Dynamics of Knowledge Creation and Transfer: The Two Person Case

Transcription

1 Dynamics of Knowlege Creation an Transfer: The Two Person Case Marcus Berliant an Masahisa Fujita June 8, 2006 Abstract This paper presents a micro-moel of knowlege creation an transfer for a couple. Our moel incorporates two key aspects of the cooperative process of knowlege creation: (i) heterogeneity of people in their state of knowlege is essential for successful cooperation in the joint creation of new ieas, while (ii) the very process of cooperative knowlege creation a ects the heterogeneity of people through the accumulation of knowlege in common. The moel features myopic agents in a pure externality moel of interaction. In the two person case, we show that the equilibrium process tens to result in the accumulation of too much knowlege in common compare to the most prouctive state. Equilibrium paths are foun analytically, an they are a iscontinuous function of initial heterogeneity. JEL Classi cation Numbers: D83, O3, R This research owes much to the kin hospitality an stimulating environment of the Kyoto Institute of Economic Research, where most of our arguments an o-si-o s occurre. The rst author is grateful for funing from the Institute, from Washington University in St. Louis, an from the American Philosophical Society. The secon author is grateful for Grants Ai for COE Research 09CE2002 an Scienti c Research Grant S from the Japanese Ministry of Eucation an Science. We thank Alex Anas, Gilles Duranton, Bob Hunt, Koji Nishikimi, Diego Puga, Tony Smith, Takatoshi Tabuchi, an seminar participants at Washington University, New York University, the Spring 2004 Miwest Economic Theory an International Trae conference, the Feeral Reserve Bank of Philaelphia, an the 2004 North American Summer Meetings of the Econometric Society for helpful comments. Eviently, the authors alone are responsible for any remaining errors an for the views expresse herein. Department of Economics, Washington University, Campus Box 208, Brookings Drive, St. Louis, MO Phone: (-34) , Fax: (-34) , berliant@artsci.wustl.eu Institute of Economic Research, Kyoto University, Yoshia-Honmachi, Sakyo-ku, Kyoto, Japan Phone: (8-75) , Fax (8-75) , fujita@kier.kyotou.ac.jp

2 Introuction. The Research Agena In this paper we examine the microynamics of knowlege creation an transfer by using a simple moel. With a focus on the two person case, the basic principles of this complex process can be uncovere. Although the two person moel amittely has its limitations, particularly in the ynamic choice of partners for knowlege creation an transfer, it has its avantages in analytical tractability because we can solve the equilibrium ynamics explicitly. Our major research questions are as follows. How o knowlege creation an transfer occur? How o they perpetuate themselves? How o agents change uring this process? Are the equilibrium ynamics e cient? As people create an transfer knowlege, they change. Thus, the history of meetings an their content is important. If two people meet for a long time, then their base of knowlege in common increases, an their partnership eventually becomes less prouctive. Similarly, if two persons have very i erent knowlege bases, they have little common groun for communication, so their partnership will not be very prouctive. For these reasons, we attempt to moel enogenous agent heterogeneity, or horizontal agent i erentiation, to look at the permanent e ects of knowlege creation an growth. Thus, we are examining how social capital is accumulate at a micro level. Our moel is analytically tractable, so we o not have to resort to simulations; we n each equilibrium path explicitly. The moel is also at an intermeiate level of aggregation. That is, although it is at a more micro level than large aggregate moels such as those foun in the enogenous growth literature, we o not work out completely its microfounations. That is left to future research. The analogy between partner ancing an working jointly to create an exchange knowlege is useful, so we will use terms from these activities interchangeably. Knowlege creation, exchange, an iniviual prouction all occur simultaneously at each point in time. The ancers can work alone or with their partner. They work together if an only if they both agree that it is useful. Prouction always occurs at a rate proportional to the agent s current stock of knowlege, as oes knowlege creation when an agent ances alone. Elsewhere, in Berliant an Fujita (2006), we provie extensions of the moel an results to the context of the general case of any number of potential partners, but to maintain analytical tractability, we o not consier asymmetric states or knowlege transfer. 2

3 The suitability of their ance partner epens on the stock of knowlege they have in common an their respective stocks of exclusive knowlege. The fastest rate of knowlege creation occurs when these factors are in balance. Our results are summarize as follows. First, in a two person moel where myopic agents can ecie whether or not to work with each other, there exist many sink points in the interaction game, epening iscontinuously on initial heterogeneity. The most interesting of these features too much homogeneity relative to the most prouctive state. For simplicity, we employ a eterministic framework. It seems possible to a stochastic elements to the moel, but at the cost of complexity. It shoul also be possible to employ the law of large numbers to a more basic stochastic framework to obtain equivalent results. Next we compare our work to the balance of the literature. Section 2 gives the moel an notation, Section 3 analyzes equilibrium in the case of two participants or ancers, Section 4 examines welfare in the two person moel, whereas Section 5 provies our conclusions an suggestions for future ancing. Two appenices provie the proofs of key results. The basic framework that employs knowlege creation as a black box riving economic growth is usually calle the enogenous growth moel. Here we make a moest attempt to open that black box. The literature using this black box inclues Shell (966), Romer (986, 990), Lucas (988), Jones an Manuelli (990), an many papers builing on these contributions. There are two key features of our moel in relation to the enogenous growth literature. First, our agents are heterogeneous, an that heterogeneity is enogenous to the moel. Secon, the e ectiveness of the externality between agents working together can change over time, an this change is enogenous. Fujita an Weber (2003) consier a moel where heterogeneity between agents is exogenous an iscrete. They examine the e ects of immigration policy on the prouctivity an welfare of workers. They note that progress in technology in a country where workers are highly traine is in small steps involving intensive interactions between workers an a relatively homogeneous work force, whereas countries that specialize in prouction of new knowlege have a relatively heterogeneous work force. This motivates our examination of how enogenous worker heterogeneity a ects inustrial structure, the spee of innovation, an the pattern of worker interaction. Di erentiation of agents in terms of quality (or vertical characteristics) of knowlege is stuie in Jovanovic an Rob (989) in the context of a search 3

