Knowledge Creation through Multimodal
Communication
Marcus Berliantyand Masahisa Fujitaz
May 1, 2024

Abstract
Knowledge creation either in isolation or joint with another person,
using either face to face or internet contact and incorporating internet search ability is analyzed. Both a conceptual phase and a technicalphase of research are analyzed, allowing workers to choose endogenously
their mode of communication. In addition to formal knowledge, tacitknowledge plays an essential role in the knowledge production process.
Lead time for face to face communication plays a key role in the optimal choice of communication mode. The sink point is ine¢ cient. Our
framework is applied to pandemic restrictions on face to face communication.
JEL codes: D83, L86 Keywords: Knowledge creation; Tacit knowledge; Multimodal communication; Pandemic restrictions

We thank the Whiteley Center in Friday Harbor at the University of Washington for theirkind hospitality during our work on this paper. The second author is grateful for Grants
Aid for Scienti…c Research Grant 23K01348 from the Japanese Ministry of Education andScience. We are grateful to two anonymous referees, Toshitaka Gokan, Nobuaki Hamaguchi,David Nagy, Kentaro Nakajima, and Tony Smith for helpful comments. The authors retain
responsibility for any errors.
yDepartment of Economics, Washington University, MSC 1208-228-308, 1 Brookings
Drive, St. Louis, MO 63130-4899 USA. Fax: (1-314) 935-4156, e-mail: berliant@wustl.edu
zInstitute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto,606-01 Japan. Phone Fax: (81-75) 753-7120, e-mail: fujita@kier.kyoto-u.ac.jp1

1

Introduction

1.1

Motivation

How should the process of knowledge creation be modeled? How does theknowledge creation process function when there are multiple ways for people
to communicate, for example face to face or using the internet? Relative tothe one communication channel case, what di¤erent patterns of joint researchamong knowledge workers emerge, and how is the productivity of research worka¤ected? Under what conditions are the conceptual and technical phases ofknowledge production best accomplished through each of the communication
channels?
When analyzing the dynamics of knowledge creation, as we do here, theconcept of tacit knowledge among the people creating new knowledge arises organically. As tacit knowledge is the part of knowledge that is not manifested
in the …nal product, readers will recognize it as what is learned by authors in
the academic research setting that is not explicitly embedded in a publishedpaper, including this one. What are the e¢ ciency consequences of tacit knowledge? What kind of innovation policy is appropriate in the presence of tacitknowledge? Can arti…cial intelligence have tacit knowledge? To address thesequestions, we must …rst build a model that incorporates tacit knowledge. We
are not aware of any formal models of either multimodal communication ortacit knowledge in the prior literature.

1.2

Preview of the results

To address our motivating questions, as depicted in Figure 1, a person, say i,with knowledge Ki can develop new ideas in isolation while interacting with
the Server. Or person i may create new ideas jointly with another person, sayj, by interacting through the Net or by working F2F (Face to Face). Using
this extension of our earlier models, that are discussed below, we can examine
the impact of rapidly developing ICT (including AI) on knowledge creation
activities.

Figure 1
This paper also aims to provide a theoretical framework for the study of
speci…c recent issues such as the impact of the Covid-19 pandemic on knowl2

edge creation, or the e¤ect of urban structure on the productivity of knowledge
workers as well as on the pattern of knowledge work in large cities.
In contrast with our previous work, here we separate the knowledge creation
process into conceptual and technical phases, and allow researchers to choose
the mode of communication, F2F or the net, that suits them best in eachphase.
Our main …ndings are as follows. First, we …nd that a crucial parameterin the model is the lead time for F2F communication. What we mean by leadtime is the …xed cost for communicating F2F, such as the cost of commuting
to the o¢ ce. Second, in contrast with our previous work, the steady state
will not, in general, be the state with highest productivity. The net e¤ect isthat achieving and maintaining the highest productivity pro…le of knowledge
in common and di¤erential knowledge requires more heterogeneity or larger
research groups than we found in our previous work. Third, the e¤ect oftacit knowledge on knowledge productivity is not internalized by the knowledge workers. Fourth, as net and transport technology improves, knowledge
composition becomes more important than geographical distance in the choiceof research partners. We shall provide more discussion of these points in the
conclusions.
Applying this framework to pandemic restrictions, we show, for example,
how the productivity of knowledge workers with longer commutes to work isa¤ected less than those with shorter commutes when pandemic restrictions on
face to face work are implemented. This application requires the introductionof multimodal communication to our model.

1.3

Related literature

Through a sequence of four related papers, Berliant and Fujita (2008, 2009,2011, 2012), we have developed a model of knowledge creation based on theinteractions among a group of heterogeneous people. These papers form thebasis for the analysis here. In Berliant and Fujita (2008), we develop the basicmodel and analyze dynamic interactions among a group of knowledge workers under the assumption of symmetry of the state of knowledge. Berliant
and Fujita (2009) investigates further the case of two knowledge workers, relaxing the symmetry assumption and allowing knowledge transfer in addition
to knowledge creation, whereas Berliant and Fujita (2011) embeds the basicmodel in a growth framework to analyze macroeconomic dynamics and thee¢ ciency properties of equilibrium. Finally, Berliant and Fujita (2012) con3

structs a two region version of the model to examine the emergence of culturesin the knowledge production community. All of these papers feature only one
channel of communication, whereas here we consider both face to face as wellas electronic communication. None of these papers touch on tacit knowledge.
This paper is at a more micro level than the earlier work, necessitated byaddressing questions concerning multiple modes of communication and tacitknowledge during the knowledge creation process. In the earlier work, we
considered knowledge creation to be opening up boxes, each containing one
idea, either alone or with another person. Opening up a box and exploring
the content, in other words creating a new idea, takes time, considered to be
an opportunity cost. In contrast, here we model, to an extent, the process of
opening a box.1 Analogous to producing a car, we model the construction of
the various components of an idea, including both the conceptual and technicalaspects, and then their assembly into a …nal product. For each phase, agents
optimize over the type of communication medium used for joint work. The
construction of components generates tacit knowledge, which is accumulated
for knowledge creation in the future.
Part of our framework here is based on the insightful empirical paper by Lin
et al (2022). They break down the knowledge creation process into conceptualand technical phases. The early conceptual phase involves tacit knowledge
deployment, whereas the later technical phase involves explicit knowledge. Using a large data set, they …nd that face to face communication is more e¤ectivein the conceptual phase, whereas remote teams can be e¤ective in the technical phase.2 Our model allows agents to select endogenously the mode ofcommunication they employ in each phase.
From a wider perspective, in related work, Aghion et al. (2017) address theinteresting ways in which arti…cial intelligence can impact economic growth at
the macro level using a Cobb-Douglas production technology. In contrast, thework here addresses how internet communication and search a¤ects knowledge creation at a very micro level. Krugman (1991) points out that face
to face contact can promote knowledge externalities between agents. Atkinet al (2022) verify this empirically. Ceci et al (2020) conduct a case study
in the aerospace industry, …nding that both face to face and electronic communication are important to knowledge creation in a company, but they areused di¤erentially by employees depending on both the relationship between
1

In ongoing work, we tackle the completely microfounded process of knowledge creation
using arti…cial intelligence.
2
Finke et al (1992) present a similar abstract framework called Geneplore.

4

communicators and the activity. Panahi et al. (2012) hypothesize that social
media are a channel for tacit knowledge sharing.
The interesting work of Bloom et al (2020) looks at idea creation froman endogenous growth perspective. They …nd that research productivity invarious sectors as well as the aggregate economy has been declining, sinceideas are becoming harder to …nd. The phenomenon has been o¤set by anincreasing number of researchers and research e¤ort.3 Tacit knowledge could
play a big role here, as it is not part of the empirical productivity accounting
or theoretical models.

1.4

Outline of the paper

Our analysis proceeds as follows. In section 2, we develop our model of two
knowledge workers using multiple modes of communication and generating
tacit knowledge in addition to patents, and analyze the steady state. In
section 3, we examine the comparative static e¤ect of lead time cost on the
optimal mode of communication. Section 4 considers knowledge growth undersymmetry, whereas section 5 introduces tacit knowledge and analyzes dynamics
in the two person case. Section 6 presents our conclusions and suggestions for
future research. The three appendices provide extensions and technical details,
including an application of our model to pandemic restrictions in Appendix A.

2

The model with two persons in the stationary state

In this section, we consider two persons/researchers, i and j, and extend themodel of Berliant and Fujita (2008, 2009) by incorporating multiple modes ofinteraction. Wherever possible, we use the same notation as in Berliant andFujita (2008, 2009).
Following on our earlier work, we model knowledge creation as a processof opening up boxes containing ideas. The labels on the boxes, that describe
their contents, are known to all, but it takes time to understand the contentsof the boxes. An example of such a box of knowledge is the creation of thispaper by its authors. The title is its label. Another example is a new recipe
3

They also note the public good nature of ideas, as we do in Berliant and Fujita (2011),
that can overcome decreasing returns and imply exponential growth in per capita income.

5

for curry rice. So the discovery of a new idea is analogous to opening a box
that contains its details.
There is a countable in…nity of boxes, each with its own unique label.
They are numbered k = 1; 2; ::: so that the state of a knowledge worker attime t 2 [0; 1) is given by a (countably in…nite) vector of zeroes and ones,
where a zero represents an unopened box to that person, and a one represents
an opened box, in other words an idea that has been discovered by that person.
At any …nite time, the number of ones in the vector is …nite. Moreover, ideasthat have been discovered by others but not by a researcher are assigned zeroto that researcher.4 We model potential states of knowledge, represented bythis vector, as in…nite, since the potential for new discoveries is unbounded.
The knowledge state of each person at a given time is thus represented by avertex of the Hilbert cube.
The rate at which the boxes can be opened depends on the stock alreadyopened by a particular person, either alone or with someone else. When working in isolation, the total stock of knowledge or boxes already opened by that
person a¤ects the rate at which new boxes are opened. When working jointly,
both the total number of boxes opened previously and their pro…le matter.
Whether they were opened together, and thus become mutual knowledge, or
independently, and thus become exclusive knowledge, determines the relative
heterogeneity of agents and the productivity of joint work. Notice that theheterogeneity of knowledge workers is endogenous to the model.
As there is an in…nite number of boxes or potential ideas, we assume thatthe probability that knowledge workers who are not working together open thesame box is zero.
In what follows, in contrast with our earlier work, we allow more channelsof communication between knowledge workers, namely face to face and internet communication. Moreover, we decompose knowledge creation into moreelementary units or phases, to be described formally in this section. These
phases involve internet search and thinking on one’s own, both when creating knowledge alone and when preparing to work with someone else. When
working with someone else, there will also be time spent communicating either
face to face or over the internet. We allow each person to optimize over the
allocation of time or frequency to these various activities. For example, whencreating a new recipe for curry rice, a chef might search the internet for recipes
4

In other work, for example Berliant and Fujita (2011), we have examined public knowledge transmission.

6

good and bad (respecting reviews on line), and then might engage in trial anderror. The allocation of time is chosen by the chef. Much of what the chef
learns becomes tacit knowledge beyond the …nal recipe; this …nal recipe might
be secret and used by his restaurant, or sold. The rate at which new recipes
are created will depend on the chef’s experience with this type of knowledge
work, but also on the tacit knowledge accumulated through the creation process.
To elaborate our model, let us consider a speci…c time, t 2 [0; 1), and let
the following variables represent the state of each person’s knowledge at timet (whenever clear, dropping t for simplicity):

ncij

nk : the size of person k’s knowledge (or number of ideas
known by person k at time t); k = i; jcnji : the size of knowledge that i and j both know,or the common knowledge for i and jdnij : the size of knowledge known by i but not known by j,or the di¤erential knowledge of i from j,dnji : the size of knowledge known by j but not known by i,or the di¤erential knowledge of j from i.

By de…nition,ni = ncij + ndij ; nj = ncij + ndji :Letnijncij + ndij + ndji = ni + nj

ncij

be the size of total knowledge that is known either by i or j.
Next, we de…ne the proportion of each type of knowledge in the total sizeof knowledge nij :
ncij
;
nij
ndij
ndji
d
=
;
m
=
;
ji
nij
nij

mcij =
mdij
implying thatmcij + mdij + mdji = 1;(1)

and henceni = nij (1
or

ni
=1
nij

mdji ); nj = nij (1
nj
=1
nij

mdji ;
7

mdij :

mdij );(2)

Using this notation, we describe next the two alternative ways of creatingknowledge (at time t).
In what follows, consistent with the notation introduced above, lower caseletters such as i and j represent persons, whereas upper case letters represent
activities. Examples of the latter include I, representing isolated or independent activity, and J, representing joint activity or activity for the purposeof joint knowledge creation. Likewise, S represents search activity, to be
explained next.
What we mean by search is to search the web by oneself to prepare forknowledge creation activity that will occur either in isolation or jointly withanother. Examples are using Google or ChatGPT. The activity is a form of
directed search, in contrast with undirected search. That is, the search terms
or questions guide the use of the web in an important and nonrandom way, andare progressively re…ned over the time used for search. Just a few years ago,
when beginning a project, an economics researcher would search the Econlit
database, for example, using search terms that de…ne the new project. The
purpose would be to …nd related papers at the frontier of knowledge and to see
similar work in terms of assumptions, implications, models, and empirics. For
example, aside from the references we knew about from previous joint work,
to compose this paper we searched on key phrases such as “R&D during theCovid 19 pandemic.” Re…nement of search terms, as well as digesting material,
takes time and e¤ort. Nowadays, with Google Scholar and ChatGPT, thee¤ectiveness of search has improved dramatically over primitive times. We
will parameterize this e¤ectiveness of directed search in our model.
Given two persons i and j, as noted previously, a person, say i, can develop
new knowledge in alternatively ways:(i) knowledge creation in isolation(ii) joint knowledge creation of i and j working together
Next, we describe in detail each type of knowledge creation, starting withthe simpler one.

(i) Knowledge creation in isolation
Figure 2 depicts the activity tree for knowledge creation by person i in isolation.
Figure 28

The top node in Figure 2 represents the …nal output AIii , which is produced
by appropriately combining the outputs of the two basic activities in the secondIS
tier: aITi (the intermediate output of Independent Thinking by i), and ai (theintermediate output of Independent Search by i). The output, or the numberof new intermediate ideas created per unit of time by each basic activity isgoverned by the following equations:
aIT
i =

IT

ni ;

( IT > 0)

(3)

aIS
i =

IS

ni ;

( IS > 0)(4)where the positive constant IT represents the e¤ectiveness of thinking alone,whereas the positive constant IS represents the e¤ectiveness of search. In
both basic activities, productivity depends on what a person already knows.5
The …nal output of knowledge creation activity by person i in isolation perunit of time is governed by the following equation:
AIii =I! IT aITi! IS aISi

IT

IS( I > 0)(5)where AIii is the number of ideas produced.6 On the right hand side, IS ; IT
0 are …xed parameters that weight the search and thought activities; we assume
that IS + IT = 1. Knowledge creation activity in a time period is divided
into two parts: search in isolation with frequency ! IS
0 and thinking in
isolation with frequency ! IT
0, where ! IS + ! IT = 1. The positive constant I represents the overall productivity of knowledge creation in isolation.
Notice that given the Cobb-Douglas functional form, individual thinking andindividual search are not perfect substitutes.
Knowledge worker i optimizes the choice of frequency over the two activities:
max AIii ! IT + ! IS = 1; ! IT

0; ! IS0yielding the optimal choices of frequencies ! IT and ! IS :! IT =IT ; ! IS =

5

IS :

“What you can learn from Wikipedia depends on what you already know.”
We note here, at the …rst introduction of a knowledge production function, that we use
extensively the Cobb-Douglas functional form thoughout this paper. Although use of thisparticular function has drawbacks, for instance in generality of the results, its big advantage
is that the functional form makes calculations vastly simpler than they would be otherwise,and permits analytical tractability.
6

9

Thus, the optimized value of knowledge output per unit of time for person iin isolation is:
AIii

=

I

=

I

IT

aIT
i

IT

IS

aIS
i

IS

(6)
(7)

ni

where
I

I

( IT

IT )

IT

( IST

I

nj

IS )

IS

:

Likewise, for person j, we have
AIjj =(8)

where I represents the same function of parameters as above. We may
note that the search productivity parameter IS depends on several factors.
First, it depends on the search technology at the time the work is done. For
example, Google (1998) preceded Google Scholar (2004) and more recently
ChatGPT. Second, it depends on the knowledge stock at the time, for example
in Wikipedia. Third, it depends on the learning capacity of person i. Similarly
for the parameter IS . In the long run, parameters might change, but here,
for simplicity, we take the parameters as …xed.
(ii) J oint knowledge creation of i and j through multimodal
communication
Next, we turn to the main focus of this section, that is, the modeling of joint
knowledge creation by two persons in communication through the Net andF2F. Our framework is motivated by the empirical work of Lin et al (2023).
We divide the knowledge production process into conceptual and technicalphases, allowing researchers to choose the best mode of communication, F2F
or Net, for each phase. Lin et al (2023) …nd that remote collaboration is lesse¤ective in the conceptual phase of research, particularly where tacit knowledge
is useful, whereas remote collaboration is used in the later stage of technicaltasks. These will be outcomes of our model.
Figure 3 represents the activity tree for joint knowledge creation by two
persons, i and j. In comparing …gures 2 and 3, we can see that the process ofjoint knowledge production is much more complex than the case of knowledge
creation in isolation.
Figure 3

