BGP growth rates, but affects transitional dynamics. For example,
the United States and Europe are in the same stage of development when data
economy emerges, but the difference in privacy protection (different $s$)
makes them tread different paths, reaching BGP at different times.

First, as the economy just starts to grow, data can significantly facilitate
the transition to BGP, as illustrated in Figure 3: Economies 1 and 2 have the
same parameters but start at different stages in their transitions to BGP,
with Economy 1 trapped in near zero growth for a prolonged period of time. An
intervention to enhance data facilities and digital infrastructure or to relax
privacy regulation (increasing $s$) can help escape the growth trap earlier.
However, even when such interventions fully relax (3), i.e., $s\to\infty$,
they compensate the lagging behind Economy 2 only partially.

Figure 3: Different pre-BGP Paths Given Different Starting Points and
Different Privacy Policies

Notes: These panels show the transitional dynamics of two economies with the
same parameters but different starting points. The full line and the dash line
show the paths of the two economies with $s=0$, the dotted line shows the path
of Economy 1 when its privacy policy becomes loose ($s=0.078$), and the dash
dotted line shows the path of Economy 1 when privacy regulation is completely
relaxed ($s\to\infty$).

In the long run, because the knowledge accumulation parameter $\zeta$ is set
to be smaller than one, it becomes less effective to provide more data for
creating new varieties of intermediate goods as the existing varieties
accumulate. Thus, the growth rate of data provision decreases after some
periods of increase. As data become less productive in the innovation process,
the economy substitutes the use of data with more labor to focus on better
exploitation of data. Finally, as the economy matures (transition to BGP), the
benefit of using data is diminishing but individual’s privacy concerns remain,
which reduce each consumer’s endogenous data contribution. In that regard,
privacy issues in the long run may not be as severe as current debates
indicate. Instead, regulatory policies could focus on the overuse of data in
the R&D sector, as discussed in Section 3.4.

##  5 Conclusion

We develop an endogenous growth model against the backdrop of the rise of big
data and digital economy. Although a decentralized economy on the balanced
growth path grows at the same rate as in the socially optimal allocations,
data are inefficiently overused and R&D is understaffed. Consumers suffer
since they are inadequately compensated for potential information leakage and
privacy violation. When consumers own data, data privacy concerns become
allayed in the long run because the use of data eventually declines. However,
less developed economies with low growth at the dawn of the data economy may
face a new form of poverty trap that potentially warrants interventions.

For the first time, we treat data as an input factor besides labor in the
process of creating new varieties of intermediate goods, which subsequently
fuel the production of final good and long-run growth. We highlight data’s
endogenous generation as by-products of economic activities, nonrivalry in a
dynamic environment, and flexible ownership. For tractability and focus, we
necessarily leave out certain aspects of the data economy such as final goods
differentiation. Therefore, our findings should be taken as first-order
benchmark results rather than foregone conclusions.

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Proofs and Extended Discussions

##  6 Proofs of Lemmas and Propositions

###  6.1 The Growth Rate of Labor Allocations in BGP

###### Lemma 6.1

In BGP, the ratios of labor employed in production sector and R&D sector both
converge to constants, thus their growth rates are both zero.

###### Proof 6.2

Proof of Lemma. From the labor market clearing condition, we have
$l_{E}(t)+l_{R}(t)=1$, and $0\leq l_{E}(t),l_{R}(t)\leq 1$. Since we need the
growth rates of all endogenous variables to be constant in BGP, if $l_{E}(t)$
or $l_{R}(t)$ grows in BGP, it becomes larger than 1 eventually, which
contradicts the definitions. Meanwhile, if one of them grows at a negative
rate in BGP, from the market clearing condition, the other one should grow
positively, which also contradicts with the definition. Thus, we can conclude
that, the only possible situation is that the two variables both remain
constant in BGP, i.e.,

$\frac{\dot{l}_{E}(t)}{l_{E}(t)}=\frac{\dot{l}_{R}(t)}{l_{R}(t)}=0.$

Q.E.D.

This is a very general conclusion, which will be used for many times in the
following proofs. Thus, we use it directly as given from now on. Notice that
we can only use this conclusion in BGP, while in other non-BGP states, the
growth rates of these two ratios may not be constant.

###  6.2 Proof of Proposition 1: BGP Growth Rates in Decentralized Model

###### Proof 6.3

Proof of Proposition 1. Combine the free-entry condition of labor (19) and the
wage determining equation from the final good producer side (15) together, and
change it into the form of growth rate, we have

$\zeta\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{\varphi}(t)}{\varphi(t)}+\frac{\dot{V}(t)}{V(t)}=\frac{\dot{N}(t)}{N(t)}.$
(30)

Recall that in BGP, $r(t)=r^{*}$, then the patent value shown in (9) is
reduced to

$\displaystyle V(t)=$
$\displaystyle\int_{t}^{\infty}e^{-r^{*}(s-t)}\pi(s)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{t}^{\infty}e^{-r^{*}(s-t)}e^{n(s-t)}\pi(t)\mathrm{d}s$
$\displaystyle=$ $\displaystyle\frac{\pi(t)}{r^{*}-n},$

where the second equation comes from (13). Thus, the growth rate of $V(t)$ is

$\frac{\dot{V}(t)}{V(t)}=\frac{\dot{\pi}(t)}{\pi(t)}=n.$ (31)

Plug (31) into (30), we have

$(\zeta-1)\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{\varphi}(t)}{\varphi(t)}+n=0.$
(32)

Meanwhile, the per capita output growth rate of the economy $g(t)$ can be
represented by the growth rate of final good production or variety (from
(14)), that is

$g(t)=\frac{\dot{y}(t)}{y(t)}=\frac{\dot{Y}(t)}{Y(t)}-n=\frac{\dot{N}(t)}{N(t)}.$

Also, from the innovation possibility frontier (17), we have

$g(t)=\frac{\dot{N}(t)}{N(t)}=\eta
N(t)^{\zeta-1}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t).$ (33)

Next, we want to pin down the growth rate of $N(t)$ and $\varphi(t)$. Consider
the free-entry condition of data (18) and rewrite it into the form of growth
rate, and plug in (31), we have

$\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}=\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-1)\frac{\dot{\varphi}(t)}{\varphi(t)}+\frac{\dot{V}(t)}{V(t)}=\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-1)\frac{\dot{\varphi}(t)}{\varphi(t)}+n.$

Then, from the Euler equation of consumption (4) and the Euler equation of
data (5), we have

$\displaystyle\frac{1}{\gamma}\left(\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}-(\sigma-1)\frac{\dot{\varphi}(t)}{\varphi(t)}\right)$
$\displaystyle=$
$\displaystyle\frac{\zeta}{\gamma}\frac{\dot{N}(t)}{N(t)}+\frac{\xi-\sigma}{\gamma}\frac{\dot{\varphi}(t)}{\varphi(t)}+\frac{1}{\gamma}n=\frac{\dot{c}(t)}{c(t)}=\frac{\dot{y}(t)}{y(t)}=\frac{\dot{N}(t)}{N(t)},$

where the second equation comes from the fact that in BGP, the growth rate of
consumption is the same as that of output. Rewrite the above equation, we get
the following expression

$\left(\zeta-\gamma\right)\frac{\dot{N}(t)}{N(t)}+(\xi-\sigma)\frac{\dot{\varphi}(t)}{\varphi(t)}+n=0.$
(34)

Then, we can derive the growth rate of $\varphi(t)$ and $N(t)$ from (32) and
(34), that is

$\frac{\dot{N}(t)}{N(t)}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n$

and

$\frac{\dot{\varphi}(t)}{\varphi(t)}=-\frac{1}{\xi}n+\frac{1-\zeta}{\xi}\frac{\dot{N}(t)}{N(t)}=\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n.$

Now, from (33), we have

$g^{*}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n.$

Q.E.D.

###  6.3 Proof of Proposition 2: Labor Allocations in BGP in Decentralized
Model

###### Proof 6.4

Proof of Proposition 2. From (33) we know that in BGP,
$N(t)^{\zeta-1}\varphi(t)^{\xi}L(t)$ is constant among all the periods $t$,
thus

$g^{*}=\eta N(t)^{\zeta-1}\varphi(t)^{\xi}(l_{R}^{*})^{1-\xi}L(t),\forall t.$

Combine (19) and (15), then plug in (9) and (13), we have

$(1-\xi)(1-\beta)g^{*}((l_{R}^{*})^{-1}-1)=r^{*}-n$

Meanwhile, from (4) we can also derive $g^{*}$, then combine it with the above
equation, we have

$\displaystyle g^{*}=\frac{1}{\gamma}(r^{*}-\rho)=$
$\displaystyle\frac{1}{\gamma}\left[(1-\xi)(1-\beta)g^{*}((l_{R}^{*})^{-1}-1)+n-\rho\right]$
$\displaystyle\Rightarrow\quad l_{R}^{*}=$
$\displaystyle\frac{(1-\xi)(1-\beta)}{\gamma+(1-\xi)(1-\beta)+\frac{\rho-n}{g^{*}}}.$

Let

$\Theta_{D}=\frac{g^{*}\gamma+\rho-n}{g^{*}(1-\xi)(1-\beta)},$

where, $g^{*}$ is shown in (20) in the main text. Then, the ratio of labor
employed in R&D sector is

$s_{D}=\frac{1}{1+\Theta_{D}}.$

Q.E.D.

###  6.4 Proof of Proposition 3: BGP Growth Rates in Social Planner’s Problem

###### Proof 6.5

Proof of Proposition 3. We should first solve the static problem before we get
to the dynamic problem. Taking other variables as given, the net output
$\widetilde{Y}(t)$ can be maximized by choosing the optimal intermediate good
usage $x_{S}(v,t)$ as

$x_{S}(v,t)=\left(\frac{\psi}{1-\beta}\right)^{-\frac{1}{\beta}}L_{E}(t).$
(35)

Then, substitute this into the definition of net output, we get the optimal
net output which is shown in (23). After that, the social planner’s problem is
set from (24) to (27). To solve this problem, we first set up the current-
value Hamiltonian equation as

$\mathcal{H}=\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}+\lambda(t)\left[\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}}\beta
N(t)l_{E}(t)-c(t)\right]+\mu(t)\eta
N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t),$

where $\lambda(t)$ and $\mu(t)$ are the shadow prices corresponding to the
constraints (26) and (25), respectively. Now, write the markup as

$A=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}},$

and derive the necessary conditions with respect to $c(t)$, $\varphi(t)$,
$l_{E}(t)$ and $N(t)$, we then have the following equations:

$\frac{\partial\mathcal{H}}{\partial c(t)}=c(t)^{-\gamma}-\lambda(t)=0,$ (36)
$\frac{\partial\mathcal{H}}{\partial\varphi(t)}=-\sigma\varphi(t)^{\sigma-1}+\mu(t)\xi\eta
N(t)^{\zeta}\varphi(t)^{\xi-1}l_{R}(t)^{1-\xi}L(t)=0,$ (37)
$\frac{\partial\mathcal{H}}{\partial l_{E}(t)}=\lambda(t)\beta
AN(t)-\mu(t)\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}L(t)=0$ (38)

and

$\frac{\partial\mathcal{H}}{\partial N(t)}=\lambda(t)A\beta
l_{E}(t)+\mu(t)\eta\zeta
N(t)^{\zeta-1}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t)=-\dot{\mu}(t)+(\rho-n)\mu(t).$
(39)

This system can be expressed by the seven equations from (25) to (27) and from
(36) to (39).

