Institute for Economic Studies, Keio University
Keio-IES Discussion Paper Series

Marshall meets Bartik: Revisiting the mysteries of the trade
Yasusada Murata, Ryo Nakajima
28 September, 2023
DP2023-015
https://ies.keio.ac.jp/en/publications/23108/

Institute for Economic Studies, Keio University
2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan
ies-office@adst.keio.ac.jp
28 September, 2023

Marshall meets Bartik: Revisiting the mysteries of the trade
Yasusada Murata, Ryo Nakajima
Keio-IES DP2023-015
28 September, 2023
JEL Classification: R12, O31, J61, C26
Keywords: knowledge spillovers, knowledge sharing, Bartik instruments, mysteries of
the trade, idea-generating process

Abstract
We identify a causal effect of top inventor inflows on patent productivity of local
inventors by combining the idea-generating process described by Marshall (1890) with
the Bartik (1991) instruments involving state taxes and commuting zone characteristicsof the United States. We find that the local productivity gains go beyond organizationalboundaries and co-inventor relationships, which implies the partially nonexcludable
good nature of knowledge in a spatial economy and pertains to the mysteries of the
trade in the air. Our counterfactual experiment suggests that the spatial distribution of
inventive activity is substantially distorted by the presence of heterogeneity in statetaxes.

Yasusada Murata
College of Economics, Nihon University
1-3-2 Kanda-Misakicho, Chiyoda-ku, Tokyo 101-8360, JAPANmurata.yasusada@nihon-u.ac.jp
Ryo Nakajima
Department of Economics, Keio University
2-15-45 Mita, Minato Tokyo 108-8345, JAPAN
nakajima@econ.keio.ac.jp

Marshall meets Bartik:
Revisiting the mysteries of the trade
Yasusada Murata∗

Ryo Nakajima†

September 28, 2023

Abstract
We identify a causal eﬀect of top inventor inﬂows on patent productivity of local
inventors by combining the idea-generating process described by Marshall (1890) with
the Bartik (1991) instruments involving state taxes and commuting zone characteristicsof the United States. We ﬁnd that the local productivity gains go beyond organizationalboundaries and co-inventor relationships, which implies the partially nonexcludable goodnature of knowledge in a spatial economy and pertains to the mysteries of the trade in
the air. Our counterfactual experiment suggests that the spatial distribution of inventive
activity is substantially distorted by the presence of heterogeneity in state taxes.
Keywords: knowledge spillovers; knowledge sharing; Bartik instruments; mysteries of
the trade; idea-generating process
JEL codes: R12; O31; J61; C26∗†

College of Economics, Nihon University. E-mail: murata.yasusada@nihon-u.ac.jp
Department of Economics, Keio University. E-mail: nakajima@econ.keio.ac.jp1

1

Introduction

Knowledge spillovers have been central to various ﬁelds of economics such as trade, growth,and geography. However, little is known about the idea-generating process through knowledge
spillovers despite Marshall’s (1890) simple explanation:
if one man starts a new idea, it is taken up by others and combined with suggestionsof their own; and thus it becomes the source of further new ideas.
While intuitive, testing for this channel has been challenging. The main diﬃculty lies inpossible endogeneity: those who generate new ideas tend to cluster together.
To uncover this idea-generating process, we identify a causal eﬀect of top inventor inﬂows
on the patent productivity of local inventors at the commuting-zone level in the United States.
In doing so, we use the PatentsView database, which is an open data platform supported by
the United States Patent and Trademark Oﬃce (USPTO). Since top inventor inﬂows are likely
endogenous, we predict those ﬂows by constructing Bartik (1991) instruments: the predictedprobability that a top inventor migrates from origin to destination constitutes a share, andinventors in the origin correspond to a shift.
We consider two types of outcomes: the productivity gains of all local inventors andthose of external inventors who are not directly connected to incoming inventors throughorganizations or co-inventor relationships. Our baseline results suggest that the former andlatter gains from an additional top inventor inﬂow are 6% and 4%, respectively. The former
are interpreted as local productivity gains from both internal knowledge sharing and external
knowledge spillovers. The latter gains go beyond organizational boundaries and co-inventor
relationships and thus pertain to the most frequently quoted passage from Marshall (1890):“The mysteries of the trade become no mysteries; but are as it were in the air.”
Our identiﬁcation strategy consists of main three steps. We ﬁrst analyze the impact of
spatial and temporal variation in individual income tax rates on the migration probabilityof top inventors for any pair of origin and destination commuting zones while controlling2

for origin-destination characteristics. We then aggregate, for each destination commuting
zone, the estimated bilateral location choice probabilities across origin commuting zones toconstruct a Bartik instrument, which depends on the tax diﬀerences between that destination
and all other commuting zones. We ﬁnally employ an instrumental variable (IV) approach,where we use the Bartik instrument in the ﬁrst-stage regression to proxy for top inventor
inﬂows and estimate a structural equation with the outcome being local patent productivity.
The identifying assumption is that the patent productivity in the destination does not directly
depend on the tax diﬀerences between that destination and all other commuting zones.1
Our novelty lies in the construction of the Bartik instrument. Since we derive the predictedmigration probability from a location choice problem of a top inventor who faces spatial andtemporal diﬀerences in individual income tax rates, our framework can be used to examine
to what extent those tax diﬀerences distort the spatial distribution of inventive activity. To
illustrate this, we run a counterfactual experiment by setting individual income taxes to theiraverage and ﬁnd that the existence of tax diﬀerences aﬀects local patent productivity up to−64.8% to 72.3% with considerable spatial heterogeneity. We further decompose those gains
and losses into two: the direct gains from tax changes; and the indirect gains from tax changes
through top inventor migration. We ﬁnd that the former eﬀect is 26.5% while the latter is
73.5%.
The contribution of our paper to the literature is threefold. First, we shed new light on
the idea-generating process described in Marshall (1890) using Bartik (1991) instruments.
Our work thus contributes to the agglomeration and innovation literature, where according
to Carlino and Kerr (2015) “there is very little insight into how knowledge is transmitted
among individuals living in close geographic proximity.” Our paper addresses this important
yet unexplored issue in an environment where tax-induced migration of top inventors brings
about new knowledge to local inventors in their destination.
1
For example, the assumption holds if local patent productivity in commuting zone 37500 (Santa Clara–
Monterey–Santa Cruz, CA) does not directly depend on the sum of the tax diﬀerence between commuting
zones 37500 and 19600 (Bergen–Essex–Middlesex, NJ), the tax diﬀerence between commuting zones 37500
and 24300 (Cook–DuPage–Lake, IL), and so forth.

3

Second, we disentangle productivity gains due to external knowledge spillovers from thosedue to internal knowledge sharing, which allows us revisit the mysteries of the trade in the air.
The theoretical background of this distinction between pure knowledge spillovers and marketmediated knowledge transfer dates back at least to Griliches (1979) and Romer (1990). To
the best of our knowledge, however, this partially nonexcludable good nature of knowledge
has not been studied in a spatial framework using modern causal inference methods. Since
the existence of the gains from external knowledge spillovers constitutes a rationale for spatial agglomeration of inventive activity, our analysis, which leverages spatial and temporalvariation in tax rates, contributes to the innovation policy literature that considers both the
beneﬁts and costs of entrepreneurial clusters (e.g., Chatterji, Glaeser, and Kerr, 2014).
Last, we derive the Bartik instruments from a location choice model involving policy
variables à la Moretti and Wilson (2017). The theory-based Bartik instruments can be readily
used for counterfactual experiments and applied to various contexts where origin-destination
ﬂows are aﬀected by the diﬀerence in policy variables across locations. Our counterfactual
analysis illustrates this type of experiment and bridges the gap between the tax and innovation
literature (e.g., Stantcheva, 2021; Akcigit et al., 2022; Akcigit and Stantcheva, 2022) and thetax and migration literature (e.g., Kleven et al., 2020). Since our analysis allows not only
for the direct productivity gains from tax changes but also for the indirect gains throughtax-induced migration of top inventors, it complements these two strands of literature.
The remainder of the paper is organized as follows. In Section 2, we explain the data andshow descriptive statistics. In Section 3, we analyze how tax diﬀerences aﬀect the migrationof top inventors and construct the Bartik instruments. Section 4 presents our baseline resultsby employing the IV approach. In Section 5, we check the robustness of the main resultsin terms of time, space, and aggregation. We estimate an event study model, consider the
spatial extent of local productivity gains, and disaggregate local inventors according to theirproductivity. Section 6 discusses the underlying mechanisms through which local productivity
gains materialize by focusing on two key channels: patent citations and trade secrets. We
conduct the counterfactual experiment in Section 7 and conclude the paper in Section 8.
4

2

Data and descriptive statistics

Our main dataset is the PatentsView database, which is an open data platform supportedby the USPTO and provides various administrative data on issued patents and patent applications. The data are based on the disambiguation process and contain, for each issued
patent, patent inventors, assignees, residential addresses of patent inventors, and patent citations. Our sample period is from 1977 to 2009, during which there were 3, 011, 262 patentapplications by 1, 280, 374 unique inventors (see Appendix A for a more detailed descriptionof the main and secondary data sources).2
Since our objective is to estimate the impact of top inventor migration on the productivityof local inventors in the destination, we need to determine (i) who qualify as top inventors;(ii) under what condition we detect the migration of a top inventor; and (iii) who in thedestination potentially gain from top inventor inﬂows.
To this end, we ﬁrst deﬁne the productivity of an inventor as the number of patents applied
for by that inventor weighted by the number of co-inventors. We then identify, for each year,
the top ﬁve inventors by productivity over the last ten years and call them the top inventors
for short.3 We detect the migration of a top inventor if the commuting zone of residence ofan inventor recorded in the patent application data diﬀers in two consecutive years and ifthat inventor was a top inventor in the ﬁrst year.4 In our sample, the total number of top
inventor migrations is 9, 178, and the number of unique top inventors who migrated at least
once is 6, 182. Thus, on average, each top inventor moved 1.485 times, conditional on moving
at least once.5
2

The sample period is dictated by data availability and consistent with that in Moretti and Wilson (2017).
This deﬁnition of top inventors is similar to Moretti and Wilson (2017).
4
We assume that the migration occurs at the end of the ﬁrst year and use the deﬁnition of commuting
zones as of 1990. When an inventor applied for more than one patent in a year, the most frequently observed
commuting zone is regarded as that inventor’s place of residence in that year. In case of a tie, we use thecommuting zone observed for the ﬁrst time in that year. We exclude commuting zones in Alaska and Hawaii.
5
While the disambiguation algorithm adopted in PatentsView is known to be highly accurate, it is not
error-free (see Toole et al., 2021). Thus, diﬀerent inventors with the same name might be mistakenly recognized
as the same inventor, which could generate seemingly frequent moves. To cope with possible overdetection of
moves, we focus on the top inventors who moved fewer than eight times. This still leaves us with more thanninety-nine percent of the inventors who applied for patents in two consecutive years and who qualiﬁed as
35

Since we analyze the impact of top inventor migration on the productivity of local inventors
in the destination, we aggregate migration ﬂows at the destination level. Figure 1 depicts
the geographic distribution of all 9, 178 top inventor inﬂows by commuting zone, and Table 1
summarizes top 10 commuting zones by top inventor inﬂows.6
When assessing the impact of top inventor migration into a commuting zone, we focus
on the local inventors who lived in that commuting zone at that time while excluding the
top inventors who moved in that commuting zone. In our sample, the number of those local
inventors is 1, 274, 192 and they applied for 2, 027, 777 patents (weighted by the number of coinventors) from 1977 to 2009. Figure 2 illustrates the geographic distribution of local patentproductivity (in logs), and Table 2 summarizes the top 10 commuting zones by local patentproductivity.
As seen from Figures 1 and 2, their spatial patterns are quite similar. The correlation
between the top inventor inﬂows and local patent productivity (the log of local patent productivity) is 0.97 (0.52) and their rank correlation is 0.85. However, since a high correlation
does not always imply causation, we take an IV approach to examine the causal eﬀect of the
top inventor inﬂows on the productivity of local inventors. As explained below, we use thevariation in the individual income average tax rates (ATRs) by state and year to constructpredicted ﬂows of top inventors by commuting zone and year.7 We also consider corporate
income tax rates (CITRs), investment tax credits (ITCs), and R&D tax credits (RTCs) that
can aﬀect top inventor migration.8 Table 3 presents summary statistics for the top inventor
inﬂows and the local patent productivity at the commuting zone × year level. We show in
Appendix A the summary statistics for other commuting zone-level variables, as well as statetaxes and tax credits, that we use in the subsequent analysis.
top inventors at least once.
6
We aggregate migration ﬂows at the commuting zone level because it captures stronger commuting ties
and thus more inventor interactions within labor market areas and because knowledge spillovers tend to belocalized at short distances (see Murata, Nakajima, Okamoto, and Tamura, 2014). We check the robustnessof the result regarding geographic space in Section 5.
7
We assume that top inventors are taxpayers at the ninety-ﬁfth (ninety-ninth) percentile of the U.S. income
distribution as a baseline (as a robustness check).
8
All data on state taxes and tax credits for the years 1977 to 2009 are provided by Moretti and Wilson(2017).

6

Figure 1: Geographic distribution of top inventor inﬂows.

(500,800](400,500](300,400](200,300](100,200](50,100](30,50](20,30](10,20](5,10]
(4,5]
(3,4]
(2,3]
(1,2]
[0,1]
No top inventor inflows

Notes: The inﬂows are deﬁned as the number of top inventors who migrated into each
commuting zone from 1977 to 2009.
Figure 2: Geographic distribution of local patent productivity (in logs).

(11,12](10,11](9,10]
(8,9]
(7,8]
(6,7]
(5,6]
(4,5]
(3,4]
(2,3]
(1,2]
(0,1]
[−1,0]
No patents

Notes: The productivity in each commuting zone is deﬁned as the number of patents by
local inventors from 1977 to 2009, weighted by the number of co-inventors. It excludes thenumber of patents by top inventors who moved in.

7

Table 1: Top 10 commuting zones by top inventor inﬂows.
rank cz number counties
state inﬂows
137500 Santa Clara–Monterey–Santa CruzCA
724
237800 Alameda–Contra Costa–San Francisco
CA557
338300 Los Angeles–Orange–San BernardinoCA408
419600 Bergen–Essex–Middlesex
NJ
372
20500 Middlesex–Worcester–Essex
5
MA
335
6
38000 San Diego
CA
266
7
19400 Kings–Queens–New YorkNY
240
819700 Philadelphia–Montgomery–Delaware
PA220
924300 Cook-DuPage–Lake
IL
219
10
20901 Hartford–Fairﬁeld–New Haven
CT
195
Notes: The inﬂows are deﬁned as the number of top inventors who migrated
into each commuting zone from 1977 to 2009.

