When individuals' types of knowledge are too diverse, a match is less likely to generate significant innovations. What are the consequences of knowledge spillovers on the promotion and concentration of economic activity. We believe that heterogeneity (in terms of different types of knowledge) plays a role in the acquisition and promotion of knowledge.
First, it is valuable in the endogenous growth context as we provide explanations behind the determinants of knowledge exchange. We demonstrate that economies with a higher population size will have more specialized patterns of knowledge exchange and higher welfare.5. This enables us to identify the effects of the level of agglomeration on the economy's endogenous patterns of knowledge exchange.
C. Production and Tastes
It implies that meeting attendance rates are higher in economies with higher population density.
D. Matching
Encounters occur between any two agents with flow probability "(U), but only a subset of encounters result in matches. Thus, the flow probability of a match is given by the flow probability of an encounter multiplied by the proportion of unmatched agents selected for knowledge exchange The selection strategy that determines which agents are accepted for matches will depend both on the effectiveness of knowledge exchange and on primitives of the economic environment, such as the ability of individuals to meet in the economy.
For example, as it becomes easier for unmatched agents to meet, individuals are expected to be more selective in the set of agents they accept for knowledge exchange.
E. Asset Values
We could endogenize the separation rate following Jovanovic (1979) if the productivity of each match is not known in advance and agents update their beliefs about the productivity of a match over time. This extension is not likely to add further insight to the basic questions we are studying. Until the breach occurs, agents are assumed to exchange information and produce.
In contrast, with probability (1- 0)), agents will remain matched and thus have an expected discounted lifetime utility of VMt+) (k,k';U) from period (t+). This implies that the flow value of matches is the sum of the flow output produced based on new. It is important to note that the selection strategy will be chosen by recognizing the trade-off between a higher number of matches (a contact rate effect) and more effective matches (a knowledge efficiency effect) because.
F. Steady-State Populations
Steady-State Equilibrium
A non-degenerate, symmetric, steady-state equilibrium (SSE) is a tuple {R(k)}k06,^*,U satisfying the following conditions: a tuple {R(k)}k06,^*,U, that meet the following conditions: Based on the model's parameters, such as matching rates, agents determine the maximum distance. By deriving the range of individuals that agents match, we can determine the degree of diversity of knowledge exchange in this economy in steady-state equilibrium.
Once equilibrium knowledge spreads, the steady-state population of matched and unmatched^* is determined. In establishing the existence of a stable equilibrium for the economy, we seek to understand how the pattern of information flows, as reflected in knowledge diffusion, responds to the degree of agglomeration, as measured by the exogenous population size in the basic model. Proposition 1: (Effect of the size of the agglomeration on the knowledge exchange pattern) Economies with a larger population mass have a smaller equilibrium knowledge distribution.
It can be noted that as knowledge itself becomes more specialized, knowledge exchange with agents in other fields becomes less effective. 17See chapter 6 of Fujita (1989) for details of the conventional model that leads to cities that are overpopulated in equilibrium relative to the social optimum. In the presence of either a fixed setup cost [Abdel-Rahman (1990)] or a free-rider effect [Palivos and Wang (1997)], the equilibrium city size may be too small.
A higher level of technology increases the productivity of each match, but also increases the cost of greater knowledge diffusion because agents lose the output that would be gained by more efficient knowledge sharing with other agents. The remaining details regarding the comparative statics in the closed city model can be found in the appendix.
Endogenous Migration
There are some models that generate cities that are underpopulated in equilibrium relative to the optimum. In relation to migration decisions, individuals must take into account the costs associated with living in the city under consideration. This cost (which is claimed to depend positively on N) can be seen, for example, as the (present discounted) value of city taxes levied on each immigrant.
A higher N may be associated with congestion costs or destruction of existing structures, thus requiring the government. For comparative statics exercises, we consider any increase in costs associated with staying in the city as an increase in 'access costs' and represent this as a shift in the access cost function. The horizontal axis of Figure 4 represents different values of the population mass, N, and the vertical axis provides the different values for an unmatched agent's expected lifetime utility unmatched and entry cost for each value of N.