Consider (38), move the second term to the right hand side, change the
equation into the form of growth rate, and substitute the growth rate of
$\lambda(t)$ and $\mu(t)$ derived from (36) and (37), we have131313Here, we
expect that $l_{E}$ and $l_{R}$ converge to a constant in BGP, as has been
shown in Online Appendix 6.1.

$\displaystyle\frac{\dot{\lambda}(t)}{\lambda(t)}+\frac{\dot{N}(t)}{N(t)}=\frac{\dot{\mu}(t)}{\mu(t)}+\zeta\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{\varphi}(t)}{\varphi(t)}+n$
$\displaystyle\Rightarrow\quad$
$\displaystyle-\gamma\frac{\dot{c}(t)}{c(t)}+(1-\zeta)\frac{\dot{N}(t)}{N(t)}-\frac{\dot{\mu}(t)}{\mu(t)}-\xi\frac{\dot{\varphi}(t)}{\varphi(t)}-n=0$
$\displaystyle\Rightarrow\quad$
$\displaystyle-\gamma\frac{\dot{c}(t)}{c(t)}+\frac{\dot{N}(t)}{N(t)}-\sigma\frac{\dot{\varphi}(t)}{\varphi(t)}=0.$

Notice that the growth rate of $c(t)$ is what we want to derive, which can be
defined as $g^{*}$, then

$\frac{\dot{c}(t)}{c(t)}=g^{*}=\frac{1}{\gamma}\left(\frac{\dot{N}(t)}{N(t)}-\sigma\frac{\dot{\varphi}(t)}{\varphi(t)}\right).$
(40)

Next, we pin down the growth rate of $N(t)$ and $\varphi(t)$. From (38), we
have

$\frac{\lambda(t)}{\mu(t)}=\frac{\eta(1-\xi)}{\beta
A}N(t)^{\zeta-1}l_{R}(t)^{-\xi}\varphi(t)^{\xi}L(t).$

Then, (39) can be reformed as

$\displaystyle\frac{\lambda(t)}{\mu(t)}A\beta l_{E}(t)+\eta\zeta
N(t)^{\zeta-1}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t)=-\frac{\dot{\mu}(t)}{\mu(t)}+(\rho-n)$
$\displaystyle\Rightarrow\quad$
$\displaystyle\left[\eta(1-\xi)l_{E}(t)+\eta\zeta
l_{R}(t)\right]N(t)^{\zeta-1}\varphi(t)^{\xi}l_{R}(t)^{-\xi}L(t)=\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-\sigma)\frac{\dot{\varphi}(t)}{\varphi(t)}+\rho.$
(41)

Since the right hand side is constant in BGP, the left hand side should also
be constant. Thus, we have

$(\zeta-1)\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{\varphi}(t)}{\varphi(t)}+n=0.$
(42)

Meanwhile, we can derive the production function (in per capita form) in the
social planner’s problem as

$y_{S}(t)=AN(t)l_{E}(t)$

Thus, the growth rate of the economy is

$g^{*}=\frac{\dot{N}(t)}{N(t)}.$ (43)

Combine (43) with (40) and also (42), we get

$g^{*}_{S}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$ (44)

which is the same as that in the decentralized model. The growth rates of
other related variables can be derived similarly. Q.E.D.

We can come back to the following illustrations after reading Appendices 6.5
and 7.1. We should notice that although the result of growth rate here equals
to that in the decentralized model, in Online Appendix 6.5, the result of
labor share in R&D sector requires the growth rate to be within a certain
range. From (45) in Online Appendix 6.5 we know that $g^{*}_{S}$ should
satisfy the following condition:

$0<g^{*}_{S}<\frac{1}{1-\zeta}\left(n+\frac{\xi}{\sigma-\xi}\rho\right).$

Plug in the expression of $g^{*}_{S}$, we have

$0<\zeta<1-(1-\gamma)\left[\frac{n(\sigma-\xi)}{\rho\sigma}+\frac{\xi}{\sigma}\right].$

When $\gamma>1$, from (49) in Online Appendix 7.1 we have
$0<\zeta<1-(1-\gamma)\frac{\xi}{\sigma}$, which includes the above condition.
However, when $0<\gamma<1$, we have
$0<\zeta<1-(1-\gamma)\left(\frac{n}{\rho}+\frac{\xi}{\sigma}\right)$, the
condition above is not always satisfied. Since in literature, we usually think
that $\gamma>1$, the social planner’s problem is always meaningful.

###  6.5 Proof of Proposition 4: Labor Allocations in BGP in Social Planner’s
Problem

###### Proof 6.6

Proof of Proposition 4.

Similar to the decentralized economy, in social planner’s problem we also have

$g^{*}=\eta N(t)^{\zeta-1}\varphi(t)^{\xi}(l_{R}^{*})^{1-\xi}L(t),\forall t.$

Combine (41) and (42), and plug in the above equation, in BGP we have

$\displaystyle\left[(1-\xi)(1-l_{R}^{*})+\zeta
l_{R}^{*}\right]g^{*}(l_{R}^{*})^{-1}=\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-\sigma)\frac{\dot{\varphi}(t)}{\varphi(t)}+\rho$
$\displaystyle\Rightarrow\quad$ $\displaystyle\left[(1-\xi)(1-l_{R}^{*})+\zeta
l_{R}^{*}\right]g^{*}(l_{R}^{*})^{-1}=\left[1-\frac{\sigma}{\xi}(1-\zeta)\right]\frac{\dot{N}(t)}{N(t)}+\left(\frac{\sigma}{\xi}-1\right)n+\rho$
$\displaystyle\Rightarrow\quad$
$\displaystyle\left[(1-\xi)(1-l_{R}^{*})+\left(\frac{\sigma}{\xi}-1\right)(1-\zeta)l_{R}^{*}\right]g^{*}(l_{R}^{*})^{-1}=\left(\frac{\sigma}{\xi}-1\right)n+\rho.$

As a result, we have

$l_{R}^{*}=\frac{1}{1+\Theta_{S}},$

where,

$\Theta_{S}=\frac{(\sigma-\xi)n+\xi\rho}{\xi(1-\xi)g^{*}_{S}}-\frac{(\sigma-\xi)(1-\zeta)}{\xi(1-\xi)},$
(45)

and $g^{*}_{S}$ is given in (44). Q.E.D.

##  7 Extended Discussions

###  7.1 The Existence and Uniqueness of BGP

###### Proposition 7.1

In this data-driven innovation model, there exists a unique BGP which is given
in Proposition 1, if the following condition is satisfied

$0<\zeta<\begin{cases}1-\frac{\xi}{\sigma}(1-\gamma),&\text{if }\gamma>1;\\\
1-(1-\gamma)\left(\frac{n}{\rho}+\frac{\xi}{\sigma}\right),&\text{if
}0<\gamma<1.\end{cases}$

###### Proof 7.2

Proof of Proposition 7.1. To ensure the existence of BGP, the growth rate of
the economy should satisfy two conditions. The first one requires $g^{*}>0$,
which makes

$\displaystyle(1-\zeta)\sigma-\xi(1-\gamma)>$ $\displaystyle 0$
$\displaystyle\Rightarrow\quad\zeta<$ $\displaystyle
1-\frac{\xi}{\sigma}(1-\gamma).$ (46)

The second one requires that the growth rate should satisfy the transversality
condition (TVC), i.e.,

$\lim_{t\to\infty}\left[\exp{\left(-\int_{0}^{t}r(s)\,\mathrm{d}s\right)}\int_{0}^{N(t)}V(v,t)\,\mathrm{d}v\right]=0.$
(47)

In BGP, the TVC is simplified into

$\displaystyle\lim_{t\to\infty}e^{-r^{*}t}N(t)V(t)=$ $\displaystyle 0$
$\displaystyle\Rightarrow\quad\lim_{t\to\infty}e^{(g^{*}+n-r^{*})t}N(0)V(0)=$
$\displaystyle 0,$

which leads to the condition $g^{*}+n-r^{*}<0$. Meanwhile, from (4) we can
know that $r^{*}=\gamma g^{*}+n+\rho$, then

$\displaystyle g^{*}+n-\gamma g^{*}-n-\rho<$ $\displaystyle 0$
$\displaystyle\Rightarrow\quad g^{*}<$ $\displaystyle\frac{\rho}{1-\gamma}$
$\displaystyle\Rightarrow\quad\zeta<$ $\displaystyle
1-(1-\gamma)\left(\frac{n}{\rho}+\frac{\xi}{\sigma}\right).$ (48)

Combine (46) and (48), and $\zeta>0$ (positive technology spillover), we have

$0<\zeta<\begin{cases}1-\frac{\xi}{\sigma}(1-\gamma),&\text{if }\gamma>1;\\\
1-(1-\gamma)\left(\frac{n}{\rho}+\frac{\xi}{\sigma}\right),&\text{if
}0<\gamma<1,\end{cases}$ (49)

which is what we require the spill-over effect $\zeta$ to be. Q.E.D.

###  7.2 Data Accumulation

In our baseline model, data are set to be fully depreciated in every period.
One might think that in reality, data involved in the innovation possibility
frontier can accumulate. This is always the case, since time-series data
usually contain useful information. To model this formally, we let the
innovation possibility frontier be:

$\dot{N}(t)=\eta N(t)^{\zeta}\Phi(t)^{\xi}l_{R}(t)^{1-\xi}L(t),$

where $\Phi(t)$ denotes the total quantity of data accumulated in R&D
activities at time $t$. Its law of motion is then:

$\dot{\Phi}(t)=\varphi(t)-\kappa\Phi(t),$ (50)

where, $\varphi(t)$ is the same as that in the previous sections, and $\kappa$
here represents the depreciation rate of data, i.e., the rate that data become
out of date every period. Other settings are still the same as the baseline
model. By introducing this adjustment into the baseline model, we have the
following proposition.

###### Proposition 7.3

In the case that data involved in the innovation possibility frontier can
accumulate, the growth of the economy in BGP remains similar as that in the
case without data accumulation.

