ISSN (Print) 0473-453X

Discussion Paper No. 1244

ISSN (Online) 2435-0982

Inventor Mobility, Knowledge Diffusion,and Growth

Yasutaka Koike-Mori
Toshitaka Maruyama
Koki Okumura

March 2024

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Inventor Mobility, Knowledge Diffusion, and Growth
Yasutaka Koike-Mori

Toshitaka Maruyama

Koki Okumura

UCLA

UCLA

UCLA *March 25, 2024

Abstract
This paper develops an endogenous growth model that incorporates a frictional inventor market and examines the allocation of inventors across firms, knowledge diffusion,and its impact on growth. In our model, inventors play dual roles: they engage inin-house R&D and transfer knowledge from previous employers to new ones whenchanging jobs. Using an administrative panel dataset on German inventors matchedto their employing establishments and patents, we find that, relative to general workers, inventors are more likely to transition to less productive establishments and suffera higher wage growth via the transition. We also find that the knowledge base of
establishments measured by patents grows faster when a significant proportion of
their inventors originate from establishments possessing a larger knowledge base. We
then calibrate the model to reflect these empirical findings and examine the effects ofinnovation policy. While subsidies to frontier firms discourage knowledge diffusionfrom these firms to technologically lagging firms, these subsidies also encourage innovation within frontier firms. The former negative effect dominates in the short term,
but the latter positive effect dominates in the long run.
* We thank Andy Atkeson, David Baqaee, Serguey Braguinsky, Ariel Burstein, Pablo Fajgelbaum, Hugo
Hopenhayn, Ryo Horii, Oleg Itskhoki, Lee Ohanian, Koki Oikawa, Liyan Shi, Takayuki Tsuruga, Jon Vogel,
Conor Walsh, Pierre-Olivier Weill and seminar participants at UCLA, SWET 2023, and The 2023 Moriguchi
Prize for valuable comments. We also thank the Institute for Employment Research for generously providing
their data. This paper was awarded the 2023 Moriguchi Prize by the Institute of Social and Economic
Research, Osaka University.

11

Introduction

Inventors play an essential role in both innovation within firms and knowledge diffusion
between firms, which are important sources of economic growth. As Arrow (1962) stated“mobility of personnel among firms provides a way of spreading information,” the mobility of inventors between firms has been considered an essential source of knowledge
diffusion between firms.1 Thus, policies related to labor markets for inventors are likely
to have a significant impact on the firm productivity and economic growth.
This study provides an endogenous growth model to analyze the market for inventors
and its impact on firm productivity and economic growth. In our model, inventors play
dual roles: (i) they participate in in-house R&D efforts, enhancing the firm’s technology,
and (ii) they facilitate the transfer of knowledge from their former employers to their
new ones when they change jobs. To quantify the model, we utilize data on inventors
and patents linked to administrative labor market career information about inventors andtheir employing establishments in Germany. With these data, we document three novel
empirical observations regarding inventors’ job transitions, wage changes accompanying
these transitions, and the influence of inventor inflows on the future innovation activityof recruiting firms. Finally, we discipline the model to align with these empirical findings
and explore the consequences of inventor labor market policies.
In the theoretical part, we introduce an endogenous growth model that features thelabor market where inventors and firms interact, and knowledge spills over across firms
via inventor job transitions. Heterogeneous firms offer job openings, considering theknowledge diffusion from the inventors’ prior employers, and inventors and firms match
randomly in a frictional labor market. We focus on on-the-job search given our interestin how the inter-firm mobility of inventors influences knowledge spillover. This modelis the first endogenous economic growth model that considers the endogenous job flows
of inventors across firms and the knowledge diffusion through the job flows of inventors.
The model’s strength lies in its ability to endogenously generate both net and gross job flowof inventors and knowledge spillovers resulting from these flows, which are responsive
to economic conditions and policy changes.
The empirical section documents novel findings for the job flows of inventors andtheir consequences. First, we examine the patterns of the mobility of inventors — defined1

For evidence from recent studies, see, Jaffe et al. (1993); Almeida and Kogut (1999); Song et al. (2003);
Hoisl (2007); Rosenkopf and Almeida (2003); Breschi and Lissoni (2009); Singh and Agrawal (2011); Kaiser
et al. (2015); Rahko (2017); Braunerhjelm et al. (2020).

2

as workers who have created patents — using inventor biography data from Germany,
which links labor market biographies and their employing establishments recorded in
the German social security data to patent register data. We find that a large proportionof inventors move to less productive establishments. This result is robust to the use ofdifferent productivity measures: establishment size, average wage, and number of patentcitations. This job flow pattern of inventors is in contrast to general workers, who are
more likely to move to more productive firms, as established in previous literature (e.g.,
Haltiwanger et al. (2018)). Moreover, we find that the wages of inventors grow morethan those of general workers when they change jobs. This finding suggests that firms
compensate for knowledge diffusion when they hire new inventors.
Then, we investigate how inventors’ job flows influence the knowledge base of establishments, as measured by patents. We find that when a larger proportion of inventors
comes from establishments with a more extensive knowledge base, the knowledge baseof establishments grows faster over the next three to five years. Furthermore, we apply an
instrumental variable to inventor flows and obtain significant results with the same sign
as in the OLS specification. These results suggest the presence of knowledge diffusion
through inventor flows.
In the quantitative section of this paper, we calibrate the model to match the key
characteristics of the joint distribution of German inventors and firm dynamics observedin the microdata. We show that the calibrated model fits the target and non-target moments
well, confirming that the model is well-suited to study counterfactual exercises.
We initially apply the calibrated model to conduct comparative statics analyses onmatching efficiency. The model suggests that a decrease in matching efficiency, followed
by a reduction in inventor mobility, leads to a decline in the economic growth rate.
According to INV-BIO data, inventor mobility in Germany has been diminishing sincethe 1990s. Similarly, Akcigit and Goldschlag (2023a) report a decline in inventor mobility
in the U.S. beginning in the early 2000s. Consequently, our model offers a framework
for understanding the relationship between the observed decrease in inventor mobility
and the deceleration of aggregate productivity growth in developed countries over recent
decades.
Finally, we analyze the transition dynamics in our model to evaluate the effects oflabor market policies on inventors. A key issue for those overseeing innovation policy is
identifying which firms should be granted subsidies. In this context, we investigate thetransition from an initial Balanced Growth Path (BGP) without subsidies to a new BGP

3

with subsidies directed at technologically frontier firms. Frontier firms are characterized
as those ranking in the upper half of the productivity distribution, with weighting basedon the number of inventors. In the short term, subsidies to frontier firms reduce aggregate
output by impeding the mobility of inventors from these leading firms to less advanced
ones, thus hampering knowledge transfer. In contrast, over the long term, this policy
boosts aggregate output by accelerating the growth rate at the technological frontier.
Therefore, the impact of targeted subsidies on specific groups of firms hinges on whether
policymakers focus on short-term or long-term economic effects.
Related Literature. Our paper is related to the literature on endogenous growth theory,
particularly the diffusion of technology and knowledge, including Luttmer (2007), Lucas(2009), Lucas and Moll (2014), Perla and Tonetti (2014), Akcigit et al. (2018), Buera andOberfield (2020), Shi and Hopenhayn (2020), Benhabib et al. (2021), and Prato (2022). Bueraand Lucas (2018) surveys this topic. Perla and Tonetti (2014) and Lucas and Moll (2014)
advanced the literature by modeling agents who choose to invest in technology diffusion.
This approach enables the investigation of incentives, externalities, and welfare-improving
policies. Our formulation of the knowledge diffusion function is based on the semiendogenous growth model proposed by Buera and Oberfield (2020), which investigatesinternational knowledge diffusion. Their model provides a micro-foundation for theknowledge diffusion function and expresses knowledge diffusion as a synergy of novel
ideas and insights drawn from others. Similar to ours, Benhabib et al. (2021) and Shi
and Hopenhayn (2020) address the interaction between R&D innovation and knowledge
diffusion. In particular, our model applies the firms’ innovation process formulated by
Benhabib et al. (2021) to generate a realistic stationary productivity distribution. In thetechnology diffusion literature, our work is most closely related to Akcigit et al. (2018),
who explicitly model inventors and analyze their role in knowledge diffusion among
inventors. We depart from this literature by focusing on knowledge diffusion among firms
due to inventor mobility. Moreover, we introduce a new perspective to this literature byincorporating labor market frictions, emphasizing the interaction between inventors andfirms.
An expansive body of empirical research supports the concept of knowledge spillovers
facilitated by job transitions of inventors. In one of the first such studies, Almeida and
Kogut (1999) show that locations with greater intraregional labor mobility between firms
tend to have more localized knowledge flows. Song et al. (2003) illustrate that mobile
inventors build upon ideas from their previous firm more often than other inventors at the4hiring firm. Rosenkopf and Almeida (2003) analyzes firm pairs, showing that those withhigher labor mobility also have greater subsequent knowledge flow. These pioneeringstudies have inspired further research to facilitate our understanding of the connection
between job transitions of inventors and knowledge spillovers (Hoisl, 2007; Breschi and
Lissoni, 2009; Singh and Agrawal, 2011; Kaiser et al., 2015; Rahko, 2017; Braunerhjelm
et al., 2020). Mawdsley and Somaya (2016) provide a review of these studies2 . Whilemost studies in this field struggle with limitations related to drawing causal inferences,some papers address the endogeneity problem. Singh and Agrawal (2011) employ adifference-in-differences approach to compare pre-move and post-move citation rates forpoached inventors’ previous and comparable control patents, concluding that acquiringfirms intensify their use of inventions from the inventors’ previous employers. Kaiser
et al. (2015) use lagged mobility and industry mobility averages as instrumental variables
for inventor mobility, uncovering a significantly positive impact of incoming inventors
on their new employers’ patent activity. Our paper is the first to integrate these insights
into an endogenous growth model, emphasizing the interaction of inventor mobility andknowledge diffusion across firms. Furthermore, our study is novel in that it compares thepatterns of job changes and the associated wage changes between inventors and general
workers, providing evidence that suggests knowledge transmission and compensation
for it.
Our paper also relates to the literature on frictional labor markets. In particular, ourstudy benefits from recent developments in the modeling of multi-worker firms and onthe-job search, including Schaal (2017), Elsby and Gottfries (2021), and Bilal et al. (2023).
The contemporary presence of on-the-job search and a non-constant return to scale revenue
function in employment makes, in general, the firm problem intractable because we needto track the distribution of wages within each firm. To address the intractability, we
assume that a firm posts a privately efficient number of vacancies, following Bilal et al.
(2023). This assumption reduces the state variables to firm productivity and the numberof inventors, thereby rendering the model tractable. Based on Bilal et al. (2022), Bilal
et al. (2023) presented an endogenous growth model where the productivity distributionof incumbents determines the productivity of entrant firms. This model introduces an
2

A related area of study is the relationship between geography and knowledge diffusion. Early research
by Jaffe et al. (1993) suggested a higher probability of cited patents originating from the same location as theciting ones. Breschi and Lissoni (2009) further improved this approach by introducing inventor mobility
as a control, revealing that spatial proximity’s effect on knowledge diffusion is cut by more than half. Thissuggests that the critical role of geography in knowledge transfer primarily results from inventors seldom
relocating across regions.

5

endogenous growth rate akin to Luttmer (2007). However, their model abstracts away theknowledge spillovers through worker mobility and its implication for economic growth.
Herkenhoff et al. (2018) and Shi (2023) explore models wherein knowledge diffuses
among firms or colleagues via worker mobility. However, these studies consider models
where firms employ only one or two workers at most. In contrast, our model allows
firms to hire an arbitrary number of inventors unless it is profitable. Furthermore, whilethese papers examine more generalized workers, we restrict our focus to inventors andinvestigate the impact on economic growth.
The rest of the paper proceeds as follows. Section 2 describes the theory. Section 3
introduces the data and empirical results. Section 4 presents the calibration of the modeland the quantitative policy counterfactual. Section 5 concludes.