In our analysis, the agent's unbalanced value function is expressed by N by substituting U from the steady state population condition (7). From the equilibrium entry condition, we can obtain the locus * and N, where individuals are indifferent between migrating and not migrating.
A. Steady-State Equilibrium with Endogenous Migration
Proposition 2: (Interactions between the pattern of knowledge exchange and the degree of agglomeration) Under endogenous migration, any factor that promotes knowledge exchange will also promote a larger population mass. As a result, agents may choose to be more selective and focus on finding more effective knowledge sharing opportunities. This immediately implies that there are more gains to be made by migrating, as there is less delay between meetings, resulting in a larger population mass.
Because of the beneficial aspects of population density for matching, agents in turn decide to become more specialized in their matching. Similar results emerge regarding the effect of different parameter values on the effectiveness of knowledge sharing. According to our closed economy analysis, a higher heterogeneity penalty (a1) induces agents to become more selective in compliance.
Due to the negative effects of the penalty on migration incentives, fewer agents will decide to migrate. An appendix contains a proof and further details of comparative statics for the open city model. Second, we consider the case where a city planner chooses to maximize the net welfare of only the individuals living in the city; the city does not yet exist when the planner solves the optimization problem.
One can also consider the case where the city exists and the social planner maximizes the well-being of current residents and potential immigrants. This means taking into account redistributive issues that detract from our main interest: the interactions between the social inefficiencies arising from knowledge exchange and the external congestion.
B. Socially Optimal Knowledge Spread and Population Mass
Thus, the social planner chooses * and N simultaneously to maximize the net welfare of a representative resident. Theorem 3: (Social inefficiency) Compared to the social optimum, a decentralized equilibrium SSEEM can be underpopulated and underspecialized in knowledge exchange or overpopulated and overspecialized. By not considering their effect on the overall population mass, the city may be overpopulated relative to the social optimum.
Note that the first term in (9) corresponds to the individual's choice of knowledge distribution in a decentralized equilibrium. However, the social planner takes into account that greater knowledge distribution reduces the mass of unmatched actors in the economy (what we call the matching externality effect). Denote VU(*^; N) as the expected lifetime utility of an unmatched agent, given the personal choice of knowledge distribution.
In addition, let VSP(**, N) denote the noncomparable expected lifetime utility of the agent following the planner's choice of knowledge dissemination. We first consider the case where the matching externality is minimal, so that the planner's knowledge spillover choice is not too much less than in equilibrium. Since the matching externality is not too strong in this case, the agent's VU function will not be much lower than the lifetime utility that would occur (for any value of N) following the planner's knowledge spillover (VSP) choice.
For the social optimum, however, the population level N^ is found where the slope of the incomparable value function with respect to N has the same slope as the input cost function, N*. In contrast, when the matching externality is strong, the equilibrium value of knowledge spillover is small relative to the optimum.
Concluding Remarks
Thus, according to Assumption 1, a break in the value function occurs in the region where the unleveled value of the agent is negative and is not important for determining the spread of individual knowledge. Differentiating VU(*;U) with respect to * provides a first-order condition for obtaining equilibrium knowledge diffusion. It is easy to see that the locus of knowledge diffusion is a downward sloping curve as shown in Figure 4.
Recall that the equilibrium input locus is a curve of values of knowledge distribution and population mass where VU(*;U) = <(N). Consider the first-order condition for knowledge diffusion and rewrite it in terms of VU(*;U). We prove the existence of the equilibrium knowledge spread in the steady state using the following argument for the mean value.
This causes individuals to try to match with more types of agents and increases the equilibrium knowledge distribution. Because each match produces better ideas (q0 higher), individuals choose to meet more types of agents because the marginal benefit of diverse knowledge distribution is greater. This increases the costs of remaining in relatively less productive interactions, decreasing equilibrium knowledge distribution.
As in Section III, we analyze the determinants of knowledge diffusion when population mass is given exogenously. Although the knowledge creation hypothesis is different from the one pursued in Section III, the basic determinants of knowledge diffusion in the economy remain the same. This example does not allow for a closed solution for the relationship between the spread of individual knowledge and the mass of the population.
Our analysis seems to suggest that for a given mass of unmatched means, a higher value of is associated with a smaller knowledge spillover.