10

The top in Figure 3 represents the …nal output AJij , which is obtained byappropriately combining the outputs of four main branches of activities. The
left branch corresponds to the joint task of conceptual development for new
patents (or papers). Each of the two branches in the middle of the second tiercorresponds to the task of further development of intermediate ideas for the
purpose of joint knowledge creation independently by each person. The rightbranch corresponds to the task of joint technical development of new patents(or papers).
The middle two branches of independent development are included to re‡ect the fact that when two persons are developing a new patent or paper,
they spend time not only working jointly F2F or through the net, but each
person also spends a signi…cant amount of time working independently for the
purpose of joint development of the …nal output. It also needs to be notedthat over the time of actual development of a new patent or paper, the four
branches of activities depicted in Figure 3 may be repeated sequentially manytimes.
Next, we describe each branch of activity in detail.
(ii 1) J oint conceptual development
As shown in the third tier of the left branch in Figure 3, Conceptual deF(Conceptual development
velopment is based on four basic activities: aCTij
CT N(Conceptual development by Thinking toby Thinking together F2F ), aij
CSIgether through the Net), ai(Conceptual development through Search Independently by person i), and aCSI
(Conceptual development through Searchj
Independently by person j).
Each basic activity is governed by the following equations:
C

F
aCT
=
ij

CF

(ncij )1C

(ndij ndji ) 2 ; ( CF > 0; 0 <

N
aCT
=
ij

CN

(ncij )1C

(ndij ndji ) 2 ; ( CN > 0)

aCSI
=
i

CS

ni ;

aCSI
=
j

C

CS

nj ; ( CS > 0)

C < 1)

(9)
(10)
(11)

Equation (9) states that when two persons are thinking jointly for conceptualFdevelopment F2F, the output aCTof conceptual intermediate ideas is generij
ated at a rate proportional to the normalized product of their knowledge incommon, ncij , the di¤erential knowledge of i from j, ndij , and the di¤erential
knowledge of j from i, ndji . The rate of creation of new intermediate ideas ishigh when the proportions of knowledge in common, knowledge exclusive toperson i, and knowledge exclusive to person j are in balance. The parameter11

C represents the weight on di¤erential knowledge as opposed to knowledge incommon in the production of new ideas in the conceptual development phase.
Knowledge in common is necessary for communication between the two persons, whereas knowledge exclusive to one person or the other implies moreheterogeneity or originality in the collaboration. The positive constant CF
represents the e¤ectiveness of conceptual development by thinking jointly F2F.
Equation (10) gives the rate of creation of conceptual intermediate ideas
thinking jointly through the Net. This equation is same as equation (9)except that e¤ectiveness parameter CF is replaced with CN . Although thetwo equations are of the same form, the di¤erence between the two parameters,
CF and CN , plays the essential role in the analysis later.
Each of the two equations in (11) states that the rate of creation of conceptual intermediate ideas through search by each person is proportional to
the size of their knowledge, where the e¤ectiveness parameter CS is common
for the two persons.
The outputs of the four basic activities at the bottom of the left branch
are combined in two steps, yielding the output aC
ij of conceptual development.
CT N
CT F
First, intermediate conceptual ideas, aij and aij , generated through thinking jointly through F2F and Net are combined as follows:
aCT
ij =

CF

F
aCT
+
ij

CN

N
aCT
;
ij

(12)

N
F
,
and aCT
where CF and CN denote the frequency of basic activities aCT
ij
ij
respectively. The two persons can jointly choose CF and CN freely, subject
to the following constraint:(1 + "F )CF +CN = 1;

CF

0;

CN

(13)

0

where "F > 0 represents the lead time of joint thinking for conceptual development F2F. In practice, "F re‡ects the time cost of preparing for a F2F
meeting, such as commuting time to the common CBD o¢ ce (or common university), or travel time between two cities or two countries where each of thetwo persons reside separately.
Alternatively, we may generalize the constraint (13) as follows:(1 + "F )CF + (1 + "N )CN = 1;

CF

0;

CN

0;

where "N represents the lead time of joint thinking for conceptual development
through the net. In practice, "N is much smaller than "F . Furthermore, tore‡ect the nature of net-technology, the value of "N is essentially independent12

of “geographical distance” between the pair of communicators. Hence, giventhat we focus in this paper on the obstacle to communication caused by “geographical distance”between the pair of communicators, and also for simplicity
of notation, we set "N = 0 and use the time constraint given by (13).7
Based on the constraint (13), the two persons jointly choose the optimal
combination of f CF ; CN g by solving the this problem:
max aCT(1 + "F )ij

CF +CN = 1;

0;

CF

CN

0

which yields the following solution:
when

CF
CN

when

CF
CN

1
1 + "F1
1 + "F

> 1;
1;

CF =

1
;
1 + "FCF = 0;

CN = 0(14)(15)

CN = 1

Using (14) and (15), the optimized value of the output of joint conceptual
thinking over a unit of time is given by
aCT
=
ij

CF

F
aCT
+
ij

CN

N
aCT
:
ij

(16)

In the second step, the output of joint thinking, aCTij , and the output of
CSI
CSI
independent search by each person, ai and aj , in (11) are combined toyield the …nal output aC
ij of the conceptual development phase as follows:
aC
ij =

C
ij

CT
[! CT aCT[! CS (aCSI+ aCSI
)] CS
ij ]
i
j

(17)

where the positive constant C
ij represents the overall e¤ectiveness of conceptual
development activities. Each of CT and CS represents a given positive weighton each type of basic activity where CT + CS = 1. Each of ! CT
0 and
! CS
0 denotes the frequency of the corresponding activity over a unit oftime, where ! CT + ! CS = 1. The two persons choose jointly the optimal
combination of f! CT ; ! CS g by solving the next problem:
maxfaC
ij ! CT + ! CS = 1; ! CT

0; ! CS0g

yielding the solution:! CT =CT ;! CS =

(18)

CS :

7

Actually, if we denote the original parameter "F as "oF , and de…ne the parameter "F by
the relation:
1"oF "N
1 + "N
=
;
that
is,
"
;
F
1 + "oF
1 + "F
1 + "Nthen we can see that there is no loss of generality in setting

13

N = 0.

Substituting (16) and (18) into (17), the optimized value of the …nal output
of conceptual development phase is obtained as follows:
aC
ij

=

C
ij

F
+
[ CT ( CF aCT
ij

N
)] CS
+ aCSI)] CT [ CS (aCSI
aCT
j
i
ij

CN

(19)
=

C
ij

[ CT ( CF

[ CS
where

CF andCSCF +(ncij )

CN )

CN

1

C

(ndij ndji )

C
2

] CT
(20)

(ni + nj )] CS ;CN are de…ned through (14) and (15).

(ii 2) Independent development by each person f or the purposeof joint creation
As noted before, when two persons are jointly creating a new patent or paper, each person spends a signi…cant amount of time working independently
to develop intermediate ideas for the …nal joint output. The modeling of thisphase of independent development by each person is rather similar to Knowledge creation in Isolation explained in (i) using Figure 2. Here, focusing on
the middle left branch of Independent development by person i in Figure 3,
the output of each basic activity at the bottom of this branch is given by thefollowing equations:JaIT= IT J ni ( IT J > 0)
(21)
i
aISJ
=
i

ISJ

ni

(22)

( ISJ > 0)J
represents the number of new basic ideas created per unit time
where aITithrough Independent Thinking by person i for Joint creation purposes, andrepresents that basic ideas created through Independent Search for JointaISJi
purposes by i. These outputs of basic activities are combined as follows fordeveloping intermediate ideas:aIii =IJ

J! IT J aITi

IT J! ISJ aISJ
i

ISJ

(23)

where aIii is the number of new intermediate ideas developed per unit time.
Each multiplier, IT J and ISJ , represents the …xed weight on the output of
the corresponding basic activity, where IT J + ISJ = 1. The frequency ofeach basic activity, ! IT J and ! ISJ , is chosen optimally by person i as follows:
max aIii ! IT J + ! ISJ = 1; ! IT J

0; ! ISJ

0 ;yielding the optimal choice of frequencies ! IT J and ! ISJ :! IT J =IT J ;! ISJ =

14

ISJ :

(24)

Thus, the optimized value of intermediate output per unit of time for person
i in this phase of Independent development is obtained by substituting (21),(22) and (24) into (23) as follows:
aIii

=

IJ

=

IJ

J
aIT
i

IT J

IT J

ISJ

aISJ
i

ISJ

(25)
(26)

ni

where
IJ

IJ

( IT JIT J )

IT J

[ ISJ

ISJ ]

ISJ

(27)

:

Likewise, for Independent development by person j in the middle rightbranch of Figure 3, we haveaIjj =
where

IJ

(28)

nj

IJ represents the same function of parameters as above.

(ii 3) J oint technical development
As shown in the third tier of the right branch in Figure 3, technical development
is based on four basic activities: aijT F (T echnical development by Thinking together F2F ), aijT N (T echnical development by Thinking together through theNet), ai SI (T echnical development through Search Independently by person i,and aj SI (T echnical development through Search Independently by person j).
These basic activities in the right branch of Technical development are inparallel to those in the left branch of Conceptual development. Furthermore,
the form of each basic activity in Technical development is essentially the sameas the corresponding one in Conceptual development, where each upper caseletter C (indicating Conceptual development) is replaced with (indicating
T echnical development). Hence, the explanation of each speci…cation below
shall be short.
The basic activities in Technical Development are governed by the following
equations:aijT F =F(ncij )1

(ndij ndji ) 2 ; (F > 0;

aijT N =N(ncij )1

(ndij ndji ) 2 ; (N > 0)

ai SI =

Sni ; aj SI =S

nj ; (

S > 0)

0<

< 1)

(29)
(30)
(31)

Although the three equations above are of the same form as, respectively, (9),(10) and (11), di¤erences in parameters F and N from parameters CF
and CN in (9) and (10) play an essential role in the choice of communication
15

mode later. In (29) and (30), the parameter
represents the weight ondi¤erential knowledge as opposed to knowledge in common in joint Technicaldevelopment. This parallels the parameter C .
To get the …nal output of the Technical development phase, …rst the intermediate concept ideas, aijT F and aijT N , are combined as follows:
aijT =

F

aijT F +N

aijT NThe two persons can jointly choose the frequencies
to the following constraint:(1 + "F )F +N = 1;

F

0;

(32)

F and

N freely, subject

(33)

0

N

where "F is the same lead time parameter for F2F meetings introduced in (13).
The optimal combination of f F ; N g is obtained by solving the following
problem:
max aijT (1 + "F )F +N = 1;

F

0;

0

N

which yields the solution:
when

F
N

when

F
N

1
1 + "F1
1 + "F

> 1;

F =

1;

1
;
1 + "FF = 0;

N = 0

(34)
(35)

N = 1

Using (34) and (35), the optimized value of joint technical thinking over a unitof time is given by
aijT = F aijT F + N aijT N :(36)
Next, the output of joint thinking, aijT , and the output of independent
search by each person, ai SI and aj SI , in (31) are combined to yield the …nal
output of the technical development phase as follows:
aij =

ij

[! T aijT ][! S (ai SI + aj SI )]

T

S

(37)

The two persons jointly choose the optimal combination of frequencies, f! T ; ! S g,
by solving the following problem:
maxfaij ! T + ! S = 1; ! T

0; ! S

0g

yielding the solution:
! T =

T;

! S=

16

S:

(38)

Substituting (36) and (38) into (37), we obtain the optimized value of the…nal output of the technical development phase as follows:
aij

=

ij

[

T

(

F

aijT F +N

aijT N )][T(ai SI + aj SI )]

S

S

(39)
=

where

F

ij

[

[

S

and

T

F +

(

F

S

(ni + nj )]

N)

N
S

1
(ncij )

(ndij

ndji ) 2 ]

T

(40)

;

N are de…ned through (34) and (35).

(ii 4) T he f inal output of joint work
As indicated in Figure 3, the …nal output AJij of joint work for knowledge creation is generated by combining the outputs of the four branches of activities,(20), (25), (39) and (40), as follows:C
AJij = [ C aC
[ I (aIii + aIjj )] I [
ij ]

(41)

aij ]

where each of C , I and N represents a given positive weight on each taskof branch-activity such that
C +

I +

= 1;

whereas each of C , I anddenotes the frequency of each branch-activity
over a unit of time. The two persons can jointly choose the optimal combination of f C ; I ; g by solving the following problem:
maxfAJij

C +

I +

= 1;

C

0;

I =

I;

=

0;

I

0g;

which yields the following solution:
C =

C;

(42)

:

Substituting (42) into (41), the maximized value of AJij is given byC
AJij = [ C aC
[ I (aIii + aIjj )] I [
ij ]

(43)

aij ]

To be more speci…c, substitution of (20), (26), (28) and (40) into (43) leads toAJij=[ni + nj ] I + CT C + T
hi CT C h
c 1 C
d
d 2C
(nij )(nij nji )(ncij )1

J

[ CF

CF +

CN

CN ]

CT

17

C

[

F

(ndij
F +

N

ndji ) 2
N]

i

T

T

(44)

where CF , CN ,and (35), whereasJ

N are de…ned respectively through (14), (15), (34)

F and

J represents a function of parameters given by

= ( C

C C
ij )

(

ij )

I

I

I

IJ

( IT JIT J )( CT ) CT C ( CS

IT J

CS )

I

CS

( ISJ

ISJ )

ISJ

I

(

S)

S

(45)

C

S

Finally, recalling the de…nitions in the …rst part of this section, we canrewrite (44) in terms of mcij , mdij and mdji as follows:
AJij

+

+

= nij
(1 mdji ) + (1 mdij ) I CT C TJhi CT C hC(mcij )1 C (mdij mdji ) 2(mcij )1
[ CF

CF +

CN

CN ]

CT

C

[

F

F +

(mdij mdji ) 2
N

N]

T

i

T

(46)

where J is a function of parameters given by (45).
Our model of the knowledge creation process is analogous to the manufacturing of an automobile. The elementary components are …rst synthesizedfrom raw materials. Then the elementary components are assembled into
complex parts. Lastly, the …nal product is created from the parts. This iswhat the customers see as sold in the market.

3E¤ect of F2F lead time on the mode of communication and joint knowledge productivity

In this section, using the model developed in Section 2 (ii), we examine how
the communication mode and joint knowledge productivity are a¤ected by achange in the value of "F . Recall that the parameter "F represents the leadtime necessary for realizing actual F2F communication by a pair of knowledge
workers. In the context of intra-city communication, "F re‡ects the commuting
time to the common workplace such as a CBD o¢ ce, research institute oruniversity. In the context of inter-city or inter-country communication, "Fre‡ects the travel time between two cities or countries for realizing research
work in F2F mode. Here, travel by each researcher between two cities orcountries is assumed to take place with equal frequency, so cost is symmetric.
Recall parameter CF introduced in equation (9), which represents thee¤ectiveness of conceptual development by thinking jointly F2F, and parameter CN in (10), representing the e¤ectiveness of conceptual development by
18

thinking jointly through the Net. Recall also parameters F and N introduced respectively in (29) and (30), representing respectively the e¤ectivenessof technical development by thinking jointly F2F and through the Net. In
the following analysis, we assume the following relations hold among these fourparameters:
Assumption 1. The following relations hold among communication parameters
f CF ; CN ; F ; N g:
CF

> 1;

CN

F

> 1; and

N

CF

F

>

CN

:

(47)

N

That is, in both the phase of conceptual development and the phase oftechnical development, thinking jointly F2F is more e¤ective than thinking
jointly through the Net. Furthermore, the relative e¤ectiveness of F2F relative
to the Net is greater in conceptual development than in technical development.
This assumption is based on two ideas. First, tacit knowledge is bestdeveloped and exploited face to face. Nonaka and Takeuchi (1995) provide
case studies, including the invention of the …rst automatic home bakery machine by Matsushita and the development of the City car model by Honda.
Both involved the exploitation of tacit knowledge. Second, empirical evidence
developed by Lin et al (2022) is consistent with this assumption.
Based on Assumption 1, Figure 4 explains the endogenous formation of
three ranges of communication mode, (a), (b) and (c), depending on the parameter value of "F when CF = CN = 2 and F = N = 1:5.
Figure 4
In this …gure, the horizontal axis displays values of the parameter "F ,whereas the vertical axis gives the productivity of F2F interaction relative
to Net interaction. The upper curve shows how the relative e¤ectiveness of
F2F relative to the Net in conceptual development changes with parameter"F ; the lower curve shows the same for relative e¤ectiveness in technical development. Let "CF (respectively "F ) represent the value of "F where the upper
curve (respectively the lower curve) meets the horizontal line with height 1.
That is, "C
F and "F are values of "F such that
CF
CN
F
N

1
1 + "CF
1
1 + "F= 1; i.e., "CF == 1; i.e., "F =
19

CF

1;

(48)

1:

(49)

CN
F
N

Next, let (a), (b) and (c) represent respectively the three ranges of parametervalue "F as shown in Figure 4. Then, recalling (14), (15), (34) and (35), andusing Figure 4, we can readily obtain the following result:
Proposition 1. Under Assumption 1, let "C
F and "F be the values of parameter "F de…ned by (48) and (49). Then, the combination of optimal frequencies,f CF ; CN ; F ; N g, associated with the four basic joint-thinking activities,
NF; aijT F ; aijT N g, forms the following three ranges in terms of the
; aCT
faCT
ij
ij
parameter value of "F :(a) for "F 2 [0; "F ) ;(b) for "F 2