4 moel. In contrast, our moel examines (enogenous) horizontal heterogeneity of agents an its e ect on knowlege creation, knowlege transfer, an consumption. 2 The Moel - Ieas an Knowlege In this section, we introuce the basic concepts of our moel of ieas an knowlege. An iea is represente by a box. It has a label on it that everyone can rea (the label is common knowlege in the game we shall escribe). This label escribes the contents. Each box contains an iea that is escribe by its label. Learning the actual contents of the box, as oppose to its label, takes time, so although anyone can rea the label on the box, they cannot unerstan its contents without investing time. This time is use to open the box an to unerstan fully its contents. An example is a recipe for making uon nooles as in Takamatsu. It is labelle as such, but woul take time to learn. Another example is reaing a paper in a journal. Its label or title can be unerstoo quickly, but learning the contents of the paper requires an investment of time. Prouction of a new paper, which is like opening a new box, either jointly or iniviually, also takes time. Suppose we have an in nite number of boxes, each containing a i erent piece of knowlege, which is what we call an iea. We put them in a row in an arbitrary orer. There are 2 persons, i an j, in the economy. We assume that each person has a replica of the in nite row of boxes introuce above, an that each copy of the row has the same orer. Our moel features continuous time. Fix time t 2 R + an consier person i. A box is inexe by k = ; 2; ::: Take any box k. If person i knows the iea insie that box, we put a sticker on it that says ; otherwise, we put a sticker on it that says 0. That is, let x k i (t) 2 f0; g be the sticker on box k for person i at time t. The state of knowlege, or just knowlege, of person i at time t is thus e ne to be K i (t) = (x i (t); x 2 i (t); :::) 2 f0; g. The reason we use an in nite vector of possible ieas is that we are using an in nite time horizon, an there are always new ieas that might be iscovere, even in the preparation of uon nooles. Given K i (t) with only nitely many non-zero components, there is an in nite number of ieas that coul be create in the next step. In this paper, we will treat ieas symmetrically. All that will matter is the 4

5 number an agent knows at a particular time. Extensions to iea hierarchies an knowlege structures will be iscusse in the conclusions. Given K i (t) = (x i (t); x 2 i (t); :::), n i (t) = X x k i (t) () k= represents the number of ieas known by person i at time t. Next, we will e ne the number of ieas that two persons, i an j, both know. Assume that j 6= i. De ne K j (t) = (x j(t); x 2 j(t); :::) an n c ij(t) = X x k i (t) x k j (t) (2) k= So n c ij(t) represents the number of ieas known by both persons i an j at time t. Notice that i an j are symmetric in this e nition, so n c ij(t) = n c ji(t). De ne n ij(t) = n i (t) n c ij(t) (3) to be the number of ieas known by person i but not known by person j at time t. Knowlege is a set of ieas that are possesse by a person at a particular time. However, knowlege is not a static concept. New knowlege can be prouce either iniviually or jointly, an ieas can be share with others. But all of this activity takes time. Next we escribe the components of the rest of the moel. Consier rst a moel with just two agents, i an j. At each time, each faces a ecision about whether or not to meet with the other. If both want to meet at a particular time, a meeting will occur. If either oes not want to meet, then they o not meet. If the agents o not meet at a given time, then they prouce separately an also create new knowlege separately. If the two persons o ecie to meet at a given time, then they share oler knowlege together an create new knowlege together. So consier a given time t. In orer to explain how knowlege creation, knowlege exchange, an commoity prouction work, it is useful for intuition (but not technically necessary) to view this time perio of xe length as consisting of subperios of xe length. Each iniviual is enowe with a xe amount of labor that is supplie inelastically uring the perio. In the rst subperio, iniviual prouction takes place. We shall assume constant returns to scale in physical prouction, so it is not bene cial for iniviuals 5

6 to collaborate in prouction. Each iniviual uses their labor uring the rst subperio to prouce consumption goo on their own, whether or not they are meeting. We shall assume below that although there are no increasing returns to scale in prouction, the prouctivity of a person s labor epens on their stock of knowlege. Activity in the secon subperio epens on whether or not there is a meeting. If there is no meeting, then each person spens the secon subperio creating new knowlege on their own. Eviently, the new knowlege create uring this subperio can i er between the two persons, because they are not communicating. They open i erent boxes. Since there is an in nity of i erent boxes, the probability that the two agents will open the same box (even at i erent points in time), either working by themselves or in istinct meetings, is assume to be zero. If there is a meeting, then the secon subperio is ivie into two parts. In the rst part, the two persons who are meeting spen their time (an labor) sharing ol knowlege, boxes they have opene in previous time perios that the other person has not opene. In the secon part, they create new knowlege together, so they open boxes together. We wish to emphasize that the ivision of a time perio into subperios is purely an expositional evice. Rigorously, whether or not a meeting occurs etermines how much attention is evote to the various activities at a given time. What o the agents know when they face the ecision about whether or not to meet at time t? Each person knows both K i (t) an K j (t). In other wors, each person is aware of their own knowlege an is also aware of the other s knowlege. Thus, they also know n i (t), n j (t), n c ij(t) = n c ji(t), n ij(t), an n ji(t) when they ecie whether or not to meet at time t. The notation for whether or not a meeting actually occurs at time t is: ij (t) ji (t) = if a meeting occurs an ij (t) ji (t) = 0 if no meeting occurs at time t. Meetings only occur if both persons agree that a meeting shoul take place. Next, we must specify the ynamics of the knowlege system an the objectives of the people in the moel in orer to etermine whether or not they ecie to meet at a particular time. In orer to accomplish this, it is easiest to abstract away from the notation for speci c boxes, K i (t), an to focus on the ynamics of the quantity statistics relate to knowlege, n i (t), n j (t), n c ij(t) = n c ji(t), n ij(t), an n ji(t). Since we are treating ieas symmetrically, in a sense these quantities are su cient statistics for our analysis. The simplest piece of the moel to specify is what happens if there is no meeting an the two people thus work in isolation. Let a i (t) be the rate of 6

