Essays on economics networks

This thesis belongs to the growing field of economic networks. In particular, we develop three essays in which we study the problem of bargaining, discrete choice representation, and pricing in the context of networked markets. Despite analyzing very different problems, the three essays share the common feature of making use of a network representation to describe the market of interest.

In Chapter 1 we present an analysis of bargaining in networked markets. We make two contributions. First, we characterize market equilibria in a bargaining model, and find that players' equilibrium payoffs coincide with their degree of centrality in the network, as measured by Bonacich's centrality measure. This characterization allows us to map, in a simple way, network structures into market equilibrium outcomes, so that payoffs dispersion in networked markets is driven by players' network positions. Second, we show that the market equilibrium for our model converges to the so called eigenvector centrality measure. We show that the economic condition for reaching convergence is that the players' discount factor goes to one. In particular, we show how the discount factor, the matching technology, and the network structure interact in a very particular way in order to see the eigenvector centrality as the limiting case of our market equilibrium.

We point out that the eigenvector approach is a way of finding the most central or relevant players in terms of the “global” structure of the network, and to pay less attention to patterns that are more “local”. Mathematically, the eigenvector centrality captures the relevance of players in the bargaining process, using the eigenvector associated to the largest eigenvalue of the adjacency matrix of a given network. Thus our result may be viewed as an economic justification of the eigenvector approach in the context of bargaining in networked markets.

As an application, we analyze the special case of seller-buyer networks, showing how our framework may be useful for analyzing price dispersion as a function of sellers and buyers' network positions.

Finally, in Chapter 3 we study the problem of price competition and free entry in networked markets subject to congestion effects. In many environments, such as communication networks in which network flows are allocated, or transportation networks in which traffic is directed through the underlying road architecture, congestion plays an important role. In particular, we consider a network with multiple origins and a common destination node, where each link is owned by a firm that sets prices in order to maximize profits, whereas users want to minimize the total cost they face, which is given by the congestion cost plus the prices set by firms. In this environment, we introduce the notion of Markovian traffic equilibrium to establish the existence and uniqueness of a pure strategy price equilibrium, without assuming that the demand functions are concave nor imposing particular functional forms for the latency functions. We derive explicit conditions to guarantee existence and uniqueness of equilibria. Given this existence and uniqueness result, we apply our framework to study entry decisions and welfare, and establish that in congested markets with free entry, the number of firms exceeds the social optimum.

Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.