"F ; "CF ;(c) for "F 2

"C
F;1 ;

CF =

1
;
1 + "FF =CN = 0;1;
1 + "F

N = 0

(50)
1
; CN = 0; F = 0; N = 1 (51)CF =1 + "F(52)
N = 1:
F = 0;
CN = 1;
CF = 0;That is, within range (a) of Figure 4, in both the phase of conceptual development and the phase of technical development, joint work is conducted solely
F2F, re‡ecting the high e¤ectiveness of F2F communication and the small
obstacle of F2F lead time. Within range (b), joint thinking for conceptual development is conducted F2F whereas joint thinking for technical development
is conducted through the Net. This re‡ects the fact that the F2F lead time isa relatively small obstacle for conceptual joint work whereas it is relatively aheavy burden for technical joint work. In contrast, within range (c), both thephase of conceptual development and the phase of technical development are
conducted solely through the net, re‡ecting the big obstacle of F2F lead time.
Next, we examine how the productivity of joint work is a¤ected by theparameter value "F . For this purpose, let us de…ne
AJij ("F )the value of AJij at "F 2 [0; 1) ;where AJij is the optimized output of joint work given by equation (44). The
values of frequencies, CF , CN , F and N , are respectively given by (50),(51) and (52), depending on the value of "F . Next, leth("F )AJij ("F )[= CF
JAij (0)

CF +

CN

CN ]

CT

C

[ CF ] CT C

[
[

F +

F
F]

N

N]

T

T

(53)
20

be the relative productivity of joint work at "F in comparison with the productivity of joint work at "F = 0, where the right side of equation (53) has beenobtained from (50) since CF = 1, CN = 0, F = 1 and N = 0 at "F = 0.
Then, based on Proposition 1, we can readily obtain the relative productivity function, h("F ), for each range of "F as follows:(a) for 0 "F < "F ,"F < "F ) = (1 + "F ) ( CT C +h("F j0ha ("F )(b) for "F"F < "CF,

hb ("F )h("F "F(c) for "CF"F < 1,hc ("F )h("F "CF"F < "CF ) = (1 + "F )"F < 1 ) = [ CF = CN ]

CT

CT

[

C

C

F=

[

F=

T

N]

N]

)

(54)

;

;

T

T

(55)

; (56)

Denoting by “ 0 ”the derivative of a function with respect to parameter "F , itfollows that:(a) for 0 "F < "F ,h0a ("F )=ha ("F )(b) for "FCT

T

1 + "F< 0; h00a (") > 0;

(57)

"F < "CF,
h0b ("F )=hb ("F )(c) for "CFC +"F < 1,

implying thatCTC

1 + "F< 0; h00b (") > 0;h0c ("F )= 0;
hc ("F )h0a ("F )>ha ("F )h0b ("F )h0 ("F )> c= 0:
hb ("F )hc ("F )

(58)

(59)

(60)

Figure 5 illustrates the relative productivity curve, h("F ), over the threeranges of parameter "F when CF = CN = 2, F = N = 1:5 and CT C == 0:25.
T
Figure 521As can been observed in the …gure, in the (a) and (b) ranges, the relative
productivity curve is decreasing and strictly convex, whereas it is ‡at overrange (c). From Proposition 1, in range (a) joint work is conducted throughF2F communication in both conceptual development and technical development, whereas in range (b) joint work is conducted through F2F interaction in
the conceptual development phase, whereas joint work is conducted throughthe Net in the technical development phase. Moreover, increasing "F meansa bigger obstacle for F2F communication, whereas the cost of Net communication is independent of "F . Hence, the relative productivity h("F ) decreases
with "F more sharply to the left of the boundary "F than to the right of "F ,making the curve kinked at "F . In contrast, in range (c), since joint work is
conducted solely by using the Net, the relative productivity is not a¤ected by
the value of "F and hence it is ‡at; however, given that Net communication isless e¤ective than F2F communication, the relative productivity is the lowest
in range (c). The curve is also kinked at the boundary "CF.
~
In Figure 5, the broken curve, ha ("F ), to the right side of boundary "Fshows the extension of curve ha ("F ) to the right of boundary "F . That is,~ a ("F ) = (1 + "F ) ( CT C +
h

T

)

; 0

"F < 1:(61)

Recall that function ha ("F ), has been derived by setting the communication
mode so that the communication in joint work is conducted solely F2F in boththe phase of conceptual development and the phase of technical development,
without utilizing Net communication at all. In other words, the broken curve
represents the relative productivity at each "F"F in the …ctitious situation
when no net-technology was available. In reality, however, in range (b), optimalcommunication requires use of the net, not F2F communication, in joint work
~ a ("F )for technical development. Hence, in range (b), the extended curve his based on non-optimal communication mode choice, and hence it is lower
than the curve hb ("F ). Finally, since the optimal communication mode inrange (c) requires use of the net, not F2F communication, in both conceptual~ a (")
development and technical development, the extended productivity curve hover range (c) is far lower than the curve hc ("F ).
We may summarize the observations above as follows:
Proposition 2. Let h("F ) be the relative productivity function de…ned byequation (53), which represents the relative productivity of joint work at each"F
0 in comparison with that at "F = 0, in other words no lead time cost.
Under Assumption 1, let (a), (b) and (c) represent the three ranges of "F as22

de…ned in Figure 4. Then, the relative productivity function h("F ) exhibits
the following features, as illustrated in Figure 5:(i) Over the ranges (a) and (b), the relative productivity curve is decreasing
and strictly convex, and kinked at the boundary "F . Given that in range(a) both conceptual joint work and technical joint work are conducted F2F,
whereas in range (b) only conceptual joint work is conducted F2F, the relative
productivity decreases with "F faster in range (a) than in range (b).
(ii) In range (c), since the joint work communication is conducted solely
through the net, the relative productivity is insensitive to the value of "F ,resulting in a ‡at curve. However, since no e¤ective usage of F2F communication is possible in this range, the relative productivity is the lowest among
the three ranges.
~ a ("), represents the extension of the relative(iii) In Figure 5, the broken curve, hproductivity curve ha (") to the right of boundary "F , showing the relative
productivity at each "F under the …ctitious restriction that no usage of the
Net is permitted. The di¤erence between the real relative productivity curve
and this …ctitious curve at each "F > "F shows how much higher productivity
in joint work is due to the e¤ective use of net communication. Speci…cally,in range (c), although the broken curve decreases toward 0 as "F increases
to 1, the actual relative productivity hc ("F ) stays relatively high due to theavailability of net communication.
Figure 5 can be interpreted as representing the impact of the development of
net-communication technology, where the broken curve represents the relative
productivity at each "F when no net-technology is available, whereas the real
curve represents the relative productivity under the e¤ective use of modern nettechnology. The case where no net-technology is available is studied by Inoueet al (2022), since the internet didn’t exist during the Spanish ‡u epidemic.
Without this substitute for F2F, the e¤ect on knowledge creation was severe.
In order to examine more closely the impact of the advances in net-communication
technology in joint work, we add one more curve to Figure 5 and obtain Figure 6. In this new …gure, the top solid curve and the bottom broken curve
are respectively the same as those in Figure 5, where the horizontal axis in
Figure 6 is compressed to half of that in Figure 5. The new middle curvein Figure 6 represents the relative productivity curve when old net-technology
was available. The middle curve has been obtained under the new set ofcommunication-e¤ectiveness parameters, f CF ; oCN ; F ; o N g, re‡ecting theold net technology where the following assumption is satis…ed:
23

Assumption 2. Let f CF ; CN ; F ; N g be the set of communication parameters used in Assumption 1, satisfying the relations in (47). Let f CF ; oCN ; F ;be the communication parameters under the old Net technology, satisfying thefollowing relations:
CF
o
CN

F

> 1;

o

> 1; and

N

CF
o
CN

F

>

o

(62)

:

N

and
CN >

o
CN ;

N >

o

(63)

N:

Next we explain this assumption. In the two sets of communication parameters above, CF and F remain the same, re‡ecting that the e¤ectiveness ofF2F communication in joint work by itself is not a¤ected by the developmentof Net technology. The conditions in (62) correspond to those in (47), whereas
relations in (63) imply the advancement of net technology in conceptual development and in technical development. For both cases in Assumption 2, theweight parameters, CT C and T
are assumed to remain the same.
oo CLet "F and "F respectively be the values of "F such that
CF
o
CN

1
1+
1

F
o
N

o "C
F

1+

o"

= 1; i.e., o "CF == 1; i.e., o "F =

F

CF
o
CN

1;

(64)

F

1:

(65)

o
N

implying from (48), (49) and (63) thato Co"F > "CF ; "F > "F :(66)

The optimal communication frequencies f CF ; CN ; F ; N g under the old
net technology for each of new ranges, (ao ), (bo ) and (co ), are obtained byo Coreplacing "F and "C
F by "F and "F respectively in (50), (51) and (52). The
relative productivity function, ho ("F ), for each range of "F is obtained byoo C
replacing "F and "C
F by "F and "F respectively in (54), (55) and (56).
As illustrated in Figure 6, with the advance of net technology from theold to the modern, the F2F-dominant range (where F2F communication is
exclusively used in both conceptual development and in technical development)
shrinks from range (ao ) to range (a); the range of mixed communication mode(in which conceptual development is through F2F interaction whereas technicaldevelopment is through the Net communication) also shrinks towards the origin

24

o

Ng

from (bo ) to (b).8 In contrast, the net-dominant range expands towards theorigin from (co ) to (c).
With such changes in the optimal communication mode with the advancement of Net technology, the productivity of joint work increases signi…cantly
in a wide range of parameter "F . To be precise, the relative productivity increases more steeply as parameter "F rises from "F toward o "CF ; and the relative
proportion of productivity improvement reaches the maximum at o "CF , and ito Cremains the same beyond "F .
We may summarize the observations above as follows:
Proposition 3. Suppose that the set of communication e¤ectiveness parameters changes from the old set f CF ; oCN ; F ; o N g to the new set f CF ; CN ; F ;where all the conditions of Assumption 1 and Assumption 2 are satis…ed; conditions (63) represent the advancement of Net technology. Then, as illustrated
in Figure 6, the relative productivity curve under the new set of parameters
shifts upward from that under the old set in a wide range of parameter "F .
Speci…cally:(i) With the advancement of Net technology, the three ranges of optimal communication mode change as follows: The F2F-dominant range shrinks toward
the origin from range (ao ) to (a) as shown in Figure 6. The range of mixed
communication mode also shrinks toward the origin from (bo ) to (b). In
contrast, the net-dominant range expands toward the origin from (co ) to (c).
(ii) Such changes in the optimal communication mode with the advancement of
Net technology imply that the productivity of joint work increases signi…cantly
in a wide range of "F : To be speci…c, as shown in Figure 6, relative productivity
increases more steeply as parameter "F moves from the new outer boundary"F of F2F-dominant range toward the old inner boundary o "CF of the netdominant range: The relative proportion of productivity improvement reaches
o C
a maximum at o "CF , and it remains the same beyond "F .
As is apparent from Figure 6, although the relative productivity of joint
work in the net-dominant range is always lower than in the F2F-dominant
range, the former improves rapidly with the advance of net technology. Furthermore, with the advance of transport technology such as the appearance
and improvement of rapid railways (e.g., the Shinkansen in Japan and China)
8

In Figure 6, the range, (bo ), signi…cantly shrinks to (b). However, in order to know precisely how much range (bo ) shrinks, we need to know more speci…cally the relative magnitudeof parameters in (63).

25

N g,

and jet airplanes, the value of "F has been shrinking for the same geographical distance. This suggests that with the advancement of net technology andtransport technology, the location of potential partners for joint work expands,
as we will discuss further in Section 4.
Finally, we may reinterpret Figure 6 in a di¤erent context. Let us imagine
that there are two groups of knowledge workers in terms of internet skills. One
group consists of those workers who are have poor internet skills such as the“old generation”or people in living in environments that lack internet access,such as rural America. The other group consists of those workers who havemastered modern net technology. Then, we can naturally reinterpret Figure
6. The lower solid curve represents the relative productivity curve for workers
with poor computer skills, whereas the upper solid curve represents that forworkers with better skills. In this context, it is not di¢ cult to imagine that
restrictions on F2F communication such as those introduced during the Covid
19 pandemic will exert di¤erent impacts on the two groups, as we discuss
further in Appendix A.

26

4Knowledge composition, productivity, andchoice of work mode

Given the joint output function AJij de…ned in equation (46), we have focusedin Section 3 on the parameter "F specifying F2F lead time, and examined
how communication mode and joint knowledge productivity are a¤ected by achange in the value of "F , …xing the knowledge composition of joint workers.
Conversely, in this section, we …x the value of parameter "F , and examine howknowledge composition of the pair of workers a¤ects joint-work productivity.
Furthermore, we specify the objective of each person in terms of knowledge
creation, and de…ne the rule used by each person to decide whether they create
new ideas jointly or in isolation as a result of optimization of their objective.
To start, let us notice that in the joint output function AJij given by (46),the optimal combination of communication-mode frequencies, f CF ; CN ; F ;is left unspeci…ed. To replace these unspeci…ed variables with explicit functions of parameters, we suppose in this section that Assumption 1 holds, andhence Proposition 1 is valid. In this context, if we let AJij ("F ) be the value ofAJij at each "F 2 [0; 1), it follows from (53) that

N g,

AJij ("F ) = AJij (0) h("F );where h("F ) is the relative productivity function of joint work at parametervalue "F . Furthermore, using (50) for "F = 0, we have:[ CF (0)CF += [ CF ] CT C [CN (0)
F]

CN ]

CT

C

[

F (0)

F +

N (0)

N]

T

:

T

Hence, the optimized joint output function (46) can be expressed explicitly as
a function of parameter "F as follows:
AJij ("F ) = nijhJ(mcij )1

(1
C+

mdji ) + (1(mdij

[ CF ] CT C [F]+mdij ) I CT C T
i CT C hCmdji ) 2(mcij )1

(mdij mdji ) 2

h("F );

T

where from (54), (55) and (56), the function h("F ) is given by:
8
(
+
)
>
for 0 "F < "F ;< (1 + "F ) CT C T
h("F ) =(1 + "F ) CT C [ F = N ] T
for "F "F < "CF;>:C[ CF = CN ] CT C [ F = N ] T for "F "F < 1;
27

i

T

(67)

(68)

where "CF and "F are de…ned respectively by (48) and (49).
Given that parameter "F is …xed for the pair of joint workers, i and j,unless necessary, we express AJij ("F ) as just AJij for notational simplicity.
Before going further, we must specify the objective of each person in terms
of knowledge creation. Speci…cally, we need to clarify how the output of
the creation process, AJij , is split between i and j. In this connection, we
introduced in our earlier papers two di¤erent speci…cations. In Berliant andFujita (2008, 2009), assuming that AJij contributes directly to increasing each
person’s felicity (or instantaneous utility) at that time, AJij is not split between
the two persons. It is a public good. In contrast, in Berliant and Fujita(2011, 2012), a …xed proportion of every collection of ideas created is assumed
to become new patents, which are sold at the given market price at that time.
The revenue from new patents is split evenly if persons i and j produce newideas together. Although both speci…cations would lead to similar results, inthis paper we adopt the latter speci…cation of Berliant and Fujita (2011, 2012).
To go further, we must de…ne the rule used by each person to decide whether
they create new ideas jointly or in isolation. The rule is derived from optimization. For this purpose, we assume that income for each person derives
from selling new ideas created as patents. We assume that the revenue fromnew patents is split evenly if persons i and j produce new ideas together.
Let ij (t) 2 f0; 1g be a function such that if
ij (t) =ji (t) = 1 for j 6= i;(69)then joint knowledge creation occurs at time t. In contrast, if
ii (t) =(70)

jj (t) = 1;

then each person creates new knowledge in isolation. Notice that since thetwo persons must agree either to work jointly or to work in isolation each,either (69) or (70) can occur exclusively at time t.
Let yi (t) be the income of each person at time t, and let (t) be the priceof patents at time t. Then suppressing t for notational simplicity:
yi =

ii

Since person i chooses

ii andAIii + ij AJij =2

for j 6= i.

(71)

ij so as to maximize yi and similarly for person28

j, it follows that9
ij =

J
I
ji = 1 () Aij =2 > Aii

ii =J
jj = 1 () Aij =2

and AJij =2 > AIjj ,AIii or AJij =2

AIjj .

(72)(73)

To study this system in greater detail, next we focus on the special casewhere persons i and j are pairwise symmetric in terms of knowledge heterogeneity. Speci…cally, suppose at time t that (suppressing t for notational
simplicity):ni = nj n.
(74)
By de…nition,nc
Then since ndij = n

ncij = ncji .

nc , it follows thatndij = ndji

nd ,

(75)

and
n = nc + nd ; nij = nc + 2nd = n + nd ;(76)

implying that
md
mc

nd
; mc
nij
nc
mcij = mcji = ij = 1nmdij = mdji =mcij = mcji =

mc
;
nij

(77)

2md .

It also follows from (2) thatn = nij (1

md );

(78)

n:

(79)

Furthermore, from (7) and (8),AIii = AIjj =I

Hence, relations (72) and (73) can be restated asij =Jji = 1 () Aij =2 >

I

n;

(80)

ii =

J
jj = 1 () Aij =2

I

n;

(81)

9

Recall from equation (46) that AJij = AJji by de…nition. In equations (72) and (73),
we can use either strict inequality or weak inequality. However, since the case of ties is notimportant in the following analysis, for convenience we use strict inequality in (72) whereas
we use weak inequality in (73).