Table 2: Top 10 commuting zones by local patent productivity.
rank cz number counties
state
productivity
137500 Santa Clara-Monterey-Santa CruzCA
143,069.321
238300 Los Angeles-Orange-San BernardinoCA116,303.629
337800 Alameda-Contra Costa-San FranciscoCA81,828.603
420500 Middlesex-Worcester-Essex
MA
80,486.767
519600 Bergen-Essex-Middlesex
NJ
77,093.518
624300 Cook-DuPage-LakeIL
75,095.386
711600 Wayne-Oakland-Macomb
MI63,437.324
819400 Kings-Queens-New YorkNY
54,570.527
921501 Hennepin-Ramsey-Dakota
MN
50,587.528
1039400 King-Pierce-SnohomishWA
49,760.915
Notes: The productivity in each commuting zone is deﬁned as the number ofpatents by local inventors from 1977 to 2009, weighted by the number of coinventors. It excludes the number of patents by top inventors who moved in.

Table 3: Summary statistics (main variables)
total
mean
sd
min
max
Local patent productivity (overall)
2,027,776.570 85.821 386.089 0.000 10205.625
Local patent productivity (internal) 1,061,199.866 44.913 252.589 0.000 8,232.745
Local patent productivity (external)
966,576.704 40.908 152.466 0.000 3,108.842
Top inventor inﬂows (overall)
9,178.000
0.388
2.139 0.000
81.000Top inventor inﬂows (intrastate)
2,271.000
0.096
0.929 0.000
37.000Top inventor inﬂows (interstate)
6,907.000
0.292
1.453 0.000
48.000
Number of observations
23,628
Number of commuting zones
716
Number of years
33
Notes: Summary statistics are based on the data described in Section 2 for the years 1977
to 2009. The local patent productivity can be decomposed into two: One is by the internal
inventors who share the same assignee as the migrating top inventors and/or who are coinventors of the migrating top inventors; and the other is by the external inventors. The
top inventor inﬂows can be decomposed into intrastate and interstate migration. Of the 722
commuting zones, four have no patents and two have only one patent during the sample
period. We thus use 716 commuting zones in our regression analysis with ﬁxed eﬀects.

8

We further classify local inventors into internal and external inventors. Local inventors
are internal if they share the same assignee as the migrating top inventors and/or if they areco-inventors of the migrating top inventors. All the other local inventors are external because
they are not directly linked to the migrating top inventors. In our sample, 42.20% of local
inventors are internal, whereas the remaining 57.80% are external. The knowledge of the
migrating top inventors can be shared with internal inventors within the same organization
and/or through co-inventors relationships (“knowledge in the lab”), or it can spill over toexternal inventors within the same commuting zone (“knowledge in the air”). We call theformer internal knowledge sharing and the latter external knowledge spillovers.

3

Tax diﬀerences and the migration of top inventors

We ﬁrst analyze how tax diﬀerences across states aﬀect the migration of top inventors fromorigin commuting zone o to destination commuting zone d in Sections 3.1-3.4, which reproduce
the results in Moretti and Wilson (2017). We then present in Section 3.5 a new method toconstruct a Bartik instrument by using the predicted ﬂows of top inventors.
Let σ(o) and σ(d) denote the states to which origin and destination commuting zones
belong, respectively. In the beginning of period t, the inventors in o, whose number is denoted by Iot , observe individual income tax rates in origin and destination states, τσ(o)t and{τσ(d)t }d̸=o . By the end of period t, they decide whether to migrate to d or to stay in o. The
number and share of inventors who migrate from o to d in period t is deﬁned as Modt andPodt = Modt /Iot , respectively. Similarly, the number and share of inventors who stay in o inperiod t is deﬁned as Moot and Poot = Moot /Iot .
3.1

Inventors

In each period, inventors choose the location that gives them the highest utility. The utility of
an inventor i, who lived in commuting zone o in the previous period and moves to commuting
zone d in the current period t, is given by Uiodt = α ln(1 − τσ(d)t ) + α ln wdt + Zd − Cod + εiodt ,9where wdt is the wage in d; Zd captures consumption amenities and the cost of living in d; Codis the cost of migration measured in utility; and εiodt represents time-varying idiosyncratic
preferences for location. The utility of an inventor i who stays in o is given by Uioot =α ln(1−τσ(o)t )+α ln wot +Zo −Coo +εioot , where we assume that Coo = 0. Taking the diﬀerence
between Uiodt and Uioot yields the utility change for inventor i, conditional on moving fromo to d. Assume that εiodt is independent and identically Gumbel distributed. Let Podt /Poot
denote the share of inventors who move from o to d relative to the share of inventors whostay in o. The log odds ratio is then given byln(Podt /Poot ) = α ln(1 − τσ(d)t ) − ln(1 − τσ(o)t ) + α [ln wdt − ln wot ] + [Zd − Zo ] − Cod . (1)
3.2

Firms

In each period, ﬁrms choose the location that yields the maximum proﬁt. The proﬁt of ﬁrm j,which is located in commuting zone o in the previous period and moves to commuting zone′′
d in the current period t, is given by ln πjodt = β ln(1 − τσ(d)t) − ln wdt + Zd′ − Cod+ ε′jodt ,′stands for state policies such as CITRs, ITCs, and RTCs in σ(d); Zd′ represents
where τσ(d)t′
production amenities in d; Codis the cost of migration for a ﬁrm; and ε′jodt represents thetime-varying idiosyncratic productivity match between a ﬁrm and a commuting zone. As in
the case with inventors, assume that ε′jodt is independent and identically Gumbel distributed.
′′
Let Podt/Poot
denote the share of ﬁrms that move from o to d relative to the share of ﬁrms

that stay in o. The log odds ratio for ﬁrms is then given by


′
′
′
′
′
ln(Podt/Poot) = β ln(1 − τσ(d)t) − ln(1 − τσ(o)t) − [ln wdt − ln wot ] + [Zd′ − Zo′ ] − Cod
.
3.3

(2)

Equilibrium

In equilibrium, labor demand equals labor supply in each commuting zone in each year.
To derive the equilibrium, we ﬁrst solve (2) for ln wdt − ln wot . We then plug the resulting′′
expression into (1) and use the equilibrium condition ln(Podt/Poot) = ln(Podt /Poot ) to obtain10the equation that we estimate as follows (see Appendix B for the derivation):




′
′
) − ln(1 − τσ(o)t)
ln(Podt /Poot ) = η ln(1 − τσ(d)t ) − ln(1 − τσ(o)t ) + η ′ ln(1 − τσ(d)t+γd + γo + γod + uodt ,

(3)

αβ
α
where η = 1+αand η ′ = 1+α
are parameters governing inventor (and ﬁrm) mobility; γd =11
[Zd + αZd′ ] and γo = − 1+α[Zo + αZo′ ] are origin and destination ﬁxed eﬀects that account1+α1′[Cod + αCod
] stands for ﬁxed eﬀects
for consumption and production amenities; γod = − 1+αthat are speciﬁc to each pair of commuting zones to capture the cost of migration for inventors
and ﬁrms; and uodt is an error term.
3.4

Estimation

When estimating (3), we proxy for τσ(d)t with the ATR for a hypothetical taxpayer at theninety-ﬁfth or ninety-ninth percentile of the U.S. income distribution because, as in Moretti′
as consistingand Wilson (2017), we do not observe top inventors’ income.9 We regard τσ(d)tof the CITR, ITC, and RTC. We use year ﬁxed eﬀects or region pair × year ﬁxed eﬀects andreport robust standard errors that allow for three-way clustering by commuting zone pair,origin-state × year, and destination-state × year.
Table 4 shows that the interstate migration result in Moretti and Wilson (2017) can be
replicated fairly well at the commuting zone level: The elasticity of the migration of top
inventors with respect to the diﬀerence in ATRs between origin and destination is positiveand signiﬁcant in all cases. In what follows, we use the result in Column (2) of Table 4 since
the speciﬁcation is most closely related to the baseline case in Moretti and Wilson (2017).

9
We report the result at the ninety-ﬁfth percentile in the main body and the result at the ninety-ninth
percentile in Appendix C.1 as a robustness check.

11

Table 4: The impact of tax diﬀerences on the migration of top inventors∆ ln(1 − ATR)
∆ ln(1 − CITR)∆ ln(1 + ITC)
∆ ln(1 + RTC)

(1)
7.357
(1.611)-0.435(1.058)0.172(0.737)0.323(0.443)(2)
6.902(1.420)-0.195(0.999)-0.083(0.688)0.311(0.395)(3)
6.406(1.292)-0.300(0.812)0.118(0.993)0.377(0.321)(4)
6.586(1.124)-0.140(0.717)-0.034(0.689)0.178(0.281)

Yes
No

Yes
No

No
Yes

No
Yes

Yes
No

No
Yes

Yes
No

No
Yes

CZ pair FE
Origin CZ FE anddestination CZ FE
Year FE
Region pair × Year FE

Observations
4866
4866
7226
7225
2
R (total)
0.893
0.904
0.907
0.917
2
R (within)
0.400
0.458
0.411
0.013
Notes: The dependent variable in each column is the logodds ratio in (3). ATR, CITR, ITC, and RTC stand for individual income average tax rate at the ninety-ﬁfth percentile,
corporate income tax rate, investment tax credit, and R&D
tax credit, respectively. Cluster-robust standard errors are inparentheses.

3.5

Bartik instrument

Our aim here is not to establish a causal relationship between inventor mobility and income
taxes. Instead, we construct a Bartik instrument from the estimated parameters in Column(2) of Table 4 as follows:
Bdt =

X

Pbodt Iot ,o̸=dwhich is used in the ﬁrst-stage regression that we explain in the next section. Observe thatPP
Bdt is the prediction of the top inventor inﬂows deﬁned as Mdt = o̸=d Modt = o̸=d Podt Iot .
We derive the predicted probability Pbodt that top inventors moved from o to d in year t by
the following three steps. First, equation (3) implies that for any pair of commuting zones c

12and d, Poct and Podt must satisfy
n
o
′
′
exp
η
ln(1
−
τ
)
+
η
ln(1
−
τ
)
+
γ
+
γ
c
oc
σ(c)t
σ(c)t
Poct
o.
n
=
Podt
′
exp η ln(1 − τσ(d)t ) + η ′ ln(1 − τσ(d)t) + γd + γod
Second, let C denote the set of commuting zones. SincePc∈C Poct = 1 holds, we have

n
o
′
′
exp
η
ln(1
−
τ
)
+
η
ln(1
−
τ
)
+
γ
+
γ
c
oc
σ(c)t
c∈C
σ(c)t
n
o .
′
exp η ln(1 − τσ(d)t ) + η ′ ln(1 − τσ(d)t) + γd + γod

PPc∈C Poct

Podt

=

1
=
Podt

Last, using the estimated parameters ηb, ηb′ , {bγc }c∈C , and {bγod }o,d∈C , we obtainno′′
exp ηb ln(1 − τσ(d)t ) + ηb ln(1 − τσ(d)t ) + γ
bd + γbodno,Pbodt = P
′ ln(1 − τ ′
exp
η
b
ln(1
−
τ
)
+
η
b
)
+
γ
b
+
γ
b
c
oc
σ(c)t
c∈C
σ(c)t

(4)

where ηb, ηb′ , {bγc }c∈C , and {bγod }o,d∈C are the estimated parameters.

4

The migration of top inventors and local patent productivity

We analyze the impact of top inventor inﬂows on local patent productivity. In this section, we
present a baseline case using a static framework. We then extend it to a dynamic setting in
the next section to check the robustness of our results. In both cases, we consider two typesof outcomes: (a) productivity gains of all local inventors and (b) those of external inventors.
The former will be interpreted as aggregate productivity gains from both internal knowledge
sharing and external knowledge spillovers. The latter will be viewed as the gains from external
knowledge spillovers because the gains go beyond organizational boundaries and co-inventor
relationships. Our main focus is on the latter since they pertain to what Marshall (1920)
referred to as the mysteries of trade in the air.

13

4.1

Empirical speciﬁcations

We start with the ﬁxed eﬀect (FE) model:

ln Ydt = ϕMdt + ξXdt + εdt ,(5)where Ydt is the productivity of inventors in destination commuting zone d in period t(measured by the number of applied for patents weighted by the number of inventors),P
Mdt =o̸=d Modt is the number of top inventors who migrate to destination commuting
zone d in period t, ϕ measures the productivity eﬀect, Xdt captures time-varying factors indestination commuting zone d and taxes in state σ(d), as well as commuting zone and time
ﬁxed eﬀects,10 and εdt is an i.i.d. shock. We assume that E(εdt |Xdt ) = 0, i.e., εdt is mean
zero conditional on Xdt .11 When estimating (5), we cluster standard errors at the commuting
zone level.12
However, the top inventor inﬂows Mdt may be correlated with εdt due to reverse causality or the existence of omitted variables that have direct impacts on both the top inventor
inﬂows and local patent productivity. Reverse causality arises when greater local patent productivity attracts top inventors, whereas omitted variables exist when there are unobserved
consumption and production amenities that have been studied since Roback (1982).
To address these endogeneity issues, we employ an IV regression, which consists of thestructural equationln Ydt = ϕs Mdt + ξ s Xdt + εsdt ,(6)

10
In the baseline case, we use the ATR at the ninety-ﬁfth percentile of the U.S. income distribution. As
robustness checks, we consider the ATR at the ninety-ninth percentile in Appendix C.1 and manufacturing
employment in commuting zone d and other taxes and tax credits in state σ(d) in Appendix C.2.
11
One may worry that the conditional mean assumption E(εdt |Xdt ) = 0 fails due to omitted variables. To
address this concern, we also consider a speciﬁcation that allows Xdt to include state-year ﬁxed eﬀects.
12
When estimating (5), we replace ln Ydt with ln(1 + Ydt ) in the baseline case to accommodate commuting
zone × year observations with no patents. As a robustness check, we drop such observations and estimate (5)while retaining ln Ydt . As reported in Appendix C.3, the results are quite similar to those in the baseline case.