###### Proof 7.4

Proof of Proposition 7.3. From (50), in BGP, we still need the growth rate of
$\Phi(t)$ to be constant, i.e.,

$\frac{\dot{\Phi}(t)}{\Phi(t)}=\frac{\varphi(t)}{\Phi(t)}-\kappa,$

which makes the right hand side of the equation be constant as well. Then, the
growth rate of information in every period $\varphi(t)$ is the same as that of
total data accumulation $\Phi(t)$, i.e.,

$\frac{\dot{\Phi}(t)}{\Phi(t)}=\frac{\dot{\varphi}(t)}{\varphi(t)}$

In the baseline model, all the derivations only involve the growth rate of the
endogenous variables, except the innovation possibility frontier, the only
place where the level value of $\varphi(t)$ appears. Then, we can still derive
the similar results simply by replacing the growth rate of $\varphi(t)$ with
$\Phi(t)$ in appropriate places. Q.E.D.

###  7.3 Derivation of the Differential Equation System in Section 4

For simplicity, here we assume that $\psi=1-\beta$. First, combining (37) and
(38), we have

$\displaystyle-\frac{\xi(1-\xi)}{\sigma-\xi}\frac{\dot{l}_{R}(t)}{l_{R}(t)}+\xi\frac{\dot{l}_{R}(t)}{l_{R}(t)}$
$\displaystyle=$
$\displaystyle\left(\zeta-1+\frac{\xi\zeta}{\sigma-\xi}\right)g_{N}(t)+\left(1+\frac{\xi}{\sigma-\xi}\right)g_{\mu}(t)+\left(1+\frac{\xi}{\sigma-\xi}\right)n-g_{\lambda}(t).$

Consider that

$\frac{\dot{l}_{R}(t)}{l_{R}(t)}=-\frac{\dot{l}_{E}(t)}{1-l_{E}(t)},$

and from the resource constraint (26) and (36), we have

$g_{c}(t)=g_{N}(t)+\frac{\dot{l}_{E}(t)}{l_{E}(t)}=-\frac{1}{\gamma}g_{\lambda}(t).$
(51)

Then, the law of motion of $l_{E}(t)$ can be derived as

$\dot{l}_{E}(t)=\frac{\left(\gamma+\zeta-1+\frac{\xi\zeta}{\sigma-\xi}\right)g_{N}(t)+\left(1+\frac{\xi}{\sigma-\xi}\right)g_{\mu}(t)+\left(1+\frac{\xi}{\sigma-\xi}\right)n}{\left(\frac{\xi(1-\xi)}{\sigma-\xi}-\xi\right)\frac{1}{1-l_{E}(t)}-\frac{\gamma}{l_{E}(t)}}$
(52)

and the growth rate of $\varphi(t)$ can be represented as

$g_{\varphi}(t)=\frac{1}{\sigma-\xi}\left[g_{\mu}(t)+\zeta
g_{N}(t)-(1-\xi)\frac{\dot{l}_{E}(t)}{1-l_{E}(t)}+n\right].$ (53)

From (38) and (39), we derive the law of motion of $g_{\mu}(t)$ as

$\dot{g}_{\mu}(t)=(g_{\mu}(t)-\rho+n)\left[(\zeta-1)g_{N}(t)+\xi
g_{\varphi}(t)+\xi\frac{\dot{l}_{E}(t)}{1-l_{E}(t)}+n+\frac{(1-\xi-\zeta)\dot{l}_{E}(t)}{\zeta+(1-\xi-\zeta)l_{E}(t)}\right].$
(54)

Similarly, the law of motion of $g_{N}(t)$ is

$\dot{g}_{N}(t)=g_{N}(t)\left[(\zeta-1)g_{N}(t)+\xi
g_{\varphi}(t)-(1-\xi)\frac{\dot{l}_{E}(t)}{1-l_{E}(t)}+n\right].$ (55)

In conclusion, the system of differential equations of the three state-like
variables are (55), (54), and (52). After deriving the transitional dynamics
of these three state-like variables, other variables such as $g_{c}(t)$ and
$g_{\varphi}(t)$ can be derived from (51) and (53), respectively.

###  7.4 Additional Results on Transitional Dynamics

To demonstrate the robustness of our findings, we show in Figure 4 and Figure
5 the transitional dynamics when $\sigma$, the disutility parameter, takes
other values with and without the data provision constraint, respectively.

(a) $\sigma=2.0$ (without constraint)

(b) $\sigma=2.0$ (without constraint)

(c) $\sigma=2.5$ (without constraint)

(d) $\sigma=2.5$ (without constraint)

(e) $\sigma=3.0$ (without constraint)

(f) $\sigma=3.0$ (without constraint)

Figure 4: Key variables along transitional path with different values of
$\sigma$, without data provision constraint

Notes. These figures show the transitional dynamics of the social planner’s
problem when $\sigma=2.0,2.5\text{ and }3.0$, without the constraint of data
provision. The economies undergo long but relatively steady states before
finally reaching BGP at the end points.

(a) $\sigma=2.0$ (with constraint)

(b) $\sigma=2.0$ (with constraint)

(c) $\sigma=2.5$ (with constraint)

(d) $\sigma=2.5$ (with constraint)

(e) $\sigma=3.0$ (with constraint)

(f) $\sigma=3.0$ (with constraint)

Figure 5: Key variables along transitional path with different values of
$\sigma$, with data provision constraint

Notes. These figures show the transitional dynamics of the social planner’s
problem when $\sigma=2.0,2.5\text{ and }3.0$, with the constraint of data
provision. The economies undergo long but relatively steady states before
finally reaching BGP at the end points.

We further illustrate the evolution of data provision in Figure 6 by plotting
the level of and cumulative level of usage/provision. The initial value of
data provision is set to $\varphi(0)=1$ by taking $\sigma=1.5$, we observe a
peak of data provision which decreases rapidly to a low level while the
cumulative data provision remains on a plateau. As we have discussed
previously, the effect of data in our model still exists in the form of new
varieties of intermediate goods over time.

(a) Provision Level (without constraint)

(b) Cumulative Level (without constraint)

(c) Provision Level (with constraint)

(d) Cumulative Level (with constraint)

Figure 6: Flow and cumulative levels of data provision with $\sigma=1.5$, with
and without data provision constraint.

Notes. These figures show the level change and cumulative level change of data
provision before reaching BGP when $\sigma=1.5$, with and without the
constraint of data provision.

##  8 Model Extensions: Proofs and Further Discussion

###  8.1 Policy Implications on BGP

First, as we prove in Online Appendix 8.3.1, reducing $s$ through privacy
policy regulation would reduce growth rate, and would not lead to welfare
improvement without sacrificing growth, if it improves welfare at all. Solving
the case explicitly is difficult, we therefore restrict our discussion here to
policy interventions that preserve growth rates.

####  8.1.1 Ineffectiveness of Taxing Data Usage

###### Proposition 8.1

Levying a tax on the usage of data alters the transitional dynamics but is
ineffective in bringing the equilibrium allocations in decentralized economy
closer to the social planner’s solution because the decentralized economy
eventually returns to the original BGP path.

###### Proof 8.2

Proof of Proposition 8.1.

First, we should prove that the optimal tax rate on data usage should be
constant to ensure the BGP growth rates not deviate from the level in baseline
model. If we add a tax term on the free-entry condition of data provision,
i.e.,

$\eta\xi
N(t)^{\zeta}\varphi(t)^{\xi-1}l_{R}(t)^{1-\xi}V(t)=\tau(t)p_{\varphi}(t)$ (56)

where $\tau(t)$ is the tax rate. Rewrite (56) into the form of growth rate, we
have

$g_{p_{\varphi}}(t)+g_{\tau}(t)=\zeta
g_{N}(t)+(\xi-1)g_{\varphi}(t)+(1-\xi)g_{l_{R}}(t)+g_{V}(t).$

Then, combine the above equation with (5), we have (in BGP)

$g_{\tau}(t)=\zeta g_{N}(t)+(\xi-\sigma)g_{\varphi}(t)-r^{*}+n+\rho.$ (57)

Also, recalling (4), (57) can be further written as

$g_{\tau}(t)=\zeta g_{N}(t)+(\xi-\sigma)g_{\varphi}(t)-\gamma
g_{c}(t)+n=(\zeta-\gamma)g^{*}+(\xi-\sigma)g^{*}_{\varphi}+n.$

If we want the growth rates $g^{*}$ and $g^{*}_{\varphi}$ to be the same as
social optimum, we can derive that

$g_{\tau}(t)=(\zeta-\gamma)\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+(\xi-\sigma)\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+n=0.$

Thus, tax rate $\tau(t)$ should be a constant in BGP. In other words, as long
as the tax rate is a constant over time, BGP growth rate remains the same.

Then, we want to show the optimal tax rate that can make the allocations in
decentralized economy close to the solution in social planner’s problem.
Imagine a decentralized economy in BGP, the government suddenly announces a
fixed rate of tax on data collection by intermediate producers from consumers,
$\tau$, in order to push the economy close to social optimum, i.e., make the
labor allocations in both sectors and data provision level the same as social
optimum. In this case, we have to derive the labor share in the two sectors in
another way, as shown below. Consider that in BGP,

$g^{*}=\eta N(t)^{\zeta-1}\varphi(t)^{\xi}(l_{R}^{*})^{1-\xi}L(t)$

and the free-entry condition of data (in BGP)

$\eta\xi
N(t)^{\zeta}\varphi(t)^{\xi-1}(l_{R}^{*})^{1-\xi}V(t)=\tau(t)p_{\varphi}(t).$

Combining the above two equations, and plug in the BGP value of $V(t)$, we
have

$\displaystyle g^{*}\xi N(t)\varphi(t)^{-1}V(t)L(t)^{-1}$ $\displaystyle=\tau
p_{\varphi}(t)$ $\displaystyle\Rightarrow\quad g^{*}\xi
N(t)\varphi(t)^{-1}\frac{\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}L_{E}(t)}{r^{*}-n}L(t)^{-1}$
$\displaystyle=\tau p_{\varphi}(t)$ $\displaystyle\Rightarrow\quad
g^{*}\xi\frac{\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}l_{E}^{*}}{\gamma
g^{*}+\rho-n}N(t)\varphi(t)^{-1}p_{\varphi}(t)^{-1}$ $\displaystyle=\tau$ (58)

Notice that in BGP, $N(t)\varphi(t)^{-1}p_{\varphi}(t)^{-1}$ converges to
constant since

$g_{N}^{*}-g^{*}_{\varphi}-g^{*}_{p_{\varphi}}=0.$

Thus, we can rewrite this term as
$N(t_{0})\varphi(t_{0})^{-1}p_{\varphi}(t_{0})^{-1}$, where $t_{0}$ is one of
the periods in BGP. Go on with (8.2), we finally derive the labor share in R&D
sector as

$l_{R}^{*}=1-\frac{(\gamma g^{*}+\rho-n)\tau(1-\beta)^{1-\frac{2}{\beta}}}{\xi
g^{*}\psi^{1-\frac{1}{\beta}}\beta}N(t_{0})\varphi(t_{0})^{-1}p_{\varphi}(t_{0})^{-1}$
(59)

On the other hand, notice that $l^{*}_{R}$ should equal to the result coming
from the free-entry condition of labor (19), then, (The derivation process is
still the same as we do in Online Appendix 6.3, with no influences from adding
the tax term of data collection.)