29

AJij ("F ) is now given by

where equation (67) for AJij
AJij

= nij

J

(1

I + CT2 (1md )

2md )1

(md )

C+

T

T

(1

2md )1[ CF ] CT C [CF](md ) CCT

C

h("F ):

T(82)
Using (78) and rearranging terms, let us de…ne the knowledge growth rate
per person for joint creation,gJ (md )

AJ
ij =2
n

=
=

1
(1 md ) I + CT C + T(1 2md )(1T(md ) C CT C +h("F )d a
d b(1 m ) (1 2m ) (md )c h("F );

C)

CT

C +(1

(83)
where
J

a

I +

2a [ CF ] CT C [
CT

C +

b

(1

C)

CT

c

C

CT

C +

F]

T

(84)

;

1;

T
C + (1
T

)

T

;

:

We note that the knowledge growth rate de…ned by equation (83) represents the “public aspect”of the knowledge creation and accumulation process.
That is, AJij in equation (83) represents newly created “formal knowledge”tobe accumulated in the “Server” in Figure 1 as public documents, which areaccessible by any person in the future. In contrast, as discussed in Section
5, “inside the brain”of each person i and j, the tacit knowledge generated in
the process of creating AJij is also accumulated for future knowledge creation.
Furthermore, in equation (83), AJij is divided by 2 to account for the numberof “patents”created per person.
By de…nition,a + b + c = 0:
Since the function gJ (md ) contains only md as a variable, di¤erentiating gJ (md )yieldsc (b + 2c) md
gJ0 (md ) = gJ (md ):(85)
md (1 md ) (2 md )Let us de…ne
1
mB
(86)
2 + cb1=(1 C ) CT C +(1
) T
2+
+
C
CT

30

C

T

)

T

which is smaller than 1=2 since

C < 1 and< 1. Thus, (85) implies thatgJ0 (md ) R 0 as md Q mB for md 2

0;

1
2

(87)

Hence, gJ (md ) is strictly quasi-concave on [0; 1=2], achieving its maximal value
at mB , which we call the “Bliss Point.”
Next, for the case of knowledge creation in isolation, from equations (7)and (8), the knowledge growth rate for each person is obtained as follows:
AIjj
AIii
=
=
ni
nj

(88)

I;

which is constant for the two persons.
Overall, the knowledge growth rate per person is given by:g(md ) = maxdI ; gJ (m )(89)

Using function gJ (md ), the selection rule (72) and (73) can be restated asij =dji = 1 () gJ (m ) >

I

(90)

ii =

d
jj = 1 () gJ (m )

I

(91)

For illustration, let us set:
C =

2
= ,
3

CT

C =

T

1
= ,
4

F = 0,and choose the rest of the parameters in appropriately so that
Then we have thatmB = 0:4 and gJ (mB ) = 1:0,

= 1:07.

so we obtain the shape of the function gJ (mB ) as depicted in Figure 7.

Figure 7
In Figure 7, the horizontal line with heightI = 0:5

represents the knowledge growth rate when i and j are working separately inisolation, which is obtained from the set of parameters:
I =

IS =

IT = 1,

31

IS =

IT =

1
.
2

This horizontal line intersects the gJ (md ) curve at points E and H. Hence,
ij

=

ii

=

d
E
H
ji = 1 for m 2 (m ; m );

(92)

d
E
H 1
jj = 1 for m 2 [0; m ] [ [m ; ]:

2

(93)

At this point, let us recall that the knowledge growth function de…ned byequation (83) contains parameter "F . Hence, focusing on parameter "F , let
us represent the knowledge growth rate function as gJ (md ; "F ) for "F 2 [0; 1).
In this general context, relation (85) becomesgJ0 (md ; "F ) = gJ (md ; "F )c (b + 2c) md:
md (1 md ) (2 md )Since the last term of the equation above is the same as that of equation(85), we can conclude that all knowledge growth rate curves, gJ (md ; "F ) for"F 2 [0; 1), share the same bliss point given by (86). This is not surprising
if we observe in equation (83) that the value of function h("F ) simply shifts
each knowledge growth rate curve vertically in the same proportion.
In Figure 8, three representative knowledge growth rate curves are depicted. The top curve, gJ (md ; 0), depicts the knowledge growth rate curve
corresponding to "F = 0. The bottom curve, gJ (md ; "CF ), shows the knowlCC
edge growth rate curve at "F = "F , where "F represents the inner boundaryof net-dominant range, given by (48). Since gJ (md ; "F ) = gJ (md ; "CF ) for allC"F"F , the bottom curve shows the knowledge growth rate curve for all "Fin the net dominant range in Figure 5.

Figure 8
As "F increases from 0 to "CF , the knowledge growth rate curve shifts down
continuously from the top curve to the bottom curve. The middle curve,gJ (md ; "F ), corresponds to a representative value of "F between 0 and "CF.
Comparing the knowledge growth rate curves in Figure 8, we can make a
few important observations. First, in terms of vertical proportion, all knowledge growth rate curves gJ (md ; "F ) for "F 2 [0; 1) locate within a relatively
small range, i.e., between h(0) and h("CThis re‡ects the fact that theF ).
relative productivity curve h("F ) in Figure 5 locates within a relatively narrow belt. Second, in contrast, any knowledge growth rate curve in Figure 8,

32

say gJ (md ; "F ), exhibits a huge variation in terms of horizontal axis parameter md . Furthermore, as noted previously in connection with Proposition 3,
with the advancement of net technology and transport technology, the di¤erence between h(0) and h("CF ) shrinks. This suggests that in seeking the mostproductive partner in knowledge creation, knowledge composition becomes increasingly more important than the geographical distance due to technologicalimprovements.
For example, in the context of Figure 8, let us suppose that researcher i has
three potential partners, j1 , j2 and j3 . In terms of "knowledge composition
md " and "distance "F ", researcher i has the following combination with eachpotential partner; (mdij1 ; 0), (mdij2 ; "F ) and (mdij3 ; "CF ) such thatmdij1 < mdij2 < mdij3 and 0 < "F < "CF,
as shown in Figure 8. Then, the knowledge growth rate is lowest at [1] wheni partners with person j1 , who locates nearest at "F = 0 and has the lowest
degree of knowledge di¤erentiation at mdij1 . Conversely, the knowledge growthrate is highest at [3] in Figure 8, when i partners with person j3 , who locates
farthest from i at "C
F and has the largest degree of knowledge di¤erentiationd
at mij3 . Thus, provided that j3 also agrees to work jointly with i, researcher
i will choose j3 , who locates farthest from i.
In the next section, we treat md as an endogenous variable, and examine
how md changes over time when a pair of researchers continue to work jointly.
And we show that when a pair of researchers continues working jointly, theproportion of their di¤erentiated knowledge tends to shrink over time.
We may summarize the observations above as follows:
Proposition 4. Keeping Assumption 1, we further assume that persons i andj have the same size of knowledge, n, at the time in question. In this context,
let gJ (md ; "F )gJ (md ) be the knowledge growth rate for joint creation perperson, de…ned by equation (83). Then, the knowledge growth rate function
gJ (md ; "F ) exhibits the following features, as illustrated in Figure 8:(i) For each "F 2 [0; 1), gJ (0; "F ) gJ (1=2; "F ) = 0, and gJ (md ; "F ) is strictly
quasi-concave on [0; 1=2].
d(ii) As "F increases from 0 to "CF , the knowledge growth rate curve gJ (m ; "F )shifts vertically down in the same proportion as the value of h("F ), where h("F )is the relative productivity function de…ned by (68), and where "C
F represents
the inner boundary of the net-dominant range given by (48). For all "F "CF,

33

gJ (md ; "F )gJ (md ; "CF ), and hence the bottom curve in Figure 8 shows theknowledge growth rate curve for all "F in the net dominant range in Figure 5.
(iii) Consequently, all knowledge growth rate curves achieve their maximum
at the same Bliss Point mB given by (86).
(iv) In terms of vertical proportions in Figure 8, all knowledge growth rate
curves gJ (md ; "F ) for "F 2 [0; 1) locate within a relatively small range, i.e.,
between h(0) and h("CIn contrast, any knowledge growth rate curve inF ).
Figure 8 exhibits a huge variation in terms of horizontal axis parameter md .
Furthermore, as noted in connection with Proposition 3, with the advancementof net and transport technology, the di¤erence between h(0) and h("CF ) shrinks,
whereas the horizontal variation of each knowledge growth rate curve in termsof md remains huge. This suggests that when a researcher seeks their best
partner, their knowledge composition becomes increasingly more important
than their geographical distance as technology improves.

5

Dynamics of the two-person system

Thus far, through Sections 2 to 4, the size of each person’s knowledge as well as
the relative composition of two persons’knowledge have been treated parametrically. In this section, we examine how the size of each person’s knowledge
as well as the relative composition of two persons’knowledge change endogenously over time when they continue interacting by sequentially choosing either
to work alone or to work together for knowledge creation.
Next we shall de…ne tacit knowledge and then explain how it arises inour context. The the ideas behind “tacit knowledge” originate with Polanyi(1958). Polanyi (1966, p.4) famously states, “We can know more than we
can tell.” Nonaka and Takeuchi (1995, p. viii) develop these ideas further,de…ning tacit knowledge:“...we classify human knowledge into two kinds. One is explicit
knowledge, which can be articulated in formal language including
grammatical statements, mathematical expressions, speci…cations,manuals, and so forth. This kind of knowledge can be transmitted
across individuals formally and easily. This has been the dominant mode of knowledge in the Western philosophical tradition.
However, we shall argue, a more important type of knowledge istacit knowledge, which is hard to articulate with formal language.
It is personal knowledge embedded in individual experience and34

involves intangible factors such as personal belief, perspective, andthe value system. Tacit knowledge has been overlooked as a criticalcomponent of collective human behavior.”
Borrowing one of their examples,10 when designing and building a new automobile, certainly a large portion of the value is embodied in the …nal productsold to consumers. However, in the process of innovating the new design, themembers of the team creating the vehicle acquire new tacit knowledge related
to design and manufacturing that cannot be easily articulated. It comes withthe experience of creation, either together or alone, and it is a part of humancapital. Tacit knowledge of a researcher expands with their innovation overtime.
Here we attempt to construct a model of (explicit and) tacit knowledge
dynamics. Explicit knowledge can be bought and sold. It is di¢ cult tomarket tacit knowledge, except as a package with a worker. Here we donot model the labor market in the context of tacit knowledge, leaving that tofuture work. Given our assumptions, tacit knowledge represents a dynamic
externality in the following sense. The choice of a knowledge worker at agiven time, in our case whether to work alone or in tandem with another, willa¤ect their tacit knowledge acquisition during this work. This choice of one
worker will a¤ect the future payo¤ of that worker as well as others since tacitknowledge acquisition will a¤ect future knowledge productivity either working
alone or with others. In fact, since agents are assumed to be myopic, they do
not account for the e¤ect of tacit knowledge acquisition on their own future
payo¤s, much less the payo¤s of others. So ine¢ cient equilibria are to be
expected. In contrast, Berliant and Fujita (2011) …nd that, under the same
framework but with neither multimodal communication nor tacit knowledge,
equilibria can have e¢ ciency properties.
To study dynamics of the two-person system, in what follows we focus on
the special case where persons i and j have the same size of knowledge at theinitial time, t = 0; namelyni (0) = nj (0)n(0).

ncij (0) = ncji (0)nc (0);(94)

It holds by de…nition that10

The particular example is the innovation involved in designing and producing the Honda
City car.

35

thus as shown in equations (74), (75) and (76), condition (94) meansndij (0) = ndji (0)nd (0);(95)

mdij (0) = mdji (0)md (0):(96)

andGiven that the initial state of knowledge is symmetric for persons i and j as
above, as seen below, it turns out that the equilibrium con…guration at any
time also maintains pairwise symmetry between the two persons; at any timet 2 [0; 1),11ni (t) = nj (t) n(t);(97)
and hence,ndij (t) = ndji (t)nd (t);(98)

mdij (t) = mdji (t)md (t);(99)

ncij (t) = ncji (t)nc (t):(100)

whereas, by de…nition,
In this context of a pairwise-symmetric equilibrium path, our main goal is
to examine how the knowledge growth rate per person changes in the long-run.
As de…ned in equation (89), given md 2 [0; 21 ], the knowledge growth rate perperson, g(md ), is given byg(md ) = maxdI ; gJ (m )

:

(101)

To make the analysis interesting, let us consider the casegJ (mB ) >

I:

(102)

Then, as depicted in Figure 7, on the horizontal axis md , there exist mE andmH such that 0 < mE < mB < mH < 12 andgJ (md ) >dEHI for m 2 (m ; m );10; mE [ (mH ; ]2dd
Let m (0) be the initial value of m at time 0. First, let us assume thatddI > gJ (m ) for m 2

gJ (md (0)) >
11

I:

(103)

This can be seen from the fact that when condition (97) is met at any time t (say, t = 0),
the right hand side of the di¤erential equation specifying n(t)_(also n_ d (t) and m_ d (t)) does
not involve any variable that is speci…c to i or j. For a more precise explanation of thispoint, please see Berliant and Fujita (2011).

36

That is, md (0) locates between mE and mH in Figure 7. Then, the two persons
will continue to work jointly (i.e., ij = ji = 1) as long as
gJ (md (t)) >

I:

(104)

In order to know which direction md (t) moves and how long the two persons willcontinue working jointly, let us examine the dynamics of md when condition(104) holds. To do so, recalling the de…nition md nd =nij and dropping t for
simplicity, we have that:
n_ d
nij

nd n_ ij
(nij )2= (nij ) 1 [n_ d n_ ij md ];m_d =Since nij

nc + 2nd , it follows thatm_ d = (nij ) 1 [n_ d(n_ c + 2n_ d ) md ](105)

(n_ c + 2n_ d ) md R 0:

(106)

implying thatm
_ d R 0 , n_ d

Hence, to identify the sign of m
_ d , we must know the values of n_ d and n_ c .
That is, when md (t) locates between mE and mH in Figure 7, the knowledge
growth rate gJ (md (t)) is realized for each person, and therefore we must calculate how much di¤erential knowledge (n_ d (t)) for each person and how muchcommon knowledge (n_ c (t)) for the two persons has been generated at that moment, t. We calculate the values of n_ d (t) and n_ c (t) through a sequence ofsteps as follows:
In the …rst step, let us recall Figure 3, which represents the structural relationship among all basic activities and intermediate activities used to generate
the …nal output, AJij , as a tree. Table 1 provides further description of thisactivity tree.

Table 1

The activity tree depicted in Figure 3 consists of three tiers of knowledge
creation activities. We may note that in describing a tree in mathematics,
the term “level”is used instead of “tier.” In the present context, however, thediagram shown in Figure 3 resembles the structure of a supply chain that produces a …nal industrial product (e.g., automobiles) by sequentially assembling37

various parts produced through multiple tiers of many factories. Thus, here
we use the term “tier.” A major di¤erence between “automobile production”
and the present “knowledge production” must be noted. That is, all parts
used in the production process of a car “disappear” into the car at the endof the assembly process. In contrast, the tacit knowledge created and usedin the knowledge production process is accumulated for knowledge creation in
the future.
In Figure 3, each node denoted by a black squareor a black dot
represents a joint activity by persons i and j, whereas each node denoted bya white circle represents an independent activity conducted by either i or jalone. The symbol attached to a node means that the frequency variables(!’s, ’s and/or ’s) have been chosen optimally.
In the second step, the output of each activity is represented by an equation,
which is listed in the third column of Table 1. Although each equation in thiscolumn of Table 1 represents new tacit knowledge generated by each activity,di¤erent pieces of new tacit knowledge cannot simply be added up because
each one has di¤erent importance for the creation of new knowledge in the
future. Thus, in the third step, we attach a “weight”to each speci…c output,
say x. For the weight on x, following the tradition of microeconomics, we use
the value @AJij =@x, representing the marginal contribution of x to the …nal
output AJij . Then, we calculate the (total) contribution of x to the …nal
output AJij by
@AJij
x;
(107)
@x
which we call the imputed value of x, using the terminology of microeconomics.
Here, it must be noted that in each calculation of (107), the pairwise symmetry
stated in equation (97), namely ni (t) = nj (t)n(t), is taken into account.
The imputed value of each output is listed in the last column of Table 1.
(Please see Appendix B for the actual calculation of imputed values.)
Next, using the terms listed in the last column of Table 1, we calculate thevalues of n_ d (t) and n_ c (t). Before doing so, however, let us observe from thelast column of Table 1 thatthe sum of imputed values in the second tier
= AJij= AJijI+ AJij

2
( I+

F +

I

2

+ AJijJ
F + AijNN)= AJij ;

(108)

38

sinceI +F +Likewise, it can be readily con…rmed thatN = 1.

the sum of imputed values in the third tier
= AJij :(109)

Therefore, when the …nal output AJij is created, the same size of tacit knowledge (in terms of imputed values) is generated in tier 2 as well as in tier 3.12
In calculating the values of n_ d (t) and n(t),_we may take into account all
imputed values in tier 2 and tier 3. However, the contents of tacit knowledge
generated in tier 2 seem rather close to those in tier 3. Thus, in order toavoid double counting of similar pieces of tacit knowledge, we consider only
the tacit knowledge generated in tier 3.13 In this context, the value of n_ dfor each person, say i, is obtained by summing up the imputed values of all
third-tier activities by person i that are identi…ed by circles in the second
column of Table 1 as follows:I

n_ d = AJij

JIS

AJij
=
2

( I+

I

+ AJij

2
F

JIT

2

ISF +

N

F

+ AJij

ISF

2+ AJij

N

ISN

2
(110)

ISN );

using the identity JIS + JIT = 1. This equation represents the size of thedi¤erential knowledge generated by each person at the time. Next, the valueof n_ c is obtained as the sum of …nal output AJij and the imputed values ofthird-tier joint activities by i and j as follows:
c

n_

=

AJij

CF

+ AJij

C

CT
F

+AJij

T
F

=

AJij

(1 +

C

CT +

F
+
aCT
ij
F
aCT
+
ij

CF
aijT F +aijT F +

T ):

N
N

CN

CN
aijT N
aijT N

N
aCT
ij
N
aCT
ij

(111)

This equation represents the size of common knowledge generated by the joint
work of i and j.
12

If we recall the microeconomics of production, this result is not surprising. That is,“all production functions / sub-production functions” in each tier are composed of CobbDouglas functions, and each function displays constant returns to scale. Hence, the sumof all imputed values in each tier equals the value of the …nal output, the price of which isnormalized to be 1.
13
Even if we take into account all pieces of tacit knowledge in tier 2 and tier 3, the main
results below would not change qualitatively.

