# The Role of Externalities in Social and Economic Networks

Möhlmeier P (2017)

Bielefeld: Universität Bielefeld.

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*Bielefeld Dissertation*|

*English*

Author

Supervisor

Dawid, Herbert

^{UniBi}; Rusinowska, AgnieszkaDepartment

Abstract

Throughout this thesis, we are going to introduce various (specific and generalized) models that capture positive and negative externalities in social and economics networks. Our main contribution is to indicate the structural implications induced hereby. We are going to derive results about pairwise stability, asymptotic pairwise stability and strong efficiency. Additionally, we draw specific conclusions depending on the underlying framework and compare our results to the related literature.

The thesis consists of 5 chapters.

Chapter 1 provides a short introduction.

In chapter 2 we develop a modification of the connections model by Jackson and Wolinsky (1996) that takes into account negative externalities arising from the connectivity of direct and indirect neighbors, thus combining aspects of the connections model and the co-author model. We consider a general functional form for agents’ utility that incorporates both the effects of distance and of neighbors’ degree. Consider a situation in which people are involved in projects. They generate some kind of knowledge by themselves and receive some from others. If an agent is involved in many projects, he will have less time to generate output by himself. However, the more connections he has, the more knowledge he will receive from neighbors, neighbors of neighbors and so on. Consequently, we introduce a framework that can be seen as a degree-distance-based connections model with both, negative and positive, externalities. Our analysis shows how the introduction of negative externalities changes certain results on stability and efficiency in comparison to the original connections model. In particular, we see the emergence of new stable structures, such as a star with links between peripheral nodes. Our analysis focuses mainly on structures with short diameters, but also considers cases with extreme levels of decay. We also identify structures, for example, certain disconnected networks that are efficient in our model, but which cannot be efficient in the original connections model. While our results are proved for the general utility function, some of them are illustrated by using a specific functional form of the degree-distance-based utility.

In chapter 3 we deal with network formation frameworks, where payoffs reflect an agent’s ability to access information from direct and indirect contacts. We integrate negative externalities due to connectivity associated with two types of effects: competition for the access to information, and rivalrous use of information. We consider two separate models to capture the first and the second situation, respectively. In the first model we assume that information is a non-rivalrous good, but that there is competition for the access to information, for example because an agent with many contacts must share his time between them and thus has fewer opportunities to pass on information to each particular contact. In the second model we do not assume that there is competition for the access to information, but rather that the use of information is rivalrous. In this case, it is assumed that when people are closer to the sender than an agent, the harmful effect is greater than when others are at the same distance to the sender as that agent. In both models we analyze pairwise stability and examine if the stability of a structure is preserved when the number of agents becomes very large. This leads to a new concept that we call asymptotic pairwise stability. We show that there exists a tension between asymptotic pairwise stability and efficiency. The results allow us to compare and contrast the effects of two kinds of competition for information.

While in chapter 2 the connections model by Jackson and Wolinsky (1996) is modified to a degree-distance-based variation, in chapter 4 we present another modification of the connections model that is closely related and takes account of negative externalities by overall connectivity. The idea for this approach goes back to Jackson and Wolinsky (1996) who mention in their seminal paper that “... one might have a decreasing value for each connection (direct or indirect) as the total amount of connectedness increases.” (p. 53.). Taking this as a starting point, we add a weighting factor depending on overall connectivity to the functional form of the original connections model. This weighting factor is independent of own links, but benefits received from direct and indirect connections are reduced by increasing overall connectivity of the other nodes in the network. In this context, we solve for pairwise stable and asymptotically pairwise stable networks and analyze strongly efficient networks. We compare the results and indicate the similarities and differences of the connections model with purely positive link externalities and the adjusted version with negative externalities by overall connectivity. What appears to be striking in the overall connectivity model is the role of the star network. It turns out to be pairwise stable, asymptotically pairwise stable and to be a very well performing structure in terms of strong efficiency. The reason for this is that it combines short distances between all nodes with a minimal number of links. Hence, all nodes receive many spillovers with low decay and relatively low punishment by overall connectivity.

Chapter 5 is more applied and we investigate a duopoly with horizontal product differentiation, in which firms strategically form costly links to customers. Such a link to a customer may be interpreted as the firm granting access to trade its product. Altering the network of links changes the structure of competition. This results in externalities and influences the equilibrium quantities and profits. We investigate in how far the degree of substitutability of the firms’ products and the costs of link formation influence equilibrium profits and thus the incentives to form or delete links. We illustrate which networks are locally and Nash stable for which regions of costs/substitutability combinations. For networks with an arbitrary number of customers we analyze local stability regions for selected networks and determine their limits as the number of customers becomes large. We also relate local and Nash stability for selected networks with n customers. For networks with three customers we entirely characterize locally stable networks. In particular, existence is guaranteed for any degree of substitutability and any cost value.

The thesis consists of 5 chapters.

Chapter 1 provides a short introduction.