$1-\frac{(\gamma g^{*}+\rho-n)\tau(1-\beta)^{1-\frac{2}{\beta}}}{\xi
g^{*}\psi^{1-\frac{1}{\beta}}\beta}N(t_{0})\varphi(t_{0})^{-1}p_{\varphi}(t_{0})^{-1}=\frac{1}{1+\frac{g^{*}\gamma+\rho-n}{g^{*}(1-\xi)(1-\beta)}}.$

As a result, the term $\tau
N(t_{0})\varphi(t_{0})^{-1}p_{\varphi}(t_{0})^{-1}$ should always be constant.
This means that changes in tax rate $\tau$ are always cancelled by changes in
$N(t)$, $\varphi(t)$ and $p_{\varphi}(t)$ together, while the labor share in
the two sectors remain unchanged. Thus, levying tax on data collection will
not push the BGP results in decentralized economy close to the social
planner’s problem. Definitely, levying tax on data collection changes the
usage of data and labor employment in the transitional state, but the economy
will return to the previous path in BGP. Q.E.D.

####  8.1.2 Optimal Subsidy on Labor Employment in R&D Sector

Although levying a tax directly on data usage comes out to be ineffective, we
can still adjust the BGP path of the decentralized economy through indirect
ways. First, we consider subsidizing the wage rate of labor employed in R&D
sector. If $\tau(t)$ denotes the gross subsidy rate141414Here, $\tau(t)$ is
set together with wage $w(t)$, which is a kind of cost. Then, it is a subsidy
rate if $0<\tau(t)<1$, while it is a tax rate if $\tau(t)>1$. In the next
section, we set $\tau^{\prime}(t)$ together with profit $V(t)$, which is a
kind of revenue. In this occasion, the meaning becomes opposite: It is a
subsidy rate if $\tau^{\prime}(t)>1$, while it is a tax rate if
$0<\tau^{\prime}(t)<1$., the free-entry condition of labor employment (19)
becomes

$\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}V(t)=\tau(t)w(t).$ (60)

Then, we have the following proposition.

###### Proposition 8.3

In order to push labor allocations in decentralized economy close to the
solution in the social planner’s problem without deviating from the BGP growth
rates, the subsidy rate on labor employment in R&D sector $\tau(t)$ should be
constant, which is determined by

$\tau(t)=\tau^{*}=\frac{(1-\beta)[(\sigma-\xi)n+\xi\rho-(\sigma-\xi)(1-\zeta)g^{*}]}{\gamma
g^{*}+\rho}<1,$

where, $g^{*}$ is BGP growth rate derived in Proposition 21.

###### Proof 8.4

Proof of Proposition 8.3. First, we prove that the optimal subsidy rate should
be constant to ensure that the BGP growth rates do not deviate from previous
levels. Rewrite the adjusted free-entry condition (60) into a form of growth
rates, we have

$\zeta g_{N}(t)+\xi g_{\varphi}(t)-\xi
g_{l_{R}}(t)+g_{V}(t)=g_{\tau}(t)+g_{w}(t).$

Also, from the determining equation of wage in production sector (15), rewrite
it into growth rate form:

$g_{w}(t)=g_{N}(t).$

In BGP, $g_{l_{R}}(t)=0$, $g_{V}(t)=n$. Thus, combining the above two
equations, we derive the growth rate of subsidy rate as

$g_{\tau}(t)=(\zeta-1)g_{N}(t)+\xi g_{\varphi}(t)+n.$

If we want the growth rates $g^{*}$ and $g^{*}_{\varphi}$ to be the same as
the solution in the social planner’s problem, we derive that

$g_{\tau}(t)=(\zeta-1)\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+\xi\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+n=0.$

Thus, the subsidy rate should also be constant.

Then, we determine the optimal subsidy rate. Go on with the derivation process
in Online Appendix 6.3 by replacing the free-entry condition (19) with (60),
we finally derive the labor share in the R&D sector with subsidy as

$(s_{D})^{\prime}=\frac{1}{1+(\Theta_{D})^{\prime}},$

where,

$(\Theta_{D})^{\prime}=\frac{\tau\left(\gamma+\frac{\rho-n}{g^{*}}\right)}{(1-\xi)(1-\beta)},$

here, $g^{*}$ is shown in (20). After that, we derive the optimal subsidy rate
of wage in R&D sector by letting the labor share in the decentralized model
equal to that in the social planner’s problem, i.e.,

$\tau(t)=\tau^{*}=\frac{(1-\beta)[(\sigma-\xi)n+\xi\rho-(\sigma-\xi)(1-\zeta)g^{*}]}{\gamma
g^{*}+\rho-n}.$

Q.E.D.

####  8.1.3 Optimal Subsidy on Profits from Intermediate Good Productions

An alternative way of pushing the decentralized model close to the solution of
social planner’s problem is to subsidize intermediate producers directly.
Formally, denote $\tau^{\prime}(t)$ as the subsidy rate of profit, then in
BGP, total value of a patent

$V(t)=\frac{\pi(t)}{r^{*}-n}$ (61)

becomes

$V(t)=\frac{\tau^{\prime}(t)\pi(t)}{r^{*}-n}.$ (62)

Then, we propose the following proposition.

###### Proposition 8.5

In order to push labor allocations in decentralized economy close to the
solution in social planner’s problem without deviating from the BGP growth
rates, the subsidy rate on profit from intermediate good productions
$\tau^{\prime}(t)$ should be constant, which is determined by

$\tau^{\prime}(t)=(\tau^{\prime})^{*}=\frac{\gamma
g^{*}+\rho-n}{(1-\beta)[(\sigma-\xi)n+\xi\rho-g^{*}(\sigma-\xi)(1-\zeta)]}>1,$

where, $g^{*}$ is BGP growth rate derived in Proposition 21.

###### Proof 8.6

Proof of Proposition 8.5. Similarly, we should first prove the constancy of
subsidy rate $\tau^{\prime}(t)$. Combining the free-entry condition of labor
(19) and the wage determining equation in production side (15), and then
plugging it into the BGP form of $V(t)$ shown in (62), we have

$\displaystyle\eta(1-\xi)N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}V(t)$
$\displaystyle=w(t)=\beta\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}N(t)$
$\displaystyle\Rightarrow\quad\eta(1-\xi)N(t)^{\zeta-1}\varphi(t)^{\xi}l_{R}(t)^{-\xi}\frac{\tau^{\prime}(t)\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}L_{E}(t)}{r^{*}-n}$
$\displaystyle=\beta\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}$
(63)

Rewrite this equation into the form of growth rate, we have

$(\zeta-1)g_{N}(t)+\xi g_{\varphi}(t)+g_{\tau^{\prime}}(t)+n=0.$

If we want the growth rate $g^{*}$ and $g^{*}_{\varphi}$ to be the same as the
solution in the social planner’s problem, we derive

$g_{\varphi^{\prime}}(t)=(1-\zeta)g^{*}-\xi g^{*}_{\varphi}-n=0.$

Thus, the subsidy rate should be constant.

Then we determine the optimal subsidy rate. Go on with (8.6), we have

$\displaystyle
g^{*}(1-\xi)(l_{R}^{*})^{-1}\frac{\tau^{\prime}\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}l_{E}(t)}{r^{*}-n}$
$\displaystyle=\beta\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}$
$\displaystyle\Rightarrow\quad
g^{*}(1-\xi)(l_{R}^{*})^{-1}\frac{\tau^{\prime}\frac{\psi^{1-\frac{1}{\beta}}\beta}{(1-\beta)^{1-\frac{2}{\beta}}}l_{E}^{*}}{\gamma
g^{*}+\rho-n}$
$\displaystyle=\beta\left[\frac{(1-\beta)^{2}}{\psi}\right]^{\frac{1}{\beta}-1}$
$\displaystyle\Rightarrow\quad(l_{R}^{*})^{-1}-1$ $\displaystyle=\frac{\gamma
g^{*}+\rho-n}{g^{*}(1-\xi)\tau^{\prime}(1-\beta)}.$

To make the labor employment the same as that in the social planner’s problem,
we need

$\displaystyle(l_{R}^{*})^{-1}-1$ $\displaystyle=\frac{\gamma
g^{*}+\rho-n}{g^{*}(1-\xi)\tau^{\prime}(1-\beta)}=\frac{(\sigma-\xi)n+\xi\rho}{\xi(1-\xi)g^{*}}-\frac{(\sigma-\xi)(1-\zeta)}{\xi(1-\xi)}$
$\displaystyle\Rightarrow\quad\tau^{\prime}$ $\displaystyle=\frac{\gamma
g^{*}+\rho-n}{(1-\beta)[(\sigma-\xi)n+\xi\rho-g^{*}(\sigma-\xi)(1-\zeta)]}.$

Q.E.D.

###  8.2 Data Nonrivalry and Creative Destruction

Formally, the innovation possibility frontier becomes

$\dot{N}(t)=\eta N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi},$ (64)

where, $D(t)$ is the composite data used by potential intermediate producers,
which consist of new data collected from consumers, $\varphi(t)$, and
historical data bundles bought from existing intermediate producers in the
past $M$ periods, $B(t)$. The composite data are represented as

$D(t)=\alpha\varphi(t)L(t)+(1-\alpha)B(t),$

where $\alpha$ represents the relative importance between historical and new
data in the data composite. If we rearrange the existing intermediate
producers by the order of entering time, and set $\delta$ as the depreciation
rate, we can define the historical data bundle as:

$B(t)=\left[M^{-\frac{1}{\epsilon}}\int_{t-M}^{t-\mathrm{d}t}\left(\delta^{t-s}\underline{d}(t,s)\varphi(s)L(s)\right)^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}s\right]^{\frac{\epsilon}{\epsilon-1}},$

where, $\underline{d}(t,s)$ stands for the proportion of time $t$ usage of
data that are generated at time $s$ ($s<t$), and $\epsilon$ is the elasticity
coefficient of data collected from different periods. We underline those
variables related to data generated in the past and traded in current period,
from the perspective of potential intermediate producers in the current
period. Similarly, we overline those variables related to data generated in
current period and traded in future periods.

For consistency with the baseline model, let us consider the case when
consumers own data. A potential intermediate producer can not only collect new
data from consumers, but also buy historical data from existing intermediate
producers with a depreciation rate of $\delta$. Thus, given the prices of data
generated from time $(t-M)$ to $(t+M)$, a potential intermediate producer
decides the quantity of data to collect from consumers, the quantity of data
to buy from existing intermediate producers entering in the past $M$ periods,
and the quantity of data to sell to potential intermediate producers who will
enter in the following $M$ periods.