7 creation of new ieas create by person i an let a j (t) be the rate of creation of new ieas create by j, both at time t. Let b ij (t) an b ji (t) be the rate of transfer of ieas from i to j an from j to i, respectively, at time t. 2 Then we assume that the creation of new knowlege uring isolation ( ij (t) = 0) is governe by the following equations: a i (t) = n i (t) an a j (t) = n j (t) when ij (t) = 0. (4) b ij (t) = 0 an b ji (t) = 0 when ij (t) = 0. So we assume that if there is no meeting at time t, iniviual knowlege grows at a rate proportional to the knowlege alreay acquire by an iniviual. Meanwhile, knowlege hel commonly by the two persons oes not grow. In particular, ieas are not share. If a meeting oes occur at time t ( ij (t) = ), then both knowlege exchange between the two persons an joint knowlege creation occur. When a meeting takes place, joint knowlege creation is governe by the following ynamics : 3 a ij (t) = [n c ij(t) n ij(t) n ji(t)] 3 (5) So when the two people meet, joint knowlege creation occurs at a rate proportional to the normalize prouct of their knowlege in common, the iniviual knowlege of i, an the iniviual knowlege of j. The rate of creation of new knowlege is highest when the proportions of ieas in common, ieas exclusive to person i, an ieas exclusive to person j are split evenly. Ieas in common are necessary for communication, while ieas exclusive to one person or the other imply more heterogeneity or originality in the collaboration. If one person in the collaboration oes not have exclusive ieas, there is no reason for the other person to meet an collaborate. The multiplicative nature of the function in equation (5) rives the relationship between knowlege creation an the relative proportions of ieas in common an ieas exclusive to one or the other agent. 2 In principle, all of these time-epenent quantities are positive integers. However, for simplicity we take them to be continuous (in R + ) throughout the paper. One interpretation is that the creation or sharing of an iea occurs at a stochastic time, an the real numbers are taken to be the probability of a jump in a Poisson process. The justi cation of the use of a real number instea of an integer seems to a little but complication to the analysis. 3 We may generalize equation (5) as follows: a ij (t) = max n( ") n i (t); ( ") n j (t); n c ij(t) n ij(t) n ji(t) o 3 where " > 0 represents the costs from the lack of concentration. This generalization, however, oes not change the results presente in this paper in any essential way. 7

8 Uner these circumstances, no knowlege creation in isolation occurs. During meetings at time t, knowlege transfer occurs in aition to the creation of new knowlege. Knowlege transfer is governe by the following ynamics: b ij (t) = [n ij(t) n c ij(t)] 2 (6) b ji (t) = [n ji(t) n c ij(t)] 2 So when a meeting occurs, knowlege transfer from i to j happens at a rate proportional to the normalize prouct of the number of ieas that person i has but that person j oes not have, an the ieas common to both persons. The explanation is that communication is necessary for knowlege transfer, so the two persons must have some ieas in common (n c ij(t)). But in aition, person i must have some ieas that are not alreay possesse by person j (n ij(t)). The same intuition applies to knowlege transfer in the opposite irection from j to i, represente by the secon equation in (6). The change in the number of ieas that both persons have in common ( _n c ij(t)) is the sum of knowlege transfers in both irections an the new ieas jointly create. From person i s perspective, the number of ieas that i has but j oesn t have (n ij(t)) ecreases with knowlege transfers from i to j. Finally, the change in the number of ieas possesse by person i is the sum of the ieas that are jointly create an the number of ieas transferre from j to i. The analogous statements hol for the variables associate with j. Let us focus on agent i (the equations for agent j are analogous). With a meeting, we have the following ynamics incorporating both knowlege creation an transfer: _n i (t) = a ij (t) + b ji (t) _n c ij(t) = a ij (t) + b ij (t) + b ji (t) _n ij(t) = b ij (t) Given this structure, we can e ne the rates of iea innovation an knowlege transfer at time t, epening on whether or not a meeting occurs. _n i (t) = [ ij (t)] n i (t) + ij (t) ( [n c ij(t) n ij(t) n ji(t)] 3 + [n ji (t) n c ij(t)] 2 ) _n c ij(t) = ij (t) ( [n c ij(t) n ij(t) n ji(t)] 3 + [n ji (t) n c ij(t)] 2 + [n ij(t) n c ji(t)] 2 ) _n ij(t) = [ ij (t)] n i (t) ij (t) [n ij(t) n c ji(t)] 2 8

9 Whether a meeting occurs or not, there is prouction in each perio for both persons. Felicity in that time perio is e ne to be the quantity of output. 4 De ne y i (t) to be prouction output (or felicity) for person i at time t, an e ne y j (t) to be prouction output (or felicity) of person j at time t. Normalizing the coe cient of prouction to be, we take y i (t) = n i (t) (7) so _y i (t) = _n i (t) By e nition, _y i (t) y i (t) = _n i(t) n i (t) which represents the rate of growth of income. Finally, we must e ne the rule use by each person to ecie whether they want a meeting at time t or not. Formally, (8) ij (t) = () (9) [n c ij(t) n ij(t) n ji(t)] 3 + [n ji (t) n c ij(t)] 2 > ni (t) an [n c ji(t) n ji(t) n ij(t)] 3 + [n ij (t) n c ji(t)] 2 > nj (t) To keep the moel tractable in this rst analysis, we assume a myopic rule. So a person woul like a meeting if an only if the increase in their rate of output with a meeting is higher than the increase in their rate of output without a meeting. 5 Note that we use the increase in the rate of output rather than the rate of output since in a continuous time moel, the rate of output at time t is una ecte by the ecision about whether to meet mae at time t. This completes the statement of the moel. Dropping the time epenence of variables to analyze ynamics, we obtain the following equations of motion. _y i = _n i = [ ij ] n i + (0) ij ( [n c ij n ij n ji] 3 + [n ji n c ij] 2 ) _n c ij = ij ( [n c ij n ij n ji] 3 + [n ji n c ij] 2 + [n ij n c ji] 2 ) _n ij = [ ij ] n i ij [n ij n c ji] 2 This system, with analogous equations for agent j, represents a partner ance. 4 Given that the focus of this paper is on knowlege creation an transfer rather than prouction, we use the simplest possible form for the prouction function. 5 We will see that the rule use in the case of ties is not important. 9