39

Finally, substituting (110) and (111) into equation (105) and noting thatn_ + 2n_ d = 2AJij , we have thatcm
_ d = (nij ) 1

AJij2[( I +

F

ISF +

N

ISN )

4md ] for md 2 (mE ; mH ):(112)

Let us de…ne

I +

F

ISF +

N
ISN
;
(113)
4
which represents the stationary point of m
_ d . Then, it follows from (112) thatm~m
_ d R 0 as md Q m~ for md 2 (mE ; mH ):(114)

Notice that both the Bliss Point mB and the stationary point m~ are independent of parameter "F . It is also clear from (113) that m~ < 1=4.
Although we have identi…ed the sign of m
_ d when the two persons work
jointly, we must also consider the case when two persons work independently.
Fortunately, this case is rather simple. First, by de…nition of md
nd =nij ,the dynamics of md are given by equation (105) also in the present context ofeach person working in isolation. Hence, to identify the sign of m
_ d , we needto know the values of n_ d and n_ c in the present context. To do so, we focus
on person i. Then, the activity tree for the knowledge created by person i in
isolation can be represented as in Figure 2.
The top node in Figure 2 represents the …nal output created by person
and aISi in isolation. The second tier consists of two basic activities: aITi .
i
From (6), the imputed value of each intermediate output can be obtained asfollows:
@AIii
@AIii
IT
I
I
a
=
A
;
aIS
(115)
IT
IS :
i
i = Aii
ii
IS
@aIT
@a
i
i
implying that:the sum of imputed value in the second tier
= AIii=I
IT + AiiIS

AIii ;(116)

since IT + IS = 1.
By assumption, since person i is working in isolation (from j), the formal
output AIii and imputed values of second-tier activities by person i become40di¤erential knowledge of person i (from j).14
person i is obtained as follows:n_ d = AIii + AIii=

Hence, the value of n_ d forI
IS + Aii

IT

2AIii ;

(117)

whereas no common knowledge is generated for the two persons:n_ c = 0:(118)

Substituting (116) and (117) into (105) yieldsm_ d = (nij ) 1 2AIii (112md ) > 0 for md 2 [0; ):
2

(119)

Therefore, whenever each person is working in isolation, the proportion ofdi¤erential knowledge, md , will keep increasing.
Now, combining the two situations of working jointly and working in isolation by two persons, we can derive the overall dynamics of md . As explained inAppendix C, depending on the relative position of m
~ and mB on the md -axis,
we have three di¤erent diagrams for the dynamics of two-person system, which
are depicted in Figure C. However, given that the three diagrams in Figure C
are qualitatively rather similar, we focus in the following on the representative
dynamics presented in Figure 9.

Figure 9

The diagram in Figure 9 is based on the assumption thatmE < m~ < mB :(120)

As depicted in this diagram, if the initial position, md (0), at time 0 is to theleft of m,~ then md (t) gradually increases towards m.
~
If m~ < m(0) < mH ,then md (t) gradually decreases towards m.
~
Only when md (0) < mH , doesmd (t) move away from m~ toward 1=2. Hence, whenever md (0) < mH , md (t)approaches the sink point, m,~ which is to the left of bliss point mB .
14

The formal output AIii by person i will be registered in the knowledge stock in theServer. Given that the size of the total stock of knowledge in the Server is almost in…nite,
it can be safely assumed that the probability of person j …nding AIii in the Server is nearlyzero (and vice versa for i). Even if we assume that a part of AIii and AIij is transferred toeach other by some mechanism, it will not change the sign of m
_ d below.

41

Notice that the value of m
~ de…ned by equation (113) is always less that 1=4.
On the other hand, as explained in Appendix C, the bliss point mB is close to
1=2 when the weights on di¤erential knowledge in K-subproduction functions(9) and (29) are su¢ ciently large. In such a case, as illustrated in Figure 9,
the K-knowledge growth rate at the sink point m~ is much lower than that at
the bliss point. In other words, the research partners are in a low-productivity
sink point trap. Thus, it is important to ask: When the research partners arein a low-productivity sink point trap, what possible mechanism could enable
the partners to escape from such a trap and attain much higher productivity?
Before examining this question further in the last section, however, we needto ask another fundamental question here.
Recall the discussion near the end of Section 4 stating that AJij in equation (83) represents newly created formal knowledge to be accumulated in theServer as public documents. However, we have just seen that the knowledge
production process also yields a large amount of tacit knowledge which will
be accumulated inside the brain of each person for future knowledge creation
activity. Thus, we need to know the growth rate of total knowledge per person, including both formal knowledge and tacit knowledge. Furthermore, wemust examine the relationship between the knowledge growth rate de…ned byequation (83) and the growth rate of total knowledge for each person.
To answer these questions, let us …rst calculate the growth rate of total
knowledge, n_ i , for person i when working in isolation. From (117) and (118),
we have thatn_ i = n_ d + n_ c
= 2AIii :(121)

Thus, recalling (88), the growth rate of total knowledge for person i whenworking in isolation is given by
2AIii
n_ i
=
=2
ni
ni

GI

I.

(122)

Next, when person i is working jointly with j, from (110) and (111) wehave thatn_ i = n_ ci + n_ di = n_ c + n_ d
= AJij(1 += AJij(1 +F

JT F +1+F

N

JT F +JT N ) +
N

JT N

2
42

AJij
2
);

( I+

F

ISF +

N

ISN )

(123)

by using the relations CT + CS = 1 and T + S = 1. Hence, the growthrate of total knowledge for person i when working jointly with j is given by
GJ (md )AJij
1 + F JT F + N JT N
n_ i
=
(1 +
)
ni
ni
2
AJij =2=(3 + F JT F + N JT N )ni= gJ (md ) (3 + F JT F + N JT N );(124)

recalling de…nition (83) and setting ni = n.
Putting together (122) and (124), the growth rate of total knowledge perperson is given by:(
2 I when I gJ (md );G(md ) =gJ (md ) (3 + F JT F + N JT N ) when gJ (md ) > I :(125)
Let us recall equation (89) which de…nes the growth rate of public knowledge per person. By comparing equations (89) and (125), we can see that thetwo functions are signi…cantly di¤erent, but they also have a close relationship.
First, we can observe that the upper term in equation (125) is twice the corresponding term in equation (89), re‡ecting the fact that the total knowledge
newly created in isolation includes the same amount of tacit knowledge (givenby (116)) as formal knowledge. The lower term in equation (125) is morethan the triple of the corresponding term in equation (89), re‡ecting the factthat the total knowledge newly acquired by person i through joint work withperson j includes the full amount of newly created formal knowledge, AJij , andthe tacit knowledge represented by equations (110) and (111).15
Next, observe from equation (124) that since function GJ (md ) is the product of gJ (md ) and a constant, it follows thatmB

the bliss point of function gJ (md )= the bliss point of function GJ (md );(126)

where mB is de…ned by (86), meaning that the maximum values of two functions, gJ (md ) and GJ (md ), are attained at the same point, mB 2 (0; 1=2).
We are now ready to summarize the dual dynamics of formal-K and tacit-K
in the two-person system. Notice that along the symmetric equilibrium path,
15

Recall that in the de…nition of the function gJ (md ) in (83), the formal K-output AJij
created by joint work is divided by 2 to account for the number of “patents” created perperson. In contrast, in deriving equation (122), the full amount of knowledge embodied inthese “patents,” AJij , is considered a part of new knowledge acquired by each person.

43

at any t 2 [0; 1), if values of ni (t) = nj (t) n(t) > 0 and md (t) 2 (0; 1=2) are
known, then values of all the rest of structural variables, nc (t) ncij (t) = ncji (t),n(t) + nd (t), mc (t) = 1 2md (t), AJij (t),nd (t)
ndij (t) = ndji (t), nij (t)AIii (t), gJ (md (t)), gI (md (t)) are uniquely determined, together with the valuesof choice rules,ii (t) =jj (t) = 1 ifI

gJ (md (t)),

ij (t) =ji (t) = 1 ifd
I < gJ (m (t)):(127)
Hence, given the initial values, n(0) > 0 and md (0) 2 (0; 1=2), solving thesystem of di¤erential equations, (112) and (119) for m_ d , and (121) and (123)for n_ = n_ i = n_ j , together with the choice rule (127), the equilibrium path of
md (t) and n(t), together with all the rest of the structural variables, is uniquelydetermined for t 2 [0; 1).
For example, let us take the case of diagram (a) depicted in Figure C, whichis qualitatively the same as Figure 9. In the context of diagram (a), Figure
10 synthesizes the dual dynamics of formal-K and total-K in the two-person
system.
Figure 10
In the bottom part of Figure 10, diagram (a) in Figure C is duplicated,
representing the dynamics of formal-K for the case of mE < m~ < mB . The
top part of Figure 10 depicts the dynamics of total-K. Comparing the bottom
part and the top part of Figure 10, we can recognize similarities and di¤erences
between the two dynamics. Since the upper curve GJ (md ) is the product of
the lower curve gJ (md ) with the constant (3 + F JT F + N JT N ), the two
curves share the same bliss point, mB , and the same sink point, m,~ on thedm -axis. Hence, in the phase of joint work by i and j, and hence when mE <md (t) < mH , the two dynamics are essentially parallel, leading respectively
to the sink points S and S at the same m.
~ However, since the constant ofmultiplication, (3 + F JT F + N JT N ), is more than 3, the upper curve
GJ (md ) is much higher than the lower curve gJ (md ). This indicates that byworking jointly for knowledge creation, the accumulation of total knowledge
by each person is much higher than that of formal knowledge created jointly.
In contrast with joint work, during the phase of knowledge creation in
isolation by each person independently, the dynamics with and without tacitknowledge are signi…cantly di¤erent. In the bottom part of Figure 10, formalK dynamics without tacit knowledge happens when md (0) < mE . The switch
44

from working in isolation to joint work occurs at the point E, where the horizontal line of height I crosses the gJ (md ) curve. Since mE on the md -axis
represents the real switching point based on the switching rule noted in (127),in the top part of Figure 10 the switch to joint work is realized at point E 0
when the growth rate of total knowledge, 2 I , from working in isolation ismuch lower than that from working together. For e¢ cient growth of total
knowledge in joint work, the switch should occur much earlier at E 00 , where
the horizontal line with height 2 I crosses the GJ (md ) curve. The opposite
situation happens at point mH on the md -axis. Hence, from the viewpoint of
total knowledge accumulation, the myopic switching rule (127) is not optimal.
To see more clearly why the switching rule (127) is not optimal from theviewpoint of total knowledge growth, let us compare switching at E withswitching at E 00 in Figure 10. Based on per capita production of formal-K,
switching occurs at point E where I = gJ (mE ). Since AIii (md ) = I n for
all md 2 [0; 12 ], recalling (83) we can rewrite this relationship at E as follows:
AIii (mE ) = AJij (mE )

1
:
2

(128)

In contrast, for achieving e¢ cient growth of total-K, switching should occur atpoint E 00 where 2 I = GJ (m),^ that is, when
2AIii (m)^ =AJij (m)
^
(3 +
2

or
AIii (m)^ = AJij (m)
^

(3 +

C

C

CT +

CT +

T)

T)

:
(129)
4
Switching rule (128) means that switching from work in isolation to joint
work should occur at point E where one half of joint output is equal to the
output of work in isolation. On the other hand, switching rule (129) meansthat switching from the work in isolation to joint work should occur at pointm^ where (3 + C CT +T )=4 of joint output is equal to the output ofwork in isolation.
The di¤erence between the two switching rules becomes clear when we
consider this situation in the context of the academic world. An evaluationcommittee always faces the di¢ cult problem of how to evaluate the contribution of each author when papers are written jointly. A simple, drastic rulewould be that when a paper is written by 2 authors, the contribution of eachauthor just equals one half of the contribution of the paper. Equation (128)represents this drastic rule. In the actual academic world, however, an evaluation committee tends to adopt a more generous rule which is represented
45

qualitatively by equation (129). That is, if we consider e¢ cient growth of
the total knowledge of each (young) researcher in the long-run, when a paper
is written by 2 persons, the evaluation committee is justi…ed to allocate much
more than one half of the paper’s contribution to each author. That is because
tacit knowledge accumulates during the knowledge production process; suchknowledge is invisible in the …nal product, namely the paper itself.
To understand more intuitively the suboptimality of the switching rule
based on equation (128), let us suppose as indicated by (128) that at a given
time, one isolated researcher writes one paper per unit time, whereas two
joint-researchers write two papers per unit time. In this situation, each of
the two joint-researchers accumulates its formal-K (from two papers) at a
rate twice that of isolated research; furthermore, each of the joint-researchers
accumulates informal-K (from developing two papers) much faster than anisolated researcher [to be precise, (1 + C CT +T ) times that of theisolated researcher]. Therefore, when relation (128) holds, the total-K of eachjoint-research accumulates much faster than that of an isolated researcher.
Returning to our case where each researcher gets its revenue from the salesof newly created patents (per person), the optimal switching rule (129) justi…esa government (or public agent) subsidy for a signi…cant portion of the revenueof each joint researcher; more precisely, the subsidy should be targeted in thefollowing proportion:
3+

C

CT +

4

T

1
1+
=
2

C

CT +

4

T

:

(130)

We may summarize the observations above as follows:
Proposition 5. To study the dynamics of knowledge creation in the context
of a two-person system, let us focus on two persons i and j who share the same…xed value of F2F lead time "F . Suppose further that persons i and j havethe same quantity of knowledge at initial time 0 as indicated by (94). Then,it turns out that the equilibrium knowledge con…guration of the two persons
maintains pairwise symmetry as described by the relations (97), (98), (99) and(100) at any time t 2 [0; 1). In this context of symmetry with two persons,the equilibrium dynamics of knowledge creation exhibit the following features:(i) When md (t) is between mE and mH in Figure 7, and hence the two persons
continue working jointly, the value m_ d is given by equation (112). This has
been obtained by using the identity (105) and taking into account the …nal
output AJij and the imputed value (listed in the last column of Table 1) of
the output of each basic activity in the third-tier of Figure 3. Likewise, when46

person i is working in isolation (from j), the value of m
_ d is given by equation(119). The dynamics of joint work represented by equation (112) has a uniquestationary point m~ given by equation (113). From equation (119), the valuedof m_ working in isolation is always positive for md 2 [0; 1=2).
(ii) Based on equations (112) and (119), representative dynamics for md (t) are
shown in Figure 9. This corresponds to the case where mE < m~ < mB . In
this case, as shown in Figure 9, whenever md (0) < mH , md (t) approaches thesink point m.
~(iii) On the one hand, the value of m
~ de…ned by equation (113) is always
less than 1=4. On the other hand, the bliss point mB is close 1=2 when eachweight on the di¤erential knowledge in K-subproduction functions (9) and (29)
is su¢ ciently large. In such a case, as shown in Figure 9, the K-growth rate
at the sink point m~ is much lower than that at the bliss point. In other words,
the two research partners eventually fall into a trap of a low-productivity sinkpoint.
(iv) In the knowledge growth rate function gJ (md ) de…ned by equation (83),
AJij represents newly created formal knowledge to be accumulated in the Server
as public documents. However, the knowledge production process also yields
a large amount of tacit knowledge created (and maintained) at each node
identi…ed by or in Figure 3. Thus, we need to know the growth rate of total
knowledge per person, including both formal knowledge and tacit knowledge.
Similar to deriving equations for m_ d , the rate of increase in total knowledge ofeach person, say i, is given by equations (121) and (123), that is:n_ i = 2AIii when i is working in isolation,
1 + C CT +T
when i is jointly working with j.
n_ i = AJij
1+
2
Hence the growth rate of total knowledge for i is given by equations (122) and(124), that is:n_ i= 2 I when i is in isolation,ni
n_ i
GJ (md ) == gJ (md ) (3 + C CT +
ni
GJ =

T)

(v) Thus, keeping the context of Figure 9 or diagram (a) in Figure C, thedual dynamics of formal-K and total-K for the two-person system can be
represented by Figure 10, as explained previously. Since the upper curve
GJ (md ) is the product of the lower curve with the constant (3 + C CT +EdHT ), the two dynamics are essentially parallel when m < m (t) < m ,47

leading respectively to the sink points S and S at the same m.
~
However,dEwhen m (0) < m , based on the switching rule (127), the switch from working
in isolation to joint work occurs at point E; in terms of the growth of total
knowledge represented by the upper curve, the switching is realized at point
E 0 when the growth rate of total knowledge, 2 I , from working isolation ismuch lower than that from working together. For e¢ cient growth of total
knowledge in joint work should occur much earlier at E 00 when the horizontal
line which height 2 I crossed the GJ (md ) curve. Hence from the view-point of
total knowledge accumulation, the myopic switching rule (127) is not optimal.
(vi) To remedy the ine¢ ciency caused by the myopic switching rule (127),
when the income of each researcher comes from the sales of newly createdpatents per person, a public agent may subsidize a signi…cant portion of therevenue, in a proportion given by (130).