The consumer’s utility maximization problem now becomes

$\max_{c(t),\varphi(t)}\int_{0}^{\infty}e^{-(\rho-n)t}\left[\frac{c(t)^{1-\gamma}}{1-\gamma}-\varphi(t)^{\sigma}\int_{t}^{t+M}\delta^{s-t}\bar{d}(s,t)\mathrm{d}s\right]\mathrm{d}t,$
(65)

where $\bar{d}(s,t)$ means the proportion of data generated at time $t$ and
sold to intermediate producers at time $s$ ($s>t$). It is taken as given by
consumers but is endogenously determined by the corresponding intermediate
producers. The budget constraint (2) and the data provision constraint (3)
remain the same. Meanwhile, it is straightforward that firms use all the new
data they collect from consumers, so we need $d(t,t)=1$. We can expect
$\bar{d}(s,t)$ to be constant in BGP since $\bar{d}(s,t)\in[0,1]$.151515Refer
to the conclusion that labor employment share in R&D and production sectors is
constant in BGP, which is shown in Online Appendix 6.1. Then, the term
$\int_{t}^{t+M}\delta^{s-t}\bar{d}(s,t)\mathrm{d}s$ should also be a constant
in BGP. Thus, the necessary conditions with respect to $c(t)$ and $\varphi(t)$
are still the same as our baseline model in BGP, which are shown in (4) and
(5).

####  8.2.1 The Decentralized Economy

The production process is still the same as that in our baseline model. Now,
consider the R&D process of potential intermediate producers. Their profit
maximization problem becomes

$\displaystyle\max_{L_{R}(t),\left\\{d(s,t)\right\\}_{s=t+\mathrm{d}t}^{t+M},\atop\left\\{d(t,s)\right\\}_{s=t-M}^{t-\mathrm{d}t},\varphi(t)}$
$\displaystyle\eta
N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi}\left[V(t)-\int_{t+\mathrm{d}t}^{t+M}e^{-\lambda(s,t)(s-t)}V(s)\mathrm{d}s\right]-w(t)L_{R}(t)$
$\displaystyle-
p_{\varphi}(t)\varphi(t)L(t)+L(t)\int_{t+\mathrm{d}t}^{t+M}\bar{p}_{\varphi}(s,t)\bar{d}(s,t)\varphi(t)\mathrm{d}s$
$\displaystyle-\int_{t-M}^{t-\mathrm{d}t}\underline{p_{\varphi}}(t,s)\underline{d}(t,s)\varphi(s)L(s)\mathrm{d}s,$
(66)

where $\underline{p_{\varphi}}(t,s)$ is the price of data collected at time
$s$ and sold to potential intermediate producers at time $t$, ($t>s$),
$\bar{p}_{\varphi}(s,t)$ is the price of data collected at time $t$ and sold
to potential intermediate producers at time $s$, ($t<s$), and $V(t)$ is the
total discounted production profit of the intermediate good invented at time
$t$. $\varphi(t)$ and $p_{\varphi}(t)$ are new data collected from consumers
at time $t$ and the corresponding price, respectively. Selling data to future
intermediate producers makes the seller face the risk of creative destruction:
Sellers’ own patents become valueless according to a Poisson process with an
arrival rate $\lambda(s,t)$, which is assumed to have the following form:

$\lambda(s,t)=c_{0}\bar{d}(s,t)^{2}.$

Notice that intermediate producers will never collect new data again if they
succeed in entering the market, thus the data they can sell to future
intermediate producers are always within the range they collect from consumers
at current period, i.e.,

$\bar{d}(s,t)\in[0,1],\forall s,t.$

Also, we assume that only incumbent intermediate producers can sell historical
data to entrant ones, that is, consumers cannot resell their historical data.

The profit maximization problem of a potential intermediate producer (8.2.1)
consists of five terms. The first term is the total benefit that it can get if
it succeeds in R&D and enters the market, minus the expected total loss of
creative destruction coming from selling data to future intermediate
producers. The second and third terms are the same as those in our baseline
model: the wage cost and the cost of new data collected from consumers. The
fourth term is the revenue from selling data to future intermediate producers.
The last term is the cost of buying historical data from the intermediate
producers who entered in earlier periods.

From the potential intermediate producers’ profit maximization in (8.2.1), we
know that the first order condition with respect to $L_{R}(t)$ is similar to
the baseline model, i.e.,

$\eta(1-\xi)N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{-\xi}V(t)=w(t).$ (67)

The first order condition with respect to $\varphi(t)$ is

$\displaystyle\eta\xi N(t)^{\zeta}D(t)^{\xi-1}L_{R}(t)^{1-\xi}V(t)\alpha
L(t)=p_{\varphi}(t)L(t)-L(t)\int_{t+\mathrm{d}t}^{t+M}\bar{p}_{\varphi}(s,t)\bar{d}(s,t)\mathrm{d}s$
$\displaystyle\Rightarrow\quad$ $\displaystyle\alpha\eta\xi
N(t)^{\zeta}D(t)^{\xi-1}L_{R}(t)^{1-\xi}V(t)=p_{\varphi}(t)-\int_{t+\mathrm{d}t}^{t+M}\bar{p}_{\varphi}(s,t)\bar{d}(s,t)\mathrm{d}s.$
(68)

For any $s\in(t,t+M]$, the first order condition with respect to
$\bar{d}(s,t)$ is

$\displaystyle L(t)\bar{p}_{\varphi}(s,t)\varphi(t)=2\eta
N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi}e^{-c_{0}\bar{d}(s,t)^{2}(s-t)}c_{0}\bar{d}(s,t)(s-t)V(s).$
(69)

For any $s\in[t-M,t)$, the first order condition with respect to
$\underline{d}(t,s)$ is

$\displaystyle(1-\alpha)\eta\xi
N(t)^{\zeta}D(t)^{\xi-1}L_{R}(t)^{1-\xi}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}\left[\delta^{t-s}\underline{d}(t,s)\varphi(s)L(s)\right]^{-\frac{1}{\epsilon}}...$
$\displaystyle\delta^{t-s}\varphi(s)L(s)\left[V(t)-\int_{t+\mathrm{d}t}^{t+M}e^{-\lambda(s,t)(s-t)}V(s)\mathrm{d}s\right]=\underline{p_{\varphi}}(t,s)\varphi(s)L(s)$
(70)

In equilibrium, the demand and supply of historical data are equal, i.e.,

$\underline{d}(t,s)=\bar{d}(t,s),\forall t>s.$

Also, the prices of historical data follow:

$\underline{p_{\varphi}}(t,s)=\bar{p}_{\varphi}(t,s),\forall t>s.$

Combining (69) with (8.2.1), we solve for $d(t,s)$ and their corresponding
prices given other variables, i.e., ($s<t,t-s\leq M$)

$\displaystyle(1-\alpha)\eta\xi
N(t)^{\zeta}D(t)^{\xi-1}L_{R}(t)^{1-\xi}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}\left[\delta^{t-s}\underline{d}(t,s)\varphi(s)L(s)\right]^{-\frac{1}{\epsilon}}\delta^{t-s}\varphi(s)L(s)...$
$\displaystyle\left[V(t)-\int_{t+\mathrm{d}t}^{t+M}e^{-\lambda(s,t)(s-t)}V(s)\mathrm{d}s\right]=2\eta
N(s)^{\zeta}D(s)^{\xi}L_{R}(s)^{1-\xi}e^{-c_{0}\bar{d}(t,s)^{2}(t-s)}c_{0}\bar{d}(s,t)(t-s)V(t)$
$\displaystyle\Rightarrow\quad$ $\displaystyle
d(t,s)^{1+\frac{1}{\epsilon}}e^{-c_{0}d(t,s)^{2}(t-s)}=\frac{(1-\alpha)\xi}{2c_{0}(t-s)}\delta^{\left(1-\frac{1}{\epsilon}\right)(t-s)}e^{\bar{g}_{D}(s-t)}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}[\varphi(t)L(t)]^{1-\frac{1}{\epsilon}}...$
$\displaystyle\left[1-\int_{t+\mathrm{d}t}^{t+M}e^{-\lambda(s,t)(s-t)}\frac{V(s)}{V(t)}\mathrm{d}s\right]$
$\displaystyle\Rightarrow\quad$ $\displaystyle
d(t,s)^{1+\frac{1}{\epsilon}}e^{-c_{0}\bar{d}(t,s)^{2}(t-s)}=\frac{(1-\alpha)\xi}{2c_{0}(t-s)}\delta^{\left(1-\frac{1}{\epsilon}\right)(t-s)}e^{\bar{g}_{D}(s-t)}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}[\varphi(t)L(t)]^{1-\frac{1}{\epsilon}}...$
$\displaystyle\left[1-\int_{t+\mathrm{d}t}^{t+M}e^{-(c_{0}d(s,t)^{2}-n)(s-t)}\mathrm{d}s\right],$
(71)

where $\bar{g}_{D}=\zeta g_{N}+\xi g_{D}+(1-\xi)n.$

Note that in BGP, $g_{D}=\max{\\{g_{\varphi}+n,g_{B}\\}}$, $g_{V}=n$, and

$\displaystyle g_{B}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\left[M^{\frac{1}{1-\epsilon}}+\frac{\epsilon}{\epsilon-1}\ln{\left(\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-s}d(t,s)\varphi(s)L(s))^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}s\right)}\right]$
$\displaystyle=\frac{\epsilon}{\epsilon-1}\frac{\mathrm{d}}{\mathrm{d}t}\left[\ln{\left(\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-s}d(t,s)\varphi(t-M)L(t-M)e^{(g^{*}_{\varphi}+n)(s-t+M)})^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}s\right)}\right]$
$\displaystyle=\frac{\epsilon}{\epsilon-1}\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\epsilon-1}{\epsilon}(\ln{\varphi(t-M)}+\ln{L(t-M)})+\ln{\left(\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-s}d(t,s)e^{(g^{*}_{\varphi}+n)(s-t+M)})^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}s\right)}\right]$
$\displaystyle=g_{\varphi}+n+\frac{\epsilon}{\epsilon-1}\frac{\mathrm{d}}{\mathrm{d}t}\left[\ln{\left(\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-s}d(t,s)e^{(g^{*}_{\varphi}+n)(s-t+M)})^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}s\right)}\right]$
$\displaystyle=g_{\varphi}+n.$

The last equation follows from the fact that the term
$\delta^{t-s}d(t,s)e^{(g^{*}_{\varphi}+n)(s-t+M)}$, $s\in[t-M,t)$ is not time
varying. Thus, we have $g_{D}=g_{\varphi}+n$, and $\bar{g}_{D}=\zeta g_{N}+\xi
g_{\varphi}+n$.