14and the ﬁrst-stage regression
Mdt = ψ f Bdt + ξ f Xdt + εfdt

with Bdt =X

Pbodt Iot ,

(7)

o̸=d

where Iot is the number of top inventors in commuting zone o in the beginning of period t.
Since Pbodt is the ﬁtted value from the regression in (3), it captures the estimated probability
that an inventor, located in o at the beginning of period t, moves to d by the end of period t.
When estimating (6), we cluster standard errors at the commuting zone level.13
Recall that Bdt is a Bartik instrument, which consists of the shares Pbodt and the shifts Iot . It
is derived from a simple theory of migration, namely that the ﬂow of inventors to destination dis the sum of the products of these two elements. We follow the shares perspective (GoldsmithPinkham et al., 2020), i.e., it is the shares Pbodt that provide the exogenous variation satisfyingE(εdt Pbodt |Xdt ) = 0. Indeed, as seen from (4), the shares Pbodt depend not only on the tax′}, but also on the distribution of tax rates acrossrates in the destination state, {τσ(d)t , τσ(d)t′′states, {τσ(c)t , τσ(c)t}c∈C . Thus, the state taxes, {τσ(c)t , τσ(c)t}σ(c)̸=σ(d) , other than those in the′}, have an indirect eﬀect on the destination productivity Ydt only
destination, {τσ(d)t , τσ(d)t

via Pbodt . This indirect eﬀect, rather than the “pull” eﬀect captured by the destination taxes′}, works as an exclusion restriction for identifying the productivity eﬀect ϕs .14{τσ(d)t , τσ(d)t

We describe the conditions to ensure the share exogeneity E(εdt Pbodt |Xdt ) = 0 in Appendix E.
We further consider two variants of the Bartik instruments to assess the sensitivity of ourP
σresults. One is the prediction of the between-state top inventor inﬂows, Bdt= o∈σ(d)Pbodt Iot ,/and the other is the predicted top inventor inﬂows into commuting zone ν(d) in the nearestPνneighborhood of d, Bdt= o̸=d,ν(d) Pboν(d)t Iot .
13
When estimating (6), we replace ln Ydt with ln(1 + Ydt ) in the baseline case to accommodate commuting
zone × year observations with no patents. As a robustness check, we drop such observations and estimate (6)while retaining ln Ydt . As reported in Appendix C.3, the results are quite similar to those in the baseline case.
14
Although our identiﬁcation relies on the exclusion restriction that the tax in one state does not directly
aﬀect the local productivity in other states, state tax competition may induce their interstate correlation.
To alleviate this concern, we examine the possibility of strategic interactions among state governments by
estimating a reaction function such that the income tax in one state responds to the income taxes in its
neighboring states (see, e.g., Brueckner, 2003). As shown in Appendix D, we ﬁnd no evidence of state tax
competition, which is consistent with the exclusion restriction.

15

4.2

Main results

Table 5 reports the estimation results for the FE and IV regressions. Column 1 reports
the FE case. Columns 2-6 are the results for diﬀerent IV regressions. Column 2 considersPPσ
Bdt = o̸=d Pbodt Iot in (7). Columns 3 adds to Column 2 an instrument, Bdt= o∈σ(d)Pbodt Iot ,/which captures top inventor ﬂows only from other states. Column 4 further adds to Column 3Pν
an additional instrument, Bdt= o̸=d,ν(d) Pboν(d)t Iot , which involves top inventor ﬂows into
commuting zone ν(d) in the nearest neighborhood of d. Columns 5 and 6 report the results
when controlling for time-varying state-speciﬁc unobservables by year × destination state FE.
In both Panels (a) and (b), we exclude the patents by top inventors who moved in from thedependent variable.
As seen from Table 5, the semi-elasticities of local patent productivity with respect to top
inventor inﬂows, as well as the elasticities of local patent productivity with respect to ln(1 −ATR), are virtually identical for all IV regressions within each panel.15 Panel (a) in Table 5shows that an inﬂow of a top inventor raises the patent productivity of all local inventors
by approximately 6%. Panel (b) shows that an inﬂow of a top inventor raises local patentproductivity by approximately 4% when we focus on external inventors who are not directly
connected to the migrating top inventors. The latter result can be interpreted as evidence forthe existence of the mysteries of trade in the air as the number reﬂects neither knowledge ﬂows
within the same assignee nor those between co-inventors. Our results obtained from the IV
regressions thus diﬀer from Zacchia (2018), who ﬁnds no city-wide spillover eﬀect of inventor
inﬂows. Our 4 percent patent productivity gains for local inventors due to an additional
inventor inﬂow could be compared with the 12 percent gains for incumbent plants’ TFP due
to a new plant opening studied in Greenstone, Hornbeck, and Moretti (2010). However, given
the diﬀerence between inventor arrival and ﬁrm entry, it is not surprising that the former
eﬀect is smaller.16
15
The semi-elasticities of local patent productivity with respect to top inventor inﬂows are somewhat smaller
for the FE regression. One possible explanation for this is the presence of urban costs such as land rents andcommuting costs that are speciﬁc to commuting zones and can vary over time (see Duranton and Puga, 2020).
16
Our paper also diﬀers from De la Roca and Puga (2017) and Moretti (2021) in that these papers focus

16

Table 5: The impact of top inventor inﬂows on local patent productivity(a) All local inventors
Top inventor inﬂows
ln(1 − ATR)

(1)

(2)

(3)

(4)

(5)

(6)

0.043
(0.006)6.041(1.038)0.062(0.013)5.915(1.040)0.060(0.012)5.899(1.038)0.059(0.011)6.017(1.038)0.059(0.013)0.060(0.012)37.75537.41823.10915.06212.03933.37731.93019.892
13.09410.53133.04034.73421.38913.90111.09335.13031.21419.47312.83910.33634.99732.98920.36413.272
10.6100.042(0.010)4.684(0.851)

0.040(0.009)4.641(0.848)0.041(0.009)4.616(0.842)0.036(0.010)0.038(0.010)37.75537.41823.10915.06212.03933.37731.92119.88713.091
10.52933.04034.73821.39213.902
11.09435.13031.20319.46712.83510.33334.99732.98820.36413.272
10.610

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
No
Yes

Yes
No
Yes

Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%(b) External inventors
Top inventor inﬂows
ln(1 − ATR)

0.027
(0.004)4.781(0.848)Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%CZ FE
Year FE
Year × state FE

Yes
Yes
No

Observations
23,628 23,628 23,628 23,463 23,562 23,397
Notes: The coeﬃcient on top inventor inﬂows is converted to semi-elasticity. ATR
stands for the individual income average tax rate at the ninety-ﬁfth percentile.
The coeﬃcient on ln(1 − ATR) is converted to elasticity. Cluster-robust standard
errors are in parentheses. Column 1 does not control for endogeneity of top
inventor inﬂows. Column 2 uses Bdt as an instrument. Column 3 uses Bdt and
σ
σ
ν
Bdt
as instruments. Column 4 uses Bdt , Bdt, and Bdt
as instruments. Columns
5 and 6 replace ln(1 − ATR) in Columns 3 and 4 with Year × state FE.

We check the robustness of these results using the speciﬁcation curve analysis as in Simonsohn, Simmons, and Nelson (2020). We consider diﬀerent speciﬁcations of the IV regressions
by focusing on ﬁve dimensions.17 Figure 3 plots the speciﬁcation curve for ϕs with 90% andon the impacts on those who migrate.
17
We consider (i) whether to use the ATR at the ninety-ﬁfth or ninety-ninth percentile; (ii) whether to useσσνln(1 + Ydt ) or drop commuting zones with Ydt = 0; (iii) whether to use {Bdt }, {Bdt , Bdt} or {Bdt , Bdt, Bdt};(iv) whether to include state-year FE; and (v) whether to include each of the other controls. Since the usual
caveat on weak instruments is applicable here, we adopt only speciﬁcations for which the null hypothesis of
weak instruments is rejected.

17

Figure 3: Speciﬁcation curve analysis.

.01 .02 .03 .04 .05 .06 .07 .08 .09.01 .02 .03 .04 .05 .06 .07 .08 .09.1(b) external inventors.1(a) all local inventors

95% CI

90% CI

baseline estimate

other estimates

95% CI

90% CI0other estimates0baseline estimate

Notes: Panel (a) (Panel (b)) illustrates the impacts of inventor inﬂows on local patentproductivity without (by) excluding co-inventors and inventors in the same assignee. The
speciﬁcation curve is depicted using 409 alternative speciﬁcations explained in footnote 17.
The vertical axis is the value of ϕs .
95% conﬁdence intervals. As seen from Panels (a) and (b), the productivity gains of approximately 6% and 4% are fairly robust for 409 alternative speciﬁcations, thus verifying that thebaseline estimates do not come from data mining.18 In what follows, we use the speciﬁcationin Column 3 of Table 5 as a baseline unless otherwise stated.
4.3

Relevance and validity of empirical strategy

Our empirical strategy relies on relevance and validity of the Bartik instruments. We now
discuss these two assumptions.
To assess the relevance of the Bartik instruments, we ﬁrst plot in Figure 4 the relaP
tionship between the actual migration ﬂows Mdt = o̸=d Modt and the Bartik instruments
P
constructed from the ﬁtted migration ﬂows. For the latter, we consider Bdt = o̸=d Pbodt Iot ,Pσbbodt Iot , and B ν = P= o∈σ(d)PBdtdto̸=d,ν(d) Poν(d)t Iot , in Panels (a), (b), and (c), respectively.
/
18
We summarize the results for some representative cases of these alternative speciﬁcations in Appendix
C.1-C.3. In particular, we check the robustness of our results by using an alternative ATRs in Appendix C.1,
by including other controls in Appendix C.2, and by dropping commuting zone × year observations with nopatents in Appendix C.3. As seen from Tables C2, C3, and C4, the results are fairly robust.

18

Figure 4: Actual versus ﬁtted ﬂows of top inventors.

0

20

40

60

80

60
40
20
0

0

0

20

20

40

40

60

60

80

(c) ﬂows to neighboring cz

80(b) interstate ﬂows

80(a) all ﬂows

0

20

40

60

80

0

20

40

60

80

Notes: In each panel, the vertical and horizontal axes are the actual migration ﬂows and theBartik instruments constructed from the ﬁtted migration ﬂows, respectively. The actual ﬂows
in Panels (a), (b), and (c) are deﬁned as Mdt . The Bartik instruments in Panels (a), (b), andσν(c) are Bdt , Bdt, and Bdt
, respectively.
There is a positive relationship in each panel, and the correlation coeﬃcients for Panels (a),(b), and (c) are given by 0.78, 0.74, and 0.38, respectively.
We further apply a test for weak instruments developed by Montiel Olea and Pﬂueger(2013) to these Bartik instruments. The test is robust to heteroskedasticity, autocorrelation,and clustering (see also Andrews, Stock, and Sun, 2019). The bottom panel of Table 5 reports
the eﬀective F statistic, which is a scaled version of the nonrobust ﬁrst-stage F statistic.
Following their baseline, we set the threshold at τ = 10% and the signiﬁcance at 5%. In allcases, the eﬀective F statistic exceeds the critical value reported at τ = 10%, thus rejecting
the null hypothesis of weak instruments.
To address the validity of the Bartik instruments, we follow the shares perspective andfocus on the exogeneity of the shares Pbodt conditional on the characteristics of the commuting
zones Xdt . We assess the plausibility of the exogeneity assumption in two steps. We ﬁrst use
the decomposition result in Goldsmith-Pinkham et al. (2020) to rewrite the overall estimate
Pof the productivity eﬀect as ϕbs = o∈C ω
bos ϕbso , which consists of the origin-speciﬁc weight ωbosand the origin-speciﬁc productivity eﬀect ϕbso . The former is referred to as the Rotemberg

19weight (Rotemberg, 1983) and measures to what extent the bias originating from commuting
zone o contributes to the overall bias. We then assess the exogeneity of the shares {Pbodt }d∈C
for commuting zones o with the top ﬁve highest Rotemberg weights ωbos by considering thecorrelation between the predicted shares {Pbodt }d∈C and the characteristics of the commuting
zones {Xdt }d∈C .
We summarize the results in Appendix F. Table F1 presents the summary of the Rotemberg weights. The origin commuting zones with the top ﬁve highest weights are Bergen-EssexMiddlesex, Cook-DuPage-Lake, Kings-Queens-NewYork, Philadelphia-Montgomery-Delaware,and Allegheny-Westmoreland-Washington.19 Table F2 reports the destination commuting
zones to which top inventors moved from these top ﬁve origin commuting zones. The resultthat most migrations are interstate is in line with the assumption that the main source ofidentifying variation comes from inventor mobility caused by personal income tax diﬀerences
between states. Table F3 reports, for each top ﬁve origin commuting zone, the correlation
between the share and the log of manufacturing employment conditional on the controls andﬁxed eﬀects.20 Reassuringly, the correlations are low, and the regression coeﬃcients on thelog of manufacturing employment are not statistically signiﬁcant at the ﬁve percent level inall speciﬁcations.

5

Robustness

In this section, we examine the robustness of our main results in terms of time, space, andaggregation. We ﬁrst extend our static framework to a dynamic setting, which allows us toassess the impacts on local patent productivity before and after top inventor inﬂows. We
then check the robustness of our main results in terms of the geographic extent of produc19
The name of each commuting zone shown here is a list of three counties with the largest numbers of
inventors (in descending order) in that commuting zone.
20
Table C3 shows that manufacturing employment is related to local patent productivity. We thus use
manufacturing employment as an observable confounder. In principle, we could examine the correlation of
the share with other commuting zone characteristics that aﬀect local patent productivity. However, in practice,such time-varying data on commuting zone characteristics are not readily available for the time period that
we consider.

20

tivity gains. We ﬁnally consider a disaggregated case where local inventors are classiﬁed bytheir patent productivity. In Appendix C, we further conduct additional robustness checks
by considering: the case where the ATR for a hypothetical taxpayer is assumed to be at
the ninety-ninth percentile of the U.S. income distribution; the case where we include othercontrols in our baseline speciﬁcation; and the case where we drop commuting zone × year
observations with no patents in our main analysis.
5.1

Dynamic analysis

As a ﬁrst robustness check, we conduct event study analysis to examine whether the baseline
results are sensitive to the inclusion of lead and lag eﬀects of top inventor inﬂows. To thisend, we extend (6) as follows:

ln Ydt =jX

ϕℓj Mdt−j + ξ ℓ Xdt + εℓdt ,

(8)

j=j+1

which is a distributed lag model in levels with a binning window [j + 1, j]. Thus, when j = −1
and j = 0, the model degenerates into the static model (6). Schmidheiny and Siegloch (2023)
show that equation (8) is equivalent to the event study model given by21

ln Ydt =

j
X

(j)

es
es
µes
j ∆Mdt + ξ Xdt + εdt ,

(9)

j=j

where

 P
j

(Mdt−k − Mdt−k−1 ) if j = j < 0 k=−∞(j)∆Mdt =
Mdt−j − Mdt−j−1
if j < j < j .



 P∞
if j = j > 0
k=j (Mdt−k − Mdt−k−1 )

(10)

es
es
es
es
Our aim is to estimate {µesj , µj+1 , ..., µj−1 , µj } with normalization µ−1 = 0. The event study
escoeﬃcients capture the cumulative eﬀect of the event of top inventor inﬂows, i.e., µes
j = µj−1 +21
Unlike in standard event study models with a single treatment of identical intensity, we consider a moregeneral case with multiple treatments of varying intensities. See Schmidheiny and Siegloch (2023) for thedetailed classiﬁcation of event study models.