48

6Conclusions

Building on our earlier work, we have developed a model of knowledge creationin the context of two persons when multiple modes of communication areavailable, and knowledge workers can independently use the internet for the
purpose of search. As in the empirical literature, we separate the developmentof new ideas into conceptual and technical phases, allowing researchers a choice
between modes of communication, face to face or remote, in each phase. We
have examined how the communication mode and joint knowledge productivity
are a¤ected by a change in the lead time necessary for realizing actual F2F
communication by a pair of knowledge workers. It has been shown that withthe advancement of net technology, joint knowledge creation can be conducted
over a wide range of geographical area without losing much productivity. Our
framework has been applied to examine the impact of pandemic restrictions
on face to face communication and knowledge production.
Departing from our earlier model based on modeling knowledge creation
as a single activity, we have elaborated the entire process of joint knowledge
creation as an activity tree depicted in Figure 3. The intermediate ideas
produced in the third-tier and second-tier are not only used as inputs for the
production of the …nal output AJij , but also are accumulated as a part of tacitknowledge for knowledge creation in the future. In this way, the concept oftacit knowledge among the people creating new knowledge arises organically.
Tacit knowledge, missing from earlier models, plays a huge role in our
analysis of the dynamics of the system. In the context of diagram (a) in Figure C, Figure 10 synthesizes the dual dynamics of formal-knowledge and tacitknowledge. Since total knowledge includes both formal knowledge and tacitknowledge, the top-curve GJ (md ), representing newly created total knowledge, is much higher than the bottom-curve gJ (md ), representing newly created formal knowledge. In the context of Figure 10, from the viewpoint of thelong-run e¢ ciency of total knowledge growth, two types of ine¢ ciency can berecognized.
First, when md (0) < mE , the switch from working in isolation to joint
work occurs at the point E, where the per capita output of patents (i.e., formal
knowledge creation) by joint work equals the output of patent production fromisolated work. However, e¢ cient growth of total knowledge requires that theswitch from working in isolation to joint work should occur much earlier, atpoint E 00 , where the growth rate of total knowledge per person in joint work
is equal to the growth rate of total knowledge in isolated work. To remedy
49the ine¢ ciency caused by a myopic switching rule based on per capita output
of patent production, a public agent may subsidize a signi…cant portion of therevenue from patent-sales, in a proportion given by (130).
Second, a more interesting and more important ine¢ ciency in the context
of Figure 10 is that the growth rate of formal-K at the sink point S on thegJ (md )-curve (or, the growth rate of total-K at the sink point S on the GJ (md )curve) is signi…cantly lower than at the Bliss Point B (or, Bliss Point B 0 on theGJ (md ) curve). This ine¢ ciency can be observed more clearly in the context
of Figure 9. As explained before, the value of sink point m~ is always lessthan 1=4. On the other hand, as explained in Appendix C, the bliss point
mB is close to 1=2 when the weight on di¤erential knowledge in K-production
is su¢ ciently large. In such a case, as illustrated in Figure 9, the knowledge
growth rate at the sink point m~ is much lower than that at the bliss point.
Thus, it is essential to ask: When the research partners are in a low-productivity
sink point trap, what possible mechanism could enable the partners to escapefrom this trap and attain much higher productivity?
In our earlier work, we proposed a square dance as such a mechanism.
That is, a small group of researchers rotate through a limited set of research
partners sequentially over time. The purpose of the square dance is to prevent
overaccumulation of common knowledge within each pair of researchers. It
has been shown that when the group size is optimal, a square dance leads theknowledge composition of each pair of researchers to the knowledge composition at the Bliss Point. However, in the present context of joint knowledge
creation based on a complex set of activities (as shown in Figure 3) and multimodal communication, it is not easy to apply a square dance, for switchingpartners rapidly among a group of researchers would not be easy in practice.
Fortunately, however, in the present context of multimodal communication,
we may be able to develop a more practical mechanism for escaping from alow productivity sink point trap. For example, let us imagine two regions (orcountries), A and B, that are far apart. Let us focus on two researchers, iand j, in region A as in Section 2; likewise, focus on two researchers, k and l,
in region B. Suppose further that at present, i and j (resp. k and l) are in a
low-productivity sink point trap in region A (resp. region B). Now, supposethat the four researchers, i, j, k and l, form a research group. Then, theknowledge composition of i and j (resp. between k and l) is similar, whereas
the knowledge compositions of i and j are di¤erent from that of l and k. In
this context, suppose that i and j (resp. k and l) jointly work on conceptual50

development F2F in region A (resp. region B), whereas i and j in region A
jointly work with k and l in region B on technical development through netcommunication. Then, it can be reasonably expected that the four researchers
together will be able to escape from a low-productivity sink point, attaining amuch higher K-productivity jointly.16
The conjecture above is just an example. It is our hope, more generally,that the model of knowledge creation through multimodal communication developed in this paper can be extended and applied in a broader context such ase¢ cient development of international academic societies in the age of rapidly
developing ICT and AI.
In the future, it would be interesting to let vary endogenously with theparticular knowledge pro…le of the two agents who are meeting to produce
new knowledge. That is, the agents who meet know their respective pro…lesof common and di¤erential knowledge, but choose a project summarized by
to maximize their output. Then, the output curve would be the upperenvelope over of productivity at the bliss points.
It would be of interest to allow other forms of agent heterogeneity, suchas large and small city residents, or CBD and suburban residents. Intergenerational transmission of knowledge and improvements in internet search
productivity due to the increasing stock of knowledge over time should be
investigated.
To sum up, there is much further work to be done to analyze the microeconomic dynamics of knowledge creation in settings with tacit knowledge andmultiple modes of communication.

16

To examine this conjecture precisely, we need to extend formally the model with two
persons in Section 2 to a model with four persons or to N persons. Such an extension isleft as an important task for the future.

51

7

Appendix A: The impact of regulating communication modes on joint knowledge productivity“The World Health Organization o¢ cially declared Covid-19 a pandemic on March 11. Within a few weeks, an estimated 16 million
U.S. knowledge workers had switched to working remotely to ‡atten the curve of the health crisis, according to a new survey by
Slack.
This amounts to nearly one-quarter of all knowledge workers in the
U.S., and that proportion has climbed even higher as more states
have urged citizens to stay home.”
Hanson (2020)

During the Covid-19 pandemic, the intensity of use of communication channels between researchers changed, as many switched to work from home, forinstance. How does the use of electronic and face to face communication
change from an exogenous event, and how does it a¤ect the patterns and volume of knowledge creation? How do pandemic restrictions change the use ofcommunication modes as well as the characteristics of the knowledge creation
process?
In this appendix, we analyze the e¤ect of restrictions on face to face communication in our model.
There is a large and rapidly expanding literature on the economic e¤ects of
the Covid19 pandemic. Here we focus on the e¤ects of the pandemic on knowledge creation activity when multiple channels of communication are present,
and in particular on the productivity of researchers under pandemic restrictions. How do they change their choice of joint or individual work, and how
do they change their modality of joint work (face to face or internet communication) with the imposition of pandemic restrictions? And how does thisinteract with the di¤ering commuting cost faced by workers living at various
distances from work?
There is some interesting empirical work associated with these questions.
Morikawa (2020) …nds signi…cant e¤ects, as well as signi…cant heterogeneityin e¤ects, of pandemic restrictions on workers in Japan. For example, worker
productivity was reduced by 30-40% when working from home as opposed to52

commuting to work. We shall discuss in more detail below how further empirical results from this paper support our theory. Inoue et al (2022) examine
the e¤ect of the Spanish ‡u pandemic from the early 20th century on patentproductivity in industries where face to face communication was important,and …nd a huge e¤ect. Of course, this pre-dates the internet. Finally, Yamauchi et al (2022) …nd a signi…cant negative shock to patent applications as
a result of the Covid-19 pandemic. Interestingly, they …nd that shocks are …netuned even to the timing of the waves of the pandemic in Japan,17 suggesting
upheavals in the modes of collaborations used in R&D. Gibbs et al (2023) …nd
a big change in worker productivity due to an abrupt imposition of work fromhome restrictions.
A number of papers consider the e¤ect on productivity of Covid restrictions
requiring remote work, related to our theoretical application below where wemodel this e¤ect. Yang et al (2022) study the e¤ect of such restrictions on
Microsoft employees, and …nd that information sharing decreased. Liu andSu (2023) conclude that working from home reduces the urban wage premiumalong with agglomeration externalities in cities, consistent with a reduction in
returns to interpersonal skills. Thus, a reduction in face to face contact in turn
reduces productivity in the conceptual phase of knowledge creation, where tacitknowledge is important. Monte et al (2023) examine how, in a monocentric
spatial model with multiple steady state equilibria, a shock to commuting suchas Covid can shift the economy from one steady state equilibrium to another,
thereby resulting in long term e¤ects. The multiple equilibria are caused bya coordination problem: two workers must be in the o¢ ce to interact, and if
only one shows up, there is no interaction. Thus, there could be equilibria
where nobody shows up at the o¢ ce, and where everyone does. In contrast, we
use a myopic core solution concept, so there is no prospect of a noncooperative
coordination failure. Thus, the ine¢ ciencies we identify are distinct fromtheirs. Using micro data that includes the Covid time period, Emanuel et al(2023) …nd that face to face interaction increases long run human capital at theexpense of short term payo¤s. Physical proximity is more important to womenand mentor-mentee relationships. Gibbs et al (2023) …nd that productivity
fell during the pandemic, as communication over the internet was more costly
than face to face communication. In an empirical study, Xiao et al (2021)…nd that commuting distance decreases the productivity of inventors.
In contrast with this other work, we attempt here to model microfounda17

See their Figure 3.

53

tions for this literature, breaking down research into conceptual and technicalphases. By making the communication mode an endogenous choice, and disaggregating this choice into two phases, we are able to address the subtletiesof creative activity, such as the role of tacit knowledge.
In the formulation (44), the combination of communication modes, f CF ; CN ;has been freely chosen. In reality, however, there exist numerous restrictions
on the use of communication modes; some are explicit whereas others are implicit. For example, old metropolises, such as London, New York and Tokyo,
have big CBDs that formed a long time ago when commuting railways werepredominant. In these metropolises, before the Covid-19 pandemic, a large
proportion of workers were forced to commute to CBD o¢ ces to ease F2F communications. Likewise, professors and students at traditional universities havebeen forced to commute to their main campuses for ease of F2F communications. Conversely, during the recent Covid-19 pandemic, many governments
introduced regulations that forced a large proportion of workers and students
in big cities to work at their homes through the net while discouraging F2F
meetings. Speci…cally, in the early period of the Covid-19 pandemic, theJapanese government asked o¢ ces in large cities to reduce worker commuting
by 80%, switching to WFH (working from home) to avoid transmission of thecorona virus through F2F communications.
Given such examples of restrictions on communication mode in the realworld, let us try to apply the model we have developed to study the impact ofsuch restrictions on joint knowledge productivity. For this purpose, focusing
on the optimal combination of communication frequencies ( CF ; CN ; F ; N )in equation (44), we represent the left side of equation (44) explicitly as:
AJij ( CF ;

CN ;

F;

N ):

CN ;

F;

N ):

(132)

This represents the …nal output of joint work under the combination of communication frequencies f CF ; CN ; F ; N g. Then, taking the ratio of (132)

54

N g,

(131)

And, given any feasible combination of communication frequencies, f CF ; CN ; F ;let us substitute this general combination for the optimized combination f CF ; CN ;on the right side of equation (44), and denote the resulting equation by
AJij ( CF ;

F;

N g,
F;

Ng

to (131), de…ne:H( CF ;CN ;AJij ( CF ; CN ; F ; N )N)
AJij ( CF ; CN ; F ; N )[ CF CF + CN CN ] CT C [=[ CF CF + CN CN ] CT C [

F;

F

F +

N

N]

T

F

F +

N

N]

T

;
(133)

representing the productivity under the combination f CF ; CN ; F ; N g relative to the optimal combination of communication frequencies. In the restof this Appendix, we adopt Assumption 1. In order to reduce the number ofvariables in equation (133), using the feasibility constraints (13) and (33) on
frequencies, let us rewrite the terms inside of each bracket in (133) as follows:
CF +

CF

F

CN

F +

CN

N

N

1
1 + "FCN
fC ( CF ; "F );
CF

=

1

CF

CN

(134)

1
1 + "FNf ( F ; "F );F=(1 + "F ) +

CN

1

F

(1 + "F ) +

N

N

(135)

whereas from Proposition 1,
CF +

CF

F

CN =

CN

F +

N

CF ("F )N =F ("F )CF +F +CF ("F )

(136)

CN

N ("F )

N

(137)

where f CF ("F ); CF ("F ); F ("F ); N ("F )g represents the optimal combination of frequencies at each "F0, as speci…ed by either (50), (51) or (52),depending on which of the three ranges "F belongs to. Substituting (134),(135), (136) and (137) into (133), the relative productivity function can be
expressed as follows:H( CF ;=[ CF ("F )F ; "F )fc ( CF ; "F ) CT C f ( F ; "F )CF + CN ("F )CN ] CT C [ F ("F )TF +N ("F )

N]

:
T

(138)
Next, in association with the optimal frequencies f CF ("F ); CF ("F ); F ("F );at "F , let
(139)
CT
C
T
F ("F )F ("F )CF ("F ) +55

N ("F )g

be the total frequency of F2F communication per unit of time that has beenrealized at "F either in conceptual development or in technical development.
Likewise, in association with any feasible combination of frequencies, f CF ; CN ;at any given "F 0, let
F

CT

CF +

C

(140)

F

T

be the total frequency of F2F communication.
For concreteness, let us imagine a large city in which a pair of knowledge
workers, i and j, reside and work. Let "F represent the commuting time of iand j to the common CBD o¢ ce where they work together F2F. Here, "F isassumed to be the same for i and j, and it is treated parametrically. Now,re‡ecting real world experience during the Covid-19 pandemic, let us suppose
the following restriction has been imposed (on this pair of workers as well as
on all CBD workers) by the government:
Restriction: Given the pair of workers i and j with common commuting time
"F , their choice of communication mode frequencies, f CF ; CN ; F ; N g,must satisfy the following restriction.18
F =

where

(141)

F ("F )

2 [0; 1) is a constant chosen by the government.

That is, the government requires the workers to reduce the total F2F frequency by (1)% from the present F ("F ), that has been chosen optimally
under no restrictions.
We are assuming throughout that communicating using the net means
working from home, whereas communicating F2F means working together atthe o¢ ce.
Given restriction (141), the workers must now solve the following problem:
max f H( CF ;018

F ; "F ;

)j CT

CF +

C

1
; 0
1 + "F

CF

F

T

F =

1
1 + "F:F ("F );

(142)

We can generalize restriction (141) as follows:
F

However, since the workers wish to makethis inequality with the equality in (141).

F ("F );
F0<1

as close to56

F ("F ) as possible, we can replace

F;

N g,

Let f ^ CF ("F ; ); ^ F ("F ; )g be the solution to problem (142). Then, using
constraints (13) and (33), we obtain the associated net frequencies as follows:^ CN ("F ; ) = 1^ N ("F ; ) = 1(1 + "F ) ^ CF ("F ; );(1 + "F ) ^ F ("F ; ):

(143)
(144)

Together,f ^ CF ("F ; ); ^ CN ("F ; ); ^ F ("F ; ); ^ N ("F ; )g(145)

represents the optimal choice of communication-frequencies under regulation.
Substituting ^ CF ("F ; ) and ^ F ("F ; ) for CF and F in (138), we denotethe maximized value of H( CF ; F ; "F ) byH( ^ CF ("F ; ); ^ F ("F ; ); "F ; )^ F ; ):
H("

(146)

Next, we study the characteristics of solution (145), which depends on
where the parameter "F belongs in terms of three ranges, (a), (b) and (c), as
de…ned in Figure 5. We start with range (c) since it is easiest. Then wemove to range (b), the next easiest. Finally, we examine range (a), the mostcomplex and interesting range.
(c) Consider the net-dominant the range, "CF ; 1 : From Proposition 1, ateach "F 2 "CF ; 1 , the optimal combination of communication mode frequencies under no restriction is given by
f CF ("F );CN ("F );F ("F );N ("F )g = f0; 1; 0; 1g:(147)

Hence, in this net-dominant range, from de…nition (139),(148)

F ("F ) = 0;

for "F 2 "CF ; 1 . Substitution of (148) into (142) yields the following equality
constraint:
CT

C

CF +

F = 0

T

for any

2 [0; 1):

which has a unique feasible (i.e., non-negative) solution such that
^ CF ("F ; ) = 0; ^ F ("F ; ) = 0;
and hence, from (143) and (144),^ CN ("F ; ) = 1; ^ N ("F ; ) = 1:57

In sum, the unique solution to problem (142) is given byf ^ CF ("F ; ); ^ CN ("F ; ); ^ F ("F ; ); ^ N ("F ; )g
= f0; 1; 0; 1g= f CF ("F );for "F 2 "CF ; 1 andconclude that

2 [0; 1).

