# Irreversible exit decisions under mean-reverting uncertainty

## Abstract

Although many economic variables of interest exhibit a tendency to revert to long-run levels, mean reverting processes are rarely used in investment and disinvestment models in the literature. Previous work by Sarkar (J Econ Dyn Control 28(2):377–396, 2003), that focuses on irreversible entry decisions, showed that mean reversion has three effects on investment: (a) the “variance effect” (mean reversion reduces the long-run uncertainty and thus brings closer the critical investment level), (b) the “realized price effect” (the lower variance resulting from mean reversion makes it less likely to reach extreme high or low price levels, thereby reducing the likelihood of reaching the investment trigger) and (c) the “risk discounting effect” (mean reversion lowers the required rate of return, which affects both the project value and the value of the real option to invest). Metcalf and Hassett (J Econ Dyn Control 19(8):1471–1488, 1995) and Sarkar (J Econ Dyn Control 28(2):377–396, 2003) showed that (a) and (b) work in opposite directions, essentially canceling each other out, however the effect of (c) depends on parameter values, making the overall effect (a–c) of mean reversion on entry decisions ambiguous and parameter-dependent. In this paper, we show that as far as irreversible exit decisions are concerned, the effect of mean reversion is negative: Mean reversion unambiguously lowers the rate of irreversible disinvestment/exit for reasonable parameter values, since the mean reversion in this case only affects the value of the real option to exit and not the value resulting from (real) option exercise.

### Keywords

Investment Uncertainty Real options Mean reversion### JEL Classification

C61 E22## 1 Introduction

Virtually all investment and disinvestment decisions are made in an uncertain environment and some of them are impossible to reverse if economic conditions change unexpectedly. As early as the works of Robichek and Van Horne (1967, 1969), Dyl and Long (1969), Bonini (1977) and Myers and Majd (1990) researchers have shown that the option to completely abandon a project or irreversibly exit a market has significant value which substantially affects the desirability and the return on investment of the venture. Berger et al. (1996) were among the first that provided empirical evidence that investors actually value abandonment or exit options when assessing going—concern corporations.

Apart from the macabre example of suicide as the ultimate irreversible exit decision [mentioned in the seminal book of Dixit and Pindyck (1994)], there are several other important economic and societal decisions that are characterised by irreversibility. At the societal level, in the context of health economics, Mahul and Gohin (1999) analyse the irreversible decision made by French health authorities to slaughter and dispose infected animal herds at the outbreak of the highly contagious foot-and-mouth disease back in the 1990s.

At the microeconomic level, firms decide to close down producing plants (see the empirical investigations in Kovenock and Phillips 1997; Anderson et al. 1998), drop brands from their portfolio of consumer products in the face of demand uncertainty (Hitsch 2006) or leave oil extraction licences expire “unexercised” due to low fuel prices and high fixed costs (e.g. see Favero et al. (1994), for empirical evidence of irreversibilities involved in oil investments). At the international level, several researchers have provided empirical evidence that firms exit foreign markets in the face of fluctuating exchange rates and sunk-cost induced hysteresis (see Requena-Silvente 2005; Bernard and Wagner 2001; Harris and Li 2010, 2011, among others).^{1}

A research stream that is known as the “real options” approach to capital investment (see Dixit and Pindyck 1994; Trigeorgis 1996, for an overview) has provided decision-makers with a plethora of optimal rules for irreversible (or partly reversible at a cost) investment and disinvestment decisions in the face of ongoing uncertainty, mainly by using the tractable geometric Brownian motion to model uncertain cash flow streams, prices, asset values or exchange rates (see McDonald and Siegel 1985, 1986; Brennan and Schwartz 1985; Pindyck 1988; Dixit 1989; Bentolila and Bertola 1990; Myers and Majd 1990, etc.). However, although the usually-employed geometric Brownian motion offers (dis)investment decision rules that are simple and intuitive, several researchers have noted that it is not a plausible price process under equilibrium conditions (see Bhattacharya 1978; Lund 1993; Bessembinder et al. 1995), and that mean-reverting processes should be preferred, for sound economic reasons.^{2}

Despite this, very few studies have offered optimal investment and disinvestment decision rules for firms facing mean-reverting price uncertainty. Notable exceptions are the studies by Metcalf and Hassett (1995), Sarkar (2003) and Tsekrekos (2010). The first two studies concentrate on irreversible entry, while the last study focuses on costly (reversible) entry and exit decisions under mean-reverting uncertainty.