21

ϕℓj =

Pj

ℓ
es
es
ℓ
h=0 ϕh for j = 0, 1, ..., j and µj = µj+1 − ϕj+1 = −P−1ℓ
h=j+1 ϕh for j = −2, −3, ..., j.

Thus, the coeﬃcients for j ≥ 0 denote cumulative productivity eﬀects from event year 0(when there are top inventor inﬂows) to year j. Since the static model abstracts from thelead and lag eﬀects, the baseline model may produce biased estimates of productivity gains.
As in the static analysis, we incorporate the Bartik instruments into the event study model.
(j)Let ∆B dt = [∆Bdt · · · ∆Bdt ]′ denote a (j + j + 1) × 1 vector of the ﬁrst time diﬀerence of the
(j)

(j)

IVs, where ∆Bdt is deﬁned in a similar way as in (10). The IV event study model consistsof the structural equation (9) and the ﬁrst-stage regression analogous to (7) as follows22(j)∆M dt = ψ ef
where ψ ef(j)= [ψ ef(j,j)· · · ψ ef

(j,j)

(j)

∆B dt + ξ ef(j)

Xdt + εef
dt

(j)

,

(11)

] is a vector of coeﬃcients.23

Figure 5 shows the results for the IV event study regressions. Panel (a) illustrates the
dynamic impact of top inventor inﬂows on the patent productivity of all local inventors,
which includes not only internal knowledge sharing within the same assignee and between
co-inventors but also external knowledge spillovers. Panel (b) corresponds to the dynamic
productivity gains of external inventors, which go beyond organizational boundaries and coinventor relationships. In both cases, we observe a substantial increase in local patent productivity in event year 0 when there are top inventor inﬂows. The post-event semi-elasticities
go up to approximately 0.05, which ensures our baseline results in Section 4.2. In contrast,
the pre-event semi-elasticities are close to zero and are not statistically signiﬁcant in any
pre-event year in Panel (b), thus suggesting no productivity gains prior to the event of top
inventor migration.24
22

In this robustness check we abstract from the possibility that treatment eﬀects can be heterogeneous.
Although several recent papers have explored under what conditions event study models provide valid averagetreatment eﬀects in the presence of heterogeneous treatment eﬀects (e.g., de Chaisemartin and D’Haultfœuille,2023), they are not readily applicable to our IV event study model with multiple treatments of varying
intensities.
23

σ(j)

When estimating the event study models with multiple instruments, we set ∆B σdt = [∆Bdt· · · ∆Bdt ]′σ(j)and ∆B νdt = [∆Bdt · · · ∆Bdt ]′ and use [∆B dt ′ ∆B σdt ′ ]′ or [∆B dt ′ ∆B σdt ′ ∆B νdt ′ ]′ as instruments in (11).
24
We report the numbers used in Figure 5 and the associated ﬁrst-stage statistics in Tables C5 and C6,
respectively. In Appendix G, we further assess robustness to possible violations of the parallel trends assumpν(j)

ν(j)

22

Figure 5: IV event study regressions.

.1(b) external inventors.1(a) all local inventors

IV ES1

IV ES2

0
−.05

−.05

0

.05

IV ES2.05

IV ES1

−5

−4

−3

−2

−1

0

+1

+2

+3

+4

+5

−5

−4

−3

−2

−1

0

+1

+2

+3

+4

+5

Notes: Panel (a) (Panel (b)) illustrates the dynamic impact of inventor inﬂows on local
patent productivity without (by) excluding co-inventors and inventors in the same assignee.
σνσ
In each panel, IV ES1 uses Bdt and Bdt
as instruments and IV ES2 uses Bdt , Bdt, and Bdt
as
instruments.

5.2

Geographic space

To check the robustness of our main results in terms of the geographic extent of productivity
gains, we replace the structural equation in (6) withln Ydt =

6
X

sr
sr
ϕsr
r(d) Mr(d)t + ξ Xdt + εdt ,(12)r(d)=1

where r(d) is the distance ring deﬁned for each destination commuting zone d and Mr(d)t is
the predicted inﬂows of top inventors in the r(d)-th ring. The ﬁrst ring r(d) = 1 stands fordestination d itself. r(d) ̸= 2, ..., 6 corresponds to commuting zones that are 0-50, 50-100,
100-150, 150-200, and 200-250 miles away from commuting zone d.25
6
Figure 6 illustrates the estimated coeﬃcients {ϕbsrr(d) }r(d)=1 . Panel (a) (Panel (b)) illustrates

the impact of top inventor ﬂows in r(d) on patent productivity in d without (by) excluding
co-inventors and inventors in the same assignee. In each panel, the impact is signiﬁcant onlytion, following the method developed by Rambachan and Roth (2023).
25
The distance between any pair of two commuting zones is calculated using the great circle formula.

23

Figure 6: Distance-ring regression.

.1(b) external inventors.1(a) all local inventors
IV3

IV1

IV2

IV3

0
−.05

−.05

0

.05

IV2

.05

IV1

same cz

0−5050−100100−150150−200

200−250

same cz

0−5050−100100−150150−200

200−250

Notes: Panel (a) (Panel (b)) illustrates the impact of inventor inﬂows on local patent productivity without (by) excluding co-inventors and inventors in the same assignee. In eachσpanel, IV1 uses Bdt as an instrument, IV2 uses Bdt and Bdt
as instruments, and IV3 usesσν
Bdt , Bdt , and Bdt as instruments.
in the ﬁrst ring for all three diﬀerent IVs, which implies that top inventor inﬂows aﬀect patentproductivity only in the commuting zone where they enter. Such localized productivity gains
are reminiscent of localized knowledge spillovers in Jaﬀe et al. (1993) and Murata et al. (2014).
We will discuss the mechanism of localized productivity gains in terms of localized knowledge
spillovers as evidenced by patent citations in Section 6.1.
We also conduct a permutation-based placebo test to assess the plausibility of our ﬁndingsthat productivity gains are localized within each commuting zone. This is done by examining
the impact of top inventor migration into commuting zone d on productivity gains in arandomly drawn hypothetical commuting zone R(d) ̸= d in state σ(d). We thus replace thestructural equation in (6) withln YR(d)t = ϕsR Mdt + ξ sR Xdt + εsR
dt

(13)

and estimate (13) for each IV speciﬁcation 1000 times with replacement to obtain the distri1000
sR
bution of {ϕsR= 0,i }i=1 . We then check if the null hypothesis of no productivity gains, ϕ24Figure 7: Placebo.

.02
.015
.01
.005
0
IV1

IV2

−.005

−.005

0

.005

.01

.015

.02

.025

(b) external inventors.025(a) all local inventors

IV3

IV1

IV2

IV3

Notes: Panel (a) (Panel (b)) illustrates the impact of inventor inﬂows on local patent productivity without (by) excluding co-inventors and inventors in the same assignee when ln Ydt
in the structural equation is replaced with ln YR(d)t , where R(d) ̸= d is a commuting zonethat is randomly drawn from state σ(d) to which commuting zone d belongs. In each panel,σσ,
as instruments, and IV3 uses Bdt , Bdt
IV1 uses Bdt as an instrument, IV2 uses Bdt and Bdt
νand Bdt as instruments.
is rejected. Figure 7 depicts the 95% conﬁdence interval and the mean of the distribution foreach IV speciﬁcation. The results in both Panels (a) and (b) show that top inventor ﬂows
into commuting zone d do not signiﬁcantly change patent productivity in randomly drawn
commuting zone R(d) ̸= d from state σ(d), thus implying that the extent of productivity
gains is geographically limited within each commuting zone.
5.3

Productivity gains by inventor productivity

We have thus far shown that top inventor inﬂows enhance patent productivity only for local
inventors. Since local inventors diﬀer in their patent productivity, we further delve into theforegoing result by addressing who gain more from top inventor inﬂows. To this end, wedisaggregate local inventors by their patent productivity and estimate the causal eﬀect foreach productivity level.
Panel (a) of Figure 8 illustrates the case with all local inventors. We observe that there

25

Figure 8: Productivity gains by inventor productivity.
(a) all local inventors(b) external inventors
IV3

IV1

IV2

IV3

.1
.05
0

0

.05

.1

.15

IV2

.15

IV1

top5

top10

top25

top50

top75

all

top5

top10

top25

top50

top75

all

Notes: Panel (a) (Panel (b)) illustrates the impact of inventor inﬂows on local productivity
gains by inventor productivity without (by) excluding co-inventors and inventors in the sameσassignee. In each panel, IV1 uses Bdt as an instrument, IV2 uses Bdt and Bdt
as instruments,σνand IV3 uses Bdt , Bdt , and Bdt as instruments.
are productivity gains at each productivity level and that more-productive local inventors
tend to gain more from top inventor inﬂows. We ﬁnd a similar pattern in Panel (b). Hence,even when we focus on external inventors who are not directly connected to the migrating top
inventors, our result suggests that more-productive local inventors beneﬁt more from theirinﬂows.

6

Mechanisms

The productivity gains estimated in the previous sections suggest that local inventors acquire
knowledge from migrating top inventors, regardless of whether local inventors are internalor external. We now discuss the underlying mechanisms through which those productivity
gains materialize. We ﬁrst focus on patent citations that have been widely used as proxy forknowledge ﬂows since Jaﬀe et al. (1993). Speciﬁcally, we count how many times local inventors
cite incoming top inventors and estimate the percentage change in the number of citations
caused by top inventor inﬂows. Furthermore, Jaﬀe et al. (1993) recognize the existence of26other knowledge ﬂows that cannot be captured by patent citations. We thus complement theforegoing analysis by using state-year variation in legal protection of trade secrets documented
by Png (2017a, 2017b) as a quasi-natural experiment. We expect that knowledge ﬂows from
migrating top inventors to local external inventors would be greater in states where legal
protection of trade secrets is weaker, so that there would be additional local productivity
gains in those states. Our results presented below are consistent with Marshall’s insight onknowledge spillovers since external inventors can not only learn patentable knowledge but
also obtain other forms of knowledge from migrating top inventors as if they were in the air.
6.1

Patent citations

To see the impact of top inventor inﬂows on local patent citations, we count how many times
the patents of the top inventors who migrated into commuting zone d in year t were cited by
the local inventors in commuting zone d in year t and denote it by Cdt . When constructing
Cdt , we focus on the patents that had been applied over the last ten years. Replacing patentproductivity Ydt in (6) with the number of citations Cdt , we consider the structural equation
for citations as follows:
ln Cdt = ϕsc Mdt + ξ sc Xdt + εsc
dt ,while retaining the same ﬁrst-stage equation (7). The coeﬃcient ϕsc gauges the magnitude ofknowledge ﬂows from migrating top inventors to local inventors.
Panels (a) and (b) of Figure 9 illustrate the estimated coeﬃcients for the case of all local
inventors and that of external inventors, respectively. In both panels, local inventors areclassiﬁed by their productivity. In Panel (a), an additional top inventor inﬂow raises thenumber of local inventors’ citations to incoming top inventors by 10-20% regardless of the
productivity of local inventors. By contrast, in Panel (b), the external inventors, especiallythose with higher productivity, tend to cite substantially more patents of the top inventors
who moved in the same commuting zone. These results suggest the existence of knowledge
ﬂows from the migrating top inventors to the local inventors, even when we focus on the27

Figure 9: Citations.

.5
.4
.3
.2
.1
0

0

.1

.2

.3

.4

.5

.6

(b) external inventors.6(a) all local inventors

top5

top10

top25

top50

top75

all

top5

top10

top25

top50

top75

all

Notes: Panel (a) (Panel (b)) illustrates the impact of an additional top inventor inﬂow on
the number of all local inventors’ (external inventors’) citations to incoming inventors (inσpercentage). In each panel, we use Bdt and Bdt
as instruments.
external inventors who are not directly connected to the migrating top inventors.26
6.2

Trade secrets

Trade secrets were formerly deﬁned and protected from misappropriation by common law in
the United States. However, these deﬁnitions and protections have been codiﬁed into law withthe enactment of federal legislation known as the Uniform Trade Secrets Act (UTSA). Whilemost states had already adopted the UTSA, there had been substantial heterogeneity in thestates’ approaches to trade secrets due to the diﬀerences in the timing of the adoption of the
UTSA and the strength of trade secrets protection during the common law era. We exploit
the state-year variation in trade secrets protection to uncover productivity gains throughknowledge ﬂows that cannot be captured by patent citations.
Given the heterogeneity in legal protection of trade secrets, inventors who migrate to a26
It is perhaps puzzling that the impact is smaller in Panel (a). One possible explanation is that internal
inventors, who account for approximately 40% of all local inventors, had already collaborated with or worked in
the same organization as the incoming top inventors and thus had already cited them prior to their migration.
In that case, we would expect less additional percentage changes in the number of internal inventors’ citations
after the arrival of the top inventors.

28

state with weaker protection would exchange knowledge more with other inventors outside
their organizations, thereby bringing about additional productivity gains to local external inventors. In contrast, the change in legal protection of trade secrets would not aﬀect knowledge
sharing within an organization, thus leaving the productivity gains of local internal inventors
unaﬀected.
We examine those diﬀerential impacts of top inventor migrations on local patent productivity. To this end, we use the state-level index of trade secrets in Png (2017a, 2017b), which
captures both the legal protection under common law and the enactment of the UTSA.27 The
index ranges between 0 and 1, and a higher score implies stronger legal protection. Let Sσ(d)tdenote an indicator variable for whether the trade secrets index in state σ(d) in year t is below
the median of the trade secrets index distribution. If Sσ(d)t = 1, the degree of trade secrets
protection is low in commuting zone d in year t, so that we expect higher patent productivity
due to a greater amount of knowledge brought about by inventor migration.
Let ϕs = [ϕs0 ϕs1 ϕs2 ] and Edt = [Mdt Sσ(d)t Mdt Sσ(d)t ]′ denote a vector of coeﬃcients and avector of endogenous variables. The structural equation is then given byln Ydt = ϕs Edt + ξ s Xdt + εsdt .

(14)

Our interest is in the coeﬃcient ϕs2 on Mdt Sσ(d)t . If ϕs2 > 0, top inventor inﬂows lead tohigher local patent productivity in commuting zones with lower trade secrets protection,
which suggests that knowledge brought about by top inventor migration is more likely to spillover to local inventors in commuting zones with lower trade secrets protection.
σ
As before, we use the Bartik instruments Bdt and Bdt
for the top inventor ﬂows Mdt tomitigate the endogeneity concern. To address the potential endogeneity of the trade secrets
indicator Sσ(d)t , we follow Png (2017b) who argues that the enactment of the UTSA is related
to the enactment of other state-level uniform laws such as the Uniform Determination of Death
Act (UDDA), Uniform Federal Lien Registration Act (UFLRA), and Uniform Fraudulent27
Png (2017a) provides the index for the years 1979 to 1998, and Png (2017b) extends it to the years 1970
to 2010.