10 3 Equilibrium Dynamics In orer to analyze our system, we rst ivie all of our equations by the total number of ieas possesse by i an j: an e ne new variables n ij = n ij + n ji + n c ij () m c ij m c ji = nc ij n = nc ji ij n ij m ij = n ij n, ij m ji = n ji n ij From (), we obtain = m ij + m ji + m c ij (2) After some etaile calculations (see Appenix a of the technical appenix for all of the steps), we obtain _m ij an _m ji as functions of m ij an m ji only, as follows. _m ij = [ ij ] f( m ij)( m ij m ji)g (3) ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g _m ji = [ ij ] f( m ji)( m ij m ji)g ij f [m ji ( m ij m ji)] 2 + m ji [( m ij m ji) m ji m ij] 3 g To stuy this more, we must stuy (9) further. Deleting time inices an iviing by n ij, ij = () [m c ij m ij m ji] 3 + [m ji m c ij] 2 > ( m ji ) an [m c ij m ji m ij] 3 + [m ij m c ij] 2 > ( m ij ) Substituting further, ij = () [( m ji m ij) m ij m ji] 3 + [m ji ( m ji m ij)] 2 > ( m ji ) an [( m ji m ij) m ji m ij] 3 + [m ij ( m ji m ij)] 2 > ( m ij ) 0

11 In other wors, meetings occur when the rate of growth of income or utility of each person is higher with a meeting than without a meeting. De ne F i (m ij; m ji) = [( m ji m ij) m ij m ji] 3 + (4) [m ji ( m ji m ij)] 2 ( m ji) F j (m ij; m ji) = [( m ji m ij) m ji m ij] 3 + [m ij ( m ji m ij)] 2 ( m ij) an M i = f(m ij; m ji) 2 R 2 + j m ij + m ji, F i (m ij; m ji) > 0g (5) M j = f(m ij; m ji) 2 R 2 + j m ij + m ji, F j (m ij; m ji) > 0g (6) whereas M = M i \ M j The function F i (m ij; m ji) represents the net bene t for i of meeting instea of isolation. Likewise for F j (m ij; m ji). The set M i represents those pairs (m ij; m ji) such that i wants to meet with j, since for these pairs, the rate of growth of i s utility or income with a meeting is higher than the rate of growth of i s utility or income without a meeting. The set M j represents those pairs (m ij; m ji) such that j wants to meet with i. Of course, the set M represents those pairs (m ij; m ji) such that both persons want to meet with each other. Thus, meetings will occur at time t for pairs in M. We represent our moel in our Figures as a function of m ij an m ji; since m ij + m ji + m c ij =, we know that m c ij = m ij +m ji, where this inequality is represente by half of the unit square (a triangle) in R 2. We put m ij on the horizontal axis an m ji on the vertical axis, omitting m c. Figure, panels (a) an (b) illustrate the sets M i an M j, respectively, for = = an for various values of. Of course, panels (a) an (b) are mirror images of each other across the 45 line. Figure 2 illustrates M, the set of pairs where both persons want to meet, an its complement, where no meetings occur, for the same parameter values. When (m ij; m ji) is close to the bounary of the triangle, meetings o not occur. The reason is that the two persons have too little in common to interact e ectively (near the ownwar sloping iagonal) or someone has too little exclusive knowlege (near the axes) to interact e ectively. Meetings only take place in the interior where the three components of knowlege are relatively balance. FIGURES AND 2 GO HERE

12 In fact we can escribe the properties of the set M in general. The set M has the shape epicte in Figure 2; see Appenix b of the technical appenix for proof. In particular, M is roughly the shape of an apple core aligne on the 45 line. As increases, the prouctivity of creating ieas alone increases, so people are less likely to want to meet to create, implying that each M i an M j shrinks as increases, as oes M. If is a little more than, M isappears. To be precise, let M() be the set M uner the parameter value. Then, whenever < 2, the set M( 2 ) is entirely containe in M( ). Thus, as shown in Figure 2, there is a unique point B containe in every M(), provie M() is nonempty. We call B the bliss point, for the point B in Figure 2 is the point where the rate of increase in income or utility is maximize for each person, as we will explain in the next section (see also Lemma A6 in Appenix c of the technical appenix). Next we iscuss the ynamics of the system. Consier rst the case where there is no meeting, so ij = 0 is xe exogenously. Then from equations (3), the ynamics are given by the following equations: _m ij = ( m ij)( m ij m ji) _m ji = ( m ji)( m ij m ji) FIGURE 3 GOES HERE Figure 3, panel (a) illustrates the graient el assuming that ij = 0. Several facts follow quickly from these erivations. First, if there is no meeting ( ij = 0), then both _m ij an _m ji are non-negative, an positive on the interior of the triangle. So if there is no meeting, the vector el points to the northeast. Furthermore, in the lower half of the triangle where m ij m ji (the other part is symmetric), we have _m ji _m ij = m ji m ij where the inequality is strict o of the iagonal. Thus, when ij = 0, the vector el points northeast but towar the iagonal. Uner the assumption of no meeting, the system tens to sink points along the iagonal line where m ij + m ji =, illustrate in Figure 3, panel (a) by a line between (0; ) an (; 0). Figure 3, panel (b) illustrates the graient el assuming that ij =. Then (3) implies: 2

13 _m ij = [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 _m ji = [m ji ( m ij m ji)] 2 + m ji [( m ij m ji) m ji m ij] 3 (7) Both of these expressions are negative on the interior of the triangle an the vector el points southwest. Consier, for convenience, the lower half of the triangle where m ij m ji; the other part is symmetric. Then _m ji _m ij = [m ji ( m ij m ji)] 2 + m ji [( m ij m ji) m ji m ij] 3 [m ij ( m ij m ji )] 2 + m ij [( m ij m ji ) m ij m ji ] 3 where the inequality is strict o of the iagonal. Thus, the vector el points southwest but towar the iagonal, as illustrate in Figure 3, panel (b). The only sink is at (0; 0), so the system eventually moves there uner the assumption of a meeting. Next, we combine the case where there is no meeting ( ij = 0) with the case where there is a meeting ( ij = ), an let the agents choose whether or not to meet. This is illustrate in Figure 4. FIGURE 4 GOES HERE The moel follows the ynamics for meetings ( ij = ) on M an the ynamics for no meetings ( ij = 0) on the complement of M. In general, there is a continuum of stable points of the system, corresponing to the points where m ij +m ji =. For these points, eventually the myopic return to no meeting ominates the returns to meetings, since eventually the two persons have almost nothing in common. These stable points, however, are not very interesting. We have not completely speci e the ynamics. This is especially important on the bounary of M, where at least one person is ini erent between meeting an not meeting. We take an arbitrarily small unit of time, t, an assume that if at least one person becomes ini erent between meeting an not meeting, but the two persons are currently meeting, then the meeting must continue for at least t units of time. Similarly, if the two persons are not meeting when one person becomes ini erent between meeting an not meeting while the other wants to meet or is ini erent, then they cannot meet for at least t units of time. So if a person becomes ini erent between meeting 3