The papers by Metcalf and Hassett (1995) and Sarkar (2003), that focus on irreversible entry decisions, collectively establish that mean reversion has three effects on investment: (a) the “variance effect” (mean reversion reduces the long-run uncertainty and thus brings closer the critical investment level), (b) the “realized price effect” (the lower variance resulting from mean reversion makes it less likely to reach extreme high or low price levels, thereby reducing the likelihood of reaching the investment trigger) and (c) the “risk discounting effect” (mean reversion lowers the required rate of return, which affects both the project value and the value of the real option to invest). Metcalf and Hassett (1995) and Sarkar (2003) showed that (a) and (b) work in opposite directions, essentially canceling each other out, however the effect of (c) depends on parameter values, making the overall effect (a–c) of mean reversion on entry decisions *ambiguous* and *parameter-dependent*. This findings are confirmed by Tsekrekos (2010), that focuses on (costly) reversible entry/exit decisions. It is shown there that the overall effect of mean reversion is significant but ambiguous in direction (depending on parameter values) on reversible entry/exit, much like in Sarkar (2003) and irreversible entry.

This paper supplements this short list by focusing on the effect of mean reversion on *irreversible exit* decisions, which are clearly relevant in real-world applications and are empirically investigated by researchers. *Irreversible* exit/abandonment decisions, are also interesting and different from the above studies of Metcalf and Hassett (1995), Sarkar (2003) and Tsekrekos (2010) for the following reason: when there is (costly or not) reversibility or irreversible entry, the firm is affected by mean reversion both before and after “real options” exercise. Both the real option and the “underlying asset” that is earned after exercise are affected by the mean-reverting process. In contrast, when a firm is contemplating irreversible exit, by exercising its put (real) option to exit the firm completely and irreversibly abandons the market, and its position is no longer affected by mean reversion. This, as shown in this paper, makes a difference in the overall effect of mean reversion on firm decisions. We demonstrate that mean reversion *unambiguously* has a *negative* effect on irreversible *exit* decisions, when compared to the usually-employed geometric Brownian motion for reasonable parameter values. This is in contrast to the *costly reversible exit* case treated in Tsekrekos (2010) or the *irreversible entry* case treated in Sarkar (2003), where depending on parameter values the effect of mean reversion on (dis)investment can be *positive* or *negative*.

The rest of the paper is organized as follows: The next section presents the assumptions needed and describes the basic model setting. Section 3 derives and analyses the optimal disinvestment firm policy under mean reversion. Section 4 compares the findings under a mean-reverting price process with firm behavior under a geometric Brownian motion, and Sect. 5 concludes.

## 2 Basic setting and assumptions

In order to formulate the problem, assume that a firm operates in a market by having made in the past a discrete unit of investment: Namely, it has initiated a single project of a given fixed size that produces one unit flow of output per period (a simple normalization). The firm incurs a flow cost of \(c\) per period of time to produce the unit of output, and it can earn an equilibrium price \(P\) for it. The firm can at any point in time decide to abandon operations by incurring a lump sum exit cost \(K\). Such a decision to exit is *completely irreversible*, and takes place instantaneously.^{3} The magnitudes \(K, c\) and the risk-free interest rate \(r\) are constant and non-stochastic. The uncertainty arises from the output equilibrium price \(P\) which is exogenous to the firm (i.e. the firm is a price-taker).