2

Model

This section introduces an endogenous growth model featuring the role of the labormarket, where inventors and firms match, and knowledge diffusion across firms due toinventors’ job flows. Time is continuous, and the horizon is infinite. Inventors play
two roles: (i) they engage in R&D activities in the firm to which they belong; (ii) when
they switch jobs, they transfer knowledge from their previous employer to the new one,
thereby enhancing productivity—this is referred to as knowledge diffusion. Inventors arehomogeneous, except in terms of which firm they belong to3 . We focus on on-the-job search
since we are interested in the effect of the inter-firm mobility of inventors on knowledge
diffusion. Inventors and firms are randomly matched in a frictional labor market. Firms
make hiring decision by internalizing the marginal benefits of their contributions throughinternal R&D and knowledge diffusion. As we will discuss later, the state variable of a firm
is summed up to the productivity of the firm, z, and the number of inventors employed,n. We construct a BGP equilibrium where aggregate variables grow at a constant rate gand inventor and firms’ productivity distributions are stationary. Section 4 presents thetransition dynamics.
3

Since this study considers constant wage contracts, wages can differ even among inventors who belong
to the same firm. However, as we will discuss later, it is not necessary to track the distribution of wages
within firms when characterizing the equilibrium

6

Household
The representative household is composed of n individuals who supply inelastically one
unit of time to the labor market for inventor. The size of the population is constant.
Individuals work as inventors and receive wage payments from their firms. There is
full insurance within the family, and thus the household problem can be split into a
choice of aggregate consumption and a stage where the consumption is distributed acrosshousehold members. The latter stage is irrelevant to labor market dynamics, so we focus
on the former. The household discounts the future at the rate ρ. It derives utility fromconsumption, which we assume is logarithmic:
Z ∞e−ρt log Ĉ(t)dt.
0

Variables with hats indicate that they are variables before detrending. We assume that thehousehold trades shares in a mutual fund that owns all firms in the economy and trades
a risk-free bond in zero net supply. As is standard, this implies that firms discount future
payoffs at a constant risk-free rate r(t) = ρ + g(t) in equilibrium on a BGP.
Production Technology
There is a unit mass of a continuum of firms. These firms produce a homogeneous
product. Each firm has heterogeneous productivity ẑ. For simplicity, firm output equals
firm productivity. As we will discuss later, in equilibrium in this model, firm productivity
support is finite, and a maximum value of firm productivity exists. Let z̄(t) denote themaximum productivity of any firm, which we interpret as the technology frontier.
Matching Technology
Each firm employs a continuum of inventors n. Firms and inventors meet in a frictional
labor market. Let Ẑ(t) denote the aggregate productivity. A firm pays a cost c(v)Ẑ(t) topost v vacancies. The cost function c(v) is increasing and concave, and satisfies c(0) = 0 andc′ (0) = 0. We focus on on-the-job search and assume that firms cannot lay off inventors,
and inventors cannot voluntarily quit their jobs. Therefore, there are no unemployed
inventors. Each vacancy randomly matches at a rate of A with an inventor who is working
at other firms. For simplicity, we assume that the vacancy matching rate A is exogenous
and does not depend on labor market tightness. An inventor meets a firm at rate Av
7

where v is the total number of vacancies. An inventor incurs no cost of the search. As
the vacancy cost is multiplied by Ẑ(t), the vacancy cost grows as the economy grows. The
rationale for this assumption is that as the economy grows, the price of resources for thevacancy (e.g., wages for human resources) also grows at the same rate.
Evolution of Firms’ Productivity
We assume that firms’ productivity changes due to the following three reasons: (i) innovation, (ii) knowledge diffusion, and (iii) leapfrog.
Innovation. The productivity of firms, with productivity ẑ and inventor count n,increases by γ(n)ẑ, where γ(·) is an increasing and concave function. Consequently, therate of productivity growth attributed to in-house R&D innovation is higher for firms that
employ a larger number of inventors.
Knowledge diffusion. When a firm with productivity ẑ poaches an inventor from afirm with productivity ẑ′ , the poaching firm’s productivity increases by α(ẑ′ /ẑ)Ẑ(t) whereα(·) is an increasing and concave function. Therefore, a firm gains more knowledge when
it poaches an inventor from a firm with higher relative productivity. While better insights
lead to higher growth, the concavity of α(·) implies that if the productivity difference
between the poaching and poached firm is large, it becomes difficult for the poachingfirms to utilize that knowledge.
Leapfrog. Finally, following Benhabib et al. (2021), we assume that firms can leapfrog
to the frontier of the productivity distribution z(t) with an arrival rate η > 0. The possibility
that firms can leap to the technology frontier represents an opportunity for the innovation
process to yield significant insights rather than just steady incremental progress. This
assumption establishes a stationary distribution with an upper bound on productivity foreach period. The existence of this upper bound in the productivity distribution is crucial,
as it ensures that the effect of knowledge diffusion does not become overly pronounced.
Contractual Environment
When a model includes random search, on-the-job search, and a non-constant return-toscale revenue function in employment, the firm’s problem generally becomes intractable.
This is because it is necessary to track the entire wage distribution within and across firms
to compute optimal retention and vacancy policies. Following Bilal et al. (2022), we maketwo assumptions regarding the contractual environment. These assumptions ensure that8the state vector consists only of the firm size and productivity.
Assumption 1. (Bertrand Competition) When a vacancy posted by a poaching firm
matches with an inventor employed at another firm, the two firms engage in Bertrand
competition through a sequential auction. First, the poaching firm makes a take-it-orleave-it wage offer. Then, the targeted firm makes a take-it-or-leave-it counteroffer to theworker. Finally, the inventor decides whether to stay or to move.
Assumption 2. (Privately Efficient Vacancy Posting) The firm posts the number of vacancies that maximizes the sum of the firm’s and its inventors’ values.
While Assumption 1 is standard in the on-the-job search literature, Assumption 2
might be viewed as somewhat stringent. This latter assumption is necessary to simplify
the analytical characterization and quantitative analysis of the model. Under these assumptions, decisions made by both the firms and the inventors are privately efficient, as
though they were maximizing their total joint value. Consequently, the state variables of
the joint value function are reduced to the firm size and productivity. Therefore, there isno need to track the wage distribution within and across firms to determine equilibrium
allocations.
Distributions and Aggregate Variables
Let F̂(ẑ, n, t) be the cumulative distribution function of firms such thatZ
1=dF̂(ẑ, n, t).

We assume that the total mass of firms in the economy is normalized to one. The distribution should also satisfy the inventor market clearing condition:
Zn=ndF̂(ẑ, n, t).

Let v̂(ẑ, n, t) be the amount of vacancy a firm (ẑ, n) post at time t. The total mass of vacancy
is given byZv(t) =v̂(ẑ, n, t)dF̂(ẑ, n, t).

9Because firms produce a homogeneous product, the aggregate output is given byZ
Ẑ(t) =

ẑdF̂(ẑ, n, t).

Let fˆ(ẑ, n, t) be a density of F̂(ẑ, n, t). Let define employment-weighted densityn fˆ(ẑ, n, t)fˆn (ẑ, n, t) ≡n
and F̂n (ẑ, n, t) the corresponding cumulative distribution. Also, let define vacancy-weighted
distributionsv̂(ẑ, n, t) fˆ(ẑ, n, t)ˆ
fv (ẑ, n, t) =v
and F̂v (ẑ, n, t) the corresponding cumulative distribution.
Condition for Successful Poaching
Define the poaching indicator function 1̂P that takes 1 if the poaching successes and takes
0 otherwise. Let Ω̂(ẑ, n, t) denote the joint value of an organization composed of a firmwith productivity ẑ and its n inventors at time t. Then, the poaching indicator function isexpressed as




1
′
′
1̂P (ẑ, n, ẑ , n , t) = 0if Ω̂n (ẑ, n, t) + α(ẑ′ /ẑ)Ẑ(t)Ω̂z (ẑ, n, t) > Ω̂n (ẑ′ , n′ , t)otherwise

The first term Ω̂n (ẑ, n, t) is the derivative of the joint value with respect to n, which
represents the change in the joint value resulting from an increase in the stock of inventors.
This term captures the marginal contribution of the inventor to the firm’s in-house R&D
activity. The term α(ẑ′ /ẑ)ZΩ̂z (ẑ, n, t) represents the change in the joint value resulting
from an increase in firm productivity when the firm hires a new inventor. This term
emerges because hiring a new inventor facilitates the transfer of ideas from the firm
where the inventor previously worked. The poaching of the inventor is successful if thetotal marginal value of the inventor for the poaching firm (ẑ, n) exceeds the value for thepoached firm (ẑ′ , n′ ).

10Hamilton-Jacobi-Bellman Equation
The following Hamilton-Jacobi-Bellman (HJB) equation determine the joint value Ω̂(ẑ, n, t):r(t)Ω̂(ẑ, n, t) −∂Ω̂(ẑ, n, t)= max ẑ − c(v̂)Ẑ(t)
v̂≥0
∂t
Z h
i+
+ Av̂
Ω̂n (ẑ, n, t) + α(ẑ′ /ẑ)Ẑ(t)Ω̂z (ẑ, n, t) − Ω̂n (ẑ′ , n′ , t) dF̂n (ẑ′ , n′ , t)|{z}
Poaching Hire

+ γ(n)ẑΩ̂z (ẑ, n, t)|{z}In-house R&Di+ η Ω̂(z, n, t) − Ω̂(ẑ, n, t)
|
{z
}
h

Leapfrog(1)
When a firm (z, n) hires a new inventor, the total value increases by Ω̂n (ẑ, n, t) +α(ẑ′ /ẑ)ẐΩ̂z (ẑ, n, t) − Ω̂n (ẑ′ , n′ , t). The first and second term is the gain in value to thefirm and incumbent inventors due to the new hire. The third term is the value the firm
and incumbent inventors give the new inventor, which equals the highest value its former
employer would pay to retain them. As mentioned earlier, the poaching is successful if
this difference is positive.
Conversely, an incumbent inventor may quit and move to a higher marginal value firm.
The firm and remaining inventors will lose Ω̂n (z, n, t) and are thus prepared to increase
the inventor’s value by Ω̂n (z, n, t) to retain them. Knowing this, the external firm hires theinventor by offering the inventor exactly Ω̂n (z, n, t). Therefore, the joint value of the firm,
remaining inventors, and poached inventor are unchanged, and no “Poached Quit” term
appears in (1).
Kolmogorov Forward Equation
The Kolmogorov forward equation (KFE) describes the evolution of the firm’s distribution
across productivity and the number of inventors. To characterize the KFE, we derive thedrifts for changes in firm-level productivity and the number of inventors. The drift for11the firm-level productivity change is given byZµ̂z (ẑ, n, t) ≡γ(n)ẑ
|{z}+ Av̂(ẑ, n, t)Ẑ(t)|In-house R&D

1̂P (ẑ, n, ẑ′ , n′ , t)α(ẑ′ /ẑ)dF̂n (ẑ′ , n′ , t) .
{z
}

(2)

Knowledge diffusion

The first term on the right-hand side represents productivity growth due to in-house
R&D. The second term accounts for the firms’ productivity growth resulting from knowledge diffusion. When firms post v̂ vacancies, these vacancies match with Av̂ inventors.
Owing to the randomness of the matchings, the original firms of these inventors aretaken from the inventor-weighted firm distribution F̂n . If a poaching attempt is successful(1̂P (ẑ, n, ẑ′ , n′ , t) = 1), the productivity of the poaching firm increases by α(ẑ′ /ẑ)Ẑ(t). Notethat our definition of µ̂z (ẑ, n, t) does not include changes in productivity due to leapfrogging, and we need to include an additional term to incorporate leapfrogging in the KFE
equation, which we will describe below.
The drift for the change in the number of inventors is determined byZµ̂n (ẑ, n, t) ≡ Av̂(ẑ, n, t)|n1̂P (ẑ, n, ẑ , n , t)dF̂n (z , n , t) − Av
n
{z
} |
′

′

′

′

Z
1̂P (ẑ′ , n′ , ẑ, n, t)dF̂v (ẑ′ , n′ , t){z}
Poached by other firms

Poaching hire(3)
The first term on the right-hand side illustrates the increase in the number of inventors
owing to poaching hires from other firms, while the second term represents a decrease in
the number of inventors as they are poached by other firms.
Given the above definition of µ̂z (ẑ, n, t) and µ̂n (ẑ, n, t), the KFE is presented as

Z z
 ∂ 

∂ ˆ
∂ 
ˆ
ˆ
ˆ
f (ẑ, n, t) = −µ̂n (ẑ, n, t) f (ẑ, n, t) −µ̂z (ẑ, n, t) f (ẑ, n, t) −η f (ẑ, n, t) + ηfˆ(ẑ, n, t)dẑδ̂(z)
∂t
∂n
∂ẑ
0
|
{z
} |
{z
}|
{z
}
N of inventor change

Productivity change

Leapfrog(4)
where δ̂(z) is the Dirac delta function, which is zero everywhere except ẑ = z where it isR
infinite and satisfies δ̂(z)dz = 1.

