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Stochastic Shadow Pricing of Renewable Natural Resources

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Abstract

By means of stochastic optimal control, this paper aims at studying the shadow pricing of renewable natural resources in uncertainty. Two cases are considered, respectively centralized and decentralized control processes. The decentralized control is in the form of a stochastic control of the state vector distributed among several agents. In both cases, the optimal control path minimizing the cost function, which is a decreasing function of time, corresponds to the real option valuation. The latter is a cost-effective optional investment in the resource stock preservation in uncertainty. The results obtained from numerical simulations show coherence with those encountered in the literature on option pricing.

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Notes

  1. A diffusion process, as a continuous-time Markov process, is a solution to a stochastic differential equation. Brownian motion is one example of it.

  2. The Brownian motion in the dynamics of xi is the same as that in the dynamics of xj. Therefore, the probability density function of the random state X(t) defines the probability of the paths of xi(t) and xj(t).

  3. In Palmer and Milutinovic [38], the co-state distribution is valued at the point X+ = X(s), such that the integral is computed with respect to s ∈ [t, T]. This is meant to take account of the possibility of state transitions in time, where a transition is function of the control variables. The absence of transition, such as in our framework, simply gives X(t).

  4. In detail, we have \(C_{t-{\Delta } t, i} = e^{-r{\Delta } t} \left [pC_{t,i + 1}+(1-p)C_{t,i-1}\right ]\), with Ct−Δt, i the binomial value, Ct, i+ 1 and Ct, i− 1 the option values from the last two nodes, p and (1 − p) the weighting probabilities of moving up and down, and r the risk free rate.

  5. By considering several components of the state vector X, our framework can be compared to the real option valuation of multiple species preservation [27].

  6. Indeed, shadow prices correspond to what an agent is willing to pay to stabilize the resource stock for an additional unitof time. If we allow a longer time to reach equilibrium, we reduce the cost. In that case, the decrease in the value of controlis dominant, such that the cost decreases [21].

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Acknowledgments

The author would like to thank Luc Doyen (CNRS, University of Bordeaux), Shandelle Henson (Andrews University), and Pierre Lasserre (CIREQ, University of Québec at Montréal) for their valuable comments and suggestions on this work. He is also grateful to the anonymous referees and to the editor for their thorough comments and suggestions, which significantly contributed to improving the quality of the paper.

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Authors and Affiliations

  1. IRSTEA — The Center of Clermont-Ferrand, 9 Avenue Blaise Pascal, 63170, Aubière, France

    Arnaud Z. Dragicevic

  2. AgroParisTech, INRA, IRSTEA, Université Clermont Auvergne, VetAgro Sup [UMR Territoires], 63170, Aubière, France

    Arnaud Z. Dragicevic

  3. Department of Economics, Istanbul Technical University (İTÜ), 34367, Istanbul, Turkey

    Arnaud Z. Dragicevic

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Dragicevic, A.Z. Stochastic Shadow Pricing of Renewable Natural Resources. Environ Model Assess 24, 49–60 (2019). https://doi.org/10.1007/s10666-018-9599-1

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