In order to keep the exposition as close as possible to models that treat the case of entry or the costly reversible case (Dixit 1989; Sarkar 2003; Tsekrekos 2010), the firm is neither allowed to alter the scale of the investment at will, nor is its profit flow dependent on the installed capital stock.^{4}

^{5}

As discussed in the introduction, our attention in this paper is on the effect of mean reversion on the optimal irreversible exit/abandonment decisions of firms. Previous work by Metcalf and Hassett (1995) and Sarkar (2003) that focused on irreversible entry has reached contrasting conclusions regarding the effect of mean reversion.

Employing this mean-reverting process makes the findings of this paper regarding irreversible exit directly comparable to the decision rules for irreversible entry provided by Sarkar (2003). Moreover, as Sarkar (2003) and Tsekrekos (2010) establish, out of the alternative mean-reverting processes that have been used in the literature, the one in (3) is the most plausible as an equilibrium output price due to the homogeneity of degree one of its drift and diffusion terms in \(\left( P,\theta \right) \).^{6} Clearly, for \(\theta =0\) Eq. (3) becomes the geometric Brownian motion in (1) with \(\mu =-\kappa \), while for \(\kappa =0\) it becomes a geometric Brownian motion process with no drift. It is easy to verify that the price process in (3) reverts to its long-run mean in the limit, i.e. \(\mathbb E \left[ P_{\infty }\left. \right| P_{0}\right] =\lim _{t\rightarrow +\infty }\mathbb E \left[ P_{t}\left. \right| P_{0}\right] =\theta \) and has a finite conditional variance \(\mathbb V \left[ P_{\infty }\left. \right| P_{0}\right] =\lim _{t\rightarrow +\infty }\mathbb V \left[ P_{t}\left. \right| P_{0}\right] =\frac{\theta ^{2}\sigma ^{2}}{2 \kappa -\sigma ^{2}}\) that is *decreasing* in the speed of mean reversion \(\kappa \) and *increasing* in the long-run mean level \(\theta \).

## 3 Optimal irreversible exit under mean reversion

### 3.1 Deriving the optimal exit policy

Let the equilibrium output price follow the process in (3), and let \(\left( {\underline{P}},+\infty \right) \) denote the range of output prices over which the firm finds it optimal to operate in the market. Denote \(V\left( P\right) \) the expected net present value of the firm, starting with a price \(P\) in the operating state, and following the optimal exit strategy.

Over the range of prices \(\left( {\underline{P}},+\infty \right) \) where it is optimal for an active firm to continue in this state, the total return of the expected net present value of the firm, \(V\left( P\right) \), comprises of the expected capital gain \(\mathbb E [dV\left( P\right) ]/dt\), plus a cash inflow \(\left( P-c\right) \) per unit of time.

^{7}

The second line of (10) corresponds to the expected present value of operating in the market forever, starting from an initial price \(P\), while the first line corresponds to the firm’s option to exit the market if economic conditions become extremely unfavorable. It is easy to verify that while seemingly complex, the value function of the active firm under mean-reverting prices in Eq. (10) collapses to the geometric Brownian motion case, for \(\kappa =0\) (or \(\theta =0\)), since from the properties of the confluent hypergeometric function, we know that \(M\left( a,b,0\right) =1\).

The above boundary conditions highlight the difference between the effect of mean reversion on irreversible entry (or reversible exit, see Sarkar 2003; Tsekrekos 2010) and the irreversible exit case that we treat here. In boundary conditions (13) and (14), mean reversion only affects the value of the real option to exit and not the value resulting from (real) option exercise (i.e. the right-hand side of the boundary conditions). In contrast, when a firm is contemplating irreversible entry (Sarkar 2003) or reversible exit (Tsekrekos 2010), both sides of the relevant boundary conditions are affected by the mean reversion parameters \(\kappa \) and \(\theta \). As demonstrated in the sections that follow, this makes a significant difference, making the overall effect of mean reversion on irreversible firm exit decisions *unambiguously negative* for reasonable parameter values.