29

Figure 10: Trade secrets.

.15
.1
.05
0
−.05

−.05

0

.05

.1

.15

.2

(b) external inventors.2(a) all local inventors

top5

top10

top25

top50

top75

all

top5

top10

top25

top50

top75

all

Notes: Panel (a) (Panel (b)) illustrates the coeﬃcients ϕs2 on the interaction term Mdt Sσ(d)tin (14) without (by) excluding co-inventors and inventors in the same assignee. In eachσ
as instruments.
panel, we use Bdt and Bdt
Transfer Act (UFTA) because these laws were introduced to harmonize state laws. Since thethree laws are unlikely to be associated with local shocks, we use them as instruments for the
trade secrets indicator. The ﬁrst-stage equation that accompanies (14) is thus given by
Edt = ψ f Zdt + ξ f Xdt + εfdt ,

(15)

FTA ′
FLRA
σ
DDA
)] is a vector, Uσ(d)t, Uσ(d)twhere ψ f is a vector of coeﬃcients and Zdt = [(Bdt , Bdt)##(Uσ(d)tℓof IVs, with Uσ(d)tindicating whether uniform law ℓ = {DDA, FLRA, FTA} was in eﬀect instate σ(d) in year t, and ## denotes an interaction-term operator.28
Panel (a) in Figure 10 illustrates the result for all local inventors. Since the strengthof trade secrets protection is unlikely to aﬀect internal knowledge sharing within the sameassignee and between co-inventors, it is not surprising that the overall impact is insigniﬁcantregardless of the productivity of local inventors. Panel (b) in Figure 10 presents the result
for external inventors, where the impacts for top 5, top 10, and top 25 local inventors (top28
The interaction-term operator ## generates all possible combinations of elements for a given pair of sets.
For example, let S1 = {A, B} and S2 = {C, D}, where each set Si has two elements. Then, [S1 ##S2 ]′ =[{A, B}##{C, D}]′ = [A B C D AC AD BC BD]′ .

30

50 and top 75) are signiﬁcant at the ﬁve percent (ten percent) level. Hence, top inventor
migration tends to enhance the patent productivity of external inventors in commuting zones
with lower trade secrets protection.

7

Counterfactuals

We now conduct a counterfactual experiment. Using the baseline speciﬁcation in Section 4,
we consider what happens to the geographic distribution of patent productivity if all state
individual income taxes are set to their average. This experiment is useful for assessing towhat extent state tax diﬀerences contribute to patent productivity diﬀerences across space.
Recalling that the changes in state taxes aﬀect the choice probabilities Pbodt in (4) and theσBartik instruments Bdt in (7), as well as Bdt, the procedure of the counterfactual experimentcan be summarized as follows. We ﬁrst derive the counterfactual probabilities Peodt to construct
edt , Be σ }, which allow us to estimate the counterfacthe counterfactual Bartik instruments {Bdt
fdt via the ﬁrst-stage regression (7). We then deﬁne the counterfactual
tual migration ﬂows M
fdt −Mcdt
e dt = M
cdt , and M
fdt are the acchanges in the migration ﬂows as ∆M
Mdt , where Mdt , McM
dttual, ﬁtted, and counterfactual migrations ﬂows, respectively.29 We ﬁnally apply the countere dt to the structural equation (6) to construct the counfactual changes in migration ﬂows ∆M
e ln Ydt = ϕbs ∆M
e dt + ξbs ∆e ln(1 − ATRσ(d)t ),terfactual changes in the log patent productivity ∆e ln(1 − ATRσ(d)t ) = ln(1 − ATR] σ(d)t ) − ln(1 − ATRσ(d)t ) captures the counterfactual
where ∆
e ln Ydt can be decomposed into
tax changes.30 Thus, the overall impact of tax changes on ∆e ln(1 − ATRσ(d)t ) and the indirect eﬀect via the changes in migration
two: the direct eﬀect ξbs ∆e dt . The indirect eﬀect can be further decomposed into two: productivity gains
ﬂows ϕbs ∆M
due to internal knowledge sharing and those due to external knowledge spillovers.
e dt = M
fdt − Mdt . However, since theIf the actual and ﬁtted ﬂows coincide, the deﬁnition reduces to ∆Mfc
actual and ﬁtted ﬂows generally diﬀer, we compute the percentage change in the migration ﬂows Mdtc−Mdt
Mdt
based on the ﬁtted and counterfactual ﬂows in the tax induced migration model and then multiply it by theactual ﬂows Mdt .
30
e ln Ydt with ∆e ln(1 + Ydt ) as before to accomWhen computing the counterfactual change, we replace ∆modate commuting zone × year observations with no patents.
29

31

Figure 11: Counterfactual experiment (Setting state taxes to their average).

(40,80](30,40](20,30](10,20](0,10](−10,0](−20,−10](−30,−20](−40,−30][−70,−40]
No patents

Notes: The ﬁgure illustrates the counterfactual percentage change in the number of patents
at the commuting zone level when all state individual income taxes in 2009 are set to theirσaverage. We use Bdt and Bdt
as instruments.
Figure 11 illustrates the percentage change in local patent productivity when state taxes
are set to their average. The overall impact tends to be large in commuting zones in California,
Oregon, North Carolina, and New York, where state taxes and initial patent productivity are
high.31 Table 6 summarizes the top 10 commuting zones by patent productivity gains. For
instance, if state taxes were equal, the number of patents in Santa Clara–Monterey–Santa
Cruz (which is the commuting zone with the highest patent productivity in Table 2) would
be larger by 72.3%. In contrast, the overall impact tends to be small in commuting zones
in Texas, Washington, Florida, and New Hampshire, where state taxes are low and initialpatent productivity is high. Table 7 summarizes the bottom 10 commuting zones by patentproductivity gains. For instance, if state taxes were equal, the number of patents in King–
Pierce–Snohomish (which is ranked as the tenth most productive commuting zone in Table 2)
31
Notably, the counterfactual changes are heterogeneous even within states, although we equalize taxes
between states. The reason is that the counterfactual choice probabilities Peodt , which are obtained by setting
all state taxes equal in (4), include ﬁxed eﬀects at the commuting zone level.

32

Table 6: Top 10 commuting zones by patent productivity gains (%).
rank cz number counties
state gains (%)
137500 Santa Clara–Monterey–Santa CruzCA
72.291237800 Alameda–Contra Costa–San Francisco
CA53.377338300 Los Angeles–Orange–San BernardinoCA
47.115438801 Multnomah–Washington–Clackamas
OR
46.022538000 San DiegoCA
38.0421701 Wake–Durham–Orange
6NC
35.253
7
19400 Kings–Queens–New YorkNY
35.080838901 Lane–Marion–Linn
OR
31.310935801 Ada–Canyon–Elmore
ID
29.7541039203 Deschutes–Crook–JeﬀersonOR
29.206
Notes: The patent productivity gains are deﬁned as the percentage change in
the number of patents when all state individual income taxes in 2009 are set
σto their average. We use Bdt and Bdt
as instruments.

Table 7: Bottom 10 commuting zones by patent productivity gains (%).
rank cz number counties
state
gains (%)
1
39400 King–Pierce–Snohomish
WA-64.777231201 Travis–Williamson–Hays
TX-50.710332000 Harris–Fort Bend–Galveston
TX-49.002433100 Dallas–Denton–Collin
TX-46.491520600 Hillsborough–Rockingham–YorkNH-41.32667100 Palm Beach–St. Lucie–MartinFL-38.19477400 Orange–Seminole–LakeFL-35.21985202 Shelby–DeSoto–Tipton
TN-35.02096900 Sarasota–Manatee–CharlotteFL-34.585107000 Dade–Broward–MonroeFL-34.561
Notes: The patent productivity gains are deﬁned as the percentage change in
the number of patents when all state individual income taxes in 2009 are set
σto their average. We use Bdt and Bdt
as instruments.

would be smaller by 64.8%. These results suggest that the presence of state tax diﬀerences
signiﬁcantly distorts the spatial distribution of inventive activity.
To see which states are most aﬀected by the presence of tax diﬀerences, we ﬁrst deﬁne,
e dt =
for each commuting zone d, the counterfactual change in the number of patents ∆YYedt −Ybdt
e dt , where Ydt , Ybdt , and Yedt are the actual,
Ydt in the same way as we deﬁne ∆MYbdt

ﬁtted, and counterfactual numbers of patents in commuting zone d, respectively. We thene dt within each state σ to obtain the counterfactual changes in the number ofaggregate ∆Ye σt = P ∆Ye dt . Denoting by Yσt = P Ydt the actual number of patents at thepatents ∆Yd∈σd∈σ
state level, we ﬁnally compute the percentage change in the number of patents at the state33

Figure 12: Counterfactual experiment (Setting state taxes to their average)

CA
NY
MA
MI
TX
external
internal
direct

WA
−60

−50

−40

−30

−20

−10

0

10

20

30

40

50

60

Notes: The ﬁgure illustrates the counterfactual percentage change in the number of patents
for the states of California, New York, Massachusetts, Michigan, Texas, and Washington when
all state individual income taxes in 2009 are set to their average. The overall change for eachstate is decomposed into the direct eﬀect via the change in state taxes and the indirect eﬀect
via the tax-induced migration. The latter consists of the internal knowledge sharing eﬀect
σ
and the external knowledge spillover eﬀect. We use Bdt and Bdt
as instruments.
eσt 32
level ∆Y.
YσtFigure 12 illustrates the percentage change in the number of patents for selected states
when state taxes are set to their average. For instance, if state taxes were equal, the numberof patents in California (where state taxes and patent productivity are high) would be greaterby 55.1%, which can be decomposed into the direct eﬀect via the reduction in California statetaxes (15.6%) and the indirect eﬀect via the tax-induced migration (39.5%). The indirect
eﬀect can be further decomposed into the internal sharing eﬀect (31.4%) and the external
spillover eﬀect (8.1%). In contrast, the number of patents in Texas (where state taxes are low
P


e dt
P
edt −Y
bdt
e σt
∆Y
Y
Pd∈σ
P Ydt
=
Observe that ∆Y=. The latter is the weighted average of the
d∈σ
bdt
Y
d∈σ Ydt d∈σ Ydt
Yσteb
percentage change YdtYb−Ydt with weight being the share of actual number of patents P YdtYdt . If the actuald∈σ
dt
and ﬁtted numbersPof patents
coincide,thepercentagechangeinthenumberofpatents
at the state level
P
32

e

σt
reduces to ∆Y
Yσt =

e

Ydt −
d∈σP

d∈σ Ydtd∈σ Ydt.

34

and patent productivity is high) would be smaller by 45.3%, which can be decomposed into
the direct eﬀect via the rise in Texas state taxes (28.5%) and the indirect eﬀect via the taxinduced migration (16.8%). The indirect eﬀect can be further decomposed into the internal
sharing eﬀect (12.7%) and the external spillover eﬀect (4.1%).
These results suggest that the indirect eﬀect via tax-induced migration can be substantial.
To see the relative role of the direct and indirect eﬀects at the national level, we aggregate
those changes in the number of patents across all commuting zones in all states that we
consider in the paper. We ﬁnd that the share of the indirect eﬀect is 0.735 (and that of thedirect eﬀect is 0.265). Our results thus complement Akcigit et al. (2022) who assess the direct
impact of state taxes on innovation.

8

Concluding remarks

In this paper we have uncovered the idea-generating process described by Marshall (1890)
using Bartik (1991) instruments. We have identiﬁed a signiﬁcant causal eﬀect of top inventor
inﬂows on the patent productivity of all local inventors. Even when we focus on local external
inventors who are not directly connected to incoming inventors through organizations or coinventor relationships, the eﬀect remains signiﬁcant and is approximately 4%, thus implyingthat the mysteries of the trade are in the air.
Since our analysis has disentangled productivity gains due to pure knowledge spillovers
from those due to market-mediated knowledge transfer, our ﬁndings are consistent with thepartially nonexcludable good nature of knowledge, whose implications have been exploredtheoretically in the technology and growth literature. The existence of the gains from pureknowledge spillovers constitutes a rationale for spatial agglomeration of inventive activity,
thus contributing to the innovation policy literature that considers both the beneﬁts andcosts of entrepreneurial clusters.
Our counterfactual experiment suggests that the existence of tax diﬀerences across states
distort the spatial distribution of inventive activity up to −64.8% to 72.3% with considerable

35

spatial heterogeneity. The decomposition of those gains and losses reveals that not only
the direct gains from tax changes but also the indirect gains from tax changes through top
inventor migration matter. Thus, more research is needed to bridge the gap between the tax
and innovation literature and the tax and migration literature.

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39

Appendix A
A.1

Data Appendix

Data sources and data construction

Patent data. The main data come from the USPTO PatentsView website (https://patentsview.org/). It includes data on patents, inventors, inventors’ addresses, assignees,and patent citations and provides data ﬁles regarding the disambiguation of inventor andassignee names (https://patentsview.org/disambiguation/). Additional procedures areused to allocate inventors’ addresses to commuting zones. We ﬁrst use the latitude and longitude of each inventor’s address (which are taken from the PatentsView data) to identify
his/her county of residence. We then relate it to the commuting zone in which the inventor
resides based on the correspondence table between counties and commuting zones in 1990 on
the IPUMS USA website (https://usa.ipums.org/usa/volii/1990lma.shtml).
State taxes and tax credits. Data on U.S. state taxes and tax credits for 1976-2019
are distributed on the Open ICPSR web page (https://www.openicpsr.org/openicpsr/
project/113057/version/V1/view). Summary statistics are presented in the Online Appendix for Moretti and Wilson (2017).
Employment data. The employment data are taken from the County Business Patterns(CBP) database (http://fpeckert.me/cbp/). Eckert et al. (2021) provide a detailed description of the data. Since the original employment data is at the county level, we aggregate
it at the commuting zone level. We use 2012 NAICS codes 31-33 to obtain the number of
employees in manufacturing.
Trade secrets data. Data on trade secrets are compiled by Png (2017a, b). Each state has
six binary scores regarding the strength of legal protection of trade secrets under the common
law and the Uniform Trade Secrets Act (UTSA). The trade secrets index used in our analysis
is the sum of the six scores divided by six, which takes a value between zero and one.