In chapter 2 we develop a modification of the connections model by Jackson and Wolinsky (1996) that takes into account negative externalities arising from the connectivity of direct and indirect neighbors, thus combining aspects of the connections model and the co-author model. We consider a general functional form for agents’ utility that incorporates both the effects of distance and of neighbors’ degree. Consider a situation in which people are involved in projects. They generate some kind of knowledge by themselves and receive some from others. If an agent is involved in many projects, he will have less time to generate output by himself. However, the more connections he has, the more knowledge he will receive from neighbors, neighbors of neighbors and so on. Consequently, we introduce a framework that can be seen as a degree-distance-based connections model with both, negative and positive, externalities. Our analysis shows how the introduction of negative externalities changes certain results on stability and efficiency in comparison to the original connections model. In particular, we see the emergence of new stable structures, such as a star with links between peripheral nodes. Our analysis focuses mainly on structures with short diameters, but also considers cases with extreme levels of decay. We also identify structures, for example, certain disconnected networks that are efficient in our model, but which cannot be efficient in the original connections model. While our results are proved for the general utility function, some of them are illustrated by using a specific functional form of the degree-distance-based utility.

In chapter 3 we deal with network formation frameworks, where payoffs reflect an agent’s ability to access information from direct and indirect contacts. We integrate negative externalities due to connectivity associated with two types of effects: competition for the access to information, and rivalrous use of information. We consider two separate models to capture the first and the second situation, respectively. In the first model we assume that information is a non-rivalrous good, but that there is competition for the access to information, for example because an agent with many contacts must share his time between them and thus has fewer opportunities to pass on information to each particular contact. In the second model we do not assume that there is competition for the access to information, but rather that the use of information is rivalrous. In this case, it is assumed that when people are closer to the sender than an agent, the harmful effect is greater than when others are at the same distance to the sender as that agent. In both models we analyze pairwise stability and examine if the stability of a structure is preserved when the number of agents becomes very large. This leads to a new concept that we call asymptotic pairwise stability. We show that there exists a tension between asymptotic pairwise stability and efficiency. The results allow us to compare and contrast the effects of two kinds of competition for information.

While in chapter 2 the connections model by Jackson and Wolinsky (1996) is modified to a degree-distance-based variation, in chapter 4 we present another modification of the connections model that is closely related and takes account of negative externalities by overall connectivity. The idea for this approach goes back to Jackson and Wolinsky (1996) who mention in their seminal paper that “... one might have a decreasing value for each connection (direct or indirect) as the total amount of connectedness increases.” (p. 53.). Taking this as a starting point, we add a weighting factor depending on overall connectivity to the functional form of the original connections model. This weighting factor is independent of own links, but benefits received from direct and indirect connections are reduced by increasing overall connectivity of the other nodes in the network. In this context, we solve for pairwise stable and asymptotically pairwise stable networks and analyze strongly efficient networks. We compare the results and indicate the similarities and differences of the connections model with purely positive link externalities and the adjusted version with negative externalities by overall connectivity. What appears to be striking in the overall connectivity model is the role of the star network. It turns out to be pairwise stable, asymptotically pairwise stable and to be a very well performing structure in terms of strong efficiency. The reason for this is that it combines short distances between all nodes with a minimal number of links. Hence, all nodes receive many spillovers with low decay and relatively low punishment by overall connectivity.

Chapter 5 is more applied and we investigate a duopoly with horizontal product differentiation, in which firms strategically form costly links to customers. Such a link to a customer may be interpreted as the firm granting access to trade its product. Altering the network of links changes the structure of competition. This results in externalities and influences the equilibrium quantities and profits. We investigate in how far the degree of substitutability of the firms’ products and the costs of link formation influence equilibrium profits and thus the incentives to form or delete links. We illustrate which networks are locally and Nash stable for which regions of costs/substitutability combinations. For networks with an arbitrary number of customers we analyze local stability regions for selected networks and determine their limits as the number of customers becomes large. We also relate local and Nash stability for selected networks with n customers. For networks with three customers we entirely characterize locally stable networks. In particular, existence is guaranteed for any degree of substitutability and any cost value.

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### Cite this

Möhlmeier P.

*The Role of Externalities in Social and Economic Networks*. Bielefeld: Universität Bielefeld; 2017.Möhlmeier, P. (2017).

*The Role of Externalities in Social and Economic Networks*. Bielefeld: Universität Bielefeld.Möhlmeier, P. (2017). The Role of Externalities in Social and Economic Networks. Bielefeld: Universität Bielefeld.

Möhlmeier, P., 2017.

*The Role of Externalities in Social and Economic Networks*, Bielefeld: Universität Bielefeld. P. Möhlmeier,

*The Role of Externalities in Social and Economic Networks*, Bielefeld: Universität Bielefeld, 2017. Möhlmeier, P.: The Role of Externalities in Social and Economic Networks. Universität Bielefeld, Bielefeld (2017).

Möhlmeier, Philipp.

*The Role of Externalities in Social and Economic Networks*. Bielefeld: Universität Bielefeld, 2017.**All files available under the following license(s):**

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