### 3.2 Analysis of solution

When active in the market, firm value consists of (a) the present value of operating cash flows and (b) a put option to exit the market. When mean reversion is introduced (i.e. as \(\kappa \) increases, say from 0 to 0.10 in terms of Fig. 2), the reduced variance of the price process decreases the put option to exit (the “variance effect”) and the reduced systematic risk increases the present value of operating in the market (the “risk discounting” effect). Which of the two effects dominates will determine whether the value of the firm \(V\left( P\right) \) will increase or decrease as \(\kappa \) increases. Evidently from Fig. 2, for higher long-run mean equilibrium prices it is the “risk discounting” effect that dominates.^{8}

*level*of the exit threshold; the “realized price” effect simply reduces the

*likelihood*of reaching this threshold.

When the “risk discounting effect” (mean reversion reduces the systematic risk of the firm, thus firm cash flows are discounted at a lower rate) dominates the “variance effect” (mean reversion reduces long-run uncertainty, bringing closer abandonment price triggers), irreversible exit under mean reversion is postponed in comparison to the geometric Brownian motion case. This is seen in Fig. 3 as \(\kappa ,\theta \) increase: quickly reverting to a higher long-run equilibrium price makes irreversible exit more unlikely, decreasing the price threshold levels that must be reached before the firm optimally abandons the market.

Bearing in mind that the “realized price” effect reinforces this optimal postponement, by making it less likely that higher entry and lower exit thresholds are reached, one can conclude that mean reversion has an unambiguous negative effect on irreversible investment exit when the “risk discounting effect” dominates.

The “risk discounting effect” is relatively more important when interest rates are low and for long-lived investment projects with high intrinsic values (high \(\theta \), low \(K\)), one can summarize this section in the following:

**Irreversible exit:***Compared to the usually—employed geometric Brownian motion price process, mean reversion unambiguously decreases irreversible exit when the risk-free interest rate is low and when long-lived, high intrinsic value projects are considered.*

The above confirms that the findings of Sarkar (2003) and Tsekrekos (2010) on the effect of mean reversion on irreversible or costly reversible entry are also valid for irreversible investment exit decisions.

## 4 The effect of mean reversion on irreversible exit decisions under uncertainty

In order to disentangle the net impact of the “variance”, the “realized price” and the “risk discounting” effects of mean reversion, one could examine whether mean reversion increases or decreases disinvestment and abandonment rates on an aggregate industry level, compared to the geometric Brownian motion case.

Of course answering such a question would require one to move to a general (rather than a partial) equilibrium model, where the price process should be endogenized as part of the resulting equilibrium. In spite of how interesting on its own such a research question is, it goes beyond the scope of this paper.

Here we will examine the effect of mean reversion in a simplistic manner by computing the ex-ante probability of exit/abandonment within a specified time horizon for an active firm. This approach has also been used in related papers by Sarkar (2003) and Tsekrekos (2010), thus adopting it here makes the findings of this paper directly comparable to previous research. As Tsekrekos (2010) points out, “*[t]he rationale for using these probability measures is that although [...the optimal exit policy...] was derived in a single-firm setting, it happens to coincide with the optimal policy of firms under general equilibrium in a perfectly competitive industry/market, as shown by*Leahy (1993)” (Tsekrekos 2010, p. 734).

An immediate implication is that if there is a large pool of active firms, individually contemplating exit from an industry/market that is characterized by the exogenous equilibrium mean-reverting price process in (3), then the ex-ante probability that a single firm will exit by time \(T\) will be a measure of the fraction of active firms, under competitive equilibrium, that will abandon the market by this time horizon. Lower (higher) probability of exit means a smaller (larger) fraction of active firms will disinvest and optimally abandon the market, i.e. mean reversion has a positive (negative) effect on investment. This way, the net impact of all three effects of mean reversion (the “variance”, the “realized price” and the “risk discounting” ones) can be examined and compared to the log-normally distributed price process (geometric Brownian motion).

The ex-ante probability that an active firm will exit the market by time \(T\) is equal to \(\Pr \left( {\underline{\tau }}\le T\right) \), with \({\underline{\tau }}=\inf \{t \ge 0 :P_{t} \le {\underline{P}}\}\) the first passage time of process (3) from level \({\underline{P}}\).