12

Technology FrontierHere, we argue that the technology frontier is finite, and we characterize its growth rate.
If z(0) < ∞, then z(t) will remain finite for all t. This is because it evolves from the firms’
productivity growth in the interval infinitesimally close to z(t), and the firm’s growth rate
is finite. Furthermore, the growth rate of the technology frontier is determined by the
productivity growth rate of firms that possess the highest growth rate among those at thetechnology frontier. This is because these firms will be at the technology frontier in thenext instant. The following lemma formally characterizes the productivity growth rate of
the technology frontier:Lemma 1. (Growth Rate of the Technology Frontier) If z(0) < ∞, then z(t) < ∞ ∀t < ∞and′µ̂z (z(t), n, t)
z (t)
=
max
g(t) ≡
z(t) n∈{n | fˆ(z(t),n,t)>0}z(t)Normalization
In the following, we examine economies in a BGP equilibrium, where the distribution
remains constant when appropriately scaled, and aggregate output experiences constant
growth. It is convenient to transform this system into a set of stationary equations forcomputing BGP equilibria. While we could standardize using any variable that grows at
the same rate as the aggregate economy, it is expedient to normalize variables relative tothe technology frontier z(t). Define the normalized values and functions as follows:z ≡ ẑ/z(t)Z(t) ≡ Ẑ(t)/z(t)Ω(z, n, t) = Ω(ẑ/z(t), n, t) ≡ Ω̂(ẑ, n, t)/z(t)(5)

F(z, n, t) = F(ẑ/z(t), n, t) ≡ F̂(ẑ, n, t)(6)

1P (z, n, z′ , n′ , t) = 1P (ẑ/z(t), n, ẑ′ /z(t), n′ , t) ≡ 1̂P (ẑ, n, ẑ′ , n′ , t)(7)v(z, n, z′ , n′ , t) = v(ẑ/z(t), n, ẑ/z(t), n′ , t) ≡ v̂(ẑ, n, ẑ′ , n′ , t)

(8)

The technology frontier is normalized to z(t)/z(t) = 1. The above normalizations make thevalue functions, productivity distributions, and growth rates stationary.
See the Figure 1 for an illustration of the original and detrended distributions. Noth13

Figure 1: Marginal Density for Productivity of Firms(a) Original(b) Detrendedfˆ(ẑ(t), ·)f (z, ·)

ggγ(n) + knowledge diffusion≡gγ(n) + knowledge diffusion − gẑ(t)at rate ηzz(t)at rate ηz≡1

Notes: Illustration of original and detrended marginal distribution for firms’ productivity on the BGP.

ing prevents the distribution from spreading without knowledge diffusion, driving the
productivity variance to infinity. However, because of knowledge diffusion, as the distribution extends, productivity growth due to knowledge diffusion increases, and theseforces compress the distribution.
Balanced Growth Path
Now, we describe a BGP equilibrium where aggregate productivity grows at a constantrate, and distributions are stationary. Define the growth rate of aggregate productivity to
be gZ (t) ≡ Ẑ′ (t)/Ẑ(t). That is, gZ (t) = gZ and F(z, n, t) = F(z, n) for all t. Aggregate output isgiven byZ
Ẑ(t) =

ẑdF̂(ẑ, n, t)Z=z(t) zdF(z, n, t)

On a BGP, the detrended productivity distribution is constant: F(z, n, t) = F(z, n). There′
fore, gZ = Ẑ′ (t)/Ẑ(t) = z (t)/z̄(t) = g, and we obtain the following lemma:

14Lemma 2. (Growth Rate of the Technology Frontier and Aggregate Productivity) On a
BGP, the aggregate productivity growth rate equals the technology frontier’s growth rate. That is,gZ = g.
The following definition summarizes the characteristics of our BGP equilibrium.
Definition 1. (Balanced Growth Path) A BGP equilibrium consists of: (i) a joint valuefunction Ω(z, n); (ii) a vacancy policy v(z, n); (iii) a stationary distribution of firms f (z, n);(iv) vacancy- and employment-weighted distributions fv (z, n) and fn (z, n); (v) poachingindicator function 1P (z, n, z′ , n′ ); (vi) the aggregate productivity Z and the total vacanciesv, and (vii) the economic growth rate g such that
1. The joint value Ω(z, n) satisfies the HJB equationρΩ(z, n) =z − c(v(z, n))ZZ
+ Av(z, n) [Ωn (z, n) + α(z′ /z)ZΩz (z, n) − Ωn (z′ , n′ )]+ dFn (z′ , n′ )
+ γ(n) − g zΩz (z, n)+ η [Ω(1, n) − Ω(z, n)]
2. The vacancy policy v(z, n) satisfies the first order conditionZ
cv (v(z, n))Z = A[Ωn (z, n) + α(z′ /z)ZΩz (z, n) − Ωn (z′ , n′ )]+ dFn (z′ , n′ )(9)

3. A density function f (z, n) satisfies the KFE equation ∂
∂µn (z, n) f (z, n) −µz (z, n) f (z, n) − η f (z, n) + η
0=−
∂n
∂z

Z 1
f (z′ , n)dz′ δ(1)0where the drift of the number of employed inventors µn (z, n) and productivity µz (z, n)
are given byZZnµn (z, n) ≡Av(z, n) 1P (z, n, z , n )dFn (z , n ) − Av
1P (z′ , n′ , z, n)dFv (z′ , n′ )nZµz (z, n) ≡ γ(n) − g z + Av(z, n)Z 1P (z, n, z′ , n′ )α(z′ /z)dFn (z′ , n′ )
′

′

′

15

′

4. Vacancy- and employment-weighted distributions are consistent:v(z, n) f (z, n)vn f (z, n)fn (z, n) =n
fv (z, n) =5. Poaching indicator function 1P (z, n, z′ , n′ ) is given by




1
′
′
1P (z, n, z , n ) = 0if Ωn (z, n) + α(z′ /z)ZΩz (z, n) > Ωn (z′ , n′ )otherwise

6. The aggregate productivity Z and the total vacancies v rate are given byZZ=

zdF(z, n)Zv=v(z, n)dF(z, n)

7. The inventor market clearing condition is satisfied:Zn=ndF(z, n)Appendix A comprehensively derivates the normalized system. We also establish
some properties of the joint value function in the Appendix. In it, we show that thefollowing properties hold: (i) Ω is increasing in productivity: Ωz > 0; (ii) Ω is increasingin the number of inventors: Ωn > 0.
The equilibrium of the model is solved numerically in Section 4. Before that, we turnto the description of the empirical results.

3

Data and Empirical Findings

In this section, we investigate the job flows of inventors — workers who have createdpatents — between establishments using inventor biography data from Germany. The
results provide motivation for our model, and we use these results to discipline thenumerical model, as explained in Section4.
16

3.1

Data

Our analyses utilize two administrative data sets, ”Linked Inventor Biography Data 1980–
2014” (INV-BIO) and ”Sample of Integrated Labor Market Biographies” (Stichprobe derIntegrierten Arbeitsmarktbiografien — SIAB).4
The INV-BIO data combines labor market biographies recorded in the German social
security data (Integrated Employment Biographies — IEB) with patent register data fromthe European Patent Office (EPO). This data set tracks information about 152,350 inventors
who have registered their patents to the EPO from 1980 to 2014. The information includes
their unique ID, age, gender, level of education, daily wage, and the number of citations
received by the patents associated with each inventor in the EPO’s records. The dataalso contains information about the establishments employing the inventors, such as theestablishment ID, the total number of their employees, and the mean daily wage of theirfull-time employees. The advantage over patent-based datasets used in previous studies(e.g., EPO patent data by Akcigit et al. (2018)) is that we can use social security information
to keep track of inventors’ flows even when they are not creating patents.
The SIAB data is a 2% random sample from IEB. This data set contains the same
information about individuals and their employing establishments as INV-BIO, except
for patent-related information. In the absence of the patent data, we identify inventors
in SIAB using a 3-digit occupation code, as described in Section 3.3. The data set covers
3,322,316 individuals from 1980 to 2019.
Since merging datasets is not allowed, we use the two datasets separately for eachanalysis: when comparing the movement patterns of inventors and other workers in
Section 3.3, we use SIAB, which includes both, but otherwise we use INVBIO, a dataset
focused exclusively on information about inventors.

3.2

Inventor Flows in INV-BIO

First, we adopt an approach similar to Haltiwanger et al. (2018) to characterize inventor
flows using INV-BIO. We assign each establishment to a percentile rank according topatent information or productivity measure. We then compute the transition probabilities
of inventor flows between these ranks.5
4More detailed information is presented in Online Appendix B.1.
Establishments could be classified into different percentiles based on the measure each year. The ranksof the origin and destination establishments are determined based on the measure from the previous year,
preceding the movement of inventors.
517

We utilize three different measures as proxies for the knowledge quality or productivity
level.6 The first measure is based on the forward citations for patents that establishments
have created. Measuring patent quality through forward citations is widely employedin the literature about patent creation (e.g., Pakes (1986); Hall et al. (2001); Akcigit et al.
(2018)). In particular, Akcigit et al. (2018) measures the idea quality of inventor teams
based on the number of forward citations their patent receives. Similarly, our measure foran establishment e in year t, zet is given by:
0P

citationset+j

zet =

j=−2

where citationset =X

3icitationsit ×

,

(10)

nie
.
ni

citationsit denotes the count of forward citations that occur five years after year t for patenti, which is created by a team including inventors employed at establishment e. Note thatthe team developing the patent can consist of inventors from different establishments. nie
represents the number of inventors at establishment e in the team, while ni represents thetotal number of inventors in the team, including those affiliated with different establishments. We multiply citationsit by nie /ni to adjust for the contribution made by inventors
from establishments other than e.7 Therefore, citationset is the count of five-year forward
citations for patents that establishment e created, adjusting for the contributions of otherestablishments. Following Akcigit et al. (2018), we use the three-year backward averageas the measure. The other measures are the number of employees (establishment size)
and the mean wage of full-time employees, following standard practice in the literature,
as summarized by Moscarini and Postel-Vinay (2018).8
Table 1 shows the transition probabilities of inventor flows from origin to destination.9
6

We assume that the knowledge quality and productivity level are positively correlated. In fact, thethree measures are positively correlated with each other as described in Online Appendix B.1.
7
In other words, we start by dividing the count of forward citations by the total number of inventors
involved in the team for each inventor’s patents. Afterward, we aggregate these values for the inventors
who are employed at establishment e.
8On-the-job search models with heterogeneous productivity firms (e.g., Postel-Vinay and Robin (2002))
predict that more productive firms offer higher wages and attract more workers, leading to their growth insize.
9Online Appendix B.1 shows the distribution of inventors according to each of the three measures.
It reveals a notable concentration of inventors within specific establishments. Irrespective of the type ofmeasure, more than half of the inventors are found in establishments ranked above the 80th percentile, andonly approximately 10 percent belong to establishments below the 50th percentile. This aligns with Akcigit

18

Table 1: Transition Probabilities of Inventor Flows(A) Rank by Citation/Inventor
Share of flows (%)

Originestablishment
rank

≤ 50%
50-60
60-70
70-80
80-100Destination establishment rank
≤ 50% 50-60 60-70 70-80 80-100
2.3
0.2
0.3
0.4
4.3
1.7
0.2
0.2
0.3
3.0
1.9
0.2
0.3
0.3
3.6
2.2
0.2
0.2
0.4
4.2
19.5
2.0
2.4
3.4
46.7

(B) Rank by Establishment Size
Share of flows (%)

Originestablishment
rank

≤ 50%
50-60
60-70
70-80
80-100Destination establishment rank
≤ 50% 50-60 60-70 70-80 80-100
2.6
0.9
0.7
0.8
6.3
0.4
0.5
0.6
0.4
2.3
0.5
0.2
0.7
0.9
3.0
0.6
0.3
0.4
1.3
4.8
5.8
2.5
3.4
4.7
55.7

Notes: Detailed description is presented below the panel (C) in the next page.

There is a substantial movement of inventors from higher ranks to lower ranks, denotedby the red-colored cells. The sum of values in these red cells amounts to 33.7% in panel(A), 18.8% in panel (B), and 28.5% in panel (C). This suggests that a large portion of inventors move from higher-ranked establishments to lower-ranked ones.10 Online Appendix
B.1 shows that this pattern is observable even when the sample is limited to job flows
accompanied by wage increases.
This pattern is not found in previous literature on worker flows. For example, Haltiwanger et al. (2018) construct transition probabilities of worker flows based on the mean
wage of firms, and they observe a higher probability of flows to higher ranks compared tolower ranks. This discrepancy suggests that the tendency for many flows to lower ranks
is a distinctive characteristic specific to inventors.
and Goldschlag (2023b)’s finding that inventors are concentrated in large incumbents in the U.S.
10
Another observable pattern is that the values along the diagonal are considerably high, particularly inthe bottom right of each panel: 49.9% in panel (A), 60.8% in panel (B), and 50.8% in panel (C). This indicatesthat many inventors tend to move within the same rank, especially within the highest rank. This can beobserved in the literature on worker flows (e.g., Haltiwanger et al. (2018)).

19

Table 2: Inventor Flows across Establishments (cont.)(C) Rank by Mean Wage
Share of flows (%)

Originestablishment
rank

≤ 50%
50-60
60-70
70-80
80-100Destination establishment rank
≤50% 50-60 60-70 70-80 80-100
2.9
1.1
1.0
1.1
3.5
0.8
0.9
1.1
0.9
2.3
1.0
0.9
2.0
2.1
4.1
1.4
1.0
1.7
3.6
7.3
5.5
3.6
5.4
7.2
37.6

Notes: This table shows transition probabilities of inventor flows across percentiles of establishments. The
inventors staying in the same establishment are excluded. The percentile rank in panel (A) is based on
the three-year backward average of forward patent citation counts. Panel (B) is based on the numberof employees, and panel (C) is based on the mean wage of full-time employees. Establishments could
be classified into different percentiles based on these measures each year. The ranks of the origin anddestination establishments are determined based on the measure from the previous year, preceding the
movement of inventors. The sample encompasses data from 1980 to 2014. The values in the table represent
the proportion of inventor flows in each cell in relation to the total flows in INV-BIO.

3.3

Inventor Flows in Comparison with Worker Flows

Next, we compare inventor flows with worker flows. We utilize the SIAB for the comparison since INV-BIO lacks information on workers other than inventors.
To identify inventors within SIAB, we use a 3-digit occupation code. We find thatthe majority of inventors in INV-BIO are affiliated with specific occupations, each with
their corresponding shares: research and development (20.2%), machine-building andoperations (19.8%), mathematics, biology, and physics (19.1%), and mechatronics, energy,
and electronics (18.8%). These four occupations account for nearly 80% of the inventors
in INV-BIO. We thus consider workers in these four occupations within SIAB to likely beinventors.
Table 2 presents a comparison of summary statistics between the two data sets. The
mean daily wage of workers in the four occupations (identified inventors) in SIAB falls
between that of workers in SIAB and that of inventors in INV-BIO. Furthermore, theproportion of female workers among the identified inventors lies between the two groups.
These findings suggest that our identified inventors include both actual inventors and a
portion of non-inventor workers. Therefore, the result of the subsequent comparison
between workers and identified inventors should be considered conservative due to thepresence of attenuation bias.