14 or not meeting at time t, the function ij (t) cannot change its value until time t+t. Finally, when at least one person initially happens to be on the bounary of M (that is, at least one person is ini erent between meeting an not meeting), then they cannot meet for at least t units of time. Uner this set of rules, we can be more speci c about the ynamic process near the bounary of M. In terms of ynamics, if the system oes not evolve towar the uninteresting stable points where there are no meetings (an the two people have nothing in common), eventually the system reaches the southwest bounary of the set M. From there, the assumption that ij is constant over time intervals of at least length t at the bounary of M will rive the system in a zigzag process towar the place furthest to the southwest an on the iagonal that is a member of M. In other wors, this is the point J = (m ; m ) 2 M with lowest norm. It is the remaining stable point of our moel. Small movements aroun J will continue ue to our assumption about the ynamics at the bounary of M, namely that meetings or isolation are sticky. As t! 0, the process converges to the point J. The point J features symmetry between the two agents with a large egree of homogeneity relative to the remainer of the points in M an the other points in the triangle generally. So given various initial compositions of knowlege (m ij; m ji), where will the system en up? If the initial composition of knowlege is relatively unbalance, in other wors near the bounary of the triangle, the sink will be a point on the iagonal where m ij +m ji =. If the initial composition of knowlege is relatively balance, then the sink will be the point J. Using the facts about the shape of M, the point J exists an is unique as long as M 6= ;. At the point J = (m J ; m J ), m J section. 2, for reasons explaine in the next 5 Without loss of generality, we can allow ij to take values in [0; ] rather than f0; g. The interpretation of a fractional ij is that at each instant of time, a person ivies their time between a meeting ij proportion of that instant an isolation ( ij ) proportion of that instant. All of our results concerning the moel when ij is restricte to f0; g carry over to the case where ij 2 [0; ]. The reason is that except on the bounary of M, persons strictly prefer ij 2 f0; g to fractional values of ij, as each person s objective function is linear in ij. On the bounary of M, our rule concerning ynamics prevents ij from taking on fractional values, as it must retain its value from 4

15 the previous iteration of the process for at least time t > 0. So if the process pierces the bounary from insie M, it must retain ij = for an aitional time of at least t. If it pierces the bounary from outsie M, it must retain ij = 0 for an aitional time of at least t. It may seem trivial to allow fractional ij when iscussing equilibrium behavior, but allowing fractional ij is crucial to the next section, where we consier e ciency. 4 E ciency To construct an analog of Pareto e ciency in this moel, we use a social planner who can choose whether or not people shoul meet in each time perio. As note above, we shall allow the social planner to choose values of ij in [0; ], so that persons can be require to meet for a percentage of the total time in a perio, an not meet for the remainer of the perio. To avoi epenence of our notion of e ciency on a iscount rate, we employ the following alternative concepts. The rst is stronger than the secon. A path of ij is a piecewise continuous function of time (on [0; )) taking values in [0; ]. For each path of ij, there correspons a unique time path of m ij etermine by equation (3), respecting the initial conition, an thus a unique time path of income y i (t; ij ). We say that a path 0 ij (strictly) ominates a path ij if y i (t; 0 ij) y i (t; ij ) an y j (t; 0 ij) y j (t; ij ) for all t 0 with strict inequality for at least one over a positive interval of time. As this concept is quite strong, an thus i cult to use as an e ciency criterion, it will sometimes be necessary to employ a weaker concept, which we iscuss next. We say that a path ij is overtaken by a path 0 ij if there exists a t 0 such that y i (t; 0 ij) y i (t; ij ) an y j (t; 0 ij) y j (t; ij ) for all t > t 0 with strict inequality for at least one over a positive interval of time. Two types of sink points were analyze in the last section. First consier equilibrium paths that have m J as the sink point; they reach m J in nite time an stay there. Using Figure 5, we will construct an alternative path 0 ij that ominates the equilibrium path ij. FIGURE 5 GOES HERE 5

16 In constructing this path, we will make use of income changes along the upwar sloping iagonal in Figure 4. Setting m ij = m ji = m (8) y i = y j = y we use (0) an () to obtain _y(t) y(t) = _y(t) n i (t) = _y(t) n(t)[ m(t)] m(t) = [ ij (t)] + ij (t) f [( m(t) ) ( m(t) m(t) )2 ] 3 m(t) + [( m(t) ) ( m(t) m(t) )] 2 )g (9) To simplify notation, we e ne the growth rate when the two persons meet, ij =, as Thus g(m) = [( + [( m m ) ( m m )2 ] 3 (20) m m ) ( m m )] 2 ) _y(t) y(t) = [ ij] + ij g(m) (2) Figure 5 illustrates the graph of the function g(m) as a bol line for = =. We can show 6 that g(m) is strictly quasi-concave on [0; ], achieving 2 its maximal value at m B 2 [ ; 2 ]. We can also show (see Lemma A6 of the 3 5 technical appenix) that m = m B correspons to the bliss point B in Figure 2. In other wors, whenever M 6= ;, B = (m B ; m B ) 2 M, so the point J = (m J ; m J ) e ne in Figure 4 an in the previous section has the property that m J 2=5, as it is e ne to be the point in M on the iagonal an closest to the origin. We e ne the point I = (m I ; m I ) in Figure 4 to be the point in M on the iagonal an farthest from the origin. Let t 0 be the time at which the equilibrium path reaches (m J ; m J ). Let the planner set 0 ij(t) = ij (t) for t t 0, taking the same path as the equilibrium path until t 0. From this time on, the planner uses only symmetric points, namely those on the upwar sloping iagonal in Figure 4; these points comprise the horizontal axis in Figure 5. At time t 0, the planner takes 0 ij(t) = 0 until 6 See Lemma A6 of the technical appenix. 6