For a market with an equilibrium output price described by the mean-reverting process in Eq. 3, the table reports the probability that an active firm will optimally exit the market by time \(T\), calculated via Monte Carlo simulation

\(\theta = 0.40\) | \(\theta = \theta ^*\) | \(\theta = 1.40\) | |||||||
---|---|---|---|---|---|---|---|---|---|

\(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | \(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | \(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | |

| |||||||||

\(\kappa = 0\) | 1.0000 | 1.0000 | 1.0000 | ||||||

\(\kappa =0.05\) | 0.7416 | 0.6608 | 0.5882 | 0.7088 | 0.6956 | 0.6996 | 0.0234 | 0.0720 | 0.1348 |

\(\kappa =0.10\) | 0.9446 | 0.8590 | 0.7694 | 0.8914 | 0.8512 | 0.8198 | 0.0000 | 0.0144 | 0.1054 |

\(\kappa = 0.15\) | 0.9908 | 0.9508 | 0.8586 | 0.9688 | 0.9444 | 0.9934 | 0.0000 | 0.0000 | 0.0344 |

| |||||||||

\(\kappa = 0.05\) | 0.7264 | 0.6366 | 0.5522 | 0.1500 | 0.2228 | 0.3586 | 0.0000 | 0.0000 | 0.0046 |

\(\kappa = 0.10\) | 0.9394 | 0.8434 | 0.7314 | 0.0426 | 0.0838 | 0.2016 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa = 0.15\) | 0.9908 | 0.9462 | 0.8334 | 0.0048 | 0.0132 | 0.0702 | 0.0000 | 0.0000 | 0.0000 |

| |||||||||

\(\kappa = 0\) | 1.0000 | 1.0000 | 1.0000 | ||||||

\(\kappa = 0.05\) | 0.7038 | 0.5798 | 0.4010 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa = 0.10\) | 0.9348 | 0.8234 | 0.6450 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa = 0.15\) | 0.9500 | 0.9392 | 0.7878 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

The parameter values examined in this numerical investigation are in the same spirit as the ones in Tsekrekos (2010). Three \(\theta \) scenarios are selected: one (\(\theta =0.4\)) that makes the intrinsic value of operating in the market low, and in which we expect the “risk discounting effect” to be dominated by the “variance effect”, and one (\(\theta =1.4\)) where the exact opposite is true. Much like in Tsekrekos (2010), the third set of scenarios (under \(\theta =\theta ^{\star }\)) attempts to “neutralize” the opposing effects of “risk discounting” and “variance”, by setting \(\theta ^{\star }=\frac{rP_{0}}{r+\lambda \rho \sigma }\) for each different triplet \(\left( \kappa ,\rho ,\sigma \right) \) examined. It is not difficult to show that for a long-run mean equilibrium output price equal to \(\theta ^{\star }\), the firm finds it equally profitable to operate in the market for ever under both the mean-reverting price process in (3) and a geometric Brownian motion with no drift.

Turning to the results, Table 1 confirms numerically the negative effect of mean reversion on irreversible exit decisions that were identified in the previous section. This negative effect on disinvestment when compared to a logarithmic random walk with drift is more pronounced for high intrinsic value projects (\(\theta =1.4\)).

For the intermediate \(\theta =\theta ^{\star }\) scenarios, the probabilities to exit under mean reversion are lower than the corresponding geometric Brownian motion ones, for all correlation and volatility levels. The decrease in exit probabilities seems to be more pronounced the lower the correlation coefficient of the equilibrium output price \(P\) with the market portfolio. The numerical evidence seems to indicate that for reasonable parameter levels, mean reversion *unambiguously* has a *negative* effect on irreversible *exit* decisions, when compared to the usually—employed geometric Brownian motion. This is in contrast to the *costly reversible exit* case treated in Tsekrekos (2010), where depending on parameter values the effect of mean reversion on disinvestment can be *positive*.

