Dynamic Programming Models of Fishing: Competition

It is natural to consider the stock of any renewable resource as a capital stock and treat the exploitation of that resource in much the same way as one would treat accumulation of a capital stock. This has been done to some extent by Cohn Clark and Gordon Monro and by Ngo Van Long, whose papers contain a discussion of this point of view. However, the analysis is much simpler than it appears in the literature especially since the interaction between markets and the natural biological dynamics has not been made clear. Three issues which have been raised only tangentially are important for the understanding of the economics of renewable resources. The first arises from the fact that the fish population might not be able to sustain the market determined amount of fish and hence should be allocated optimally over time. Second, there is the cost externality. Here costs are affected by the size of the stock of fish which again affects the optimal allocation over time. The third issue is how central planning (typically used in capital theory) can be employed as a tool in analyzing a decentralized economy with perfect futures markets and property rights. In this paper a simple framework, using only elementary mathematical techniques, will be provided in which many important results new to the literature, as well as many familiar results, can be derived. One bonus of this approach is that much of the previous literature can be organized and rationalized. The simplicity of the mathematics has the effect of revealing the underlying economic intuition of the subject. Throughout the paper, as a matter of convenience, the selfrenewing resource will be referred to as "fish. " However, it should be clear that the analysis is perfectly general in that it can be applied to any self-renewing resource. In fact, by specializing the production function, an elementary exposition of the theory of exhaustible resources is implicit. There are two main sections to this work plus an Appendix. The first section analyzes the time path of prices, output, and resource stock under the assumption of free-entry competition, and the second section studies "optimal planning" (or perfect competition). In the first part of the discussion of each market structure, extraction costs are assumed to be independent of the size of the fish stock. In the second part of each section this assumption is relaxed. Finally the Appendix contains several of the results used in the body of the paper. The case of free-entry competition can be treated in a very simple manner. Each period's output is determined by the intersection of the demand and marginal harvesting cost curves (i.e., supply equals demand). Subtracting this output from the natural dynamics yields the dynamics of free-entry competition. Using this technique, it is easy to see the conditions under which free-entry competition leads to the exhaustion of the stock of fish. Furthermore, the free-entry case serves as a benchmark by which to evaluate the effects of the institution of property rights on the exploitation of the resource. The case of a centrally planned economy is studied next. This case is shown to be identical to a competitive outcome in which property rights are clearly defined and prices and outputs are endogenously determined so as to be consistent with demand conditions. One result, coincidentally, is that it is possible that the optimal solution might be obtained without property rights. The tech*The Hebrew University; University of Virginia; University of Illinois and Northwestern University, respectively. Research support from the National Science Foundation under grants SOC 77-27340 and SOC 7905900 and the US-Israel BSF under grant 1828-79 is gratefully acknowledged.