(149)

N ("F )g;

F ("F );CF ("F );Hence, recalling (133) and (138), we can^ F ; ) = 1 for "F 2 ["F ; 1) ; 2 [0; 1):
H("

(150)

In other words, in the net-dominant range of "F , the restriction of the total
F2F frequency in (141) has no e¤ect on the modal choice of communication.
(b) Consider the range, "F 2 "F ; "CF , of mixed communication mode: FromC
Proposition 1, at each "F 2 "F ; "F , the optimal combination of communication mode frequencies under no restriction is given by
f CF ("F );CN ("F );F ("F );F ("F ) =CT1
; 0; 0; 1
1 + "FN ("F )g =(151)

implying thatCT

CF ("F ) =CC

1 + "F(152):

Thus, the maximization problem (142) becomes:
max H( CF ;)j

F ; "F ;

0

CF

CT

CF +

C

1
; 0
1 + "F1
1 + "F

F

CT

F =

T

C

1 + "F

;

(153)

:

Observing that @H=@ CF > 0 for CF 2 (0; 1=(1 + "F )) when "F 2 "F ; "CF ,we can readily obtain the solution of this maximization problem as follows:f ^ CF ("F ; ); ^ CN ("F ; ); ^ F ("F ; ); ^ N ("F ; )g =1 + "F;1; 0; 1(154)

for "F 2 "F ; "CF and
^ F; ) =H("

2 [0; 1), which in turn implies that
+ (1 + "F ) (1

)

CT

C

CF

=

1

2

"F ; "CF(1 + "F )CN

CFfor "FCNand

2 [0; 1)58+

(1 + "F )

CN

CT

C

CF

(155)

Hence, we have that for "F 2 "F ; "CF and

2 [0; 1),

^ F ; )=@"F > 0; H("^ C ; ) = 1 for any@ H("F^ F ; )=@ > 0; H("^ F ; 0) = (1 + "F )
@ H("

CT

CN

CF

C

2 [0; 1);(156)

^ F ; 1) = 1: (157); H("

Comparing (151) and (154), we can see that in the (b) range of mixedcommunication modes, imposing a restriction on the total F2F frequency in theform of (141) reduces, at each "F 2 "F ; "CF , the F2F frequency in the conceptual development phase from the optimal frequency, 1=(1 + "F ), to =(1 + "F ).
In turn, this increases the net use frequency for conceptual development from0 to 1 . As indicated in (156) and (157), the impact of such changes in com^ F ; ) is smaller (i.e., amunication frequencies on the relative productivity H("
relatively smaller reduction in relative productivity) either as "F increases oras increases (i.e., the restriction is weaker).
(a) Consider the F2F dominant range, "F 2 [0; "F ): For simplicity of notationin the analysis of this case, let us assume that19
C =

CT

(158)

T:

T

From Proposition 1, at each "F 2 [0; "F ), the optimal combination of communication frequencies under no restriction is given by
f CF ("F );CN ("F );F ("F );N ("F )g =

1
1
; 0;
;0 ;
1 + "F
1 + "F(159)

Recalling de…nition (139) and using (157) yields
F ("F ) =2 T;
1 + "Fand hence the restriction (141) becomesT ( CF +

F) =

2 T
;
1 + "For2:
1 + "F
Thus, the maximization problem (142) becomes:
CF +

max

H( CF ;

F ; "F ;

)j

(160)

F =

CF +

F =

19

2
;0
1 + "FCF

1;
1 + "F

Without this assumption, we can conduct the analysis in this section and obtain qualitatively the same result. However, without this assumption, the notation in the analysis
becomes awfully complex.

59

0

1
;
1 + "F

F

(161)

:

To obtain the solution of this maximization problem, let us neglect for awhile the inequality constraints in (161), and consider the following simpler
problem:2:(162)
max H( CF ; F ; "F ; )j CF + F =1 + "FTo get the …rst-order condition for the solution of (162), let us set@H( CF ; F ; "F ; )@ CF

2
F = 1+"F

(163)

= 0:
CF

From straight forward calculation using (138), we can obtain the unique solution of (163) as follows:~ CF ("F ; )1
1 + "F+

q("F )
2

(164)

;

where
vC ("F ) v ("F );vC ("F ) v ("F )1CF
1; v ("F )1 + "F
CN

q("F )vC ("F )

(165)
F
N

1
1 + "F

1:(166)

From Assumption 1, we always have for "F 2 [0; "F ) thatvC ("F ) > 0; v ("F ) > 0; q("F ) > 0;(167)

and hence ~ CF ("F ; ) > 0. Furthermore, it can be readily shown that the left
hand side of equation (163) is positive (respectively, negative) for CF < ~ CF(respectively, CF > ~ CF ). Hence, the function, H( CF ; F ; "F ; ) subject to~
F = 2 =(1 + "F )CF , achieves its maximum at CF . However, in general,
for f CF ; F = [2 =(1 + "F )CF ]g to be the solution to maximization
problem (161), it must satisfy also all the inequality conditions in (161). The
last inequality condition in (161) can be rewritten as:
0

F =

2
1 + "FCF

1
1 + "F

that is,
CF

2
2
1
and
1 + "F
1 + "F

60

CF :

(168)

Based on the observations above, let ^ CF ("F ; ) and ^ F ("F ; ) be de…ned
respectively as^ CF ("F ; ) = min^ F ("F ; ) =1
2;; ~ ("F ; ) ;1 + "F 1 + "F

2
1 + "F^ ("F ; ):

(169)
(170)

where ~ CF ("F ; ) is de…ned by (164). Then, it is not di¢ cult to con…rm thatfor "F 2 [0; "F ) and 2 [0; 1), the pair f ^ CF ("F ; ); ^ F ("F ; )g is the uniquesolution of the maximization problem (161) that satis…es, of course, all theconstraints in (161), including the second inequality constraint in (168).
Although (169) and (170) together de…ne the solution to problem (161) implicitly, to obtain the explicit solution of (161), we must consider the following
three cases separately:
n
o
2
1
~
min 1+"F ; ("F ; ) ;(171)
Case 1: 1+"Fno
21
~ ("F ; ) ;Case 2: 1+"
min
;
(172)
1+"F
F
o
n
2
1
:
(173)
Case 3: ~ ("F ; ) min 1+"
;
1+"F
F

Let us study each case in turn.
(i) Case 1 means the following two inequalities hold:1
1 + "F1
1 + "F

2
1,
1 + "F2~ CF ("F ; )1
1 + "F+

Case 1 ,

max

1
;1
2

q("F )
2,1

q("F )
2

Hence,
q("F )
2(174)

On the other hand, from (169) and (170), by using (143) and (144) the optimal
combination of communication frequencies under restriction (160) for Case 1
is given by:f ^ CF ("F ; ); ^ CN ("F ; ); ^ F ("F ; ); ^ N ("F ; )g =

2
1
1
; 0;
; 2(1
1 + "F
1 + "F) :(175)

Comparing the optimal combination (159) under no restrictions and the restricted optimum (175) above, we can see thatf CF ("F ; );CN ("F ;)g = f ^ CF ("F ; ); ^ CN ("F ; )g =61

1
;0
1 + "F

(176)

1
2
1 ^
>
= F ("F ; );1 + "F
1 + "F("F ; ) = 0 < 2(1) = ^ N ("F ; ):F ("F ;

N

(177)

)=

(178)

That is, under a restriction in the form of (160), the optimal frequencycombination in the conceptual development phase is maintained; whereas in
the technical development phase, the restriction (160) reduces the F2F-frequency
while increasing the net-frequency. This result arises for the following reason:
From Assumption 1, F2F communication is more e¤ective in the conceptual
development phase than in the technical development phase. On the other1
hand, condition (174) implies that
, i.e., the restriction is not strong.
2
Thus, it is e¢ cient to absorb the impact of the total F2F-frequency restriction
solely in the technical development phase.
Next, based on (159) and (175), we have that
^ F ; ) = (2
H("

1) +

N

T

(1 + "F ) 2 (1

(179)

)

F

for "F 2 [0; "F ) under condition (174). Hence,^ F; )@ H("^ F ; 1) = 1; H("^ F ; 1) => 0; H("
@
2

T

N

<1

(1 + "F )F(180)^ F; )@ H("^> 0; H(0;) = (2
"F

1) +

N

T

2 (1

< 1; (181))

F

for < 1 as expected.
(ii) Case 2 means the following two inequalities hold:2
1 + "F2
1 + "F1,
1 + "F~ ("F ; )1;
2
1
1 + "F+

q("F )
2,

q("F ):
2

Hence,
1
q("F )
and
:
(182)
2
2
On the other hand, from (169) and (170) and using (143) and (144), the optimal
combination of communication frequencies under restriction (160) for Case 2
is given by:
Case 2 ,f ^ CF ( ; "F ); ^ CN ( ; "F ); ^ F ( ; "F ); ^ N ( ; "F )g =2
;1
1 + "F2 ; 0; 1(183)

62

Comparing the optimal combination (159) under no restrictions and therestricted optimum above, we can see that a strong restriction in the form of1(160) withelicits huge changes in communication mode: First, given that2
F2F communication can be used more e¤ectively in the conceptual development phase than in the technical development phase, the F2F frequency in thetechnical development phase is reduced from 1=(1 + "F ) ( the non restricted
optimum) to 0, whereas the net frequency in the technical development phaseis increased from 0 to 1. Such changes in the communication frequencies inthe technical development phase allows the full permissible capacity of F2F
frequency to be utilized in the conceptual development phase.
Next, using (159) and (183), we have that
^ F ; ) = 2 + (1
H("

CN

2 )

T

T

N

(1 + "F )CF(1 + "F )

;

(184)

F

for "F 2 [0; "F ) under condition (182). Thus,^ F; )@ H("^ F ; 1) => 0; H("@2^ F ; 0) =
H("

CN

N

;

F
T

N

(1 + "F )CF

T(1 + "F )

(185)

;

F

^ F; )@ H("^> 0; H(0;) = 2 + (1"F
for < 21 .
(iii) Case 3.
hold:

T

(1 + "F )

2 )

CN

T

CF

N

T

: (186)F

Recalling (164), Case 3 means the following two inequalities~ ("F ; )~ ("F ; )1,
1
1 + "F2
q("F ),
1 + "F2

q("F );
2:

Thus,
Case 3 ,q("F )
2

1

q("F )
2(187)

implying that q("F ) 12 .
On the other hand, from (169) and (170), using (143) and (144), the optimal
combination of communication frequencies under restriction (160) for Case 3
is given byf ^ CF ( ; "F ); ^ CN ( ; "F ); ^ F ( ; "F ); ^ N ( ; "F )g1
q("F )
q("F )
1+;1+;=
1 + "F2
2
1 + "F63

q("F )
2;1

q("F )
2(188)

implying that in the domain of f ; q("F )g where the strict inequalities holdin relation (187), each component in (188) is strictly positive; that is, eachfrequency combination given by (188) represents an interior solution. In contrast, the optimal combination (159) under no restriction represents a corner
solution. The constrained optimum (188) implies that in order to absorb effectively the constraint (160) on the total F2F frequency, in each of the phasesof conceptual development and technical development, only a part of F2F frequency in the optimal solution (159) has been switched to the net. When
we compare the three cases below, it will become clear that Case 3 tends tohappen when F2F communication is much more e¤ective than net communication in both the phase of conceptual development and the phase of technicaldevelopment.
Next, based on (159) and (188), we have that
^ F; ) =H("+

q("F )
2
q("F )
2CN

1(1 + "F ) +CF(1 + "F )

F
N

1

T

N

(1 + "F ) +

F

N

T

(1 + "F )

;

F

(189)
for "F 2 [0; "F ) and under condition (187). Hence,^ F; )
@ H("
> 0:
@

(190)

It is also not di¢ cult to see from (189) that
^ F; )
@ H("
> 0;
"F

(191)

under condition (187).
(iv) Synthesis. Thus far, we have examined each case separately. Putting
the results of the three cases together, Figure A shows which case happens
under what range of parameters.
Figure A
In the right half of Figure A, the ( ; q)-space is partitioned into three cases.
In the left half of Figure A, the q("F )-curve is depicted by taking two examples
of -parameters:
Example 1:
Example 2:

CN
CF
CN
CF

64

= 2;
= 3;

N
F
N
F

= 1:5

(192)

= 2:

(193)

First, let us consider Example 1. Substituting the parameter values in
Example 1 into (165) and (166) yields:
q1 ("F ) =

1=2
1+"F
3=2
1) ( 1+"
F

2
( 1+"
F

1)

;

(194)

implying that1
= 1;
2
as depicted in Figure A. Since q1 ("F ) > 1 for all "F 2 (0; 1=2), only two cases
can happen: Case 1 when 2 [1=2; 1), and Case 2 when 2 [0; 1=2). Noticethat the parameter values in Example 1 correspond to those in Figure 4.
Next, let us consider Example 2, which yields the following function:
q1 (0) = 1; q10 ("F ) > 0; q1

q2 ("F ) =

1
1+"F
3
( 1+"
F

2
1) ( 1+"
F

1)

;

(195)

implying that1
q2 (0) = ; q20 ("F ) > 0; q2 (0:268) = 1; q2 (1) = 1;2
as shown in Figure A. Since q2 ("F ) < 1 for "F 2 [0; 0:268), Case 3 canhappen for each "F 2 [0; 0:268) with intermediate values of . Notice that theparameter values in Example 2 correspond to those for Case 2 in Figure 5. As
discussed in the last paragraph of Section 3, relative values of ’s in Example 2(i.e., Case 2 in Figure 5) correspond to those knowledge workers who have poorinternet skills (or poor internet access). For such workers, in both the phasesof conceptual development and technical development, F2F communications
are much more e¤ective than net communication; hence, these workers retain
a large fraction of F2F frequencies in both phases when is an intermediate
value.
^ F ; ), has
Finally, let us recall that the relative productivity function, H("
been de…ned by (146), which shows the ratio of fthe maximized value of joint
work output at "F under the F2F frequency constraint g over fthe maximized
value of joint work output at "F without a frequency constraintg. We havederived the following set of relative productivity functions:(c) function (150) for the net dominant range, ["F ; 1) ,(b) function (155) for the mixed communication-mode range, "F 2 "F ; "CF ,(a) for the F2F dominant range, "F 2 [0; "F ) ;function (179) for Case 1,
function (184) for Case 2,
function (189) for Case 3,
65Based on these relative productivity functions, Figure B presents a set of^ F ; ) as a function of the total F2F frequency
relative productivity curves H("
constraint at each selected value of "F .

Figure B

In drawing Figure B, the -parameters are based on those in Example 2 in
(193), implying that
"F = 1; "CF = 2.
Furthermore, the values of the -parameters are set as
CT

C =

T

= 0:25.