The reason behind this difference is the following: If the firm can costly revert back to being active in the market (i.e. exit is not irreversible), when exercising its put option to exit, it acquires a call option to re-enter in the future. The “underlying asset” of this call option is the present value of future cash flows from operations (plus a put option to re-exit since the decision is reversible). This future stream of cash flows is affected by the “risk discounting effect”. In contrast, under irreversible exit, by exercising its put option to exit the firm completely and irreversibly abandons the market. Its position is then unaffected by mean reversion, and there is no “risk discounting effect” as in the reversible case. This leads firms to be less willing to make irreversible exit decisions from markets where equilibrium output prices are highly mean reverting, in comparison to cases where such decisions can be reversed, even at a cost.

For a market with an equilibrium output price described by the mean-reverting process in Eq. 3, the table reports the probability that an active firm will optimally exit the market by time \(T\), calculated via Monte Carlo simulation

\(\theta = 0.40\) | \(\theta = \theta ^*\) | \(\theta = 1.40\) | |||||||
---|---|---|---|---|---|---|---|---|---|

\(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | \(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | \(\sigma = 0.10\) | \(\sigma = 0.15\) | \(\sigma = 0.25\) | |

| |||||||||

\(\kappa = 0\) | 0.9910 | 0.9509 | 0.8785 | ||||||

\(\kappa = 0.05\) | 0.0738 | 0.1828 | 0.3416 | 0.0072 | 0.0886 | 0.2998 | 0.0004 | 0.0134 | 0.1288 |

\(\kappa = 0.10\) | 0.3964 | 0.4488 | 0.5178 | 0.0118 | 0.1660 | 0.4300 | 0.0000 | 0.0022 | 0.0640 |

\(\kappa = 0.15\) | 0.7866 | 0.7146 | 0.6734 | 0.0162 | 0.2630 | 0.5476 | 0.0000 | 0.0000 | 0.0198 |

| |||||||||

\(\kappa = 0\) | 0.9904 | 0.9466 | 0.8605 | ||||||

\(\kappa = 0.05\) | 0.0412 | 0.1078 | 0.2058 | 0.0000 | 0.0135 | 0.0851 | 0.0000 | 0.0000 | 0.0004 |

\(\kappa = 0.10\) | 0.3514 | 0.3784 | 0.4096 | 0.0058 | 0.1004 | 0.2155 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa = 0.15\) | 0.7668 | 0.6746 | 0.6012 | 0.0138 | 0.1799 | 0.3314 | 0.0000 | 0.0000 | 0.0000 |

| |||||||||

\(\kappa = 0.05\) | 0.0176 | 0.0276 | 0.0252 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa = 0.10\) | 0.3106 | 0.2960 | 0.2370 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

\(\kappa =0.15\) | 0.7496 | 0.6358 | 0.5010 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

## 5 Conclusions

Although managers and investors truly apprehend and value the flexibility inherent in being able to abandon an investment project early, far fewer research papers are devoted to the analysis of irreversible abandonment/exit decisions than to irreversible investment/entry.

In this paper we examine the optimal irreversible exit decisions of firms under mean-reverting uncertainty. The aim is to investigate whether the findings of previous research that mean reversion has an ambiguous (i.e. parameter-dependent) effect on irreversible entry decisions are directly extendable to abandonment/exit decisions that cannot be reversed.

The results of the model in this paper indicate that they are actually not fully extendable. Mean reversion has a significant impact on firm optimal policies, as measured by the probability of exit for an active firm over a certain period of time. It is shown that as far as irreversible exit decisions are concerned, the effect of mean reversion is clearly negative: In this case mean reversion lowers the rate of irreversible disinvestment/exit, since the “variance”, “realized price” and “risk discounting” effects documented in the previous literature only affect the value of the real option to exit and not the value resulting from (real) option exercise. This is in contrast to the costly reversible exit case and the irreversible entry one.