17 (m I ; m I ) is attaine, prohibiting meetings so that the ancers can pro t from ieas create in isolation. Then the planner sets 0 ij(t) = until (m J ; m J ) is attaine, permitting meetings an the evelopment of more knowlege in common. The last two phases are repeate as necessary. From Figure 5, the income paths y i (t; 0 ij) an y j (t; 0 ij) generate by the path 0 ij clearly ominate the income paths y i (t; ij ) an y j (t; ij )generate by the equilibrium path ij. Thus, the equilibrium is far from the most prouctive path. Next consier equilibrium paths ij (t) that en in sink points on the ownwar sloping iagonal in Figure 4. Our ominance criterion cannot be use in this situation, since in potentially ominating plans, the planner will nee to force the couple to meet outsie of region M in Figure 4 in early time perios. During this time interval, the ancers coul o better by not meeting, an thus a comparison of the income erive from the paths woul rely on the iscount rate, something we are trying to avoi. So we will use our weaker criterion here, that of overtaking. Given an equilibrium path ij (t) with sink point on the iagonal, the planner can construct an overtaking path 0 ij(t) as follows. The rst phase is to construct a path 0 ij(t) that reaches a point in region M in nite time. Such a path can reaily be constructe using Figures 3 an 4. 7 After reaching region M, the secon an thir phases are the same as escribe above for the construction of a path that ominates one ening with m J. Since the paths with sinks on the ownwar sloping iagonal have income growth at every time, whereas the new path 0 ij(t) features income growth that excees whenever the couple is meeting, 0 ij(t) overtakes ij (t). The most prouctive state m B is characterize by less homogeneity than the stable point m J. Of course, attaining m B requires the social planner to force the two persons not to meet some of the time. Otherwise, the system evolves towar more homogeneity. 7 Such a path can be constructe as follows. In Figure 2 or Figure 4, take the union of all close, one imensional intervals parallel to the 45 line with one enpoint on an axis an the other enpoint a member of M. Call this set M 0. From time 0, take =. Using Figure 3(b), the path hits M 0 in nite time. From this time on, take = 0. Using Figure 3(a), the path hits M in nite time. 7

18 5 Conjectures an Conclusions We have consiere a moel of knowlege creation an exchange that is base on iniviual behavior, allowing myopic agents to ecie whether joint or iniviual prouction is best for them at any given time. externality moel of knowlege creation, with no markets. This is a pure In the present context of two people, there are a continuum of sink points (equilibria) for the knowlege accumulation process. Every state where the two agents have a negligible proportion of ieas in common is attainable as an equilibrium for some initial conition. There is one aitional an more interesting sink, involving a large egree of homogeneity as well as symmetry of the two agents, an this is attainable from a non-negligible set of initial conitions. Relative to the e cient state, the rst set of sink points has agents that are too heterogeneous, while the secon sink point has agents that are too homogeneous. 8 Of course, the major limitation of this work is the use of only two people. In other work, we employ more agents, but at a cost, namely the absence of knowlege transfer while limiting ourselves to symmetric states. Here we iscuss the many alternate irections for future work an extensions of the framework. Chief among these are the introuction of foresight on the part of agents, the introuction of stochastic elements into the moel, an the introuction of sie payments. extensions of the basic framework are much easier. Though the two person moel is limite, It is apparent from the analysis in section 3 that limite foresight, in the form of short sighteness instea of perfect myopia, will not be enough to overturn our results. In orer to completely overturn the results of section 2, the agents must have long range foresight. section 3. In this case, they can construct more e cient paths as in Moreover, long range foresight in combination with sie payments coul prouce tutelage when the initial state is asymmetric. When the person with more knowlege is willing to accept payment for teaching, the equilibrium paths can look very i erent from what we have propose. In the international context, where each country is often represente by one agent, our moel might be applicable. be assigne representative agents. Two countries or two regions woul In that context, knowlege creation an transfer, especially as relate to evelope an eveloping countries, woul be of interest. Analogs of the transfer paraox woul be quite fascinating. 9 8 The proximate cause is agent myopia. 9 We refer to our companion paper, Berliant an Fujita (2006), for further iscussion of 8

19 References [] Berliant, M., Fujita, M., Knowlege Creation as a Square Dance on the Hilbert Cube. Mimeo. [2] Fujita, M. an S. Weber, Strategic immigration policies an welfare in heterogeneous countries. Discussion Paper No. 569, Kyoto Institute of Economic Research, Kyoto University. [3] Jones, L., Manuelli, R., 990. A convex moel of equilibrium growth: Theory an policy implications. Journal of Political Economy 98, [4] Jovanovic, B. an R. Rob, 989. The growth an i usion of knowlege. The Review of Economic Stuies 56, [5] Lucas, R. E., Jr., 988. On the mechanics of economic evelopment. Journal of Monetary Economics 22, [6] Romer, P., 986. Increasing returns an long-run growth. Journal of Political Economy 94, [7] Romer, P., 990. Enogenous technological change. Journal of Political Economy 98, S7 S02. [8] Shell, K., 966. Towar a theory of inventive activity an capital accumulation. American Economic Review 6, extensions. 9

20 ( a) ( b) = 0.2 = =.2 =.0 = 0.8 = 0.6 = 0.4 = = 0.6 = 0.8 =.0 =.2 Figure : The sets M i an M j uner various values of = = 0.8 = 0.6 = 0.4 = Figure 2: The set M uner various values of, an the bliss point B. 20

21 (a) = 0 (b) = 0 Figure 3: Dynamics uner a xe value of. 0 M I J 0 Figure 4: Dynamics with enogenous. 2

22 Figure 5: E ciency an the bliss point 6 Technical Appenix 6. Appenix a Theorem A: Knowlege ynamics evolve accoring to the system: _m ij = [ ij ] f( m ij)( m ij m ji)g ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g _m ji = [ ji ] f( m ji)( m ij m ji)g ji f [m ji ( m ij m ji)] 2 + m ji [( m ij m ji) m ji m ij] 3 g Proof of Theorem A: Recalling (3), m ij + m c = m ji 22