20

Table 2: Identified Inventors in SIAB and Inventors in INV-BIO
Summary statistics(1980 - 2014)
Daily wage, Euro Mean
S.D.
Age
Mean
S.D.
Females, %N of obs., thousand

Workers
59.0
47.2
38.7
12.9
45.2
21,344

SIAB
Identified inventors
78.9
52.1
38.4
12.4
14.8
2,871

INV-BIO
Inventors
156.2
30.0
42.4
9.0
5.7
420

Notes: This table compares the summary statistics between workers in SIAB and the inventors in INV-BIO.
Identified inventors in SIAB are workers who work in the following four occupations: ”research and development”, ”machine-building and operations”, ”mathematics, biology, and physics”, and ”mechatronics,
energy, and electronics.” The worker in the table includes the identified inventors. The summary statistics
are calculated using a pooled sample with daily wage, age, and gender filled in.

We estimate the following Probit model:P(Dit = 1) = Φ(β0 + β1 Iit + β2 Xit )(11)

Individual i is the job changer without an unemployment spell. Iit serves as a dummy forthe inventors, taking a value of one if individual i works in one of the four occupations inyear t, and zero otherwise. Dit equals one if individual i moves from a more productive
establishment to a less productive one in year t, and zero if the move is to a more productive
establishment. Note that we investigate the moving between establishments rather thanranks here. For constructing Dit , we use the number of employees or mean wage as a proxyfor productivity. The vector of control variables, Xit , includes age, a square of age, gender,and educational attainment. To avoid the incidental parameter problem, we estimate themodel without incorporating fixed effects.11 The function Φ is the cumulative distributionfunction of the standard normal distribution.
The coefficient of our interest is β1 . The positive β1 implies that inventors are more
likely to move to less productive establishments than other workers. Standard errors (SEs)are clustered by destination establishment and year, supposing the presence of persistentestablishment and year specific shocks.
Table 3 presents results. The first column uses the establishment size as the productivity
measure, and the second column uses the mean wage as the measure. The results show
The estimated value of β1 in the linear model with the fixed effects is also significantly positive as in
Table 3. See Online Appendix B.2.
11

21

Table 3: Estimation Result for Inventor Flows(1) P(Dit = 1)
Whole sample
Sample with wage ↑.077***.036***.052***.012***(.004)(.004)(.004)(.004)

Iit(2) ∆ log wit.017***(.005).021***(.004)

Dit-.078***(.006)-.084***(.005)

Dit × Iit-.002(.006)√√
Mean wage
859,861.13

Control
Fixed Effects
Measure for Dit
N
Adj. R2

√

√

√

√

.016***(.006)√√

Size
3,572,567.019

Mean wage
3,533,344
.016

Size
2,082,939.005

Mean wage
2,060,714
.003

Size
859,888.13

Notes: Control variables include age, a square of age, gender, and educational attainment. Fixed effects
include year, year × industry, and destination establishment fixed effects. Iit equals to one if individual iworks in one of the four occupations (”research and development”, ”machine-building and operations”,”mathematics, biology, and physics”, and ”mechatronics, energy, and electronics”) in year t, and zerootherwise. Dit equals one if individual i moves to a less productive establishment in year t, and zerootherwise. The productivity measure is based on the establishment size or the mean wage in year t − 1. The
sample spans from 1980 to 2019. SEs clustered by year and establishments are reported in parentheses. *p < 0.1, ** p < 0.05, *** p < 0.01.

that the probability of an inventor transitioning to an establishment with fewer employees
or a lower mean wage is higher than other workers, implying that inventors are more
likely to move to a less productive establishment than other workers. Moving to the thirdand fourth columns, we narrow down the sample to job changers who experience wage
increases, and the coefficients for Iit are still significantly positive. This suggests that many
inventors move to less productive establishments and experience wage increases.
To further examine the association between the direction of flows and wages, we run
the following regression:
log wit − log wit−1 = β0 + β1 Dit + β2 Iit + β3 Dit Iit + β4 Xit + α + εit(12)

The variable wit represents the daily wage of individual i after a job change, while wit−1
represents the wage before the job change. The vector of fixed effects α includes year, year
× industry, and destination establishment fixed effects. The definition of other variables

22remains the same as in the equation (11).
The last two columns of Table 3 show the estimation results. The coefficients for Iet
are significantly positive, indicating that inventors experience greater wage increases byaround 2% through job changes than other general workers.
The coefficients of Dit are negative, meaning that workers tend to experience fewer
wage increases when moving to less productive establishments. However, the coefficientsfor Dit ×Iit are significantly positive in the fifth column. The positive coefficient implies thatinventors experience fewer wage decreases by moving to less productive establishments
than general workers.
Knowledge transfer with inventor mobility has the potential to explain these results.
That is, an inventor who worked in a high-productivity establishment can transfer thatknowledge to a low-productivity establishment when changing jobs. Therefore, establishments are more willing to poach inventors from more productive establishments thanother general workers and compensate inventors for the benefits.

3.4

Empirical Evidence of Knowledge Diffusion

The result in the previous section suggests the presence of knowledge diffusion via inventor flows. This section further investigates how the inventor flows influence theproductivity growth of establishments.
Our specification is given by:log zet+j − log zet = β0 + β1 H-Shareet + β2 Xet + αe + αt + εet(13)

zet is the knowledge quality of establishment e in year t, as defined in (10) of Section 3.2.
We use three, four, or five-year forward citations for zet . The variable H-Shareet represents
the percentage share of inventor inflows from establishments with higher knowledge basemeasured by patent catitions, to total inventor inflows to establishment e. log net is thelog of the number of inventors. The vector of control variables Xet includes the log ofthe establishment size, number of inventors, mean wage, and zet . The vector of fixed
effects α includes year, year × industry, and establishment fixed effects. The equation(13) is estimated using INV-BIO from 1980 to 2019. Standard errors (SEs) are clustered bydestination establishment and year.
The results are reported in Table 4. The table shows that when more inventors comefrom productive establishments, the knowledge growth of the poaching establishments is23

Table 4: Estimation Result for Knowledge Growth

H-Shareet (%)

Control
Fixed Effects
CitationN
Adj R2j=3.0022***(.0004)log zet+ j − log zet
j=4
j=5
j=3
.0028*** .0027*** .0024***(.0004)(.0004)(.0003)j=3.0024***(.0003)

√
√

√
√

√
√

√
√

√
√

3y fwd24,625.21

3y fwd22,270.27

3y fwd19,982.36

4y fwd26,791.23

5y fwd26,451.25

Notes: Control variables are zet , log of a number of employees and mean wage. Fixed effects include year,
year × industry, and establishment fixed effects. The sample spans from 1980 to 2014. zet is the forward
citation measure (backward 3-year moving average). H-Shareet is the share of the number of inventors who
moved from establishments with a higher ze′ t−1 at t − 1. If there are no inflows, H-Shareet is set to zero. SEsclustered by year and establishment are reported in parenthesis. * p < 0.1, ** p < 0.05, *** p < 0.01, .

higher over a period of three, four, or five years.
However, the coefficient of H-Shareet is susceptible to the endogeneity problem. The
unobservable expectation for log zet+j − log zet can be correlated with the realized log zet+j −log zet and H-Shareet . To address this issue of omitted variable bias12 , we employ an
instrumental variable (IV) strategy. In this approach, we utilize the patent citation rankfor establishments in their states from the previous year (referred to as Regional Ranket−1 )as an instrument for H-Shareet . In the first stage, the Regional Ranket−1 is expected to behighly correlated with H-Shareet . A lower knowledge rank indicates a higher numberof establishments with greater knowledge level in the state. Consequently, the share of
inventors poached from these highly knowledgeable establishments (H-Shareet ) is more
likely to be higher. The instrument can be considered to satisfy the exclusion condition
when taking into account the fixed effects and control variables. To address the issue ofmean reversion in knowledge quality, we add zet−1 as one of the control variables.
Table 5 shows the result using the IV. In the first stage, we find a significant correlation
between Regional Ranket−1 and H-Shareet . Specifically, if the knowledge level is relatively
lower within the state (indicated by a higher value of Regional Ranket−1 ), H-Shareet tends to
12This omitted variable bias can have either an upward or a downward effect. If there is an expectationof high productivity growth, establishments may offer higher wages to attract more inventors from highly
productive establishments, leading to an upward bias. On the other hand, if lower productivity growthis anticipated, establishments may attempt to offset the lower growth by poaching inventors from moreproductive establishments, resulting in a downward bias.

24

Table 5: Estimation Result for Knowledge Growth using IV

H-Shareet (%)

j=3.092***(.007)

log zet+ j − log zet
j=4
j=5
j=3
.101*** .103*** .084***(.007)(.009)(.005)

j=3.088***(.006)

Control
Fixed Effects
CitationFirst Stage IV
Regional Ranket−1

√
√

√
√

√
√

√
√

√
√

3y fwd3y fwd3y fwd4y fwd5y fwd24.4***(1.7)

22.7***(1.5)

22.8***(1.8)

30.0***(1.7)

29.3***(1.7)

N
F statistic

22,213204.8

20,052232.6

17,996155.2

23,137302.7

23,609286.2

Notes: Control variables are the log of the establishment size, number of inventors, mean wage, zet , andzet−1 . The sample spans from 1980 to 2014. zet is the the forward citation measure at t. H-Shareet is the shareof the number of inventors who moved from establishments with a higher z at t − 1. Regional Ranket−1 is theestablishment’s rank of zet−1 among all establishments in the state (16 states). A higher rank means lower
zet−1 . The rank is normalized so that the maximum is equal to 1. SEs clustered by year and establishment
are reported in parenthesis. *** p < 0.01, *** p < 0.05. The first stage F value is the Cragg-Donald Wald F
statistic.

be higher.
The impact of poaching from more knowledgeable establishments is highly significant
and even larger than the results obtained from the OLS in Table 4. In the first column
in Table 5 using the three-year forward citations, 1% increase in the share of inventors
from more productive establishments increases citations by around 10% relative to theunconditional mean. In sum, our results suggest that establishments can enhance theirknowledge growth by recruiting inventors from high-productivity establishments.

4

Quantitative Analysis

This section quantifies the effects of inventor job flows and knowledge transfers on innovation and productivity and studies the effects of counterfactual policy. To do this, wecalibrate the model from Section 2 to match the data desrcibed in Section 3. We subsequently demonstrate that the calibrated model closely fits the data for targeted moments,
and we use it to examine the effects of policies related to the labor market for inventors.
25

4.1

Stochastic Process for In-House R&D Ability and Functional Forms

In this section, we introduce a stochastic process to characterize the unpredictable natureof in-house R&D ability, following the model presented in Benhabib et al. (2021). We
set the functional form of the in-house R&D function to γ(n, i) = γi nδ , where the index irepresents the in-house R&D ability, which can either be high (h) or low (l). The innovation
ability i follows a two-state Markov process. The R&D capacity is greater when in theh state than in the l state (γh > γl ). The transition intensity (the rate at which the R&D
ability changes) from the l to h state is denoted by λl , and the transition intensity from theh to l state is denoted by λh . We employ this two-state Markov process to formulate thestochastic innovation process because it allows the composition of firms at the technology
frontier to change over time, which is essential for the existence of a stationary distribution
on a BGP.
We will conduct our numerical exercises by calibrating the model with high transition
rates. The characteristics of the stochastic process with conditional draws are similar to
those with unconditional draws when the switching rates are high, as estimated in ourcalibrations.
cvϕ+1 .
For other functional forms, we assume that the vacancy cost function is c(v) = ϕ+1
The knowledge diffusion rate function is α(z′ /z) = α (z′ /z)β such that the knowledge
diffusion rate is increasing in the productivity of the poached firm relative to the poaching
firm.