Drawing on Figure B, we make the following observations:
All relative productivity curves meet at (1; 1) since = 1 means no F2F
frequency constraint.
^
In the net-dominant range "F 2 "CF ; 1 , it holds that H("F ; ) = 1 for all
2 [0; 1] since any F2F frequency constraint is not binding in the net dominant
range.
As "F decreases from "CF toward 0, the relative productivity curve rotatescounterclockwise around the pivot (1; 1), implying that
^ F; )@ H("> 0 for "F 2 0; "CF ;@"F^ F; )@ H("> 0 for 2 (0; 1) :
@

(196)
(197)

Relation (196) arises for the following reason. As can be seen from (50),the optimal values of CF and F decrease as "F increases; likewise, from(51), the value of CF decreases as "F increases. That is, when there is noconstraint on the total F2F meeting frequency, communication in joint work
depends less on F2F as "F increases. Hence, as "F increases, the negative
impact of a given total frequency constraint on the joint work output isrelatively smaller. Conversely, as "F decreases toward 0, the negative impact of a given constraint becomes relatively greater; and hence the relative^ F ; ) becomes lower as "F decreases toward 0.
productivity H("
Relation (197) arises for the simple reason that when "F is …xed, the negative
impact of F2F constraint on the joint-work output becomes weaker as66

increases (i.e., the constraint becomes less binding), and hence the relative^ F ; ) increases as increases towards 1.
productivity H("
Based on (196) and (197), we see that in Figure B, the relative productivity^
H("F ; ) decreases most signi…cantly in the F2F-dominant range (a) when thevalue of is small (i.e., the total F2F constraint is strong). Speci…cally, we^ 0)
can see from Figure B that at ( ; "F ) = (0; 0), the relative productivity H(0;equals 0:64. That is, when no F2F communication is permitted (i.e., allcommunications are forced to be through the net only), and when "F = 0 (i.e.,
no lead-time for F2F communication), the productivity of joint work decreases
36% in comparison with the situation without a total-F2F-constraint. Under^ 1=4) is 0:76,
the speci…cation ("F ; ) = (0; 1=4), the relative productivity H(0;i.e., a 24% reduction in joint-work productivity.
As shown in Figure B, in the F2F dominant range (a), there are three cases of
optimal combinations of communication-mode frequencies under a total F2F
constraint. In the area for Case 3 (below the doted boundary line curve),
optimal frequencies of F2F and net are both positive in both the phases ofconceptual development and technical development. In this way, the negative
impact of a total F2F constraint on joint-work productivity is mitigated moste¤ectively.
Recall that in the context of a metropolitan area, "F represents the commuting time to the common CBD o¢ ce. Thus, the results above indicate that
under the same total F2F restriction , workers residing near the CBD see
their joint-work productivity decrease relatively more than those residing far
from the CBD.
Figure B is based on the relative values of ’s given in Example 2 of (193).
In contrast, let the relative values of ’s be set at those given in Example 1 of
^ 0) = 0:75. Thus, in terms of(192). Then, it can be readily shown that H(0;^ 0), moves upward to 0:75
Figure B, the lowest relative productivity point, H(0;from the present 0:64. Similarly, when relative values of ’s are changed fromthose in Example 2 to those in Example 1, all relative productivity curves in
Figure B shift upward signi…cantly for all < 1. Recall that relative values of’s in Example 2 correspond to workers who have poor computer skills (or poorinternet access). In contrast, relative values of ’s in Example 1 correspondto workers who have better skills (or better internet access).
Finally, notice that the relative values of ’s in Example 1 of (192) (vs.
Example 2 of (193)) correspond to those values in Case 1 (vs. Case 2) inFigure 6. Thus, the results mentioned just above suggest that the negative67

impact of a total F2F constraint on relative productivity becomes weaker withthe advancement of net-technology.
We may summarize our observations as follows:
Proposition A. In the context of a metropolitan area, let us imagine thatthe government (facing the Covid-19 pandemic) introduces the restriction thateach pair of joint knowledge workers must reduce the total F2F frequency by(1)% from the present optimal choice. To be precise, at each "F , therestriction is de…ned by equation (141) together with (139) and (140). In this
^ F ; ) be the relative productivity function de…ned by (142) andcontext, let H("(146), which represents the ratio of fthe maximized value of joint work output
at "F under the F2F frequency constraint g over fthe maximized value of
joint work output at "F without a frequency constraintg. For illustration, let
the relative values of communication-e¤ectiveness parameters, ’s, be set as
those in Example 2 of (193), implying that the "F -parameter range is divided
into the following three sub-ranges:(c) the net dominant range: "CF ; 1 = [2; 1),(b) the range of mixed communication mode: "F ; "CF = [1; 2),(a) the F2F dominant range: [0; "F ) = [0; 1).
In this context, the impact of the total F2F frequency restriction on the relative^ F ; ), can be represented as in Figure B. Speci…cally:
productivity, H("(i) Since = 1 means no F2F frequency constraint, all relative productivity^ F ; ) for each …xed "F , meet at (1; 1).
curves, H("(ii) In the net-dominant range (c), since F2F frequency is not binding, the F2F
frequency constraint has no impact on the relative productivity, implying that
^ F ; ) = 1 for all "F 2 "C ; 1 and 2 [0; 1].
H("FC(iii) As "F decreases from "F toward 0, the relative productivity curve rotates^ F ; ) > 0 for
counterclockwise around the pivot (1; 1), implying that @ H("^"F 2 (0; "C
2 (0; 1).
F ) and @ H("F ; ) > 0 for^ F; )(iv) Hence, as can be seen in Figure B, the relative productivity H("
decreases most signi…cantly in the F2F-dominant range, (a), when the valueof is small (i.e., the total F2F constraint is strong). Speci…cally, we can see
from Figure B that at ( ; "F ) = (0; 0), the relative productivity H(0; 0) equals
0:64; that is, the productivity of joint work decreases 36% in comparison withthe situation of no total frequency constraint. For another speci…cation, at^ 1=4) equals 0:76, implying a("F ; ) = (0; 1=4), the relative productivity H(0;24% reduction in joint-work productivity.
(v) In the context of a metropolitan area, the results above indicate that under
the same total F2F restriction , workers residing near the CBD experience a68

relatively higher decrease in their joint-work productivity compared with thoseresiding far from the CBD.
(vi) Figure B is based on the relative values of ’s given in Example 2 of
(193). In contrast, let the relative values of ’s be set at those given in
Example 1 of (192). Then, it can be readily shown that in Figure B, the lowest^ 0), moves upward to 0:75 from the present
relative productivity point, H(0;0:64; furthermore, all relative productivity curves shift upward signi…cant for
all < 1. In other words, the relative productivity loss from the same F2F
frequency constraint is smaller for workers with better computer skills (orbetter internet access) than that for workers with poor computer skills (orpoor internet access).
(vii) Finally, comparing relative values of ’s in Example 1 and Example 2, we
can also reinterpret the results above as follows: the negative impact of a total
F2F constraint on relative productivity becomes weaker with the advancementof net technology.

69

8

Appendix B: Calculation of imputed values
in Table 1

F
For example, let us calculate the imputed value of aCT(the top of the thirdijtier in Table 1). Observe that

@AJij
@AJij
=
F
@aCT
@aC
ij
ij

@aC
ij
CT F
@aij

(198)

First, from (43),log AJij=C+[log C + log aC
ij ] +[log[log I + log(aIii + aIij )]I+ log aij ]:Hence,@AJij= AJij
@aC
ij

C

1
:
aC
ij

(199)

Next, from (19),log aC
ij

= log

C
ij +

CT

F
[log CT + log( CF aCT
+
ij

CN

N
aCT
)]
ij

+ CS [log CS + log(aCSI+ aCSI
)]:
i
j
Thus,
@aC
ij
= aC
ij
CT F
@aij

CF

CT
CF
CT F
aij + CN

N
aCT
ij

(200)

:

Substituting (199) and (200) into (198) yields:
@AJij
F
@aCT
ij

= AJij= AJij

C

1 C
:aij
aC
ij

CF

CT
CF
CT F
aij + CN

N
aCT
ij

CF
C

CT
CF

F
aCT
+
ij

CN

N
aCT
ij

:

Hence,@AJij
F
aCT= AJij
ij
F
@aCT
ij

C

CT
CF

CT F
CF aij
F
aCT
+ CN
ij

N
aCT
ij

F
which gives the imputed value of aCT
listed in the fourth column of Table 1.
ij
Likewise, we can obtain the imputed value of each activity output listed in
the fourth column of Table 1.

709Appendix C: Three possible dynamics of thetwo-person system

In Figure C, diagrams (a), (b) and (c) illustrate each of three possible dynamics
for md , whereas diagram (d) explains the determination of the relationshipbetween m~ and mB .20

Figure C
First, diagram (a) shows the dynamics of md for the case, mE < m~ < mB ,which is also the case illustrated in Figure 9. As depicted in this diagram, if
the initial position, md (0), at time 0 is to the left of m,~ then md (t) gradually
increases towards m.
~
If m~ < m(0) < mH , then md (t) gradually decreases
towards m.
~ Only when md (0) > mH does md (t) move away from m~ towarddHd1=2. Hence, whenever m (0) < m , m (t) approaches the sink point, m,~
Bwhich is to the left of the bliss point m .
Next, diagram (b) depicts the dynamics of md for the case mB < m~ < mH .
In this case, except when md (0) > mH , md (t) approaches the sink point m~ toBthe right of the bliss point m .
Third, diagram (c) describes the case where m~ < mE , which happens whengJ (m)~ < I . In this case, except when md (0) > mH , md (t) approaches thesink point mE , where the K-growth rate is much lower than at the bliss pointmB .
Diagram (d) in Figure C explains when each of the three possible cases ofdynamics happens. For the purpose of intuitive understanding of when eachcase happens, we here focus on the special situation where
C =

:

(C-1)

That is, both in equations (10) and (11) and equations (29) and (30) (theK-production function for joint thinking through the Net), the weight on differential knowledge has the same value. In this case, from (86), we have
20

Theoretically speaking, there exists the possibility of a fourth case where mB < mH <m
~ < 1=2. This can happen only in the extreme situation where both F and N are close
zero, and I and gJ (mB ) are nearly equal. Hence, in the following discussion, we neglect
this fourth case. Actually, it is possible that m~ = mB . But this case happens on a set ofmeasure zero, so we neglect it as well.

71

that
mB =

1
1+ 1

mB ( );(C-2)

meaning that the value of mB is uniquely determined by the value of parameter. In diagram (d) of Figure C, function mB ( ) is depicted by the bold curvefor 2 [0; 1].
In contrast, the value of m
~ is determined, as in (113), by the values of’s, independent of . Hence, here we treat m~ as a single parameter. By
de…nition, the maximum value of m
~ equals 1=4:
1
max m
~ = :
4
Given each m~ 2 (0; 1=4], as shown in diagram (d), by settingm~ = mB ( ),(C-3)

the value of (m)~ is uniquely determined by m
~ = mB .
Now, given m~ 2 (0; 1=4], we can see from diagram (d) thatm
~ < mB ( ) for2 ( (m);~ 1=2):(C-4)

That is, given m~ 2 (0; 1=4], if we take the value of su¢ ciently large so that
> (m),~ then the corresponding bliss point, mB ( ), locates to the right ofm,~ which corresponds to diagrams (a) and (c) in Figure C. By de…nition,> (m)~ means that the weight on di¤erential knowledge in knowledge subproduction functions (10) and (29) is su¢ ciently large. Furthermore, whenis close (m),~ then we have diagram (a). On the other hand, when is muchlarger than (m),~ then the corresponding bliss point mB is far to the right ofm,~ yielding diagram (c).
In contrast to (C-4), we can see from diagram (d) that for each m~ 2 (0; 1=4),m~ > mB ( ) for

2 (0; (m)):
~

(C-5)

That is, given m~ 2 (0; 1=4), if the value of is su¢ ciently small so that
< (m),~ then the corresponding bliss point, mB ( ), locates to the left of m,~which corresponds to diagram (b) in Figure C. By de…nition, < (m)~ means
that the weight on common knowledge in sub-production functions (10) and(29) is su¢ ciently large.

72

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[2] Atkin, D., Chen, M.K. and Popov, A., 2022. “The Returns to Faceto-Face Interactions: Knowledge Spillovers in Silicon Valley.” Working
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https://slack.com/intl/zh-tw/blog/collaboration/report-remotework-during-coronavirus[15] Inoue, H., Nakajima, K., Okazaki, T. and Y.U. Saito, 2022. “The Role ofFace-to-face Contact in Innovation: The Evidence from the Spanish Flu
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75

Figure 1. Knowledge creation through multiple modes of communication

AI∗
ii

a IS
i

a IT
i

Figure 2. The activity tree for knowledge creation by person 𝑖 in Isolation.

AJ∗
ij

a C∗
ij

a CTF
a CTN
a CSI
a CSI
a ITJ
ij
ij
j
i
i
Conceptual development
jointly

a ∗
ij

a I∗
jj

a I∗
ii

a ITJ
a ISJ
j
i

Independent
development
by i

a ISJ
a TF
a TN
a SI
a SI
j
ij
ij
j
i

Independent
development
by jTechnical development
jointly

Figure 3. The activity tree for joint knowledge creation:∎ the final output of joint work●

representing a joint activity with subscript 𝑖𝑗,○

representing an independent activity for the purpose of

joint creation with subscript 𝑖, 𝑖𝑖, 𝑗 or 𝑗𝑗.

 CF
 CN

 F
 N

 CF
 CN

 11

F

1
 F
 N

(a)

0

0. 5

 11

F

1

(b)

 F

(c)

F

 CF
and the three ranges of

Figure 4. The parameter value of F2F lead-time 𝜀communication mode when

2 and

1.5.

h F 1h a  F 

h b  F 

0. 82
0. 76
0. 70

h c  F 

h̃ a  F 

(a)

0

0. 5

1

(b)

 F

2 and

1. 5  F

 CF

Figure 5. The relative productivity curve ℎ 𝜀
when

(c)

1.5, 𝜌

∙𝜌

in the three ranges of 𝜀
𝜌

∙𝜌

0.25.

h F , h o  F 
1h a  F 
h b  F 

0. 76

h c  F 
h ob  F 

0. 64
0. 58
0. 5 h oa  F   h̃ a  F (a)0h̃ a  F 

 F (b)  CF
0. 5

(c)

(c)

1

2

o
F

(a o )h oc  F oCF(b o )

F(c o )

Figure 6. The impact of the advancement of net-technology on the relative productivity curve withthe comparison of two cases where

Case 1 : The relative productivity curve ℎ 𝜀
when

2 and

1.5, 𝜌∙𝜌𝜌∙𝜌

in the three ranges (a), (b) and (c), of 𝜀
0.25.

Case 2 : The relative productivity curve ℎ 𝜀in the three ranges (a ), (b ) and (c ), of 𝜀

when

0.25.

3 and

2, 𝜌

∙𝜌

𝜌

∙𝜌

revenue growth rateB

1

g J m d 

E

I

H

mB
0

mH

0.4

mE

Figure 7. The knowledge growth rate curve 𝑔 𝑚

0.5

md

and the Bliss Point 𝑚 .

revenue growth rateg J m d ; 01

g J m d ;  F [3]g J m d ;  CF 
[1]

[2]

mB
0

m dij1m dij2

Figure 8. Knowledge growth rate curves 𝑔 𝑚 ; 𝜀

0.4
for 𝜀

0

𝜀

m dij30.5

md

𝜀 , sharing the same Bliss

Point 𝑚 , and the share of differential knowledge for each of three potential partners,
𝑖, 𝑗 , 𝑖, 𝑗 , 𝑖, 𝑗 .

The knowledge growth rate1

I

g J m d 

B

E

H

mH
0

mE

̃
m

0.25

mB

Figure 9. The dynamics of two-person system when 𝑚

0.5

𝑚

𝑚 .

md

K growth rateB∗

G J m d   g J m d  3   C   CT      T 

S∗

E∗

H∗

H′

E′

2 I

B

E ′′
S
I

H ′′

g J m d HEg J m

0m

md

mE

̃
m

mB

mH

m 12

Figure 10. Dual dynamics of formal-K and total-K for the two-person system.

q2

q 1  F 

3
2Case 2

Case 1q 2  F 

1

1
2

Case 3

F

2 
F

1

1
F

3/4

1/2

0. 268
1/4

1/4

0

1/2

3/4

1

Figure A. Parameter ranges for the three cases of the impact of communication
regulation with two examples of 𝑞 𝜀function:

Example 1: 𝑞 𝜀

when

2

and

1.5

Example 2: 𝑞 𝜀

when

3

and

2.



̂  F , H
increasing

F

(c)

̂  CF ,  1H(b)

̂  2F , 0
H

Case 1̂  F , 0H
0. 9Case 3
Case 2(a)̂  1F , 0H

̂ 0, 0 0. 64
H
0. 6

≈

≈

0

Figure B.

≈

1/4

≈

1/2

≈

3/4



1

Relative productivity curve ℋ 𝜀 , 𝜂 as a function of total F2F constraint 𝜂 withparametric 𝜀such that 0

and 𝜌

𝜌

∙𝜌

∙𝜌

0.25.

𝜀 < 𝜀

1

𝜀

𝜀

2; where

3,

2

(b)

K growth rate(a)

K growth rateBB
g J m d 

g J m d 

I

I

0 mE

mB

̃
m

1
2

mH

md

md
0

(c)

K growth rate

̃
m

mE

mB

(d)

̃
m

mH

1
2

mB

B

1
2

g J m d 
m B

I

1
4

̃
max m

̃
m

mH

0

̃ mE
m

mB



md
1
2

0

̃
m

1
3

1

Figure C. The three possible cases, (a), (b) and (c), for the dynamics of two-person system; and diagram(d) for explaining the relationship between 𝑚 and 𝑚 .

tier
top

secondtier

activity
outputequationof x

imputed value of x:

■

AJ∗
ij

(43)

AJ∗
ij

●

a C∗
ij

(19)

AJ∗
ij   C

○

a I∗
ii

(25)

I
AJ∗
ij  2

○

a I∗
jj

(28)

I
AJ∗
ij  2

●

a ∗
ij

(39)

AJ∗
ij   (9) ∗CF  a CTF
J∗
Aij   C   CT  ∗ CTF ij∗ CTN
 CF  a ij   CN  a ij

CTN
● a ij(10) ∗CN  a CTNJ∗
Aij   C   CT  ∗ CTF ij∗ CTN
 CF  a ij   CN  a ij

○

a CSI
i

(11)

AJ∗
ij 

 C  CS
2

○

a CSI
j

(11)

AJ∗
ij 

C   CS
2

○

a ITJ
i

(21)

AJ∗
ij 

I  ITJ
2

○

a ISJ
i

(22)

AJ∗
ij 

 I  ISJ
2

○

a ITJ
j

(21) with j

AJ∗
ij 

 I   ITJ
2

○

a ISJ
j

(22) with j

AJ∗
ij 

 I  ISJ
2

●

a TF
ij

(29)

AJ∗
ij      T TN
● a ij(30)

 ∗N  a TNJ∗
Aij      T  ∗ TF ij∗ TN
 F  a ij   N  a ij

○

a SI
i

(31)

AJ∗
ij 

    S
2

○

a SI
j

(31)

AJ∗
ij 

    S
2

CTF
● a ij

third
tier

∂AJ∗
ij /∂x  x ∗F  a TF
ijTN∗ ∗F  a TF
ij   N  a ij

Table 1. The output x at each node of activity tree in Figure 6, and the imputed value of x.