Now, let us consider (8.2.1). First, it is obvious that in BGP,

$\frac{\dot{p}_{\varphi}(s,t)}{p_{\varphi}(s,t)}\leq\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}.$

Otherwise, after a certain period, when the price of historical data exceeds
the price at which they are newly sold, intermediate producers can get
arbitrage by buying and selling historical data. Thus, the growth rate of the
term

$p_{\varphi}(t)-\int_{t+\mathrm{d}t}^{t+M}\bar{p}_{\varphi}(s,t)\bar{d}(s,t)\mathrm{d}s$

is just the same as that of $p_{\varphi}(t)$, which is denoted as
$g_{p_{\varphi}}$.161616We can further discuss the growth rates of historical
data and new data prices. If
$\dot{p}_{\varphi}(s,t)/p_{\varphi}(s,t)<\dot{p}_{\varphi}(t)/p_{\varphi}(t)$,
then we can expect that in a certain period, the price of new data becomes
much larger than that of historical data, or we can say that the price of
historical data becomes trivial compared with that of the new data. In this
case, it is obvious that intermediate producers demand all of the historical
data since they are very cheap, i.e., $d(s,t)\to 1$. Then, we can derive the
optimal $d(s,t)$ directly and the results of other variables remain unchanged.
As a result, in the following analysis, we just consider the case that
$\dot{p}_{\varphi}(s,t)/p_{\varphi}(s,t)=\dot{p}_{\varphi}(t)/p_{\varphi}(t)$.
Now, transforming (8.2.1) into the form of growth rate and plugging in the
relation of growth rates derived in BGP, we get

$\displaystyle\zeta g_{N}+\xi g_{D}+(1-\xi)n+n$
$\displaystyle=g_{\varphi}+n+g_{p_{\varphi}}$
$\displaystyle\Rightarrow\quad\zeta
g_{N}+\xi\left(g_{\varphi}+n\right)+(1-\xi)n$
$\displaystyle=g_{\varphi}+g_{p_{\varphi}}$
$\displaystyle\Rightarrow\quad\zeta g_{N}+\left(\xi-1\right)g_{\varphi}+n$
$\displaystyle=g_{p_{\varphi}}.$ (72)

Combining (8.2.1) with the Euler equation derived from consumers’ problem (5),
we get (note that in BGP, $g_{c}=g_{N}=g^{*}$)

$\displaystyle\zeta g_{N}+\left(\xi-1\right)g_{\varphi}+n$
$\displaystyle=(\sigma-1)g_{\varphi}+\gamma g_{c}$
$\displaystyle\Rightarrow\quad(\zeta-\gamma)g^{*}+(\xi-\sigma)g^{*}_{\varphi}+n$
$\displaystyle=0.$ (73)

Meanwhile, from (67), as well as the wage determining equation from the final
good producer side (15), we have

$\displaystyle\zeta g_{N}+\xi g_{D}-\xi n+g_{V}$ $\displaystyle=g_{N}$
$\displaystyle\Rightarrow\quad(\zeta-1)g_{N}+\xi(g_{\varphi}+n)-\xi n+n$
$\displaystyle=0$ $\displaystyle\Rightarrow\quad(\zeta-1)g^{*}+\xi
g^{*}_{\varphi}+n$ $\displaystyle=0.$ (74)

Combining (8.2.1) and (8.2.1), we derive the BGP growth rates of variety and
data provision, which are the same as those in our baseline model.

$g^{*}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n$

and

$g^{*}_{\varphi}=\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n.$

Since there is no change in the evolution of labor allocations, the expression
of labor employment in R&D sector is also the same.

Now, we derive the proportion of data sharing. From (8.2.1) we have the growth
rate of the right hand side as

$\displaystyle-
g_{D}+\frac{1}{\epsilon}g_{B}+\left(1-\frac{1}{\epsilon}\right)(g_{\varphi}+n)=0.$

Thus, the left hand side is a constant determined by

$\displaystyle d_{D}^{1+\frac{1}{\epsilon}}e^{-c_{0}d_{D}^{2}(t-s)}=$
$\displaystyle\frac{(1-\alpha)\xi}{2c_{0}(t-s)}\delta^{\left(1-\frac{1}{\epsilon}\right)(t-s)}e^{\bar{g}_{D}(s-t)}M^{-\frac{1}{\epsilon}}\left[1-\int_{t+\mathrm{d}t}^{t+M}e^{-(c_{0}d_{D}(r,t)^{2}-n)(r-t)}\mathrm{d}r\right]...$
$\displaystyle
D(t)^{-1}B(t)^{\frac{1}{\epsilon}}[\varphi(t)L(t)]^{1-\frac{1}{\epsilon}}.$

The terms
$D(t)^{-1}B(t)^{\frac{1}{\epsilon}}[\varphi(t)L(t)]^{1-\frac{1}{\epsilon}}$
can be further represented by

$\displaystyle
D(t)^{-1}B(t)^{\frac{1}{\epsilon}}[\varphi(t)L(t)]^{1-\frac{1}{\epsilon}}$
$\displaystyle=$ $\displaystyle\left\\{\alpha
B(t)^{-\frac{1}{\epsilon}}[\varphi(t)L(t)]^{\frac{1}{\epsilon}}+(1-\alpha)B(t)^{1-\frac{1}{\epsilon}}[\varphi(t)L(t)]^{\frac{1}{\epsilon}-1}\right\\}^{-1}$
$\displaystyle\equiv$
$\displaystyle\left[\alpha\Delta_{D}^{-\frac{1}{\epsilon}}+(1-\alpha)\Delta_{D}^{1-\frac{1}{\epsilon}}\right]^{-1},$
(75)

where,

$\displaystyle\Delta_{D}=$ $\displaystyle B(t)[\varphi(t)L(t)]^{-1}$
$\displaystyle=$
$\displaystyle[\varphi(t)L(t)]^{-1}\left[M^{-\frac{1}{\epsilon}}\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-r}d_{D}(t,r)\varphi(r)L(r))^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}r\right]^{\frac{\epsilon}{\epsilon-1}}$
$\displaystyle=$ $\displaystyle
M^{\frac{1}{1-\epsilon}}\left[\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-r}d_{D}(t,r)e^{(g^{*}_{\varphi}+n)(r-t)})^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}r\right]^{\frac{\epsilon}{\epsilon-1}}.$
(76)

In order to derive the proportion of historical data trading, suppose that
data generated at time $t$ can only be sold to intermediate producer entering
at time $t+M$. Also, it is reasonable to guess that
$d(t+M,t)=d(t+2M,t+M)=...=d_{D}$, since they are all proportions within the
range of $[0,1]$ and have the same time interval. Then, (8.2.1) is simplified
as

$d_{D}^{1+\frac{1}{\epsilon}}e^{-c_{0}d_{D}^{2}M}=\frac{(1-\alpha)\xi}{2c_{0}M}\delta^{\left(1-\frac{1}{\epsilon}\right)M}e^{-\bar{g}_{D}M}M^{-\frac{1}{\epsilon}}\left[1-e^{-(c_{0}d_{D}^{2}-n)M}\right]\left[\alpha(\Delta_{D}^{\prime})^{-\frac{1}{\epsilon}}+(1-\alpha)(\Delta_{D}^{\prime})^{1-\frac{1}{\epsilon}}\right]^{-1},$

where,

$\Delta_{D}^{\prime}=M^{\frac{1}{1-\epsilon}}\delta^{M}d_{D}e^{-(g^{*}_{\varphi}+n)M}.$
(77)

Thus, $d_{D}$ can be derived numerically given the values of parameters. We
refer to Jones and Tonetti (2020) to set $\alpha=0.5$, $\epsilon=50$, and
$c_{0}=0.2$ and we take $M=1$ and $\delta=0.9$, $\xi=0.5$, $\zeta=0.85$,
$\gamma=2.5$, $n=0.02$, and $\sigma=1.5$. We then get $d_{D}=0$ or
$d_{D}=3.70>1$. We omit the trivial result of zeros and focus on the non-zero
result. It seems that the negative effect of creative destruction is too
trivial to make the optimal choice of incumbent intermediate producers be
within an appropriate range, if we take $c_{0}=0.2$ as Jones and Tonetti
(2020) do in their paper. However, if we increase $c_{0}$ to 4, the optimal
choice of $d_{D}$ becomes 0.92, which is now a reasonable value. We discuss
this further after deriving the corresponding result for the social planner’s
problem.

####  8.2.2 The Social Planner’s Problem

The social planner’s problem is similar to the baseline model shown from (24)
to (27), except that the consumers’ utility function (24) is now (65), and the
innovation possibility frontier (25) is now (64).

The maximization problem is shown as

$\max_{c(t),\left\\{d(s,t)\right\\}_{s=t+\mathrm{d}t}^{t+M},\atop\left\\{d(t,s)\right\\}_{s=t-M}^{t-\mathrm{d}t},\varphi(t)}\int_{0}^{\infty}e^{-(\rho-n)t}\left[\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}\int_{t}^{t+M}\delta^{s-t}\bar{d}(s,t)\mathrm{d}s\right]\mathrm{d}t,$

s.t.

$\dot{N}(t)=\eta N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi},$
$c(t)=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}}\beta
N(t)l_{E}(t),$

and $l_{E}(t)+l_{R}(t)=1$. Let

$A=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}},$

we set up the current-value Hamiltonian equation as

$\mathcal{H}=\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}\int_{t}^{t+M}\delta^{s-t}\bar{d}(s,t)\mathrm{d}s+\lambda(t)\left[A\beta
N(t)l_{E}(t)-c(t)\right]+\mu(t)\eta N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi}.$

The necessary conditions with respect to $c(t)$, $\varphi(t)$, $l_{E}(t)$, and
$N(t)$ are

$\frac{\partial\mathcal{H}}{\partial c(t)}=c(t)^{-\gamma}-\lambda(t)=0,$ (78)
$\frac{\partial\mathcal{H}}{\partial\varphi(t)}=-\sigma\varphi(t)^{\sigma-1}\int_{t}^{t+M}\delta^{s-t}\bar{d}(s,t)\mathrm{d}s+\mu(t)\alpha\eta\xi
N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi}\frac{1}{\varphi(t)}=0,$ (79)
$\frac{\partial\mathcal{H}}{\partial l_{E}(t)}=\lambda(t)\beta
AN(t)-\mu(t)\eta(1-\xi)N(t)^{\zeta}D(t)^{\xi}l_{R}(t)^{-\xi}L(t)^{1-\xi}=0$
(80)

and

$\frac{\partial\mathcal{H}}{\partial N(t)}=\lambda(t)A\beta
l_{E}(t)+\mu(t)\eta\zeta
N(t)^{\zeta-1}D(t)^{\xi}l_{R}(t)^{1-\xi}L(t)^{1-\xi}=-\dot{\mu}(t)+(\rho-n)\mu(t).$
(81)

For the necessary conditions of $\bar{d}(s,t)$ and $\underline{d}(t,s)$, we
should derive them in other ways. For certain periods $t$ and $s$ ($s<t$,
$t-s\leq M$), the Hamiltonian equation is rewritten into the following form

$\displaystyle\widetilde{\mathcal{H}}=$
$\displaystyle\frac{c(s)^{1-\gamma}-1}{1-\gamma}-\varphi(s)^{\sigma}\int_{s}^{s+M}\delta^{r-s}\bar{d}(r,s)\mathrm{d}r+e^{-(\rho-n)(t-s)}\left[\frac{c(t)^{1-\gamma}-1}{1-\gamma}-\varphi(t)^{\sigma}\int_{t}^{t+M}\delta^{r-t}\bar{d}(r,t)\mathrm{d}r\right]...$
$\displaystyle+\lambda(s)\left[A\beta N(s)l_{E}(s)-c(s)\right]+\mu(s)\eta
N(s)^{\zeta}D(s)^{\xi}L_{R}(s)^{1-\xi}+\lambda(t)\left[A\beta
N(t)l_{E}(t)-c(t)\right]...$ $\displaystyle+\mu(t)\eta
N(t)^{\zeta}D(t)^{\xi}L_{R}(t)^{1-\xi}.$

