Abstract
This chapter reviews some fundamental models related to the exploitation of a renewable resource, an important topic when dealing with regional economics. The chapter starts by considering the growth models of an unexploited population and then introduces commercial harvesting. Still maintaining a dynamic perspective, an analysis of equilibrium situations is proposed for a natural resource under various market structures (monopoly, oligopoly and open access). The essential dynamic properties of these models are explained, as well as their main economic insights. Moreover, some key assumptions and tools of intertemporal optimal harvesting are recalled, thus providing an interesting application of the theory of optimal growth.
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Notes
- 1.
In this chapter \(^{\prime }\) denotes the unit-time advancement operator, that is \(x^{\prime }\) \(=x\left( t+1\right) \).
- 2.
New dynamic phenomena can be observed when the species has a unimodal growth function, i.e., it has maximum growth at an intermediate value of the population. In fishery models, this typically occurs when the population has the tendency to decrease when its level drops below a certain threshold, known as the critical depensation level. We do not deepen further this point here for the sake of time.
- 3.
For instance this happens under Gompertz growth, considered later on in an example of this chapter.
- 4.
To simplify notation, unless otherwise stated, in the following we write \(x(t)=x\) and \(\ h(t)=h\).
- 5.
- 6.
A production function (or harvesting function) of Cobb-Douglas type with fishing effort and fish biomass as production inputs and \(\lambda =\mu ={1}/{2}\), is used by several authors, see, e.g., [11, 13, 28], as it captures two fundamental aspects of fishing activity such as gear saturation and congestion. In particular, gear saturation is expressed by decreasing marginal return to fishing effort while congestion is expressed by decreasing marginal return to stock (or fish biomass).
- 7.
Here, the best response function (also known as reaction function) gives the optimal output for a country given the output of another country.
- 8.
Notice that (5.17) can be regarded as an optimal growth model. However, two important features present in the optimal fishery model differentiate it from a standard growth model. First, the type of resource suggests a “production” function that does not satisfy the Inada conditions. Second, the profit function depends, in general, not only on the harvesting h, but also on the level of the resource x.
- 9.
As remarked in the previous note, the fishery “production” function \(f(x)=\left( \alpha +\gamma \right) x-\beta x^{2}\) does not satisfy the Inada conditions, in particular, \({\lim }_{x\rightarrow 0^{+}}f^{\prime }(x) \ne +\infty \) and \({\lim }_{x\rightarrow +\infty }f^{\prime }(x) \ne 0^{+}\).
- 10.
In the previous example we have \(h=Ex\). Here we reason directly in terms of harvesting h for the sake of simplicity.
- 11.
Notice that \({\alpha }/({2\beta })\) is indeed the golden rule level of the stock. To be more precise, from \(\dot{x}=0\), we get that at equilibrium there is \(\overline{h}=x\left( \alpha -\beta x\right) \), so that instantaneous profit is \(\overline{\pi } =m\overline{h}-b\overline{h}^{2}= m\left[ x\left( \alpha -\beta x\right) \right] -b\left[ x\left( \alpha -\beta x\right) \right] ^{2}\). Instantaneous profit \(\overline{\pi }(x)\) is maximized by \(x={\alpha }/({2\beta })\) if \(m>{\alpha ^{2}}/({4\beta })\), since \(\overline{\pi }^{\prime }\left( {\alpha }/({2\beta })\right) =0\) and \(\overline{\pi }^{\prime \prime }\left( {\alpha }/({2\beta })\right) <0\). However, for \(m<{\alpha ^{2}}/({4\beta })\), \(\overline{\pi }(x)\) has minimum at \(x={\alpha }/({2\beta })\), whereas \(\overline{\pi }(x)\) is maximized by the “golden rule” levels given by the state values in \(E_{2}\) and \(E_{3}\), namely, by \(x={\alpha \pm \sqrt{\alpha ^{2}-4m\beta }}/({2\beta })\).
- 12.
This equilibrium is a solution of system (5.30) and, being constant, satisfies a transversality condition of the form \({\lim }_{t\rightarrow +\infty }\pi ^{\prime } (h_{1})e^{-\delta t}=0\), which implies \({\lim _{t\rightarrow +\infty } }H^{c}e^{-\delta t}=0\).
- 13.
They can also be computed when countries have equal discount factors: \(r_{1}=r_{2}=r\).
- 14.
Differently from the modeling where harvesting efforts are controlled, here the regulator does not enforce any restraint.
- 15.
This assumption of perfectly elastic demand for the resource is particularly well justified whenever the resource is a staple food for the consumers or several substitutes to the resource are traded in the market, see also [11] on this point.
- 16.
More precisely, by solving the first order condition \({\partial \pi _{i}}/{\partial h_{i}}=0\), which is also sufficient for being \({\partial ^{2}\pi _{i}}/{\partial h_{i}^{2}} =-{2\gamma }/({q_{i}x})<0\).
- 17.
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Lamantia, F., Radi, D., Sbragia, L. (2016). Dynamic Modeling in Renewable Resource Exploitation. In: Bischi, G., Panchuk, A., Radi, D. (eds) Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-33276-5_5
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