4.2

Calibration

We calibrate the model along a BGP equilibrium to match features of the allocation of
inventors across establishments and characteristics of inventor job flows between establishments.
Externally Set or Normalized
We normalize or set to standard values six parameters, as summarized in the Panel A of
Table 6. The discount rate ρ implies an annual real interest rate of 5%. The first-ordercondition for vacancies implies that we cannot identify c and A separately, so we normalizec. We normalize the productivity of technology frontier z and the measure of firms m to
1 without loss of generality. We also set l-type R&D coefficient γl to zero without loss of26

Table 6: Parameter Values
Parameter
Description
Value— Panel A. Externally Set or Normalized —ρDiscount Rate
0.0041z
Frontier Productivity
1mMeasure of Firms
1cVacancy Cost Coefficient100
ϕVacancy Cost Elasticity
3.45
γll-type R&D Coefficient0— Panel B. Direct Match to Data —nMeasure of Inventors
5λhJump Intensity: h → l
0.02
λl
Jump Intensity: l → h
0.01— Panel C. SMM Calibration —β
Diffusion Curvature
0.33
αDiffusion Rate
0.0012γhh-type R&D Coefficient 0.0006
ηLeapfrog
0.0001
δR&D Curvature
0.35
A
Matching Efficiency
0.26
Notes: List of model parameters and calibrated values. In Panel C, all parameters are calibrated jointly for
the SMM calibration.

generality13 . We use the value of the vacancy cost elasticity calibrated by Bilal et al. (2022).
Direct Match to Data
We set three parameters to directly match the moments from German inventor data, assummarized in Panel B of Table 6. The measure of inventor n is determined by the average
number of inventors per establishment, given a unit measure of firms.
The transition rate of innovation ability λh and λl match the estimations from the twostate Markov transition matrix for the growth rate of the knowledge in establishments
13

See Benhabib et al. (2021)

27

near the technology frontier in the spirit of Benhabib et al. (2021). Details are as follows.
The knowledge in establishments is measured by patent citation, and we designate establishments within the top 10% of the 5-year forward citation measure, Zet in Section 3.2, as
the frontier each year. Among these, establishments exhibiting positive growth rates of
the citations are categorized as being in the high state, whereas the remaining are seen as
being in the low state. Based on this, we estimate the transition matrix.
Internal Calibration Using SMM
We estimate the six key parameters of the model listed in the Panel C of Table 6. Theseonparameters are captured by the vector: Θ = β, α, γh , η, δ, A and estimated by minimizing
the objective function
L (Θ) = (m̂ − m (Θ))′ W −1 (m̂ − m (Θ))
where m̂ is a vector of empirical moments and m (Θ) are their model counterparts. The
diagonal components of matrix W have the same weights. All non-diagonal components
are zero. In the case of distributional information, the weights are adjusted so thatthe weights of the entire distribution add up to one. For example, the unconditional
distribution of inventors is characterized by five quantile points and therefore weights
1/5. All non-diagonal components are zero. 14

4.3

Results
Table 7: Targeted Moments
Moments
Data Model
EE rate (%, monthly)
1.17
1.13
Growth rate (%, monthly)
0.16
0.13
Distribution of inventor by firm ranking
Figure 2aDistribution of inventor flow by poaching firm ranking
Figure 2b

Notes: EE rate and growth rate are both monthly frequencies; EE rate is calculated by dividing the numberof job changers by the number of inventors in the INV-BIO data.
14
Prior to estimation, we did the first 100 iterations for each of the six parameters and excluded regionsthat did not converge. Then the parameter space we explore is as follows: β ranges from 0.1 to 0.7, α ranges
from 0.001 to 0.003, γh ranges from 0.0001 to 0.001, η ranges from 0.00005 to 0.0003, δ ranges from 0.2 to
0.5, and A ranges from 0.05 to 0.3. For each of these parameter spaces, we take 20 grids and compute 2000
Halton grids.

28

Figure 2: Inventor Distributions by Firm Productivity(a) Density Weighted by the Mass of Inventors(b) Density Weighted by Inventor Inflows

0.80

0.70
Model
Data

0.60

0.60

Model
Data

0.50
0.40

0.40
0.30
0.20

0.20

0.10
0.00
< 50%

50 to 60

60 to 70

70 to 80

0.00
< 50%

80 to 100

50 to 60

60 to 70

70 to 80

80 to 100

Notes: The data in (a) and (b) are plots of the distribution in Table B.3. (A) and the marginal distribution ofpoaching firms in Table 1 (B). The corresponding model values are calculated using the productivity of thefirms, the inventor and their joint density f (ẑ, n, t) under the calibrated parameters.

Table 7 and Figure 2 summarize the target moments and parameters. Not only do themacro moments (EE rate and growth rate) in Table 7 provide a good fit, but Figure 2 also
shows that the joint distribution of inventors and firms is well replicated. Overall, despite
over-identification, the fit of the moments within the internal calibration is reasonably
good.
Although the parameters are calibrated jointly, we will discuss the most relevant moments for each parameter. First, the EE rate primarily provides information about thematching efficiency A, which governs the size of the job flow. The growth rate has information mainly on γh . Both α and β are related to knowledge diffusion, where β controls
for the sensitivity of knowledge diffusion to the difference in productivity between poaching firms and incumbents, and α adjusts the average size of knowledge diffusion. These
parameters predominantly determine poaching firms’ distribution over productivity in
Figure 2b. Finally, Figure 2a, representing the relationship between productivity andinventors, primarily informs δ and η.
Next, we discuss the fit of the calibrated model for a important non-targeted regression
result. In our empirical analysis, using equation (13), we find that the greater the shareof inventor inflows from higher knowledge firms, the greater the productivity gains of
the poaching firms. We examine whether the model can replicate this relationship. The
29

corresponding equation of the model is given byµ̂z (ẑ, n, t) = β0 + β1 H-Share + β2 log n + ε
where µ̂z (ẑ, n, t) is the growth rate of productivity and H-Share is the fraction of µ̂n (ẑ, n, t)
who transitioned from firms with higher productivity. We compute the coefficients using
weighted least squares, with the density function of the firms f (ẑ, n, t) in each grid as the
sample. The model-implied coefficient β1 is 0.003, well within the range of the OLS and IV
estimates in Table 4, thereby successfully producing a reasonable quantitative magnitude.

4.4

Quantitative Exercises

The previous section demonstrated that the calibrated model accurately aligns with thedata for both targeted and non-targeted moments. Consequently, the model is well-suitedfor conducting counterfactual analyses. Initially, we will assess the impact of changes
in matching efficiency in frictional labor markets for inventors. Subsequently, we will
investigate the ramifications of a hypothetical policy intervention. In this policy analysis,
we suggest a hypothetical policy that offers subsidies to frontier firms, aiming to foster
innovation within these entities.
Quantifying the Impact of Inventor Market Frictions
How do changes in frictions within the market for inventors impact the economy? Frictions in this market can emerge from a variety of sources, such as search and matching
frictions, regulations related to labor, and agreements between employees and firms. To
assess the impact of shifts in market frictions for inventors, we conduct a comparative
static analysis focusing on the matching efficiency parameter A.
As Figure 3 (a) shows, in our model, an increase in matching efficiency increases thevacancy posting of firms. Also, as Figure 3 (b) shows, higher matching efficiency andmore vacancy postings lead to higher job-to-job transition rates for the inventor.
As matching efficiency increases, the economic growth rate increases (Figure 3 (c))
through the following mechanism. First, as the job-to-job transition rate increases, knowledge diffusion becomes more active. As a result, the dispersion of firm productivity
decreases (Figure 4 (b)). Note that high-productivity firms grow mainly through in-house
R&D, while low-productivity firms grow mainly through knowledge diffusion (See equation (2)). Since lower variance in firm productivity reduces growth through knowledge30

Figure 3: Comparative Statics: Aggregate Variables(a) Aggregate Vacancy(b) EE Rate

0.098

1.3

0.097

1.2

0.096

1.1
0.095

1.0
0.094

0.9
0.093
0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

0.20

Matching Efficiency A

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

Matching Efficiency A(c) Economic Growth Rate
0.134
0.132
0.130
0.128
0.126
0.124
0.122
0.120
0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

Matching Efficiency A

Notes: The figure display the comparative statics of varying matching efficiency A.

diffusion, it leads to a relative increase in inventor hiring by more productive firms. As
a result, more inventors are attracted to firms in the technology frontier. As shown inLemma 2, the economic growth rate in a BGP is determined by the productivity growthrate of frontier firms. Therefore, the economic growth rate also increases (Figure 3 (c)).
Our model links the observed decrease in inventor mobility in Germany and the US
to the low economic growth in developed countries in recent years, as documented in
the secular stagnation literature (e.g., Summers (2014); Eggertsson et al. (2019); Akcigit
and Ates (2021)). In the INV-BIO data, we find a significant decline in the inventor job
31

Figure 4: Comparative Statics: Distribution(a) St.d. of log n(b) St.d. of log z

1.190

2.21

1.180

2.20

1.170

2.19

1.160

2.18

1.150

2.17

1.140

2.16

1.130

2.15

1.120
0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

Matching Efficiency A

2.14
0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

Matching Efficiency A

Notes: The figure display the comparative statics of varying matching efficiency A.

Table 8: The Change in the Second-Order Moments of the Distribution in the Data
1998
Coefficient of variation of N of inventors 0.93
Coefficient of variation of productivity
2.93

2014
1.12
0.27

Notes: The first column shows the change in the coefficient of variation of the number of inventors working
in each establishment in the INV-BIO data. We compare the values for 1998, the first year of Figure 5, and2014, the last year. Similarly, the second column shows the change in the coefficient of variation of theestablishments’ innovativeness measured by patent citation.

flow rate over time (Figure 5). Similarly, Akcigit and Goldschlag (2023a) document thatinventor mobility has decreased since the early 2000s in the US. Our model indicates thatthe decline in the matching efficiency, which subsequently leads to a reduction in job-tojob transitions, contributes to a lower economic growth rate. Thus, our model provides
a framework linking the observed decrease in inventor mobility to the slowdown inaggregate productivity growth in developed countries over the past few decades.
In our model, changes in the distribution of firms and inventors play an essential rolein the mechanism linking inventor mobility and economic growth rates, and indeed, thechanges in the distributional characteristics of the model are consistent with the data. The
first row of Table 8 shows the coefficient of variation of the number of inventors working
in each establishment. Here, we compare the values for 1998, the first year of Figure 5,
32

Figure 5: Inventor Job Flow Rate in Germany
60

Inventor Job Flow Rate (%)

50

40

30

20

10

0
1998

2000

2002

2004

2006

2008

2010

2012

2014

Year

Notes: This figure shows the job flow rate for German inventors, calculated using INV-BIO data.

and 2014, the last year. Similarly, the second row shows the coefficient of variation of
the establishments’ innovativeness measured by patent citation. The direction of thesechanges in the data is consistent with the direction of change in the variance of z and thevariance of n in our model (Figure 4) when the matching efficiency A decreases.
Policy Exercises: Subsidy for Firms Near the Technology FrontierThis section analyzes the consequences of subsidies for firms near the technology frontier.
Policy-makers may consider encouraging R&D activity by offering subsidies to technologically progressive firms. Alternatively, they may wish to subsidize technologicallylagging firms to promote knowledge diffusion through the movement of inventors. To
find the more beneficial policy, we examine the transition from an initial BGP without
subsidy to a new BGP with a subsidy aimed at technologically frontier firms.
We investigate the transition from an initial BGP with no subsidy to a new BGP witha subsidy rate 10% for frontier firms. We define frontier firms as those with productivity
z in the top half of the distribution, weighted by the number of inventors15 . We assume
subsidies are financed by a constant rate tax on the bottom half of the distribution. This
exercise imposes a 17% tax on the production of these remaining firms. Subsidies and15

In the distribution weighted by the number of inventors, the firms with productivity z in the top halfof the distribution correspond to approximately the top 10% of firms with productivity in the unweighted
distribution.

33

Figure 6: Transition Dynamics(a) Output(b) Consumption
0.05

with Subsidy/without Subsidy (%)

with Subsidy/without Subsidy (%)

0.05

0.00

−0.05

−0.10

−0.15

−0.20

−0.25

0

10

20

30

40

50

60

70

80

90

100

0.00
−0.05

−0.10

−0.15

−0.20

−0.25

−0.30

0

10

20

30

40

50

60

70

80

90

100

Year

Year

Notes: The figures display transitional dynamics on implementing a counterfactual subsidy for technologically frontier firms. Panel (a) shows the path for aggregate output relative to the old aggregate output.
Panel (b) shows the path for aggregate consumption.

taxation changes are permanent. The agents do not anticipate the policy change until
t = 0, and they are perfect foresight after t = 1.
The left panel of Figure 6 displays the output path relative to the baseline balanced
growth path (BGP). Aggregate output decreases during the initial 40 years, but shows anincrease in the long run. In the short run, this policy hinders job flows from technologyleading firms to laggard firms, thereby impeding knowledge diffusion to the latter. However, over the long term, it enhances the growth rate of the technology frontier, which ultimately fosters positive impacts on knowledge diffusion to laggard firms. Consequently,
the rate of economic growth increases in the long term.
Lastly, we calculate the policy change’s welfare effects by analyzing the economy’stransitional dynamics, applying a discount rate ρ to future periods following policy implementation. Our results indicate a decline in welfare of 0.14%, measured in terms of
consumption equivalent. This outcome is primarily driven by a short-term decline inaggregate productivity. However, this effect is largely offset by a long-term increase inproductivity. Our analysis suggests that the effectiveness of a policy is dependent on thepolicymaker’s time horizon.

34

5Conclusion

This paper explores labor markets where inventors and firms interact, focusing on theimplications for the distribution of inventors across firms, knowledge diffusion, and productivity growth. To examine these dynamics, we construct an endogenous growthmodel that incorporates frictions in labor markets for inventors. In our model, inventors(i) contribute to in-house R&D efforts, enhancing the firm’s technology, and (ii) facilitate
knowledge transfer from their previous employers to their new ones when they change
jobs. Heterogeneous firms create job openings, considering the knowledge transfer fromthe inventors’ prior employers. To quantify this framework, we utilize data on inventors
and patents connected to administrative labor market career information about individuals and their employing establishments in Germany. We find three empirical findings:(i) inventors are more likely to transition to less productive establishments compared togeneral workers, (ii) inventors face significant increases in wages when changing jobs
compared to general workers, and (iii) the number of patent citations increases more
rapidly when a higher proportion of their inventors originates from higher productivity establishments. We calibrate our model to align with these empirical findings anddemonstrate that the calibrated model closely fits both target and non-target moments,
confirming its suitability for conducting counterfactual exercises. We then examine thetransition from an initial Balanced Growth Path (BGP) without a subsidy to a new BGP
with a subsidy targeting technologically frontier firms. In the short term, this subsidy decreases aggregate output by discouraging inventor mobility from frontier firms to laggard
firms, thereby hindering knowledge diffusion. However, in the long term, the impactof the subsidy on output is reversed, as it enhances the growth rate of the technologicalfrontier.

