Knowledge produced by an agent at one point in time becomes part of that agent's knowledge stock in the future. -employees in the same group want to change partners within the group as often as possible. In the first subperiod, public knowledge generated by pairs of employees in the same region is studied.
In the second subperiod, public knowledge created by pairs of workers from different regions is studied by knowledge workers. It is natural to assume that public knowledge is better transmitted to workers in the same region where it was created.
One Region
The equilibrium path of K worker interactions and the sink point of the knowledge creation process depends on the initial state, md(0). The equilibrium path remains to the left of the bliss point, so the bliss point is never reached. Other initial conditions for the system are possible, but we refer to Berliant and Fujita for detailed investigation of the other cases.
Since we wish to examine how the knowledge creation system responds to the introduction of interaction with another region, our focus in the remainder of the paper is on case (b). Both cases specify initial heterogeneity to the left of the happiness point, so that K-workers are highly homogeneous relative to maximum productivity at the happiness point. Thus, even though workers start relatively homogeneous, they can differentiate themselves from others by working with everyone else and eventually reach relative heterogeneity and maximum happiness point productivity without any interference.
Although a K-worker works with everyone else, the state measured to the left of the happiness point, that is, with more worker homogeneity and lower than optimal productivity, is the steady state. The sink point is to the left of the happiness point, so the happiness point is never reached. In the case of Figure 1, for example, the sink point is far to the left of the happiness point; thus, g(maut) is much lower than g(mB).
This suggests that if the division of the population into two regions results in greater heterogeneity of the composition of knowledge in each region, then the rate of growth of knowledge of the economy will increase; we will discuss this in detail in the next sub-section.
Di¤erentiation Between Two Regions
Definition: A stationary state in the two-region system is called welfare improvement when the rate of growth of the associated knowledge for each individual is higher than that in the initial one-region state. By definition, no point (mB; mS) on the supreme iso-N curve can be reached as a steady-state myopic center point. In other words, when there is no discount to the interregional transfer of public knowledge (i.e., e = 1), or when there is no within-group externality for interregional interactions in knowledge creation (i.e., C= 0), it is impossible to reach a New Eden.
Intuitively, low means that the discount in productivity for working with someone in the other region compared to working with someone in the home region is large. A value close to 1 means that there is not much difference in productivity or cost of working with someone in the other region compared to the home region. This graph shows that as one goes from 0 to 1, for a given person in region A, the knowledge gap between that person and potential partners in the home and the other region becomes close.
Obviously, when = 1, there is no difference between partners in the home and remote regions, so mS =mB and this is represented by the upward sloping 45 line. When 2(0;1) varies parametrically, the isocurves defined by (64) and (65) have the following characteristics: i) All isocurves pass through the origin. ii) Each iso-curve is strictly convex and tangent to the vertical line at mB = 12; at the border. iii) Except at the origin, the isocurve shifts continuously upwards as it increases from 0 to 1; at the border, it coincides with the upward sloping diagonal line. To answer this question, note that from equation (51) for the one-region situation and from equation (52) for the two-region situation, the potential improvement in productivity K is completely determined by the magnitude of g(maut) for a case of the region with respect to the size of g(mS) = g(mB) in the case of two regions.
Referring to Figure 7 , for each point (mB; mS) in Figure 7 , we examine the ratio of knowledge productivity in New Eden compared to autarky.
The Transition Process
Given that N is large and that people only work with partners in the same region, they will work with everyone else in the region for a short period of time, the same for each partnership. The regions continue in an autarkic state until the regions are sufficiently partitioned, with equal productivity for potential partners in their own region and in the other region. At this point, a person in the region begins to participate in an interregional working group, as described in the previous subsection for …nal steady state, and with people in their own region who are not in their interregional working group, also as described in the previous subsection for ...nal steady state.
However, the size of the interregional working group, and the total time spent working with partners in the home region, will not be the same as in the stable situation. When they finally reach the happiness point for their partners from the other region in their interregional working group, they will switch to N and 'that will maintain the happiness point. We have tried to clarify a second role of spatial distance in economics, in addition to the first and obvious role, which is to create a barrier to the exchange of goods between locations.
This second role is the propagation of the differentiation of the agents themselves, in the sense that they form separate cultures. It can result in an increase in knowledge productivity throughout the economy compared to the situation where there is no spatial distance between the agents. The key to this increase is in the ability of interregional working groups to form and further differentiate agents residing in the same region due to knowledge spillovers within the interregional working group.
Faster knowledge transfer as a result of improved information technology obviously makes the dissemination of new ideas faster, but it also tends to create more homogeneity in the knowledge base of researchers.
Justi…cation of the Knowledge Absorption Function
When a person works with a partner from another region, they absorb a different part of all the new ideas created at that time through interregional interactions. At the same time, when a person works with others in their interregional workgroups, they absorb part of the new ideas created at that time within that workgroup. For all but the interregional workgroup externality, the knowledge absorbed by the externality is knowledge common to the workers in the same region.
Ideas learned through the interregional working group externality become knowledge in common for only the members of the specific working group. That is, these ideas are not learned by persons in a region who are not members of the same workgroup, and thus they become part of the distinguishing knowledge between members of that interregional workgroup and everyone else. Equation (70) represents the analogous equation for inter-regional partnerships in the second sub-period.
Equation (71) represents the analogous equation for partnerships from the person's interregional work group in the second subperiod. The term in parentheses on the left side represents new knowledge generated by K-worker i in cross-regional working group partnerships at an instant. The equation states that the rate of absorption of public knowledge by a person's cross-regional work group is proportional to their capacity to produce new ideas with work group partners from the other region. 22.
For equation (72), the attention is divided into the two sources of interregional externalities, namely the attention to the externality Cb from general interregional partners and the attention to the externality !C from partners in one's interregional working group.
Basic Dynamics Without Symmetry
Basic Dynamics Under Symmetry
To prove Theorem 2, we find the steady state of the form given in Theorem 2 that is consistent with the maximization of individual income. Since n_i from (105) is independent of person i's choice variables, Fi is the only term left from the expression fory_i that depends on person i's choice variables at the time they are chosen. Note that in each of these three cases, since we have considered a representative individual i 2 A, if 'i is a solution to the maximization problem, then the definition of the myopic core implies so.
As shown in Section 8.4.3 of the Technical Appendix, we can easily see that in cases (i) and (ii), condition (114) leads to a contradiction of the assumption about the sign of the term in the square brackets in (113) given from the special case, or in a contradiction of the definition of a steady state. By the following reasoning, it must also be the case that ' >0 in the steady state. The other relationship for determining ' and N' simultaneously can be obtained from another steady state, (92), as follows.
Having defined all the endogenous variables (as functions of the exogenous variables) in the New Eden, we now proceed to show that the New Eden is essentially myopic. In general, the path of the myopic core will depend on the initial conditions, but here we focus on the steady state in New Eden. Of course, we can …x some time and examine the payments to agents at that time since agents are myopic.
Much of the work in this subsection has been to show that, starting from the state of New Eden, if a person can choosef ijg2Nj=1 where ji. This immediately implies that no coalition of one or two persons can do better than the New Eden at some time t, asyi is independent of the choice variables of persons and the selection of f ijg2Nj=1 to maximize y_i is optimal for any personi.