In equilibrium, we have $\bar{d}(t,s)=\underline{d}(t,s)=d(t,s)$. Then, the
first order condition with respect to $d(t,s)$ is

$\displaystyle\frac{\partial\widetilde{\mathcal{H}}}{\partial d(t,s)}=$
$\displaystyle-\varphi(s)^{\sigma}\delta^{t-s}+\mu(t)(1-\alpha)\eta\xi
N(t)^{\zeta}D(t)^{\xi-1}L_{R}(t)^{1-\xi}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}[\delta^{t-s}d(t,s)\varphi(s)L(s)]^{-\frac{1}{\epsilon}}...$
$\displaystyle\delta^{t-s}\varphi(s)L(s)=0.$ (82)

Consider that in BGP, $d(t,s),\forall t,s$ should be constants, thus

$\int_{t}^{t+M}\delta^{t-s}\bar{d}(s,t)\mathrm{d}s$

should also be a constant, which we denote by $\bar{M}$. Then, from (79), we
derive the shadow price $\mu(t)$ as

$\mu(t)=\frac{\sigma\bar{M}}{\alpha\eta\xi}\varphi(t)^{\sigma}N(t)^{-\zeta}D(t)^{-\xi}l_{R}(t)^{\xi-1}L(t)^{\xi-1}.$

Substitute this into (8.2.2), we have

$\displaystyle\varphi(s)^{\sigma}\delta^{t-s}$
$\displaystyle=\frac{\bar{M}\sigma(1-\alpha)}{\alpha}\varphi(t)^{\sigma}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}[\delta^{t-s}d(t,s)\varphi(s)L(s)]^{-\frac{1}{\epsilon}}\delta^{t-s}\varphi(s)L(s)$
$\displaystyle\Rightarrow\quad d(t,s)$
$\displaystyle=\left[\frac{\bar{M}\sigma(1-\alpha)}{\alpha}\varphi(t)^{\sigma}\varphi(s)^{-\sigma}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}M^{-\frac{1}{\epsilon}}\delta^{-\frac{1}{\epsilon}(t-s)}[\varphi(s)L(s)]^{1-\frac{1}{\epsilon}}\right]^{\epsilon}$
$\displaystyle\Rightarrow\quad d(t,s)$
$\displaystyle=\left[\frac{\bar{M}\sigma(1-\alpha)}{\alpha}\delta^{-\frac{1}{\epsilon}(t-s)}e^{\bar{g}_{S}(s-t)}M^{-\frac{1}{\epsilon}}\varphi(t)^{1-\frac{1}{\epsilon}}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}L(t)^{1-\frac{1}{\epsilon}}\right]^{\epsilon},$
(83)

where,

$\bar{g}_{S}=\left(-\sigma+1-\frac{1}{\epsilon}\right)g^{*}_{\varphi}+\left(1-\frac{1}{\epsilon}\right)n.$

From (80), by moving the second term to the right hand side, changing the
equation into the form of growth rate, and substituting the growth rate of
$\lambda_{t}$ and $\mu_{t}$ derived from (78) and (79), we have

$\displaystyle\frac{\dot{\lambda}(t)}{\lambda(t)}+\frac{\dot{N}(t)}{N(t)}=\frac{\dot{\mu}(t)}{\mu(t)}+\zeta\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{D}(t)}{D(t)}-\xi
n$ $\displaystyle\Rightarrow\quad$
$\displaystyle-\gamma\frac{\dot{c}(t)}{c(t)}+(1-\zeta)\frac{\dot{N}(t)}{N(t)}-\frac{\dot{\mu}(t)}{\mu(t)}-\xi\frac{\dot{D}(t)}{D(t)}+\xi
n=0$ $\displaystyle\Rightarrow\quad$
$\displaystyle-\gamma\frac{\dot{c}(t)}{c(t)}+\frac{\dot{N}(t)}{N(t)}-\sigma\frac{\dot{\varphi}(t)}{\varphi(t)}=0.$
(84)

Notice that the growth of $c(t)$ is what we want to derive, which can be
defined as $g^{*}$, then

$\frac{\dot{c}(t)}{c(t)}=g^{*}=\frac{1}{\gamma}\left(\frac{\dot{N}(t)}{N(t)}-\sigma\frac{\dot{\varphi}(t)}{\varphi(t)}\right).$
(85)

Next, we work to pin down the growth rate of $N(t)$ and $\varphi(t)$. From
(80), we have

$\frac{\lambda(t)}{\mu(t)}=\frac{\eta(1-\xi)}{\beta
A}N(t)^{\zeta-1}l_{R}(t)^{-\xi}D(t)^{\xi}L(t)^{1-\xi}.$

Then, (81) can be reformed as

$\displaystyle\frac{\lambda(t)}{\mu(t)}A\beta l_{E}(t)+\eta\zeta
N(t)^{\zeta-1}D(t)^{\xi}l_{R}(t)^{1-\xi}L(t)^{1-\xi}$
$\displaystyle=-\frac{\dot{\mu}(t)}{\mu(t)}+(\rho-n)$
$\displaystyle\eta\left[(1-\xi)l_{E}(t)+\zeta
l_{R}(t)\right]N(t)^{\zeta-1}D(t)^{\xi}l_{R}(t)^{-\xi}L(t)^{1-\xi}$
$\displaystyle=\zeta\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{D}(t)}{D(t)}-\sigma\frac{\dot{\varphi}(t)}{\varphi(t)}+\rho-(1+\xi)n$
$\displaystyle\eta\left[(1-\xi)l_{E}(t)+\zeta
l_{R}(t)\right]N(t)^{\zeta-1}D(t)^{\xi}l_{R}(t)^{-\xi}L(t)^{1-\xi}$
$\displaystyle=\zeta g_{N}+(\xi-\sigma)g_{\varphi}+\rho-n.$ (86)

Since the right hand side is constant in BGP, the left hand side should also
be constant. Thus, we have

$(\zeta-1)\frac{\dot{N}(t)}{N(t)}+\xi\frac{\dot{D}(t)}{D(t)}(1-\xi)n=(\zeta-1)g_{N}+\xi
g_{\varphi}+n=0.$ (87)

We can easily derive that the BGP growth rate of variety $g^{*}_{N}$ is that
of the economy $g^{*}$. Combining (85) with (87), we still get the same BGP
growth rates as in our baseline model

$g^{*}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$

and

$g^{*}_{\varphi}=\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$

We then calculate the labor employment in R&D sector in BGP. Eq. (86) can be
rewritten as:

$\displaystyle\eta\left[(1-\xi)l_{E}(t)+\zeta
l_{R}(t)\right]N(t)^{\zeta-1}D(t)^{\xi}l_{R}(t)^{-\xi}L(t)^{1-\xi}$
$\displaystyle=\zeta g_{N}+(\xi-\sigma)g_{\varphi}+\rho-n.$

Then, let $g^{*}=\eta
N(t)^{\zeta-1}D(t)^{\xi}(l_{R}^{*})^{1-\xi}L(t)^{1-\xi}$, (41) can be further
rewritten as:

$\displaystyle[(1-\xi)(1-l_{R}^{*})+\zeta l^{*}_{R}]g^{*}(l_{R}^{*})^{-1}$
$\displaystyle=\zeta g^{*}+\xi(g^{*}_{\varphi}+n)-\sigma
g^{*}_{\varphi}+\rho-(1+\xi)n$
$\displaystyle\Rightarrow\quad[(1-\xi)(1-l_{R}^{*})+\zeta
l^{*}_{R}]g^{*}(l_{R}^{*})^{-1}$
$\displaystyle=\left[\zeta+\frac{(\xi-\sigma)(1-\gamma)}{\sigma}\right]g^{*}+\rho-n$
$\displaystyle\Rightarrow\quad[(1-\xi)(1-l_{R}^{*})+\zeta
l^{*}_{R}]g^{*}(l_{R}^{*})^{-1}$
$\displaystyle=\left[1+\frac{\sigma(\zeta-1)}{\xi}\right]g^{*}+\left(\frac{\sigma}{\xi}-1\right)n+\rho.$
(88)

We have the same labor allocations as those in our baseline model.

Finally, we derive the historical data trading proposition. From (8.2.2) we
have the growth rate of the right hand side as

$\displaystyle\left(1-\frac{1}{\epsilon}\right)g_{\varphi}+\left(\frac{1}{\epsilon}\right)g_{B}-g_{D}+\left(1-\frac{1}{\epsilon}\right)n=0.$

Thus, $d(t,s)$ is a constant determined by (8.2.2). The terms
$\varphi(t)^{1-\frac{1}{\epsilon}}B(t)^{\frac{1}{\epsilon}-1}L(t)^{1-\frac{1}{\epsilon}}$
can be further represented by expression similar to (8.2.1), except that
$d(t,s)$ may be different, i.e.,

$\varphi(t)^{1-\frac{1}{\epsilon}}D(t)^{-1}B(t)^{\frac{1}{\epsilon}}L(t)^{1-\frac{1}{\epsilon}}=\left[\alpha\Delta_{S}^{-\frac{1}{\epsilon}}+(1-\alpha)\Delta_{S}^{1-\frac{1}{\epsilon}}\right]^{-1},$

where,

$\Delta_{S}=M^{\frac{1}{1-\epsilon}}\left[\int_{t-M}^{t-\mathrm{d}t}(\delta^{t-r}d_{S}(t,r)e^{(g^{*}_{\varphi}+n)(r-t)})^{\frac{\epsilon-1}{\epsilon}}\mathrm{d}r\right]^{\frac{\epsilon}{\epsilon-1}}.$
(89)

Similar to the last section, a simplified version of this model can help us
understand this problem further. We also assume that data generated at time
$t$ can only be sold to intermediate producer entering at time $t+M$. Also, it
is reasonable to guess that $d(t+M,t)=d(t+2M,t+M)=...=d_{S}$ since they are
all proportions within the range of $[0,1]$ and have the same time interval.
Then, (8.2.2) is simplified as

$d_{S}^{\frac{1}{\epsilon}}=\frac{\delta^{-M}\sigma(1-\alpha)}{\alpha}\delta^{-\frac{1}{\epsilon}M}e^{-\bar{g}_{S}M}M^{-\frac{1}{\epsilon}}\left[\alpha(\Delta_{S}^{\prime})^{-\frac{1}{\epsilon}}+(1-\alpha)(\Delta_{S}^{\prime})^{1-\frac{1}{\epsilon}}\right]^{-1},$

where,

$\Delta^{\prime}_{S}=M^{\frac{1}{1-\epsilon}}\delta^{M}d_{S}e^{-(g^{*}_{\varphi}+n)M}.$
(90)

We use the same parameters as those in the decentralized economy, which leads
to $d_{S}=0$ or $d_{S}=0.53\in(0,1)$. We also focus on the non-zero result.
Notice that creative destruction does not affect the social planner’s
solution, so this result still holds when we increase the value of $c_{0}$.
Comparing the optimal selling proportion derived in these two different
situations ($d_{D}$ and $d_{S}$), we conclude that incumbent intermediate
producers are always inclined to sell more historical data to entrants than
the social optimal level, unless the negative effect of creative destruction
becomes too large.171717Given the parameters we use, we find that $d_{D}$
becomes smaller than $d_{S}$ when $c_{0}$ increases to 14.