23 an iviing by n ij yiels _y i n ij = _n i n ij = [ ij] ( m ji) + ij ( [m c m ij m ji] 3 + [m ji m c ] 2 ) _n c ij n ij = ij ( [m c m ij m ji] 3 + [m ji m c ] 2 + [m ij m c ] 2 ) _n ij n ij = [ ij ] ( m ji) ij [m ij m c ] 2 Substituting (2) for m c, _y i n ij = _n i n ij = [ ij] ( m ji) + ij ( [( m ij m ji) m ij m ji] 3 + [m ji ( m ij m ji)] 2 ) _n c ij n ij = ij ( [( m ij m ji) m ij m ji] 3 + [m ji ( m ij m ji)] 2 + [m ij ( m ij m ji)] 2 ) _n ij n ij = [ ij ] ( m ji) ij [m ij ( m ij m ji)] 2 23

24 Now _m ij = (n ij=n ij ) t = _n ij n ij _n ij n ij (n ij ) 2 = _n ij n ij n ij _nij nij n ij = [ ij ] ( m ji) ij [m ij ( m ij m ji)] 2 m ij ( _n ij n + _n ji n + _nc ij n ) = [ ij ] ( m ji) ij [m ij ( m ij m ji)] 2 m ij f[ ij ] ( m ji) ij [m ij ( m ij m ji)] 2 + [ ij ] ( m ij) ij [m ji ( m ij m ji)] 2 + ij ( [( m ij m ji) m ij m ji] 3 + [m ji ( m ij m ji)] 2 + [m ij ( m ij m ji)] 2 )g = [ ij ] ( m ji) ij [m ij ( m ij m ji)] 2 m ij f[ ij ] ( m ji) + [ ij ] ( m ij) + ij [( m ij m ji) m ij m ji] 3 g = [ ij ] f m ji 2m ij + (m ij) 2 + m ij m jig ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g = [ ij ] f( m ij)( m ji) m ij + (m ij) 2 g ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g = [ ij ] f( m ij)( m ji) m ij( m ij)g ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g = [ ij ] f( m ij)( m ji m ij)g ij f [m ij ( m ij m ji)] 2 + m ij [( m ij m ji) m ij m ji] 3 g The fourth line follows from (), that implies Symmetric calculations hol for _m ji. _n ij n ij = _n ij n ij + _n ji n ij + _nc ij n ij (22) 6.2 Appenix b Theorem A2: Suppose that (m ij; m ji) 2 M. Then (m ji; m ij) 2 M an the line segment [(m ij; m ji); (m ji; m ij)] M. In particular, if M 6= ;, then it contains a point on the iagonal segment [(0; 0); (; )]. Moreover, the iagonal intersecte with M is a convex set. In fact, every line parallel to the iagonal 24

25 intersecte with M is a convex set. Finally, every point in M \ ((0; 0); (; )) has a neighborhoo containe in M. To prove Theorem A2, we procee with a sequence of lemmata. First we nee some e nitions to make notation easier. De nitions: f(m; m 0 ) = [( m m 0 ) m m 0 ] 3 h(m; m 0 ) = [( m m 0 ) m 0 ] 2 With these e nitions, the equations e ning M i (5) an M j (6) become: f(m ij; m ji) + h(m ij; m ji) ( m ji) > 0 (23) f(m ji; m ij) + h(m ji; m ij) ( m ij) > 0 (24) Lemma A: (m ij; m ji) 2 M i an m ij m ji imply (m ji; m ij) 2 M i. (m ij; m ji) 2 M j an m ij m ji imply (m ji; m ij) 2 M j. Proof of Lemma A: f(m ij; m ji) = f(m ji; m ij). h(m ji ;m ij ) = [ m ij ] h(m ij ;m ji ) m 2, ji since m ij m ji. (m ij; m ji) 2 M i implies f(m ij; m ji) + h(m ij; m ji) ( m ji) > 0. Since h(m ji; m ij) h(m ij; m ji) an m ij m ji, f(m ji; m ij) + h(m ji; m ij) ( m ij) > 0. Hence, (m ji; m ij) 2 M i. A symmetric argument works for the secon part of the lemma. Lemma A2: Suppose that m ij m ji. Then (m ij; m ji) 2 M if an only if (m ij; m ji) 2 M i. Proof of Lemma A2: It is obvious that (m ij; m ji) 2 M implies (m ij; m ji) 2 M i. So suppose that (m ij; m ji) 2 M i. Then by symmetry of the e nitions of M i an M j, (m ji; m ij) 2 M j. By Lemma A, (m ji; m ij) 2 M i. Applying symmetry of the e nitions again yiels (m ij; m ji) 2 M j. Hence (m ij; m ji) 2 M j \ M i = M. Lemma A3: Suppose that (m ij; m ji) 2 M. Then (m ji; m ij) 2 M an the line segment [(m ij; m ji); (m ji; m ij)] M. In particular, if M 6= ;, then it contains a point on the iagonal segment [(0; 0); (; )]. Proof of Lemma A3: First, if (m ij; m ji) 2 M, then (m ji; m ij) 2 M by symmetry of the e nitions of M i an M j. Now consier the line segment [(m ij; m ji); (m ji; m ij)]. In particular, consier the case m ij m ji an the line segment between (m ij; m ji) an the point (m; m) on the iagonal, [(m ij; m ji); (m; m)] [(m ij; m ji); (m ji; m ij)] (the line segment [(m; m); (m ji; m ij)] can be covere with a symmetric argument). Since for all ( bm ij; bm ji) 2 [(m ij; m ji); (m; m)], 25