## Footnotes

- 1.
Requena-Silvente (2005) reports that close to 10 % of UK small and medium-sized exporters

*irreversibly*withdrew from the foreign markets they operated in the 1994–1998 period. This percentage does not include firms that only temporarily ceased exporting (which were dropped from his sample). Similarly, Bernard and Wagner (2001) report that between 1990–1997, about 8 % of German exporting plants left the foreign market every year on average. - 2.
The argument in Lund (1993) is that in equilibrium, a price processes should not be unbounded from above, as the logarithmic random walk with drift clearly is, since new entry or expanded production by incumbent suppliers will induce reversion to lower price levels.

- 3.
This could easily be extended to the case where the irreversible exit decision takes time to enforce, using the treatment in Majd and Pindyck (1987).

- 4.
- 5.
- 6.
See also the discussion in ((Tsekrekos 2010, p.728 and footnote 4)).

- 7.
- 8.
It should be stressed that, although not apparent in the upper panel of Fig. 2 due to the scale of the graph, the mean reversion parameters \(\theta \) and \(\kappa \) significantly affect the irreversible exit/abandonment price trigger, \({\underline{P}}\). In the upper panel, output equilibrium prices of \({\underline{P}}=\left[ 1.6398, 1.67204,1.70827,1.74946\right] \) lead to abandonment for values of mean reversion speed \(\kappa =\left[ 0, 0.02, 0.05, 0.10\right] \). Stronger reversion to low levels makes the firm optimally abandon the market at higher output equilibrium prices, which is the exact opposite of what is observed in the lower panel of the Figure.

## Notes

### Acknowledgments

This paper has greatly benefited from the valuable comments and suggestions made by the participants of the international workshop on “The Economics of Irreversible Choices” that was organized by the Lombardy Advanced School of Economic Research (LASER) and the DEFAP Graduate Business School in Public Economics and was hosted by the Università degli Studi di Brescia in Italy. Special thanks are due to David Schüller who acted as the discussant of the paper at the workshop, as well as to the organizers Giacomo Corneo, Luca Di Corato, Michele Moretto, Paolo Panteghini, Carlo Scarpa and last but not least Sergio Vergalli.