40

Data for state tax competition analysis. Data on the socio-politico-economic characteristics used in the state tax competition analysis are taken from multiple sources. The data on
population for various age and race groups are from the “U.S. Intercensal County Population
Data by Age, Sex, Race, and Hispanic Origin” web page (https://www.nber.org/research/data/us-intercensal-county-population-data-age-sex-race-and-hispanic-origin)
operated by NBER. The key economic and state ﬁnance data are from the “State Economic
and Government Finance Data” web page (https://doi.org/10.7910/DVN/CJBTGD) provided by Klarner (2015). The political party aﬃliation of each state governor is from the
National Governors Association web page (https://www.nga.org/governors/).
A.2 Other summary statistics
Table A1: Summary statistics at the commuting zone level (other variables)
mean
sd
min
max
ATR
0.238
0.030 0.164
0.330
0.315
0.410
ATR99
0.032 0.244
ATR50
0.108
0.027 0.033
0.169
CITR
0.064
0.027 0.000
0.138
ITC
0.009
0.023 0.000
0.100
RTC
0.022
0.038 0.000
0.250
TSI
0.340
0.235 0.000
0.767
22398.524 61797.154 0.000 1152493.572
EMP# CITES ALL
21.966347.065 0.000
17344.000# CITES EXT
2.098
37.218 0.000
2215.000
Number of observations
23,628
Number of commuting zones
716
Number of years
33
Notes: Summary statistics are based on the data described in Section 2 forthe years 1977 to 2009. ATR (ATR99, ATR50), CITR, ITC, RTC, and TSI
stand for individual income average tax rate at the ninety-ﬁfth (ninety-ninth,ﬁftieth) percentile, corporate income tax rate, investment tax credits, R&D
tax credits, and trade secrets index, respectively. EMP, # CITES ALL, and# CITES EXT are manufacturing employment, the number of citations by
all local inventors, and the number of citations by local external inventors.

41

Table A2: Summary statistics at the state level (other variables)
mean
sd
min
max
ATR
0.240 0.030 0.164 0.330
ATR99
0.317 0.032 0.244 0.410
ATR50
0.108 0.026 0.033 0.169
CITR
0.067 0.028 0.000 0.138
ITC
0.009 0.022 0.000 0.100
0.024 0.044 0.000 0.250
RTC
TSI
0.339 0.227 0.000 0.767
Number of observations
1,584
Number of states
48
Number of years
33
Notes: Summary statistics are based on the data described in Section 2 for the years 1977 to 2009. ATR(ATR99, ATR50), CITR, ITC, RTC, and TSI stand forindividual income average tax rate at the ninety-ﬁfth(ninety-ninth, ﬁftieth) percentile, corporate income tax
rate, investment tax credits, R&D tax credits, and trade
secrets index, respectively.

Appendix B

Derivation of equation (3)To derive (3), we ﬁrst solve (2) for ln wdt − ln wot as follows


′
′
′
′
′
ln wdt − ln wot = β ln(1 − τσ(d)t) − ln(1 − τσ(o)t) + [Zd′ − Zo′ ] − Cod− ln(Podt
/Poot
).
′
′
) = ln(Podt /Poot ), we obtain
Plugging this expression into (1) and setting ln(Podt
/Poot



ln(Podt /Poot ) = α ln(1 − τσ(d)t ) − ln(1 − τσ(o)t )
 

′
′
′
+α β ln(1 − τσ(d)t) − ln(1 − τσ(o)t) + [Zd′ − Zo′ ] − Cod− ln(Podt /Poot )+ [Zd − Zo ] − Cod ,which yields(1 + α) ln(Podt /Poot ) = α ln(1 − τσ(d)t ) − ln(1 − τσ(o)t )
 

′
′
′
+α β ln(1 − τσ(d)t) − ln(1 − τσ(o)t) + [Zd′ − Zo′ ] − Cod+ [Zd − Zo ] − Cod .
42

We thus have


α 
αβ 
′
′
ln(1 − τσ(d)t ) − ln(1 − τσ(o)t ) +ln(1 − τσ(d)t) − ln(1 − τσ(o)t
)
1+α
1+α
1
1
α
′
]
+
[Zd − Zo ] +[Zd′ − Zo′ ] −[Cod + αCod
1+α
1+α
1+α


α 
αβ 
′
′
=
ln(1 − τσ(d)t ) − ln(1 − τσ(o)t ) +ln(1 − τσ(d)t) − ln(1 − τσ(o)t
)
1+α
1+α
1
1
1
′
].
+
[Zd + αZd′ ] −[Zo + αZo′ ] −[Cod + αCod
1+α
1+α
1+α

ln(Podt /Poot ) =

αβ
α
1
1
1
′
Setting η = 1+α, η ′ = 1+α, γd = 1+α[Zd + αZd′ ], γo = − 1+α[Zo + αZo′ ], γod = − 1+α[Cod + αCod]

and adding uodt , we obtain the expression in (3).

Appendix C
C.1

Additional robustness checks

An alternative average tax rate

In the log odds regressions, we proxy for τσ(d)t with the ATR for a hypothetical taxpayer
at the ninety-ﬁfth percentile of the U.S. income distribution. We report an alternative casewhere we use the ninety-ninth percentile in Table C1. Using speciﬁcation (2) in Table C1
we construct the three types of Bartik instruments as before and check the robustness of ourmain results in Table C2. As seen from Table C2, our main semi-elasticities are virtually
identical to the baseline cases: Panel (a) shows that an inﬂow of a top inventor raises thepatent productivity of all local inventors by approximately 6%, and Panel (b) shows that an
inﬂow of a top inventor raises local patent productivity by approximately 4% when we focus
on external inventors.
Using this alternative ATR, we also conduct the dynamic analysis in Figure C1. Panel(a) illustrates the dynamic impact of inventor inﬂows on the patent productivity of all local
inventors, which includes not only internal knowledge sharing within assignees and between
co-inventors but also external knowledge spillovers. Panel (b) corresponds to the case of “themysteries of the trade in the air”, where we exclude co-inventors and inventors in the same43

Table C1: The impact of tax diﬀerences on the migration of top inventors (robustness)∆ ln(1 − ATR99)∆ ln(1 − CITR)∆ ln(1 + ITC)
∆ ln(1 + RTC)

(1)
2.340
(1.315)-0.440(1.115)-0.700(0.719)0.223(0.445)(2)
2.504(1.198)-0.251(1.063)-0.913(0.672)0.234(0.392)(3)
2.382(1.150)-0.147(0.841)-0.549(0.948)0.319(0.324)(4)
2.455(0.966)-0.018(0.750)-0.711(0.666)0.124(0.285)

Yes
No

Yes
No

No
Yes

No
Yes

Yes
No

No
Yes

Yes
No

No
Yes

CZ pair FE
Origin CZ FE anddestination CZ FE
Year FE
Year FE × Region pair

Observations
4866
4866
7226
7225
2
R (total)
0.892
0.902
0.906
0.916
2
R (within)
0.392
0.451
0.405
0.003
Notes: The dependent variable in each column is the log-odds
ratio in (3). ATR99, CITR, ITC, and RTC stand for individual income average tax rate at the ninety-ninth percentile,
corporate income tax rate, investment tax credits, and R&D
tax credits, respectively. Cluster-robust standard errors are inparentheses.

Figure C1: IV event study regressions (robustness).

.1(b) external inventors.1(a) all local inventors

IV ES1

IV ES2

0
−.05

−.05

0

.05

IV ES2.05

IV ES1

−5

−4

−3

−2

−1

0

+1

+2

+3

+4

+5

−5

−4

−3

−2

−1

0

+1

+2

+3

+4

+5

Notes: Panel (a) (Panel (b)) illustrates the dynamic impact of inventor inﬂows on local
patent productivity without (by) excluding co-inventors and inventors in the same assignee.
νσσas, and Bdt
as instruments and IV ES2 uses Bdt , Bdt
In each panel, IV ES1 uses Bdt and Bdt
instruments.
44

Table C2: The impact of top inventor inﬂows on local patent productivity (robustness)(a) All local inventors
Top inventor inﬂows
ln(1 − ATR99)

(1)

(2)

(3)

(4)

(5)

(6)

0.044
(0.007)2.580(0.693)0.064(0.013)2.625(0.699)0.062(0.012)2.604(0.697)0.062(0.012)2.658(0.699)0.059(0.013)0.060(0.012)36.60537.41823.10915.06212.03933.59831.74219.78113.02510.47833.17934.73521.38513.89411.08635.17030.97119.33012.751
10.26935.14133.11220.43113.30910.6370.044(0.010)2.178(0.621)

0.043(0.009)2.136(0.618)0.044(0.009)2.160(0.619)0.035(0.010)0.038(0.010)36.60537.41823.10915.06212.03933.59831.71319.76413.016
10.47133.17934.73821.38613.89511.08635.17030.94819.31812.74410.26335.14133.11220.43113.30910.636

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
No
Yes

Yes
No
Yes

Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%(b) External inventors
Top inventor inﬂows
ln(1 − ATR99)

0.027(0.004)2.142(0.617)

Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%CZ FE
Year FE
Year × state FE

Yes
Yes
No

Observations
23,628 23,628 23,628 23,463 23,562 23,397
Notes: The coeﬃcient on top inventor inﬂows is converted to semi-elasticity.
ATR99 stands for the individual income average tax rate at the ninety-ninth percentile. The coeﬃcient on ln(1−ATR99) is converted to elasticity. Cluster-robust
standard errors are in parentheses. Column 1 does not control endogeneity of top
inventor inﬂows. Column 2 uses Bdt as an instrument. Column 3 uses Bdt and
σ
σ
ν
Bdt
as instruments. Column 4 uses Bdt , Bdt, and Bdt
as instruments. Columns 5
and 6 replace ln(1−ATR99) in Columns 3 and 4 with Year×destination state FE.

assignee. In both cases, the results are quite similar to the baseline cases: the pre-eventsemi-elasticities are close to zero and post-event semi-elasticities increase to approximately
0.05, which ensures the results of our static model. We report the numbers that we use in
Figure C1 and the associated ﬁrst-stage statistics in Tables C5 and C6, respectively.

45

C.2

Including other controls in our baseline speciﬁcation

We further check the robustness of our main results in Table 5 by including additional controls such as the ATR at the ﬁftieth percentile (ATR50), CITR, ITC, and RTC, as well as
manufacturing employment (EMP) at the commuting zone level. Table C3 shows that ourbaseline results are virtually identical for all speciﬁcations including both the FE and IV
cases. In particular, the IV regressions reported in Panel (a) show that an inﬂow of a top
inventor raises the patent productivity of all local inventors by approximately 6%. The IV
regressions reported in Panel (b) show that an inﬂow of a top inventor raises local patentproductivity by approximately 4% when we focus on external inventors.
C.3

Dropping commuting zone × year observations with no patents

We check the robustness of our main results in Table 5 by dropping commuting zone × year
observations with no patents, instead of using log(1 + Ydt ) as in the baseline case. Table C4
shows that the results are qualitatively similar to our baseline results for all speciﬁcations
including both the FE and IV cases. In particular, the IV regressions reported in Panel (a)show that an inﬂow of a top inventor raises the patent productivity of all local inventors byapproximately 5-6%. The IV regressions reported in Panel (b) show that an inﬂow of a top
inventor raises the patent productivity of external inventors by approximately 3-4%.

46

Table C3: The impact of top inventor inﬂows on local patent productivity(a) All local inventors
Top inventor inﬂows
ln(1 − ATR)
ln(1 − ATR50)ln(1 − CITR)
ln(1 + ITC)
ln(1 + RTC)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

0.043
(0.006)
6.159(1.179)-0.181(1.388)0.021(0.645)0.127(0.409)-0.100(0.251)

0.063(0.013)
6.175(1.182)-0.285(1.376)-0.176(0.647)0.160(0.408)-0.212(0.252)0.060(0.012)
6.137(1.179)-0.325(1.377)-0.090(0.640)0.133(0.407)-0.174(0.248)0.060(0.011)6.217(1.183)-0.234(1.379)-0.134(0.654)0.120(0.407)-0.172(0.253)0.044(0.007)6.245(1.147)-0.385(1.372)-0.129(0.637)0.257(0.404)-0.095(0.250)0.118(0.025)38.55437.41823.10915.06212.03933.27231.89419.87113.081
10.52132.87834.72121.381
13.89511.0880.066(0.013)6.267(1.146)-0.510(1.358)-0.354(0.638)0.300(0.403)-0.219(0.251)
0.124(0.025)38.75137.41823.10915.06212.0390.064(0.012)6.201(1.144)-0.544(1.360)-0.245(0.632)0.262(0.402)-0.171(0.247)0.121(0.025)33.21431.95319.90513.102
10.5370.064(0.011)6.256(1.148)-0.437(1.363)-0.288(0.646)0.254(0.402)-0.165(0.251)0.122(0.025)32.81434.72421.38413.89811.091

0.042(0.010)4.802(0.983)-0.207(1.176)0.174(0.544)0.298(0.395)-0.137(0.221)

0.040(0.009)4.756(0.980)-0.255(1.175)0.228(0.541)0.277(0.394)-0.103(0.218)0.041(0.009)4.690(0.977)-0.102(1.176)0.153(0.551)0.293(0.393)-0.102(0.221)

0.028(0.004)4.847(0.958)-0.264(1.170)0.225(0.542)0.359(0.393)-0.047(0.218)0.079(0.020)38.55437.41823.10915.06212.03933.27231.88419.86513.07810.51932.87834.72621.38413.89711.0890.044(0.010)4.864(0.961)-0.359(1.164)0.055(0.543)0.392(0.393)-0.141(0.221)0.084(0.020)38.75137.41823.10915.06212.0390.042(0.009)4.805(0.958)-0.418(1.163)0.124(0.540)0.364(0.392)-0.100(0.218)0.082(0.020)33.21431.94119.89913.09810.5340.044(0.009)4.734(0.958)-0.269(1.165)0.047(0.550)0.387(0.392)-0.098(0.222)0.083(0.020)32.81434.72921.38713.90011.093ln(1 + EMP)
Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%(b) External inventors
Top inventor inﬂows
ln(1 − ATR)
ln(1 − ATR50)ln(1 − CITR)
ln(1 + ITC)
ln(1 + RTC)

0.027
(0.004)4.790(0.978)-0.127(1.181)0.325(0.543)0.272(0.395)-0.050(0.218)ln(1 + EMP)
Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%

CZ FE
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Year FE
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Year × state FE
No
No
No
No
No
No
No
No
Observations
23,628 23,628 23,628 23,463 23,628 23,628 23,628 23,463
Notes: The coeﬃcient on top inventor inﬂows is converted to semi-elasticity. ATR (ATR50) stands
for the individual income average tax rate at the ninety-ﬁfth (ﬁftieth) percentile. The other coeﬃcients are converted to elasticity. Cluster-robust standard errors are in parentheses. Column 1 does
not control endogeneity of top inventor inﬂows. Column 2 uses Bdt as an instrument. Column 3σσνuses Bdt and Bdt
as instruments. Column 4 uses Bdt , Bdt, and Bdt
as instruments. Columns 5 to 8
repeat the same speciﬁcations as Columns 1 to 4 with ln(1 + EMP) at the commuting zone level.