35

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38

Appendix
A

Theoretical Derivations

A.1

Normalizing the Distribution

Differentiate (6) with respect to ẑ and n and use the definition z ≡ ẑ/z(t),
∂ ∂
1 ∂ ∂
F(ẑ/z(t), n, t) =F̂(ẑ, n, t)∂ẑ ∂nz(t) ∂z ∂nf (z, n, t) = z(t) fˆ(ẑ, n, t)(14)

Differentiate (14) with respect to t and use the chain rule to obtain the transformation of
the time derivative:′∂ẑ z (t) ∂∂′f (ẑ/z(t), n, t) −f (ẑ/z(t), n, t) = z (t) fˆ(ẑ, n, t) + z(t) fˆ(ẑ, n, t).
∂t∂tz(t) z(t) ∂z′

Define the growth rates of the technology frontier as g(t) ≡ z (t)/z(t). Use the definition ofg(t) and the definition z ≡ ẑ/z(t),∂∂∂f (z, n, t) − zg(t) f (z, n, t) = g(t)z(t) fˆ(ẑ, n, t) + z(t) fˆ(ẑ, n, t)∂t∂z∂t
Use (14),∂∂∂f (z, n, t) − zg(t) f (z, n, t) − g(t) f (z, n, t) = z(t) fˆ(ẑ, n, t).
∂t∂z∂t
Next, we derive the law of motion for firm-level detrended productivity. Note that
d
d ẑ
z=
dt
dt z(t)d!
′ẑẑ z (t)= dt −
z(t) z(t) z(t)dẑ= dt − g(t)z
z(t)

A1

(15)

Therefore, the drift of detrended productivity isµz (z, n, t) =µ̂z (z, n, t)− g(t)zz(t)(16)

Use (2), and then, use (6), (7) , and (8)16 ,Zµz (z, n, t) = γ(n) − g(t) z + Av(z, n, t)Z(t)1P (z, n, z′ , n′ , t)α(z′ /z)dFn (z′ , n′ , t).

Similarly, we derive the law of motion for firm-level employment growth, which is afunction of detrended variables. Substitute (6), (7), and (8) into (3),µn (z, n, t) ≡µ̂n (ẑ, n, t)
Z=Av(z, n, t)n
1P (z, n, z , n , t)dFn (z , n , t) − Av
n
′

′

′

′

Z
1P (x, n, x′ , n′ , t)dFv (z′ , n′ , t).
(17)Multiply (4) by z(t) ,
z(t)

 ∂ 

∂ ˆ
∂ 
µ̂n (ẑ, n, t)z(t) fˆ(ẑ, n, t) −µ̂z (ẑ, n, t)z(t) fˆ(ẑ, n, t)f (ẑ, n, t) = −
∂t
∂n
∂ẑ
Z z
− ηz(t) fˆ(ẑ, n, t) + ηz(t) fˆ(ẑ, n, t)dẑδ̂(z)0Use (14), (16), and (17),17
 ∂


∂
∂
∂
f (z, n, t) =zg(t) f (z, n, t) + g(t) f (z, n, t) −µn (z, n, t) f (z, n, t) −µz (z, n, t) − g(t)z f (z, n, t)
∂t
∂z
∂n
∂z
Z 1
− η f (z, n, t) + η
f (z, n, t)dzδ(1)016h and ĥ such thatof variables,
R R For any set of functions
R R
R Rh(z, n) = ĥ(ẑ, n), using changeRˆh(z, n)z f (z, n)(1/z)dzdn =h(z, n) f (z, n)dzdn. Therefore, ĥ(ẑ, n)dF̂(ẑ, n) =R ĥ(ẑ, n) f (ẑ, n)dẑdn =h(z, n)dF(z, n).
∂
∂
∂
1
17
For any set of functions h and ĥ such that h(z, n) = ĥ(ẑ, n), ∂ẑĥ(ẑ, n) = ∂zh(z, n) dz
dẑ = ∂z h(z, n) · z(t) .

A2


∂
∂
Because ∂zzg(t) f (z, n, t) = zg(t) ∂zf (z, n, t) + g(t) f (z, n, t), we get
 ∂

∂
∂
f (z, n, t) = −µn (z, n, t) f (z, n, t) −µz (z, n, t) f (z, n, t) −η f (z, n, t)+η
∂t
∂n
∂z

A.2

Z 1
f (z, n, t)dzδ(1)0Normalizing the Value Function

Differentiate (5) with respect to ẑ and rearrange,∂∂Ω(z, n, t) = Ω̂(ẑ, n, t).
∂z
∂z

(18)

Differentiate (5) with respect to n and rearrange,∂∂Ω(z, n, t) =Ω̂(ẑ, n, t)/z(t).
∂n
∂n

(19)

Rearrange and differentiate (5) with respect to t,′z (t) ∂∂∂′Ω̂(ẑ, n, t) = z (t)Ω(ẑ/z(t), n, t) − ẑΩ(ẑ/z(t), n, t) + z(t) Ω(ẑ/z(t), n, t).
∂t∂tz(t) ∂z′

Divided by z(t) and use the definition of g(t) ≡ z (t)/z(t) and the definition z ≡ ẑ/z(t) ,1 ∂∂∂Ω̂(ẑ, n, t) = g(t)Ω(z, n, t) − g(t)z Ω(z, n, t) + Ω(z, n, t).
∂z∂tz(t) ∂t(20)

Divide (1) by z(t) and then substitute (18), (19), and (20), and use (6),∂r(t) − g(t) Ω(z, n, t) + Ω(z, n, t) = max z − c(v)Z(t)v≥0∂tZ+ Av [Ωn (z, n, t) + α(z′ /z)Z(t)Ωz (z, n, t) − Ωn (z′ , n′ , t)]+ dFn (z′ , n′ , t)
+ γ(n) − g(t) zΩz (z, n, t)+ η [Ω(1, n, t) − Ω(z, n, t)]A3

Use r(t) = ρ + g(t),ρΩ(z, n, t) +∂Ω(z, n, t) = max z − c(v)Z(t)
v≥0
∂t
Z
+ Av

[Ωn (z, n, t) + α(z′ /z)Z(t)Ωz (z, n, t) − Ωn (z′ , n′ , t)]+ dFn (z′ , n′ , t)
+ γ(n) − g(t) zΩz (z, n, t)+ η [Ω(1, n, t) − Ω(z, n, t)]

A.3 Ωz (z, n) > 0
Rewrite the problem in terms of x = log z. Denote with an abuse of notation, Ω(x, n) =Ω(ez , n) and v(x, n) = v(ez , n). Also, denote α̂(x, x′ ) = α(exp(x′ − x))/ exp(x). Then, the HJB
equation becomesρΩ(x, n) = exp(x) − c(v(x, n))ZZ
+ Av(x, n) [Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ )]+ dFn (x′ , n′ )(21)

+ γ(n) − g Ωx (x, n)+ η [Ω(0, n) − Ω(x, n)] .
Denote ζ(x, n) = Ωx (x, n). Differentiate the Bellman equation (21) w.r.t. x and use theenvelope theorem,
ρζ(x, n) = exp(x)∂Av(x, n)∂xZ[Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ )]+ dFn (x′ , n′ )
+ γ(n) − g ζx (x, n)− ηζ(x, n).
Let define the poaching indicator function as




1
′
′
1P (x, n, x , n ) = 0if Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ ) > 0
otherwise

A4

(22)

Then,∂∂xZ[Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ )]+ dFn (x′ , n′ )Z∂[Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ )]+ dFn (x′ , n′ )
=
∂x
Z
∂
=
1P (x, n, x′ , n′ ) [Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ )] dFn (x′ , n′ )∂x
Z=
1P (x, n, x′ , n′ ) [ζn (x, n) + α̂x (x, x′ )Zζx (x, n) + α̂(x, x′ )Zζx (x, n)] dFn (x′ , n′ )The second equality follows because for any differentiable function f (x),f ′ (x)



+ 
∂ 

f (x) = not differentialble


∂x



0

if

f (x) > 0

if

f (x) = 0

if

f (x) < 0

+
+ 



∂ 
∂ 
and at f (x) = 0, the derivative ∂xf (x) is bounded by ∂xf (x) ∈ min 0, f ′ (x) , max 0, f ′ (x) ,and the measure of (x′ , n′ ) that satisfies Ωn (x, n) + α̂(x, x′ )ZΩx (x, n) − Ωn (x′ , n′ ) = 0 is zerofor any (x, n). Therefore, (22) is rewritten asρζ(x, n) = exp(x)Z+ Av(x, n)1P (x, n, x′ , n′ )dFn (x′ , n′ )ζn (x, n)Z+ Av(x, n)Z 1P (x, n, x′ , n′ )α̂x (x, x′ )dFn (x′ , n′ )ζ(x, n)Z+ Av(x, n)Z 1P (x, n, x′ , n′ ))α̂(x, x′ )dFn (x′ , n′ )ζx (x, n)+ γ(n) − g ζx (x, n)− ηζ(x, n)Zρ + η − Av(x, n)Z!
1P (x, n, x , n )α̂x (x, x )dFn (x , n ) ζ(x, n)
′

′

′

′

= exp(x)Z+ Av(x, n)1P (x, n, x′ , n′ )dFn (x′ , n′ )ζn (x, n)

A5

′

Z
+ γ(n) + Av(x, n)Z!
1P (x, n, x , n )α̂(x, x )dFn (x , n ) − g ζx (x, n)
′

′

′

′

′

Now, define the “effective discount rate”
Z
R(x, n) = ρ + η − Av(x, n)Z

1P (x, n, x′ , n′ )α̂x (x, x′ )dFn (x′ , n′ )Define the stochastic process
(
)
Z
′
′
′
′
′
dxt = γ(nt ) + Av(xt , nt )Z 1P (x, n, x , n )α̂ (xt , x ) dFn (x , n ) − g dt
(
)
Z
′
′
′
′
dnt = Av(xt , nt ) 1P (x, n, x , n )dFn (x , n ) dt

(23)

We can now use the Feynman–Kac formula (Pham (2009)) to go back to the sequential
formulation:"Z ∞ R#t− 0 R(xτ ,nτ )dτζ(z, n) =Eeexp(x)dt | x0 = z, n0 = n, {xt , nt } follows (23)0Because exp(x) is positive, ζ(z, n) is positive. This concludes the proof.

A.4 Ωn (z, n) > 0
Denote ζ(x, n) = Ωx (x, n) and α̂(x, x′ ) = α(exp(x′ − x))/ exp(x) Differentiate the Bellman
equation (21) w.r.t. n,Zρζ(x, n) = + Av(x, n)1P (x, n, x′ , n′ )dFn (x′ , n′ )ζn (x, n)Z+ Av(x, n)Z 1P (x, n, x′ , n′ )α̂(x, x′ )dFn (x′ , n′ )ζx (x, n)+ γ(n) − g ζx (x, n)+ γ′ (n)Ωx (x, n)+ η [ζ(0, n) − ζ(x, n)]ρζ(x, n) =γ′ (n)Ωx (x, n)

A6

Z+ Av(x, n)1P (x, n, x′ , n′ )dFn (x′ , n′ )ζn (x, n)
!
Z
′
′
′
′
′
+ γ(n) + Av(x, n)Z 1P (x, n, x , n )α̂(x, x )dFn (x , n ) − g ζx (x, n)+ η [ζ(0, n) − ζ(x, n)]Define the stochastic process()
Z
dxt = γ(nt ) + Av(xt , nt )Z 1P (x, n, x′ , n′ )α̂ (xt , x′ ) dFn (x′ , n′ ) − g dt + (0 − xt ) dHt
(
)
Z
′
′
′
′
dnt = Av(xt , n) 1P (x, n, x , n )dFn (x , n ) dt

(24)

where Ht is a compensated Poisson process of intensity η. Again, we can use the Feynman–
Kac formula to go back to the sequential formulation:"Z ∞ζ(z, n) =E#
e γ (n)Ωx (x, n)dt | x0 = z, n0 = n, {xt , nt } follows (24)−ρ ′0Because γ′ (n) > 0 by assumption and Ωx (x, n) is positive from the previous proof, ζ(z, n)
is positive. This completes the proof.

B

Empirical Appendix

B.1

Data

Our analyses utilize two administrative data sets, ”Linked Inventor Biography Data 19802014” (INV-BIO) and ”Sample of Integrated Labor Market Biographies” (Stichprobe derIntegrierten Arbeitsmarktbiografien - SIAB). Both data sets are constructed by the Institutefor Employment Research (IAB).
The SIAB data is a 2% random sample from (Integrated Employment Biographies - IEB).
The IEB combines data from five different sources, each of which may contain informationfrom various administrative procedures. It comprises all individuals in Germany who
hold at least one of the following employment statuses: employment subject to social
security, marginal part-time employment, receipt of benefits according to the German
Social Code III or II, official registration as a job seeker at the German Federal Employment
Agency, and (planned) participation in programs of active labor market policies (Dauth
A7

Table B.1: Summary Statistics(A) INV-BIO
Establishment level variables
N of inventors (net )N of employees
Mean daily wage, EuroN of three-year forward citations for patents(three-year backward average, zet )Share of inventors moving from higher productivity est.
(H-Shareet ), %Total inventor inflows

Mean
4.9
688.9
121.6
11.3

S.D.
18.5
2150.6
55.5
69.2

N of est. (thus.)
119
119
119
119

61.2

49.5

119

1.67

5.30

119

(B) SIAB
Worker level variables
Dummy for moving to less productive est. (Dit )based on est. size
based on mean wage
Dummy for the identified inventors (Iit )Daily Wage, EuroAge
Share of Women, %

Mean

S.D.

