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An investment model with switching costs and the option to abandon

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Abstract

We develop a complete analysis of a general entry–exit–scrapping model. In particular, we consider an investment project that operates within a random environment and yields a payoff rate that is a function of a stochastic economic indicator such as the price of or the demand for the project’s output commodity. We assume that the investment project can operate in two modes, an “open” one and a “closed” one. The transitions from one operating mode to the other one are costly and immediate, and form a sequence of decisions made by the project’s management. We also assume that the project can be permanently abandoned at a discretionary time and at a constant sunk cost. The objective of the project’s management is to maximise the expected discounted payoff resulting from the project’s management over all switching and abandonment strategies. We derive the explicit solution to this stochastic control problem that involves impulse control as well as discretionary stopping. It turns out that this has a rather rich structure and the optimal strategy can take eight qualitatively different forms, depending on the problems data.

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Notes

  1. Using a trivial re-parametrisation, we can allow for the project to yield a constant payoff rate while it is in its “closed” mode (see Remark 1).

  2. For the same reason, it would make sense in some economic applications to allow for at least \(K_0\) to be negative, as long as \(K_1 + K_0 > 0\). However, such a relaxation would add most significant complexity and would result in a substantially longer paper.

  3. Although this setting is convenient for the problem’s formulation, switching followed by immediate abandonment is never optimal due to the strict positivity of \(K_\ell \), \(\ell = 1, 0\).

  4. The inequality \(\vartheta < n\), where n is defined by (15), is essential for the value function to be finite.

  5. In the description of the five possible regions, we characterise subsets of \(]0,\infty [\) as open or closed relative to the topology on \(]0,\infty [\) that is the trace of the usual topology on \(\mathbb {R}\), for instance, \(]0,a] = ]0,\infty [ \setminus ]a, \infty [\) and \([a, \infty [ = ]0,\infty [ \setminus ]0, a[\) are closed sets.

  6. We use the notation \(\delta _\dagger \) rather than the simpler \(\delta \) because this point will appear in assumptions that we will make in later cases.

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Acknowledgements

We thank an anonymous referee and an associate editor for comments and suggestions that enhanced our original manuscript. The research of Carlos Oliveira was supported by Fundação para a Ciência e Tecnologia through the Grant SFRH/BD/102186/2014.

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Authors and Affiliations

  1. Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Mihail Zervos & Kate Duckworth

  2. Department of Mathematics and CEMAT, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Carlos Oliveira

Authors
  1. Mihail Zervos
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  2. Carlos Oliveira
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  3. Kate Duckworth
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Correspondence to Carlos Oliveira.

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Zervos, M., Oliveira, C. & Duckworth, K. An investment model with switching costs and the option to abandon. Math Meth Oper Res 88, 417–443 (2018). https://doi.org/10.1007/s00186-018-0641-5

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