47

Table C4: The impact of top inventor inﬂows on local patent productivity(a) All local inventors
Top inventor inﬂows
ln(1 − ATR)

(1)

(2)

(3)

(4)

(5)

(6)

0.039
(0.006)6.934(1.257)0.054(0.012)
6.838(1.260)0.051(0.010)6.841(1.259)0.050(0.010)7.017(1.255)0.051(0.012)0.052(0.011)37.09637.41823.10915.06212.03932.97631.93019.892
13.09410.53132.65734.70421.372
13.89011.08534.25831.37819.56912.89710.38034.17033.00720.37713.282
10.619

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
No
Yes

Yes
No
Yes

20,03820,03820,03819,941

19,972

19,8750.023(0.004)5.450(1.054)0.034(0.009)5.380(1.056)0.032(0.008)5.357(1.054)0.033(0.008)5.340(1.044)0.029(0.009)0.031(0.009)

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
Yes
No

Yes
No
Yes

Yes
No
Yes

37.07237.41823.10915.06212.03932.97131.91719.88513.09010.52832.65134.70621.37313.89111.08534.22931.35019.55312.88810.37334.13732.99520.37013.27810.615Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%CZ FE
Year FE
Year × state FE
Observations(b) External inventors
Top inventor inﬂows
ln(1 − ATR)CZ FE
Year FE
Year × state FE
Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%Observations
19,903 19,903 19,903 19,806 19,837 19,740
Notes: The coeﬃcient on top inventor inﬂows is converted to semi-elasticity. ATR
stands for the individual income average tax rate at the ninety-ﬁfth percentile.
The coeﬃcient on ln(1 − ATR) is converted to elasticity. Cluster-robust standard
errors are in parentheses. Column 1 does not control endogeneity of top inventorσinﬂows. Column 2 uses Bdt as an instrument. Column 3 uses Bdt and Bdt
asσν
instruments. Column 4 uses Bdt , Bdt , and Bdt as instruments. Columns 5 and 6
replace ln(1 − ATR) in Columns 3 and 4 with Year × destination state FE.

48

Table C5: The dynamic impact of top inventor inﬂows on local patent productivity
Top inventor inﬂows (j = −5)
Top inventor inﬂows (j = −4)
Top inventor inﬂows (j = −3)
Top inventor inﬂows (j = −2)
Top inventor inﬂows (j = 0)
Top inventor inﬂows (j = +1)
Top inventor inﬂows (j = +2)
Top inventor inﬂows (j = +3)
Top inventor inﬂows (j = +4)
Top inventor inﬂows (j = +5)(1)-0.023(0.010)-0.019(0.014)-0.002(0.010)-0.018(0.010)0.027(0.012)0.046(0.015)0.046(0.013)0.051(0.012)0.027(0.014)0.043(0.013)(2)-0.020(0.008)-0.013(0.008)-0.001(0.009)-0.014(0.007)0.035(0.009)0.047(0.014)0.045(0.013)0.048(0.011)0.026(0.011)
0.045(0.013)(3)
0.001(0.007)-0.007(0.011)0.006(0.008)-0.004(0.009)0.033(0.011)
0.043(0.014)0.037(0.012)0.044(0.011)0.030(0.013)0.045(0.013)(4)
0.000(0.006)-0.007(0.007)0.008(0.007)-0.005(0.006)0.038(0.010)0.050(0.013)0.037(0.012)0.040(0.009)0.030(0.011)0.049(0.012)(5)-0.020(0.009)-0.021(0.011)-0.004(0.010)-0.018(0.008)0.030(0.011)
0.047(0.015)0.045(0.013)0.049(0.012)0.033(0.013)0.054(0.013)(6)-0.018(0.008)-0.016(0.008)-0.001(0.009)-0.014(0.006)0.037(0.009)0.049(0.013)0.045(0.012)0.048(0.011)
0.032(0.011)
0.055(0.013)(7)
0.004(0.007)-0.008(0.009)0.006(0.009)-0.003(0.008)0.038(0.012)0.047(0.015)0.040(0.012)0.043(0.011)0.036(0.014)0.057(0.014)(8)
0.003(0.006)-0.008(0.007)0.009(0.007)-0.004(0.006)0.042(0.010)0.054(0.013)0.042(0.011)
0.040(0.009)0.035(0.012)0.060(0.013)Observations
15752
15642
15752
15642
15752
15642
15752
15642
σ
σ
Notes: Odd number columns use Bdt and Bdt
as instruments. Even number columns use Bdt , Bdt
, and
ν
Bdt
as instruments. Columns (1) to (4) use ATR95, whereas Columns (5) to (8) use ATR99 as robustness
checks. The dependent variable in Columns (1), (2), (5), and (6) is the patent productivity for all local
inventors, whereas that in Columns (3), (4), (7), and (8) is the patent productivity for external inventors.
Cluster-robust standard errors are in parentheses.

49

Table C6: First-stage statistics for the dynamic analysis(a) Sanderson-Windmeijer (under identiﬁcation)
Top inventor inﬂows (j = −5)
Top inventor inﬂows (j = −4)
Top inventor inﬂows (j = −3)
Top inventor inﬂows (j = −2)
Top inventor inﬂows (j = 0)
Top inventor inﬂows (j = +1)
Top inventor inﬂows (j = +2)
Top inventor inﬂows (j = +3)
Top inventor inﬂows (j = +4)
Top inventor inﬂows (j = +5)(b) Sanderson-Windmeijer (weak identiﬁcation)
Top inventor inﬂows (j = −5)

(1)

(2)

(3)

(4)

340.216(0.000)156.024(0.000)177.020(0.000)163.037(0.000)257.894(0.000)442.020(0.000)208.149(0.000)255.615(0.000)126.082(0.000)271.808(0.000)575.561(0.000)284.008(0.000)308.252(0.000)223.759(0.000)446.932(0.000)477.121(0.000)421.991(0.000)465.129(0.000)266.307(0.000)417.144(0.000)391.350(0.000)160.880(0.000)217.101(0.000)211.554(0.000)204.805(0.000)372.415(0.000)194.555(0.000)277.758(0.000)130.618(0.000)216.519(0.000)671.175(0.000)248.067(0.000)325.561(0.000)254.252(0.000)397.033(0.000)463.968(0.000)381.307(0.000)503.246(0.000)229.608(0.000)338.164(0.000)30.80527.28035.43531.812(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = −4)
14.127
13.461
14.56711.758(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = −3)
16.02814.61019.65815.431(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = −2)
14.76210.60619.15512.051(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = 0)23.35121.18318.54418.818(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = +1)
40.02322.61433.72121.991(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = +2)
18.84720.001
17.61618.073(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = +3)
23.14522.04625.15023.852(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = +4)
11.41612.622
11.827
10.883(0.000) (0.000) (0.000) (0.000)
Top inventor inﬂows (j = +5)
24.61119.77119.60516.028(0.000) (0.000) (0.000) (0.000)σ
Notes: Odd number columns use Bdt and Bdt
as instruments. Even number columns useσν
Bdt , Bdt , and Bdt as instruments. Columns (1) and (2) use ATR95, whereas Columns (3)and (4) use ATR99 as robustness checks.

50Appendix D

State tax competition

We test the null hypothesis that state governments strategically determine their taxes. Basedon the previous studies (e.g., Brueckner, 2003), we estimate a reaction function where a state
responds to the choices of its neighboring states as follows:

τσt = ρX

ωσσ′ τσ′ t + βXσt−1 + γσ + γt + εσt .

(16)

σ ′ ̸=σThe dependent variable τσt is the tax in state σ in year t, and the termP

σ ′ ̸=σ ωσσ ′ τσ ′ t on theright-hand side is the weighted sum of the taxes in the neighboring states with the weight
being ωσσ′ . Xσt−1 stands for local patent productivity and a vector of socio-politico-economic
characteristics for state σ in year t − 1 (see Table D1).33 γσ and γt denote state and year ﬁxed
eﬀects, respectively, and εσt is the error term.
Our null hypothesis is that ρ > 0. If ρ > 0 were to hold, there would be strategic tax
competition between state governments, which would induce interstate correlation between
taxes and productivity. In that case, the exclusion restrictions would be violated since the
top inventor inﬂows would not mediate the interstate correlation.
However, measuring ρ is challenging because state tax decisions are simultaneous. To
Paddress the potential endogeneity problem that the main regressor σ′ ̸=σ ωσσ′ τσ′ t is correlatedwith the error term εσt , we follow Kelejian and Prucha (1998) and use the weighted sum ofneighboring states’ socio-politico-economic characteristics as an instrument. Let Xσt−1 =[x1,σt−1 , · · · , xK,σt−1 ]′ , where xk,σt−1 is the k-th characteristic. We ﬁrst generate the weighted
P
sum σ′ ̸=σ ωσσ′ xk,σ′ t−1 for each characteristic {xk,σt−1 }K
k=1 . These K instruments are thenused to estimate the predicted value of the weighted sum of neighboring states’ tax rates,Pσ ′ ̸=σ ωσσ ′ τσ ′ t , in the ﬁrst-stage regression.
Estimating ωσσ′ is diﬃcult due to lack of degrees of freedom. We thus consider several

33
When the state governments make a tax rate decision, they observe the information on the socio-politicoeconomic conditions in the previous year. We thus use the lagged variables Xσt−1 .

51

Table D1: Summary statistics
mean
sd
min
max
obs
log patent productivity
6.285 1.422 2.881 10.201 1617
log population
15.011 1.009 12.951 17.430 1617
share of black or African American population 0.114 0.121 0.002 0.705 1617
share of population younger than 20
0.292 0.028 0.205 0.414 1617
share of population older than 64
0.124 0.018 0.075 0.184 1617
unemployment rate
5.919 2.046 2.342 17.350 1617
log total income
18.015 1.185 15.010 21.186 1617
log gross state product18.191 1.170 15.106 21.403 1617
democrat governor (dummy variable)
0.511 0.500 0.000 1.000 1617
log tax revenue
15.291 1.159 12.318 18.581 1584
log government debt
15.302 1.253 10.872 18.819 1584
log government revenue
16.110 1.134 13.301 19.511 1584
Notes: Summary statistics are based on the data described in Appendix A.1 for theyears 1977 to 2009.
alternative weights and examine the sensitivity of the estimates. In the baseline cases, we
use the following two types of weights. One is the ﬁrst-order contiguity weight: ωσσ′ = 1 ifstate σ ′ is contiguous with σ and ωσσ′ = 0 otherwise. The other is constructed such that it is
proportional to top inventor ﬂows from state σ ′ to state σ.34
Table D2 reports the regression results for the sample period 1977-2009.35 Columns 1-2(3-4) use the ﬁrst-order contiguity (top-inventor-inﬂow) weights. Columns 1 and 3 show thatthe estimated values of ρ are negative and insigniﬁcant. Columns 2 and 4 include the ownstate productivity and show that it does not signiﬁcantly aﬀect the estimates of ρ. These
results imply that there is neither strategic tax competition nor a direct relationship between
tax and productivity across states. We further apply a test by Montiel Olea and Pﬂueger(2013) to each speciﬁcation in Table D2. The eﬀective F -statistics indicate that in all cases
34

In the speciﬁcation curve analysis below, we consider four other weights: (i) the second-order contiguity
weight, i.e., ωσσ′ = 1 if state σ ′ is contiguous with σ or is contiguous with the states that are contiguous withstate σ; (ii) the inverse-distance weight, i.e., ωσσ′ is inversely related to the geographical distance between σ
and σ ′ ; (iii) the inverse-distance weight with a cutoﬀ distance, i.e., the interstate eﬀect is assumed to be zero
beyond 1000 miles; and (iv) the top-inventor-inﬂow weight, i.e., ωσσ′ is proportional
to the total number ofP
top inventors who migrated from σ ′ to σ. All weights are normalized such that σ′ ̸=σ ωσσ′ = 1 for any σ.
35
We proxy for τσt with the log of one minus ATR at the ninety-ﬁfth percentile, ln(1 − ATRσt ), in state σ
in year t since it signiﬁcantly aﬀects top inventor migration as shown in Section 3. We drop Washington D.C.
from the sample because some state characteristics are unavailable.

52

Table D2: State tax competitionP
σ ′ ̸=σ(1)-0.163(0.314)

ωσσ′ τσ′ t

0.016(0.038)0.193(0.107)-0.175(0.179)0.024(0.287)-0.000(0.001)
0.003(0.032)0.029(0.015)0.001(0.001)-0.024(0.012)0.000(0.002)-0.002(0.004)(2)
0.027(0.260)0.005(0.003)0.016(0.033)0.236(0.117)-0.160(0.168)0.029(0.249)-0.000(0.001)-0.005(0.031)0.028(0.016)0.002(0.001)-0.027(0.011)-0.000(0.002)0.001(0.003)0.017(0.029)0.111(0.120)-0.192(0.165)0.051(0.222)-0.000(0.001)-0.009(0.028)0.036(0.014)0.001(0.001)-0.023(0.011)-0.000(0.002)-0.002(0.004)(4)-1.071(0.797)0.005(0.003)0.013(0.029)0.165(0.124)-0.167(0.160)0.088(0.212)-0.000(0.000)-0.013(0.029)0.036(0.014)0.002(0.001)-0.025(0.010)-0.000(0.002)-0.000(0.003)15.92426.392
15.173
9.191
7.055

18.51726.68915.3689.320
7.156

50.74225.15914.4448.745
6.715

50.30725.14314.3558.639
6.612

log patent productivity
log population
share of black or African American population
share of population younger than 20
share of population older than 64
unemployment rate
log total income
log gross state productdemocrat governor
log tax revenue
log government debt
log government revenue

Eﬀective F statistic
τ = 5%
τ = 10%τ = 20%τ = 30%(3)-1.364(0.822)

Observations
1,584
1,584
1,584
1,584
Notes: Columns 1-2 (3-4) use the ﬁrst-order contiguity (top-inventor-inﬂow) weights.
Columns 2 and 4 include the own-state productivity ln Yσt−1 .

we can reject the null hypothesis of a weak instrument at the conventional level.
We check the robustness of the results using the speciﬁcation curve analysis as in Simonsohn, Simmons, and Nelson (2020). We consider diﬀerent speciﬁcations of equation (16) byfocusing on three dimensions. First, we include four other weights discussed in footnote 34 in
the speciﬁcation curve analysis to examine the sensitivity of the estimates. Second, we estimate the model for every possible combination of the variables listed in Table D1. Last, since

53

−20

−15

−10

−5

0

5

10

Figure D1: Speciﬁcation curvebaseline estimate

other estimates

95% CI

90% CI

Notes: The speciﬁcation curve is depicted using 30, 843 alternative speciﬁcations explainedin Appendix D. The vertical axis is the value of ρ.
the model is overidentiﬁed, we use diﬀerent combinations of neighboring states’ characteristics
as instruments, which allows us to explore the sensitivity of the estimates.36
Figure D1 plots the speciﬁcation curve for ρ with 90% and 95% conﬁdence intervals. The
signs of the estimated coeﬃcient are not signiﬁcantly positive at the 10% level for 30, 843
alternative speciﬁcations, thus verifying that the baseline estimates do not come from data
mining. Hence, we may conclude that there was no strategic tax competition between states
during the sample period.