N of workers (thus.)

0.50
0.52
0.10
44.4
33.7
47.3

42.1
12.9
-

4,669
4,583
5,691
5,691
5,691
5,691

and Eppelsheimer 2020 for more detail).
The patent information contained in the INV-BIO dataset is sourced from registerdata recorded in PATSTAT, which includes bibliographical, procedural, and legal status
information on patent applications handled by the European Patent Office. Additionally,
data from DPMAregister, the online patent register of the German Patent and Trademark
Office, is incorporated to enhance the PATSTAT data extract. The DPMAregister provides
exclusive records of national patent applications that are not transferred to the European
Patent Office or filed under the PCT (Patent Cooperation Treaty) route. As a result, theINV-BIO dataset comprises inventors who are listed on patent filings at the European
Patent Office (EPO) between 1999 and 2011 and have been successfully linked with IEB(Dorner et al. 2018 for more detail).
Table B.1 shows the summary statistics for INV-BIO and SIAB, respectively. Table
B.2 shows the correlation between the three measures used as the proxy for knowledge
quality or productivity level in Section 3.2 and 3.3. We can observe positive correlations
between them.
A8

Table B.2: Correlation between Three Measures
Correlation
zet
Est. size
Mean wage

zet
1.00
0.44
0.11

Est. size

Mean Wage

1.00
0.08

1.00

Table B.3: Distribution of Inventors(A) Rank by Citation
Establishment percentile rank

≤50%

50-60

60-70

70-80

80-100Share of Inventors (%)

13.1

2.5

3.2

4.6

76.7

(B) Rank by establishment size
Establishment percentile rank

≤50%

50-60

60-70

70-80

80-100Share of Inventors (%)

9.3

3.9

5.2

7.5

74.0

(C) Rank by Mean Wage
Establishment percentile rank

≤50%

50-60

60-70

70-80

80-100Share of Inventors (%)

10.2

6.6

10.5

14.8

57.8

Notes: This table shows the distribution of inventors across percentiles of establishments. The percentile inpanel (A) is based on the three-year backward average of forward patent citation counts. Panel (B) is basedon the establishment size, and panel (C) is based on the mean wage of full-time employees. Establishments
could be classified into different percentiles based on these measures each year. The sample encompasses
data from 1980 to 2014. The values in the table represent the proportion of inventors within each percentile
in relation to the total number of inventors in INV-BIO.

B.2

Robustness Check of Empirical Analyses

The table from B.4 to B.6 shows the result of the robustness check for our empirical result.
Table B.4 shows the transition matrix of inventor flows with wage increases, suggesting
A9many flows from more productive establishments to less productive ones, even in thissample.
Instead of Probit model in Section 3.3, we estimate the following equation to control
for fixed effects,
Dit = β0 + β1 Iit + β2 Xit + αe + αt + εit(25)

Definitions of variables are the same as in Section 3.3. Table B.5 shows that inventors are
more likely to move to less productive establishments conditional on fixed effects.

A10

Table B.4: Transition Probabilities of Inventor Flows with wage increase(A) Rank by Citation/Inventor
Destination establishment rank

Share of flows (%)

≤ 50%

50-60

60-70

70-80

80-100

≤ 50%

2.7

0.2

0.3

0.4

4.1

Origin

50-60

2.1

0.2

0.2

0.3

3.1

establishment

60-70

2.3

0.2

0.3

0.4

3.6

rank

70-80

2.7

0.2

0.3

0.4

3.6

80-100

19.0

1.8

2.1

3.1

45.9

(B) Rank by Establishment Size
Destination establishment rank

Share of flows (%)

≤ 50%

50-60

60-70

70-80

80-100

≤ 50%

3.8

1.2

0.9

0.9

6.6

Origin

50-60

0.4

0.8

0.9

0.5

2.1

establishment

60-70

0.4

0.2

1.1

1.2

2.8

rank

70-80

0.5

0.3

0.4

1.9

4.9

80-100

3.5

1.5

2.1

3.1

58.2

(C) Rank by Mean Wage
Share of flows (%)

Destination establishment rank
≤ 50%

50-60

60-70

70-80

80-100

≤ 50%

4.4

1.5

1.3

1.4

4.4

Origin

50-60

0.9

1.3

1.6

1.1

2.5

establishment

60-70

0.8

0.9

2.6

2.9

4.1

rank

70-80

0.7

0.6

1.6

4.8

7.4

80-100

2.0

2.6

2.8

4.9

42.4

A11

Table B.5: Estimation Result for Inventor Flows (Linear Model)
Dit
Whole sample
Iit

Sample with wage ↑.008***.010**.009***.014***(.003)√(.004)√(.003)√(.004)
√

√

√

√

√

Size

Mean wage

Size

Mean wage

2,938,537

2,959,368

1,609,460

1,617,613

.25

.22

.21

.20

Control
Fixed Effects
Measure for Dit
N
2

Adj. RC

Quantitative Appendix

C.1

Numerical Solution to Joint Value HJB Equation

Change of variables
We want to solveρΩ(z, n, i) = max z − c(v)Zv≥0Z+ Av [Ωn (z, n, i) + α(z′ /z)ZΩz (z, n, i) − Ωn (z′ , n′ , i′ )]+ dFn (z′ , n′ , i)
+ γ(n, i) − g zΩz (z, n, i)+ η [Ω(1, n, i) − Ω(z, n, i)]+ λi [Ω(z, n, −i) − Ω(z, n, i)]cvϕ+1 . The first order condition for vacancies
As in our quantitative exercise, let c(v) = ϕ+1
isZ
ϕcv(z, n, i) Z = A [Ωn (z, n, i) + α(z′ /z)ZΩz (z, n, i) − Ωn (z′ , n′ , i′ )]+ dFn (z′ , n′ , i′ )(Av(z, n, i) =
cZ

Z

) ϕ1
[Ωn (z, n, i) + α(z′ /z)ZΩz (z, n, i) − Ωn (z′ , n′ , i′ )] dFn (z′ , n′ , i′ )+A12Consider a change of variables. Let z̃ = log z, ñ = log n. Now, define Ω̃(z̃, ñ, i) =eΩ ez̃ , eñ , i = Ω(z, n, i). Applying the chain rule to Ω(z, n, i) = Ω(logz, log n, i), and rearranging:e ñ (eΩn (z, n, i)n = Ωz, ñ, i)e z̃ (eΩz (z, n, i)z = Ωz, ñ, i)or equivalently,
e ñ (ez, ñ, i)Ωeñe z̃ (ez, ñ, i)ΩΩz (z, n, i) =ez̃Ωn (z, n, i) =As in our quantitative exercise, let α(z′ /z) = α (z′ /z)β . Then, we can rewrite it as ′ βα(z′ /z) = α ez̃ /ez̃= αeβ(z̃ −z̃)′

As in our quantitative exercise, let γ(n, i) = γi nδ . Then, we can rewrite it asγ(n, i) =γi eδñDefine F̃(z̃, ñ, i) = F(ez̃ , eñ , i) = F(z, n, i). The relationship between density of F(z, n, i) andF̃(z̃, ñ, i) is given by∂ ∂F(z, n, i)
∂z ∂n
∂ ∂
=
F̃(log z, log n, i)
∂z ∂n
1 ˜
=
f (z̃, ñ, i)zn

f (z, n, i) =When we change the variables from (z, n) to (z̃, ñ) , the Jacobian is zn. Therefore, for any
set of functions h and h̃ such that h(z, n, i) = h̃(z̃, ñ, i),ZZh̃(z̃, ñ, i)dF̃(z̃, ñ, i) =A13h(z, n)dF(z, n, i)

The total output can be expressed asZZ=

ez̃ dF̃(z̃, ñ, i)

The total mass of the inventor can be expressed asZn=
Let defineeñ dF̃(z̃, ñ, i)eñ f˜(z̃, ñ, i)f˜ñ (z̃, ñ, i) =n

and F̃ñ (z̃, ñ, i) the corresponding cumulative distribution. Then, the inventor-weighted
distribution fñ (z, n, i) can be rewritten asn f (z, n, i)n
1 1 eñ f˜(z̃, ñ, i)
= ·
z n
n
1 1 ˜
= · fñ (z̃, ñ, i)z nfn (z, n, i) =Therefore, for any set of functions h and h̃ such that h(z, n, i) = h̃(z̃, ñ, i),ZZh̃(z̃, ñ, i)dF̃ñ (z̃, ñ, i) =h(z, n)dFn (z, n, i)

Define ṽ(z̃, ñ, i) = v(ez̃ , eñ , i) = v(z, n, i). Then,(Aṽ(z̃, ñ, i) =cZZ he ñ (eΩz, ñ, i)/e + αeñβ(z̃′ −z̃)e z̃ (ee ñ (eZΩz, ñ, i)/e − Ω
z , ñ , i )/e
z̃

′

′

′

ñ

i+

) ϕ1
dF̃ñ (z̃ , ñ , i )′′′The Bellman equation can be rewritten ascṽ(z̃, ñ, i)ϕ+1 Z
ϕ+1
Z h
i
′
′ +
e z̃ (ee ñ (ee ñ (e+ Aṽ(z̃, ñ, i)Ω
z, ñ, i)/eñ + αeβ(z̃ −z̃) ZΩ
z, ñ, i)/ez̃ − Ωz′ , ñ′ , i′ )/eñ dF̃ñ (z̃′ , ñ′ , i′ )e z̃ (e+ γi eδñ − g Ωz, ñ, i)ρΩ̃(z̃, ñ, i) = ez̃ −

A14

h
i
+ η Ω̃(0, ñ, i) − Ω̃(z̃, ñ, i)hi+ λi Ω̃(z̃, ñ, −i) − Ω̃(z̃, ñ, i)
Let define poaching indicator function with transformed variable 1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) as




1
′
′ ′
1̃P (z̃, ñ, i, z̃ , ñ , i ) = 


0

′
e z̃ (ee ñ (ee ñ (ez, ñ, i)/ez̃ > Ωz′ , ñ′ , i′ )/eñ
if Ωz, ñ, i)/eñ + αeβ(z̃ −z̃) ZΩ

otherwise

The Bellman equation can be rewritten ascṽ(z̃, ñ, i)ϕ+1 Z
ϕ+1
Z


ñ
e ñ (e1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) dF̃ñ (z̃′ , ñ′ , i′ )Ω
z, ñ, i)+ Aṽ(z̃, ñ, i)/eZ hi′e z̃ (e+ Aṽ(z̃, ñ, i)αZ/ez̃
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )eβ(z̃ −z̃) dF̃ñ (z̃′ , ñ′ , i′ )Ω
z, ñ, i)Z hi′e ñ (e− Aṽ(z̃, ñ, i)
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )Ω
z′ , ñ′ , i′ )/eñ dF̃ñ (z̃′ , ñ′ , i′ )e z̃ (e+ γi eδñ − g Ωz, ñ, i)hi+ η Ω̃(0, ñ, i) − Ω̃(z̃, ñ, i)hi+ λi Ω̃(z̃, ñ, −i) − Ω̃(z̃, ñ, i)ρΩ̃(z̃, ñ, i) = ez̃ −

Z h
i
c
′
ϕ+1
e ñ (eṽ(z̃, ñ, i) Z − Aṽ(z̃, ñ, i)
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )Ω
z′ , ñ′ , i′ )/eñ dF̃ñ (z̃′ , ñ′ , i′ )ρΩ̃(z̃, ñ, i) = e −
ϕ+1
Z


ñ
e ñ (e+ Aṽ(z̃, ñ, i)/e
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) dF̃ñ (z̃′ , ñ′ , i′ )Ω
z, ñ, i)
!
Z h
i
δñ
z̃
′
′ ′ β(z̃′ −z̃)′′ ′e z̃ (e+ γi e + Aṽ(z̃, ñ, i)αZ/e
1̃P (z̃, ñ, i, z̃ , ñ , i )edF̃ñ (z̃ , ñ , i ) − g Ωz, ñ, i)hi+ η Ω̃(0, ñ, i) − Ω̃(z̃, ñ, i)hi+ λi Ω̃(z̃, ñ, −i) − Ω̃(z̃, ñ, i)z̃

A15Implicit method
We solve the Bellman equation using an implicit method. Let ∆ denote step-size and τ theiteration of the algorithm. Then given Ω̃τ−1 (z̃, ñ, i), the implicit method gives an updatei1h τΩ̃ (z̃, ñ, i) − Ω̃τ−1 (z̃, ñ, i) + ρΩ̃(z̃, ñ, i) =
∆
Z h
i
c
z̃
ϕ+1
′
′ ′
ñ′
e τ−1 (ee −ṽ(z̃, ñ, i) Z − Aṽ(z̃, ñ, i)
z
,
ñ
,
i
)/e
dF̃ñ (z̃′ , ñ′ , i′ )1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )Ω
ñ
ϕ+1
Z


ñ
e τ (e+ Aṽ(z̃, ñ, i)/e
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) dF̃ñ (z̃′ , ñ′ , i′ )Ω
ñ z, ñ, i)
!
Z h
i
′
′ ′
δñ
z̃
′
′ ′ β(z̃′ −z̃)e τ (edF̃ñ (z̃ , ñ , i ) − g Ω+ γi e + Aṽ(z̃, ñ, i)αZ/e
1̃P (z̃, ñ, i, z̃ , ñ , i )ez̃ z, ñ, i)hi+ η Ω̃τ (0, ñ, i) − Ω̃τ (z̃, ñ, i)hi+ λi Ω̃τ (z̃, ñ, −i) − Ω̃τ (z̃, ñ, i)
Rearranging this expression:1+ ρ + η Ω̃τ (z̃, ñ, i)
∆
Z


ñ
e τ (e− Aṽ(z̃, ñ, i)/e
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) dF̃ñ (z̃′ , ñ′ , i′ )Ω
ñ z, ñ, i)
!
Z h
i
δñ
z̃
′
′ ′ β(z̃′ −z̃)′′ ′e τ (e1̃P (z̃, ñ, i, z̃ , ñ , i )edF̃ñ (z̃ , ñ , i ) − g Ω− γi e + Aṽ(z̃, ñ, i)αZ/ez̃ z, ñ, i)hi− λi Ω̃τ (z̃, ñ, −i) − Ω̃τ (z̃, ñ, i)c= ez̃ + ηΩ̃τ (0, ñ, i) −ṽ(z̃, ñ, i)ϕ+1 Z
ϕ+1
Z h
i
′
′ ′
ñ′
e τ−1 (e− Aṽ(z̃, ñ, i)
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )Ω
z
,
ñ
,
i
)/e
dF̃ñ (z̃′ , ñ′ , i′ )
ñ
+

1 τ−1
Ω̃ (z̃, ñ, i)∆

A16

We now discretize ñ on an evenly spaced Nñ × 1 grid and z̃ on an evenly spaced Nz̃ × 1.
Stack these according to:



 e
z
,
ñ
,
h
1
1





 e
 z2 , ñ1 , h 


..