###  8.3 Data Ownership: Firms versus Consumers

We extend the model to allow different specifications of data ownership. As
described earlier, the firm ownership problem becomes trivial with zero data
processing cost ($\theta=0$). Also, it is not tractable to analyze the case
that consumers pay the firms for decreasing the usage of data in order to
lower their disutilities from privacy violation. For both reasons, we focus on
$\theta>0$ in (16)—a friction representing the cost of handling data by firms.

####  8.3.1 Analysis with Consumer Ownership of Data

Let us first consider what happens to our baseline model once we have non-
trivial “data processing cost” by firms. When consumers own data, the budget
constraint of consumers is still the same as (2), and so are other conditions
that we do not mention in this section. What makes the model different here is
the profit maximization problem of potential intermediate producers who now
have to pay the “data processing cost” in addition to the fees given to
consumers. Then, we have the following proposition.

###### Proposition 8.7

With $\theta>0$ and consumer ownership of data, the BGP growth rates and labor
allocations are the same as those in the baseline model, provided
$\sigma(2-\zeta)+(\xi+\phi)(\gamma-1)>0$.

###### Proof 8.8

Proof of Proposition 8.7. The condition
$\sigma(2-\zeta)+(\xi+\phi)(\gamma-1)>0$ holds easily when the knowledge
spillover effect is not so large and consumers’ EIS are rather small.

The maximization problem of intermediate producers now becomes

$\max_{L_{R}(t),\varphi(t)}\eta
N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t)V(t)-w(t)l_{R}(t)L(t)-\theta\varphi(t)^{\phi}-p_{\varphi}(t)\varphi(t)L(t).$

Free-entry conditions with respect to labor is still the same as (19), while
the other free-entry condition becomes

$\eta\xi
N(t)^{\zeta}\varphi(t)^{\xi-\phi}l_{R}(t)^{1-\xi}V(t)L(t)=\frac{p_{\varphi}(t)L(t)}{\varphi(t)^{\phi-1}}+\theta\phi.$
(91)

Since $p_{\varphi}(t)$, $\varphi(t)$ and $L(t)$ all grow in constant rates in
BGP, we go on with deriving the BGP growth rates by discussing different
situations separately.

##### Case 1.

In BGP, if $g^{*}_{p_{\varphi}}+n>(\phi-1)g^{*}_{\varphi}$, it can be derived
that the right hand side of (91) converges to the first term, i.e.,

$\eta\xi
N(t)^{\zeta}\varphi(t)^{\xi-\phi}l_{R}(t)^{1-\xi}V(t)L(t)\approx\frac{p_{\varphi}(t)L(t)}{\varphi(t)^{\phi-1}}.$

Then, the free-entry condition becomes the same as that in (18). Consequently,
the BGP growth rates become the same as those shown in Proposition 21.
Furthermore, from

$\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}=(\sigma-1)\frac{\dot{\varphi}(t)}{\varphi(t)}+\gamma\frac{\dot{c}(t)}{c(t)}=\left[\frac{\sigma+\gamma-1}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$

we can derive the condition as

$\displaystyle g^{*}_{p_{\varphi}}+n>$ $\displaystyle(\phi-1)g^{*}_{\varphi}$
$\displaystyle\Rightarrow\quad\left[\frac{\sigma+\gamma-1}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+n>$
$\displaystyle(\phi-1)\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n$
$\displaystyle\Rightarrow\quad\sigma(2-\zeta)+(\xi+\phi)(\gamma-1)>$
$\displaystyle 0,$

which is always the case given the values of the above parameters.

##### Case 2.

If $g^{*}_{p_{\varphi}}+n\leq(\phi-1)g^{*}_{\varphi}$, it can be derived that
the right hand side of (91) converges to a constant. Then, this problem
becomes the same as the case when firms own data. Then, the BGP growth rate of
$p_{\varphi}(t)$ becomes

$\frac{\dot{p}_{\varphi}(t)}{p_{\varphi}(t)}=(\sigma-1)\frac{\dot{\varphi}(t)}{\varphi(t)}+\gamma\frac{\dot{c}(t)}{c(t)}=\left[\frac{(\sigma-1)(2-\zeta)+\gamma(\xi+\phi)}{\phi(1-\zeta)-\xi}\right]n.$

And the condition becomes

$\displaystyle g^{*}_{p_{\varphi}}+n\leq$
$\displaystyle(\phi-1)g^{*}_{\varphi}$
$\displaystyle\Rightarrow\quad\left[\frac{(\sigma-1)(2-\zeta)+\gamma(\xi+\phi)}{\phi(1-\zeta)-\xi}\right]n+n\leq$
$\displaystyle(\phi-1)\left[\frac{2-\zeta}{\phi(1-\zeta)-\xi}\right]n$
$\displaystyle\Rightarrow\quad\sigma(2-\zeta)+(\xi+\phi)(\gamma-1)\leq$
$\displaystyle 0,$

which does not exist given the standard values of the parameters.

Thus, the BGP growth rates when consumers own data are

$g^{*}=\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n$ (92)

and

$\frac{\dot{\varphi}(t)}{\varphi(t)}=-\frac{1}{\xi}n+\frac{1-\zeta}{\xi}\frac{\dot{N}(t)}{N(t)}=\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$

which are the same as those shown in Proposition 21. Obviously, the labor
allocations in BGP is still the same as those shown in Proposition 3.2. Q.E.D.

In general, we assume $\sigma>0$, $\zeta\in[0,1]$, $\xi\in[0,1]$, $\phi>1$,
$\gamma>1$. Thus, the proposition exists when we take standard values of the
parameters. The BGP growth rate of consumption is now larger than that of the
data provision. However, if we make the data provision constraint (3) tighter,
i.e., let $s$ be smaller so that the constraint binds in BGP, we have lower
growth rates. Formally, if $s$ satisfies

$s<g^{*}_{\varphi}-g^{*}=\left[\frac{1-\gamma-\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n,$

we always have

$(g^{*}_{\varphi})^{\prime}=(g^{*})^{\prime}+s.$ (93)

Then, Eq. (34) derived from the free-entry condition of data (18) fails. Thus,
we combine (32) with (93) to derive the BGP growth rates in this case as

$(g^{*})^{\prime}=\frac{\xi
s+n}{1-\zeta-\xi}<\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n=g^{*},$
(94)

which means the BGP growth rate in this case is always lower than that in our
baseline model.

In addition, we consider the social planner’s problem, which leads to the
following proposition.

###### Proposition 8.9

With $\theta>0$ in the social planner’s problem, the BGP growth rate as well
as labor allocations still do not change compared with the baseline model,
provided $\gamma\sigma+(\phi-1)(\gamma-1)+1>0$.

###### Proof 8.10

Proof of Proposition 8.9.

The condition $\gamma\sigma+(\phi-1)(\gamma-1)+1>0$ holds easily when the
“data processing cost” is convex and consumers’ consumption growth is rather
insensitive to real interest rates.

In the social planner’s problem, the resource constraint now becomes

$C(t)=\widetilde{Y}(t)\equiv
L_{E}(t)^{\beta}\int_{0}^{N(t)}x(v,t)^{1-\beta}\,\mathrm{d}v-\int_{0}^{N(t)}\psi
x(v,t)\,\mathrm{d}v-\theta\varphi(t)^{\phi}.$

Then, the optimal net output becomes

$c(t)=\left(\frac{\psi}{1-\beta}\right)^{1-\frac{1}{\beta}}\beta
N(t)l_{E}(t)-\frac{\theta\varphi(t)^{\phi}}{L(t)}.$ (95)

Thus, (37) becomes

$\frac{\partial\mathcal{H}}{\partial\varphi(t)}=-\sigma\varphi(t)^{\sigma-1}-\theta\phi\frac{\lambda(t)\varphi(t)^{\phi-1}}{L(t)}+\mu(t)\xi\eta
N(t)^{\zeta}\varphi(t)^{\xi-1}l_{R}(t)^{1-\xi}L(t)=0.$ (96)

Since $c(t)^{-\gamma}=\lambda(t)$, we should first discuss the value of the
additional term in BGP. This term converges to zero when

$-\gamma\frac{\dot{c}(t)}{c(t)}+(\phi-1)\frac{\dot{\varphi}(t)}{\varphi(t)}<n.$

If the above inequality is satisfied, the BGP growth rates will be the same as
those we derived in Proposition 3.3. Then, we further derive the inequality as

$\displaystyle-\gamma\left[\frac{\sigma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n+(\phi-1)\left[\frac{1-\gamma}{(1-\zeta)\sigma-\xi(1-\gamma)}\right]n<$
$\displaystyle n$
$\displaystyle\Rightarrow\quad\gamma\sigma+(\phi-1)(\gamma-1)>$
$\displaystyle-1,$ (97)

which is always the case given standard values of the above parameters. Thus,
we can verify that Proposition 3.3 is still satisfied in this new setting.
Obviously, labor allocations in BGP are still the same as those shown in
Proposition 3.4. Q.E.D.

Similarly, this proposition also holds when we take standard values of the
parameters. Propositions 8.7 and 8.9 both show that our baseline model is
robust under general data processing cost $\theta$.

####  8.3.2 Analysis with Firm Ownership of Data

When firms own data, consumers’ budget constraint (2) now becomes

$\dot{a}(t)=(r(t)-n)a(t)+w(t)-c(t),\forall t\in[0,\infty).$

Then, we have the following proposition about the BGP growth rates and labor
allocations in this model.

###### Proposition 8.11

If firms own data, the BGP growth rate becomes

$g^{*}_{c}=g^{*}_{N}=g^{*}_{y}=g^{*}=\left[\frac{\xi+\phi}{\phi(1-\zeta)-\xi}\right]n,$
(98) $g^{*}_{\varphi}=\left[\frac{2-\zeta}{\phi(1-\zeta)-\xi}\right]n>0.$ (99)

Labor allocations in this case have the same form as those shown in
Proposition 3.2, except that the corresponding BGP growth rates take (98) and
(99) here.

###### Proof 8.12

Proof of Proposition 8.11. When firms own data, the profit maximization
problem of a potential intermediate producer becomes

$\max_{L_{R}(t),\varphi(t)}\eta