36
Since the usual caveat on weak instruments is applicable here, we adopt only speciﬁcations for which thenull hypothesis of weak instruments is rejected.

54Appendix E

Assumptions to ensure share exogeneity

We impose the following two assumptions to obtain share exogeneity in Section 4.3. First,
εdt has mean zero conditional on Xdt , i.e.,E(εdt |Xdt ) = 0,(17)where Xdt consists of the own state tax τσ(d)t and ﬁxed eﬀects δ = {δd , δt }. Second, otherstate taxes {τσ(c)t }c̸=d and εdt are independent conditional on Xdt , i.e.,τσ(c)t ⊥ εdt |Xdt

for c ̸= d and c, d ∈ C,

(18)

where A ⊥ B|C denotes conditional independence of A and B given C. Under these assumptions, we can show that E(εdt Pbodt |Xdt ) = E(εdt Pbodt |τσ(d)t , δ) = 0 as follows.
By the property of conditional expectations, we haveE(εdt Pbodt |τσ(d)t , δ) = E[E(εdt Pbodt |τ t , δ)|τσ(d)t , δ],
where τ t = {τσ(c)t }∀c∈C . The right-hand side of the above equation becomes:
E[E(εdt Pbodt |τ t , δ)|τσ(d)t , δ] = E[Pbodt E(εdt |τ t , δ)|τσ(d)t , δ]
= E[Pbodt E(εdt |τσ(d)t , δ, {τσ(c)t }c̸=d )|τσ(d)t , δ]
= E[Pbodt E(εdt |τσ(d)t , δ)|τσ(d)t , δ]
= E[Pbodt · 0|τσ(d)t , δ] = 0.
The ﬁrst equality follows because Podt is a function of τ t and δ. The second equality holds
because τ t = {τσ(d)t , {τσ(c)t }c̸=d }. The third equality is due to the conditional independence
assumption (18). The last equality comes from the conditional mean assumption (17).

55Appendix F

Decomposition of the Bartik estimator

To assess the validity of the Bartik estimator, we follow Goldsmith-Pinkham et al. (2020) andP
decompose it into the weighted sum of just-identiﬁed IV estimators, ϕbs = o∈C ωbo ϕbso . In ourcontext, the coeﬃcient ϕbso is the origin-speciﬁc productivity eﬀect using the share Pbodt as aseparate instrument. The Rotemberg weight ωbo measures how sensitive the estimate ϕbs is topossible misspeciﬁcation in each instrument.
Table F1 presents the summary statistics.37 Panel A shows that the Rotemberg weights
are positive in almost all cases. Panel B shows that the weights are highly correlated with thevariances of the shares var(Pbo ), where the variances are taken across destination commuting
zones d and times t. Panel C reports origin commuting zones with the top ﬁve highest
Rotemberg weights. These commuting zones account for 26.3 percent of the total share.
Panel D provides the heterogeneous eﬀects interpretation of the Bartik instrument. ThePP
bo vbod , where ϕbsd is the destination-speciﬁc
Bartik estimator can be rewritten as ϕbs = d ϕbsd o ω
productivity eﬀect and vbod ≥ 0 is deﬁned in the same way as in Proposition 4 in GoldsmithPinkham et al. (2020). Since negative Rotemberg weights for some origin commuting zones oP
may make o ω
bo vbod negative, the Bartik estimator ϕbs may become a nonconvex combination.
Pbo ϕbso is much larger than the
However, since Panel D shows that the positive part o|bωo >0 ωPbo ϕbso , the negative Rotemberg weights are unlikely to be a problem fornegative part o|bωo <0 ω
the LATE-like interpretation of the productivity eﬀect.
We further assess the validity of the identiﬁcation assumption. For each of the top ﬁve
origin commuting zones in Panel C of Table F1, Table F2 lists the top ﬁve destination
commuting zones by the predicted top inventor migration probability. The result that mostmigrations are predicted to be interstate aligns with the assumption that the main source ofidentifying variation comes from inventor mobility caused by individual income tax diﬀerences
across states.
37
the decomposition is applicable to a single estimator, we focus on the simple case with Bdt =P Since
Pbodt Iot .
o̸=d

56

Table F1: Summary of the decomposition of the Bartik estimator
Panel A: Negative and positive weights
Negative
PositiveSum-0.00021.0002Mean-0.0000
0.0099Share
0.00020.9998ωbo
1.00000.9204-0.0478
0.03720.7865

I¯o

ϕbso

Fbo

var(Pbo )1.0000-0.0365-0.06030.78011.0000-0.0268-0.04171.0000-0.0845

1.0000

I¯o
186
113
63
68
48

ϕbso
0.0507
0.06170.04020.05120.0358

95% CI(0.033, 0.097)(0.037, 0.170)(0.011, 0.065)(0.033, 0.170)(0.018, 0.059)

Panel B: Correlations
ω
bo
I¯o
ϕbso
Fbo

var(Pbo )Panel C: Top ﬁve Rotemberg weight commuting zones
19600 Bergen-Essex-Middlesex (NJ)
24300 Cook-DuPage-Lake (IL)
19400 Kings-Queens-New York (NY)19700 Philadelphia-Montgomery-Delaware (PA)
16300 Allegheny-Westmoreland-Washington (PA)ωbo0.08380.05350.04480.04140.0396

Panel D: Estimates of ϕbso for positive and negative weightsω
b -weightd sum
Share of overall ϕbs
Mean
Negative
0.00010.0023-0.5575
Positive
0.04170.99770.0478Notes: Panel A reports the sum, mean, and share of the positive and negative Rotemberg weights. Panel B
reports correlations between the Rotemberg weights (bωo ), the number of top inventors (I¯o ), the just-identiﬁedscoeﬃcient estimates (ϕbo ), the ﬁrst stage F -statistic of the share (Fbo ), and the variance of the shares acrossdestinations and times (var(Pbo )). Panel C reports the origin commuting zones with the top ﬁve highest
Rotemberg weights. The state of the representative county of each commuting zone is in parenthesis, where
the representative county is the one with the largest number of inventors. The ninety-ﬁve percent conﬁdence
interval is the weak instrument robust conﬁdence interval as in Chernozhukhov and Hansen (2008) over a range
P
P
from 0 to 0.5. In Panel D “b
ω -weighted sum” reports o|bωo <0 ωbo ϕbso for negative and o|bωo >0 ωbo ϕbso for positivePP
cases, and “share of overall ϕbs ” reports (1/ϕbs ) o|bωo <0 ωbo ϕbso for negative and (1/ϕbs ) o|bωo >0 ωbo ϕbso for positive
cases.

We ﬁnally examine the relationship between the top ﬁve shares Pbodt and location-speciﬁccharacteristics that may be correlated with the outcome Ydt as suggested by GoldsmithPinkham et al. (2020). For the shares Pbodt to satisfy the conditional exogeneity assumption
in Section 4.1, they should not be correlated with destination commuting zone characteristics
conditional on the controls and ﬁxed eﬀects in the structural equation (6).38 Based on the
38

Let Wdt be destination commuting zone characteristics that may be correlated with Ydt . No correlation
between the shares and destination characteristics, corr(Pbodt , Wdt |Xdt , δ) = 0, is an observable analogue ofcorr(Pbodt , εdt |Xdt , δ) = E(Pbodt · εdt |Xdt , δ) = 0 under the conditional mean zero assumption E(εdt |Xdt , δ) = 0.

57

Table F2: Destinations to which top inventors migrated from the highest-weight origins(A) 19600 Bergen-Essex-Middlesex (the highest weight)
38000 San Diego
37500 Santa Clara-Monterey-Santa Cruz
39400 King-Pierce-Snohomish
35801 Ada-Canyon-Elmore
37800 Alameda-Contra Costa-San FranciscoState of therepresentative
county
NJ
CA
CA
WA
ID
CA

(B) 24300 Cook-DuPage-Lake (the second highest weight)
7100 Palm Beach-St. Lucie-Martin
9100 Fulton-DeKalb-Cobb11304 Fairfax-Montgomery-Prince George’s
37000 Stanislaus-Merced-Tuolumne
37500 Santa Clara-Monterey-Santa Cruz

IL
FL
GA
MD
CA
CA

0.01130.00910.00810.00790.0079(C) 19400 Kings-Queens-New York (the third highest weight)
37500 Santa Clara-Monterey-Santa Cruz
18600 Albany-Saratoga-Rensselaer
37800 Alameda-Contra Costa-San Francisco
38801 Multnomah-Washington-Clackamas
39400 King-Pierce-Snohomish

NY
CA
NY
CA
OR
WA

0.01180.01140.01110.01000.0097(D) 19700 Philadelphia-Montgomery-Delaware (the fourth highest weight)
Destination CZs
39400 King-Pierce-Snohomish
28900 Denver-Jeﬀerson-Arapahoe
38000 San Diego
38801 Multnomah-Washington-Clackamas
37800 Alameda-Contra Costa-San Francisco

PA
WA
CO
CA
OR
CA

Predicted
migration
probability
0.00780.00730.00710.00700.00690.01510.01430.01310.01160.0113(E) 16300 Allegheny-Westmoreland-Washington (the ﬁfth highest weight)
PA
37500 Santa Clara-Monterey-Santa CruzCA
0.021737800 Alameda-Contra Costa-San Francisco
CA0.021138300 Los Angeles-Orange-San Bernardino
CA0.019521501 Hennepin-Ramsey-Dakota
MN
0.01807000 Dade-Broward-MonroeFL
0.0175
Notes: The top ﬁve destination commuting zones are deﬁned by the predicted migration probability of the top
inventors from each origin commuting zone. The representative county is the one with the largest number of
inventors.

58

Table F3: Relationship between the shares and manufacturing employment size
Share: the predicted probability of top inventors from
19600
24300
19400
19700
16300
BergenCookKingsPhiladelphiaAlleghenyEssexDuPageQueensMontgomeryWestmorelandMiddlesex
Lake
New York
DelawareWashington
Correlation coeﬃcient
0.0002763-0.0027881
0.0013622-0.0018730
0.0026412-0.0000013-0.0000209
0.0000071
0.0000201
Regression coeﬃcient-0.0000080(0.0000108) (0.0000128) (0.0000143)(0.0000112)(0.0000105)
Notes: The ﬁrst row reports the correlation coeﬃcient between the ﬁrst-order time diﬀerence in the share andthe ﬁrst-order time diﬀerence in the log of manufacturing employment. The second row reports the regression
coeﬃcient of the log of manufacturing employment on the share conditional on ln(1 − ATR), commuting zone
ﬁxed eﬀects, and year ﬁxed eﬀects. To control for commuting zone ﬁxed eﬀects in the regression, we take theﬁrst-order time diﬀerence of the variables in the regression. Cluster-robust standard errors are in parentheses.

result in Table C3, we regard the log of manufacturing employment in destination commuting
zone d as such a characteristic. Table F3 presents the results of this analysis. We take thetime diﬀerence of the variables to control for commuting zone ﬁxed eﬀects. Reassuringly, thecorrelations are low in all cases and the regression coeﬃcients on the log of manufacturing
employment are not statistically signiﬁcant at the ﬁve percent level, thus implying the absenceof correlation between the shares and possible confounders.

Appendix G

Robustness to possible violations of the parallel trends
assumption

We examine the robustness to possible violations of the parallel trends assumption for the
event study analysis in Section 5. Following Rambachan and Roth (2023) we specify, for each
event study regression, a set ∆ = {δ : |(δt+1 − δt ) − (δt − δt−1 )| ≤ M, ∀t} to bound the degree
to which the slope of diﬀerential trend δ can vary between consecutive periods. We use thedefault setting in the R package HonestDiD provided by Rambachan and Roth (2023), namelythat the value of M ranges from 0, which corresponds to a linear trend, to half a standard
deviation of the parameter of interest.
Figure G1 illustrates the robustness test result, where we assess the presence of a diﬀerential trend δ1 for the ﬁrst-period eﬀect µ1 under the normalization of µ−1 = 0. In Panel (a)59Table G1: Summary statistics of the “break-down” values for the placebo simulations
95th percentile 99the percentile(a) all local inventors
0.00000.0010(b) external inventors
0.00000.0010Notes: Panel (a) (Panel (b)) presents the distribution of the“break-down” values of M obtained from the permutationbased placebo analysis for the event study model, where
we consider all local inventors (external inventors). The“break-down” value is deﬁned as the value of M at which
the null hypothesis that the ﬁrst period eﬀect is zero canno longer be rejected. The summary statistics are for 200
simulation results.
(Panel (b)), the “break-down” value of M , at which the null hypothesis that the ﬁrst-period
eﬀect is zero can no longer be rejected, is 0.006 (0.006). Since the value is 40 (42.90) percentof the standard error of the estimated eﬀect µ̂1 , the parallel trends assumption holds for areasonable deviation from a linear trend.39
We further perform a permutation-based placebo analysis to see how likely the “breakdown” value of M = 0.006 is to occur. Speciﬁcally, we ﬁrst estimate each event study model
200 times by randomly reshuﬄing the commuting zones to which top inventors moved andthen apply the sensitivity analysis proposed by Rambachan and Roth (2023) mentioned above
to the estimates.
Table G1 shows the summary statistics of the simulated “break-down” values of M . In
both Panels (a) and (b), the values are zero in ninety-ﬁve percent of cases. These results
indicate that the deviation from the linear trend up to M = 0.006 occurs extremely rarely.
Therefore, we may conclude that the parallel trends assumption is unlikely to be violated.

σ39
We examine the parallel trends assumption for the IV event study regression using Bdt and Bdt
asinstruments, which corresponds to the IV ES1 case presented in Figure 5. The standard error of the ﬁrstperiod eﬀect µ̂1 for all local inventors (for external inventors) is 0.015 (0.014).

60

Figure G1: Robustness to possible violations of parallel trends assumption(a) all local inventors(b) external inventors
Original

FLCI

.06
.04
.02
0

0

.02

.04

.06

.08

FLCI

.08

Original

0

.001

.002

.003

.004

.005

.006

.007

0

.001

.002

.003

.004

.005

.006

.007

Notes: Panel (a) (Panel (b)) illustrates the sensitivity analysis proposed by Rambachan
and Roth (2023) for the event study model, where we consider all local inventors (external
inventors). In each panel, the leftmost bar is the ninety-ﬁve percent conﬁdence interval of
the estimate µ̂1 and the other bars are ﬁxed length conﬁdence intervals (FLCIs) considered
Rambachan and Roth (2023).

61