.






zNz̃ , ñ1 , h 
 e


..




.



 e
 z1 , ñNñ , h 


..




.





 e
z
,
ñ
,
h
N
N
z̃
ñ





 e
 z1 , ñ1 , l 


..




.




e
z , ñ , l 
Nz̃

Nñ

The above equation can be rewritten in vector form as:1e τ − µz Ωe τ − ΛΩ̃τ = π + 1 Ω̃τ−1+ ρ + η Ω̃τ − µn Ω
ñ
z̃
∆
∆

where
• the element of Nz̃ × Nñ × 2 vector Ω̃τ consists of Ω̃τ (z̃, ñ, i),• the element of Nz̃ × Nñ × 2 vector Ω̃τñ consists of Ω̃τñ (z̃, ñ, i),• the element of Nz̃ × Nñ × 2 vector Ω̃τz̃ consists of Ω̃τz̃ (z̃, ñ, i),• the element of Nz̃ × Nñ × 2 vector µn consists ofZñAṽ(z̃, ñ, i)/e
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ ) dF̃ñ (z̃′ , ñ′ , i′ ),• the element of Nz̃ × Nñ × 2 vector µz consists ofδñ

γi e + Aṽ(z̃, ñ, i)αZ/e

z̃

Z h

i
′
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )eβ(z̃ −z̃) dF̃ñ (z̃′ , ñ′ , i′ ) − g,

and
A17

(26)

• the element of Nz̃ × Nñ × 2 vector π consists ofcez̃ + ηΩ̃τ (0, ñ, i) −ṽ(z̃, ñ, i)ϕ+1 Z
ϕ+1
Z h
i
′
′ ′
ñ′
e τ−1 (e
z
,
ñ
,
i
)/e
dF̃ñ (z̃′ , ñ′ , i′ )− Aṽ(z̃, ñ, i)
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )Ω
ñ(Nz̃ × Nñ × 2) × (Nz̃ × Nñ × 2) matrix Λ is −λh IN ×N λh IN ×N 
z̃
ñ
z̃
ñ


Λ = 
 λI−λl INz̃ ×Nñ l Nz̃ ×Nñ
where INz̃ ×Nñ is Nz̃ × Nñ identity matrix. Let Dñ be the (Nz̃ × Nñ × 2) × (Nz̃ × Nñ × 2) matrix
that, when pre-multiplying Ω̃τ , gives an approximation of Ω̃τñ . Analogously, define Dñ :e τ = Dñ ΩeτΩn̄e τ = Dz̃ ΩeτΩz̄To compute the derivative matrices Dñ and Dz̃ , we follow an upwind scheme. That is,
we use a forward approximation when the drift of the state variable is positive and abackward approximation when the drift of the state is negative. Using these, we can write(26) as11+ ρ + η − µn Dñ − µz Dz̃ − Λ Ω̃τ = π + Ω̃τ−1 .
∆∆e τ through the above equation.
The implicit method works by updating Ω

C.2

Numerical Solution to Kolmogorov Forward Equation

The total mass of the inventor can be expressed asZv=
Let defineṽ(z̃, ñ, i)dF̃(z̃, ñ, i)ṽ(z̃, ñ, i) f˜(z̃, ñ, i)f˜ṽ (z̃, ñ, i) =v

A18

and F̃ṽ (z̃, ñ, i) the corresponding cumulative distribution. We construct the Kolmogorov
forward equation in terms of the transformed variables.
 ∂ 

∂ 
µ̃ñ (z̃, ñ, i) f˜(z̃, ñ, i) −µ̃z̃ (z̃, ñ, i) f˜(z̃, ñ, i)
∂ñ
∂z̃
Z 1
− η f˜(z̃, ñ, i) + η
f˜(z̃′ , ñ, i)dz′ δ(0)0=−0− λi f˜(z̃, ñ, i) + λ−i f˜(z̃, ñ, −i)where 18
Z

v
′
′ ′
′
′ ′ 
1̃P (z̃, ñ, i, z̃ , ñ , i ) dF̃ñ (z̃ , ñ , i ) − A
1̃P (z̃′ , ñ′ , i′ , z̃, ñ, i)dF̃ṽ (z̃′ , ñ′ , i′ )
n
Z h
i
′
δñ
z̃
µ̃z̃ (z̃, ñ, i) = γi e + Aṽ(z̃, ñ, i)αZ/e
1̃P (z̃, ñ, i, z̃′ , ñ′ , i′ )eβ(z̃ −z̃) dF̃ñ (z̃′ , ñ′ , i′ ) − gṽ(z̃, ñ, i)µ̃ñ (z̃, ñ, i) = A
eñ

Z

We can vectorize this in the same way as above, and obtain
0 = −Dñ µ̃ñ f˜ − Dz̃ µ̃z̃ f˜ − η f˜ + η f˜0 + Λ′ f˜
where• the element of Nz̃ × Nñ × 2 vector f˜ is the stacked as the value function,• the element of Nz̃ × Nñ × 2 vector µ̃ñ consists of µ̃ñ (z̃, ñ, i),• the element of Nz̃ × Nñ × 2 vector µ̃z̃ consists of µ̃z̃ (z̃, ñ, i), andR1• the element of Nz̃ × Nñ × 2 vector f˜0 consists of 0 f˜(z̃′ , ñ, i)dz′ δ(0)19 .
To construct the derivative matrices, we use a backward approximation when the drift ofthe state variable is positive, and a forward approximation when the drift of the state is
negative. This expression can be rearranged to yield−Dñ µ̃ñ − Dz̃ µ̃z̃ − η + Λ′ f˜ = −η f˜0
and the distribution of individuals is updated according to the above equation.
d log nd log zdz̃= ddtñ and dz/z
Note that dn/n
dt =dt
dt = dt = dt .
19
Due to the Dirac delta function δ(0), the elements of f˜0 can take positive value only if z̃ = 0.
18

A19

C.3

Solving the Transition Path

We illustrate how to solve the transition path. We solve the transition path in terms oftransformed variables and later recover the non-transformed values over the transition.
Finally, we explain how to calculate the consumption-equivalent welfare gain from thepolicy change.
Perfect Foresight Equilibrium
First, we define the perfect foresight equilibrium, which differs from the balanced growth
path equilibrium in that it has a time notation and, the HJB equation and KFE havetime derivative terms. Note that the perfect foresight equilibrium is detrended by the
productivity of the technology frontier z(t) at each period.
Definition 2. (Perfect Foresight Equilibrium) A perfect foresight equilibrium consists of: (i)a joint value function Ω(z, n, t); (ii) a vacancy policy v(z, n, t); (iii) a stationary distributionof firms f (z, n, t); (iv) vacancy- and employment-weighted distributions fv (z, n, t) andfn (z, n, t); (v) poaching indicator function 1P (z, n, z′ , n′ , t); (vi) the aggregate productivity
Z(t), the aggregate consumption C(t), and the total vacancies v(t), and (vii) the economic
growth rate g(t) such that
1. The joint value Ω(z, n, t) satisfies the HJB equation∂Ω(z, n, t) =∂tz − c(v(z, n, t))Z(t)Z+ Av(z, n, t) [Ωn (z, n, t) + α(z′ /z)Z(t)Ωz (z, n, t) − Ωn (z′ , n′ , t)]+ dFn (z′ , n′ , t)ρΩ(z, n, t) ++ γ(n) − g(t) zΩz (z, n, t)+ η [Ω(1, n, t) − Ω(z, n, t)]
2. The vacancy policy v(z, n, t) satisfies the first order conditionZ
cv (v(z, n, t))Z = A[Ωn (z, n, t) + α(z′ /z)ZΩz (z, n, t) − Ωn (z′ , n′ , t)]+ dFn (z′ , n′ , t)

A20

3. A density function f (z, n, t) satisfies the KFE equation
 ∂

∂
∂
f (z, n, t) = −µn (z, n, t) f (z, n, t) −µz (z, n, t) f (z, n, t) −η f (z, n, t)+η
∂t
∂n
∂z

Z 1
f (z, n, t)dzδ(1)
0

where
Z

µz (z, n, t) ≡ γ(n) − g z + Av(z, n, t)Z 1P (z, n, z′ , n′ , t)α(z′ /z)dFn (z′ , n′ , t)
Z
Z
n
′
′
′
′
1P (z′ , n′ , z, n, t)dFv (z′ , n′ , t)µn (z, n, t) ≡Av(z, n, t) 1P (z, n, z , n , t)dFn (z , n , t) − Avn
4. Vacancy- and employment-weighted distributions are consistent:v(z, n, t) f (z, n, t)v(t)n f (z, n, t)
fn (z, n, t) =n(t)fv (z, n, t) =5. Poaching indicator function 1P (z, n, z′ , n′ , t) is




1
′
′
1P (z, n, z , n , t) = 0if Ωn (z, n, t) + α(z′ /z)ZΩz (z, n, t) > Ωn (z′ , n′ , t)otherwise

6. The aggregate productivity Z(t), aggregate consumption C(t), and the total vacanciesv(t) satisfy rate satisfy
Z
Z(t) =

zdF(z, n, t)()ZC(t) = 1 −c(v(z, n, t))dF(z, n, t) Z(t)Zv(t) = v(z, n, t)dF(z, n, t)
7. The inventor market clearing condition is satisfied:Zn(t) =A21

ndF(z, n, t)

Now, we recover the non-transformed values over the transition from the transformed
values. Without loss of generality, let normalize the productivity of the technology frontier
at time t = 0 to 1: z(0) = 1. Then,Z tz(t) = exp!
g(τ)dτ

0Ẑ(t) = z(t)Z(t)Ĉ(t) = z(t)C(t)Solution Algorithm
To solve the transition path, we guess a path for optimal behavior of firms over a discretized
grid for productivity, number of inventors and time. Subsequently, we iterate on
1. Given a path for the distribution of firms and inventors, update optimal behavior offirms and inventors backwards in time;
2. Given a path for behavior, update the evolution of the distribution of firms andinventor forward in time
3. If the updated path for the distributions and behavior are close enough to the original
path, stop. Otherwise return to 1.
Recover Non-Transformed Values
Now, we recover the non-transformed values over the transition from the transformed
values. Without loss of generality, let normalize the productivity of the technology frontier
at time t = 0 to 1: z(0) = 1. Then,Z tz(t) = exp!
g(τ)dτ

0Ẑ(t) = z(t)Z(t)Ĉ(t) = z(t)C(t)Consumption-Equivalent Welfare Gains
Definition 3. (Consumption-Equivalent Welfare Gains from Policy Change) Considernoan economy without policy change and the associated consumption path Ĉ(t) . The
t≥0

A22

consumption equivalent welfare gains from policy change is the scalar L such that the connosumer is indifferent between the consumption path L × Ĉ(t)and the consumptiont≥0
path generated by the policy change.
o nLet V Ĉ(t)define the welfare:t≥0oV Ĉ(t)
n


t≥0

Z ∞
e−ρt log Ĉ(t)dt.

≡
0

Then,
n
o

V L × Ĉ(t)



Z ∞



e−ρt log L × Ĉ(t) dt
t≥0
Z0 ∞
Z ∞
−ρt
=
e log Ldt +e−ρt log Ĉ(t)dt
0
0
Z ∞
log L
+
e−ρt log Ĉ(t)dt
=
ρ
0
o 
n
log L
=
+ V Ĉ(t)t≥0ρ=This implies
n
o 
o 

n
log L
= V L × Ĉ(t)− V Ĉ(t)t≥0t≥0ρor equivalently,
n
o
h n 
L = exp ρ V L × Ĉ(t)


t≥0

o
n
− V Ĉ(t)

oi
t≥0

n
o
Let Ĉ′ (t)denote the consumption path generated by policy change. Then, the cont≥0
sumption equivalent welfare gains from policy change are calculated asoh n n
L = exp ρ V Ĉ′ (t)


t≥0

23

o
n
− V Ĉ(t)oit≥0