### References

- Abel AB, Eberly JC (1996) Optimal investment with costly reversibility. Rev Econ Stud 63(4):581–593CrossRefGoogle Scholar
- Abramowitz M, Stegun IA (1972) Handbook of mathematical functions. Dover Publications, New YorkGoogle Scholar
- Alvarez LHR (2011) Optimal capital accumulation under price uncertainty and costly reversibility. J Econ Dyn Control 35(10):1769–1788CrossRefGoogle Scholar
- Anderson DW, Murray BC, Teague JL, Lindrooth RC (1998) Exit from the meatpacking industry: a microdata analysis. Am J Agric Econ 80(1):96–106Google Scholar
- Bentolila S, Bertola G (1990) Firing costs and labour demand: How bad is Eurosclerosis? Rev Econ Stud 57(3):381–402CrossRefGoogle Scholar
- Berger PG, Ofek E, Swary I (1996) Investor valuation of the abandonment option. J Financ Econo 42(2):257–287Google Scholar
- Bernard AB, Wagner J (2001) Export entry and exit by German firms. Weltwirtschaftliches Archiv 137(1):105–123CrossRefGoogle Scholar
- Bessembinder H, Coughenour JF, Seguin PJ, Smoller MM (1995) Mean reversion in equilibrium asset prices: evidence from the futures term structure. J Financ 50(1):361–375Google Scholar
- Bhattacharya S (1978) Project valuation with mean-reverting cash flow streams. J Financ 33(4):1317–1331CrossRefGoogle Scholar
- Bonini CP (1977) Capital investment under uncertainty with abandonment options. J Financ Quant Anal 12(1):39–54CrossRefGoogle Scholar
- Brennan MJ, Schwartz ES (1985) Evaluating natural resource investments. J Bus 58(2):135–157CrossRefGoogle Scholar
- Dixit AK (1989) Entry and exit decisions under uncertainty. J Polit Econ 97(3):620–638CrossRefGoogle Scholar
- Dixit AK, Pindyck RS (1994) Investment under uncertainty. Princeton University Press, New JerseyGoogle Scholar
- Dyl E, Long H (1969) Abandonment value and capital budgeting: comment. J Financ 24(1):88–95CrossRefGoogle Scholar
- Favero CA, Pesaran MH, Sharma S (1994) A duration model of irreversible oil investment: theory and empirical evidence. J Appl Econom 9(Supplement):95–112Google Scholar
- Harris RID, Li QC (2010) Export-market dynamics and the probability of firm closure: Evidence from the United Kingdom. Scott J Polit Econ 57(2):145–168CrossRefGoogle Scholar
- Harris RID, Li QC (2011) The determinants of firm exit from exporting: evidence for the UK. Int J Econ Bus 18(3):381–397CrossRefGoogle Scholar
- Harrison MJ (1985) Brownian Motion and Stochastic Flow Systems. Krieger Publishing Company, Robert EGoogle Scholar
- Hitsch GJ (2006) An empirical model of optimal dynamic product launch and exit under demand uncertainty. Mark Sci 25(1):25–50CrossRefGoogle Scholar
- Kovenock D, Phillips GM (1997) Capital structure and product market behavior: an examination of plant exit and investment decisions. Rev Financ Stud 10(3):767–803CrossRefGoogle Scholar
- Leahy JV (1993) Investment in competitive equilibrium: the optimality of myopia. Quart J Econ 108(4): 1105–1133Google Scholar
- Lund D (1993) The lognormal diffusion is hardly an equilibrium price process for exhaustible resources. J Environ Econ Manag 25(3):235–241Google Scholar
- Mahul O, Gohin A (1999) Irreversible decision making in contagious animal disease control under uncertainty: An illustration using FMD in Brittany. Eur Rev Agric Econ 26(1):39–58CrossRefGoogle Scholar
- Majd S, Pindyck RS (1987) Time to build, option value and investment decisions. J Financ Econ 18(1):7–27CrossRefGoogle Scholar
- McDonald RL, Siegel DR (1985) Investment and the valuation of firms when there is an option to shut down. Int Econ Rev 26(2):331–349CrossRefGoogle Scholar
- McDonald RL, Siegel DR (1986) The value of waiting to invest. Quart J Econ 101(4):707–727CrossRefGoogle Scholar
- Merton RC (1973) An intertemporal capital asset pricing model. Econometrica 41(5):867–887CrossRefGoogle Scholar
- Metcalf GE, Hassett KA (1995) Investment under alternative return assumptions: Comparing random walks and mean reversion. J Econ Dyn Control 19(8):1471–1488CrossRefGoogle Scholar
- Myers SC, Majd S (1990) Abandonment value and project life. Adv Futures Options Res 4(1):1–21Google Scholar
- Pindyck RS (1988) Irreversible investment, capacity choice and the value of the firm. Am Econ Rev 78(5):969–985Google Scholar
- Requena-Silvente F (2005) The decision to enter and exit foreign markets: evidence from UK SMEs. Small Bus Econ 25(3):237–253CrossRefGoogle Scholar
- Robichek AA, Van Horne JC (1967) Abandonment value and capital budgeting. J Financ 22(4):577–589Google Scholar
- Robichek AA, Van Horne JC (1969) Reply. J Financ 24(1):96–97CrossRefGoogle Scholar
- Sarkar S (2003) The effect of mean reversion on investment under uncertainty. J Econ Dyn Control 28(2):377–396CrossRefGoogle Scholar
- Sick G (1995) Real options. In: Jarrow RA, Maksimovic V, Ziemba WT (eds) Finance (Handbooks in Operations Research and Management Science), vol 9. Elsevier Science B.V.Google Scholar
- Trigeorgis L (1996) Real options: managerial flexibility and strategy in resource allocation. MIT Press, CambridgeGoogle Scholar
- Tsekrekos AE (2010) The effect of mean reversion on entry and exit decisions under uncertainty. J Econ Dyn Control 34(4):725–742CrossRefGoogle Scholar