26 bm ij bm ji, by Lemma A2 it su ces to show that ( bm ij; bm ji) 2 M i. We must verify the equation stating that ( bm ij; bm ji) 2 M i, namely f( bm ij; bm ji) + h( bm ij; bm ji) ( bm ji) > 0 (25) Now for all ( bm ij; bm ji) 2 [(m ij; m ji); (m; m)], there exists an x 0 with bm ij = m ij x m ji + x = bm ji, since the line segment lies below the iagonal. Now f(m ij x; m ji + x) f(m ij; m ji) = [( m ji m ij) (m ij x) (m ji + x)] 3 [( m ji m ij) (m ij) (m ji)] 3 = [( m ji m ij) (m ij) (m ji) + ( m ji m ij) x (m ij m ji x)] 3 [( m ji m ij) (m ij) (m ji)] 3 [( m ji m ij) (m ij) (m ji) + ( m ji m ij) x 2 ] 3 [( m ji m ij) (m ij) (m ji)] 3 0 h(m ij x; m ji + x) h(m ij; m ji) = [(m ji + x) ( m ji m ij)] 2 [m ji ( m ji m ij)] 2 0 Finally, Hence, ( m ji x) ( m ji) f( bm ij; bm ji) + h( bm ij; bm ji) ( bm ji) = f(m ij x; m ji + x) + h(m ij x; m ji + x) ( m ji x) f(m ij; m ji) + h(m ij; m ji) ( m ji) > 0 The last line follows because (m ij; m ji) 2 M. Lemma A4: For any constant a 2 ( ; ) the intersection of the set M an the line f(m ij; m ji) 2 R 2 + j m ij + m ji, m ji = m ij ag is a convex set. Proof of Lemma A4: Since M is symmetric with respect to the iagonal m ij = m ji, let us consier a 0. Setting m ji = m ij a in (4), e ne k(m ij) F i (m ij; m ij a) = ( + a 2m ij)m ij(m ij a) =3 + ( + a 2m ij)(m ij a) =2 ( + a m ij) Since m ji = m ij a 0 an m ij + m ji = 2m ij a, the omain of the function k is a m ij + a 2 26 where 0 a <

27 By Lemma A2, the intersection of the set M an the line m ji = m ij set of points satisfying k(m ij) > 0: a is the We show that function k(m ij) is strictly concave on (a; +a ), an thus the set 2 of points satisfying the inequality is convex. Di erentiation of the function k yiels where A(m ij) 3 where k 0 (m ij) = A(m ij) + B(m ij) + ( + a 2m ij )m ij(m ij a) 2=3 6(m ij ) 2 + 2m ij( + 3a) a( + a) B(m ij) ( + a 2m 2 ij )(m ij a) =2 ( + 3a 4m ij ) The secon erivative of k is k 00 (m ij) = A 0 (m ij) + B 0 (m ij) A 0 (m ij) = = 2 (m ij) 2 ( + 3a 2 ) a( + a)( + 3a)m ij + a 2 ( + a) 2 2 h m ij( + 3a 2 ) 9 ( + a 2m ij )m ij (m ij a) 5=3 i 2 + 3a 2 (+a) 2 ( a) 2 a(+a)(+3a) 2 9 ( + a 2m ij )m ij (m ij a) 5=3 ( + 3a2 ) 4 B 0 (m ij) = ( a) 2 4 ( + a 2m ij )(m ij a) 3=2 implying that k 00 (m ij) = A 0 (m ij) + B 0 (m ij) < 0 on (a; +a ), so k is strictly 2 concave on (a; +a). Thus, 2 fm ij 2 (a; +a) j 2 k(m ij) > 0g is convex, an the proof of the lemma is complete. Lemma A5: Every point in M \ ((0; 0); (; )) has a neighborhoo containe in M. Proof of Lemma A5: This follows irectly from the e nition of M; it implies that M is an open set. Theorem A2 follows irectly from the combination of all of the Lemmata in this section. 27

28 6.3 Appenix c Lemma A6: The function g(m) e ne by (20) has the following properties: (i) g(m) is strictly quasi-concave on 0; 2. (ii) g(m) achieves its maximal value at m B 2 [ 3 ; 2 5 ]. (iii) The point (m B ; m B ) correspons to the bliss point B in Figure 2, which is the unique point containe in every M that is nonempty. Proof of Lemma A6: (i) an (ii): For m 2 0; 2, let an e ne x(m) m m or m(x) = x + x G(x) ( x)x 2 =3 + [( x)x] =2 for x 2 [0; ] (26) Then, using e nition (20) g(m) = G(x(m)) Hence, Notice that so Now where g 0 (m) = G 0 (x(m)) x 0 (m) x 0 (m) = + m ( m) 2 > 0 g 0 (m) R 0 exactly as G 0 (x(m)) R 0. G 0 (x) = C(x) + D(x) C(x) 3 [( x)x2 ] 2=3 (2 3x)x D(x) 2 [( x)x] =2 ( 2x) Taking the erivatives of C an D respectively yiels C 0 (x) = 2 ( 9 x) 5=3 x 4=3 < 0 D 0 (x) = ( 4 x) 3=2 x 3=2 < 0 Therefore, consiering that C(x) R 0 as x Q 2=3 D(x) R 0 as x Q =2 28

29 we can conclue that there exists a unique x 2 [=2; 2=3] such that G 0 (x) R 0 as x Q x meaning that G is strictly single peake an strictly quasi-concave, achieving its maximum value exactly at x. Hence, the function g(m) also is strictly single peake an strictly quasi-concave, achieving its maximum value at m B m(x ) = x 2 [=3; 2=5] + x (iii) To show that the point (m B ; m B ) correspons to the bliss point B in Figure 2, let us recall how the bliss point has been e ne. Let M() be the set M uner the parameter value > 0. Then, a point ( m ij; m ji) 2 R 2 is calle a bliss point if it hols that for any > 0, M () 6= ; =) ( m ij; m ji) 2 M () (27) To show the existence an the uniqueness of such a point, since M() is symmetric to the iagonal, let us focus on the lower half of M(), an e ne M L () = (m ij; m ji) 2 M() j m ij m ji Then, by Lemma A2, M L () coincies with the lower part of M i associate with : M L () = (m ij; m ji) 2 M i () j m ij m ji = f(m ij; m ji) 2 R 2 j m ij m ji 0; m ij + m ji ; f(m ij; m ji) + h(m ij; m ji) ( m ji) > 0g When m ij + m ji = or m ji = 0, we have f(m ij; m ji) = h(m ij; m ji) = 0, implying that M L () oes not contain any point (m ij; m ji) such that m ij + m ji = or m ji = 0. Thus, we can rewrite M L () as follows: M L () = n (m ij; m ji) 2 R 2 j m ij m ji > 0; m ij + m ji < ; f(m ij ;m ji ) + h(m ij ;m ji ) m ji m ji = f(m ij; m ji) 2 R 2 j m ij m ji > 0; m ij + m ji < ; h m ij m ji m ij m ji m ji m ji i =3 + h m ij m ji Given any (m ij; m ji) 2 M L () such that m ij > m ji, e ne m ji m ji i =2 > g (28) o > m m ij + m ji 2 29