بهره برداری انحصار چند جانبه از یک دارایی تولیدی مالکیت خصوصی

In this paper, we build a Closed-Loop Nash Equilibrium of a private property productive asset oligopoly. We compare and contrast private with common property in terms of exploitation rates and social welfare, and provide a comparative dynamic analysis with respect to the number of firms in the industry. Contrary to previous studies on oligopolistic exploitation of productive assets, before exploitation begins, the resource is parcelled out: each firm privately owns and manages the assigned parcel over the entire planning horizon. Compared with the common property regime, we find a new set of results, both in the short- and in the long-run. As for social welfare, we provide conditions on the implicit growth rate and the initial asset stock under which the socially optimal allocation of the resource implies a natural monopoly.

This paper builds a Closed-Loop Nash Equilibrium (CLNE) of a private property productive asset oligopoly. We compare and contrast private with common property in terms of exploitation rates and social welfare, and provide a comparative dynamic analysis with respect to the number of firms in the industry. To the best of our knowledge, the existing literature dealing with oligopolistic exploitation of productive assets has focussed on the case of common property/open access, exclusively. Yet, in the real world, the relevance of private ownership of productive assets can hardly be questioned (e.g. 70% of land in the United States is privately owned; approximately 50% of forest and other wooded land in Europe is privately owned; approximately one-third of global fisheries production is supplied by commercial aquaculture, much of it in Asia). Indeed, property rights are essential in determining how productive assets such as land and natural resources are used and managed, and how benefits from these resources are distributed. A number of studies have analyzed the problem of the exploitation of a common pool resource in a strategic context. In this respect, two different strands can be identified in the literature. First, there are papers analyzing competition for the exploitation of the resource, ignoring the strategic interaction in the output market. These papers typically assume that agents' payoffs do not depend on rivals' exploitation rates. The extracted resource is solely used as a consumption good (Levhari and Mirman, 1980, Benhabib and Radner, 1992, Fisher and Mirman, 1992, Fisher and Mirman, 1996 and Dockner and Sorger, 1996). Second, other papers have analyzed competition in the exploitation of the resource as well as in the output market. In this case, agents’ payoffs depend also on rivals' exploitation rates. The resource is seen as a productive asset, and it is used as the only input in the production of the final good (Reinganum and Stokey, 1985, Karp, 1992, Benchekroun, 2003, Benchekroun, 2008 and Sandal and Steinshamn, 2004). Our paper relates to the stream of literature that considers the resource as a productive asset, but departs in a fundamental way by dropping the common property assumption and drawing attention to private property: the resource is parcelled out, and the benefits from each parcel accrue only to the firm having control over it. Closest in spirit and methodology to our model is the Benchekroun (2008) model. We build on Benchekroun, 2003 and Benchekroun, 2008 seminal papers by modifying the property rights assumption. Benchekroun, 2003 and Benchekroun, 2008 assumes that there exists only one common property resource whose dynamics is governed by a given equation of motion such that today's extraction rate by a firm reduces the stock of the resource available for future exploitation by all the other firms. Instead, we assume that each firm receives a parcel of the resource, and the equation of motion governing the dynamics is such that today's exploitation rate by the firm owning the parcel does not affect the stock of the resource available for rivals' future exploitation. Unlike Benchekroun, 2003 and Benchekroun, 2008, we have as many equations of motions as firms, n≥2n≥2. The presence of multiple states complicates the solution of the game, significantly. We consider a noncooperative linear-quadratic differential game in which firms' strategies are exploitation plans. Firms use the resource as an input in the production of the final good, and compete in the same output market. We assume that, before exploitation begins, the initial asset stock of the resource is split among n≥2n≥2 firms. Each firm owns and manages the assigned parcel over the entire planning horizon. The asset grows at an exponential rate until a certain threshold is reached, after which the growth of the asset starts declining. Two classical examples are represented by the growth of a fish population, and the growth of a stand of trees in a forest. We characterize the optimal exploitation plans under private property and compare and contrast them with those under common property, which have been studied in Benchekroun (2008). Next, we evaluate the short- and long-run impact of a change in the number of firms participating in the initial division of the resource on individual and aggregate production and social welfare. We focus on the steady state solutions of the game in which firms adopt feedback (closed-loop) rules. As in Benchekroun, 2003 and Benchekroun, 2008, equilibrium strategies can give rise to multiple steady state equilibria. However, while under common property the initial asset stock of the resource does not depend on the number of firms sharing access to the resource, under private property, the size of the parcel received by each firm is affected by the number of firms participating in the division of the resource. For a given initial asset stock, in a symmetric equilibrium, the higher the number of firms, the smaller the size of the parcel assigned to each firm. Therefore, in our model, changing the number of firms crucially affects the initial conditions, leading to a whole new set of results.

This paper makes three contributions to the literature on the exploitation of productive assets. First, it provides a characterization of CLNE strategies under the assumption of private property of the resource. Contrary to previous studies on the joint exploitation of a common pool resource, the initial asset stock is split among n≥2n≥2 firms, and each firm owns and manages the assigned parcel over the entire planning horizon. To the best of our knowledge, the differential game literature has focussed on common property/open access, exclusively. We compare and contrast equilibrium strategies under private property with those under common property, which have been characterized in Benchekroun (2008). We show that, under private property, firms start exploiting the resource later. In the short-run, private property welfare-dominates common property for all stocks in excess of a certain threshold. In the long run, private property welfare-dominates common property when the resource grows slowly, or when the resource grows fast and the initial asset stock is scarce. Interestingly, when the resource grows fast and the initial asset stock is abundant, property rights do not affect exploitation rates, neither in the short- nor in the long-run. Only in this case the steady state level of the asset stock is the same under the two property right regimes. Otherwise, firms are more conservationist in the exploitation of the resource under private property, thus mitigating the “tragedy of the commons”. Second, this paper evaluates the short- and long-run impact of a change in the number of firms participating in the division of the resource on per firm and industry output. Substantively, our comparative dynamic analysis is closely related to that of Benchekroun (2008), who studied the short- and long-run impact of reducing access to a common pool renewable resource. A striking difference w.r.t. common property is that, in our setting, a change in the number of firms causes a change in the initial division of the resource, and, therefore, a change in the initial conditions of the dynamic system. Furthermore, while Benchekroun (2008) focusses only on industry output, we consider both per firm and industry output, as well as social welfare. We show that when the initial asset stock is relatively scarce and the asset stock grows slowly, a decrease in the number of firms yields a short-run decrease and a long-run increase in per firm production. Short- and long-run industry output increase. When instead the initial asset stock is relatively abundant and the asset stock grows rapidly, a decrease in the number of firms yields an increase in per firm production, both in the short- and in the long-run. Short- and long-run industry output decrease. Third, this paper identifies the conditions on the implicit growth rate and the initial asset stock under which the socially optimal allocation of the resource implies a natural monopoly. We show that for any initial number of firms in the industry, any splitting of the resource can be socially harmful. Increasing market competition turns out to be socially desirable only when the asset stock grows rapidly and the initial asset stock is abundant.