Entry and exit decisions based on a discount factor approach

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Abstract

Entry and exit decisions under price uncertainty are discussed by a discount factor approach to investment. The approach makes it possible to express the value of the firm as a function of a set of trigger prices at which investment takes place, and to determine the optimal policy by simple maximization. Starting with a standard entry–exit model by Dixit [1989. Entry and exit decisions under uncertainty. Journal of political economy 97(3), 620–638], the paper expands to construction and scrapping decisions, investment lags, diminishing production capacity, and limits to the number of lay-up periods. Economic implications are investigated using experimental data and empirical shipping data.

Introduction

Models of entry and exit decisions under price uncertainty, pioneered by Mossin (1968) and generalized by Brennan and Schwartz (1985) and Dixit (1989), are main tools in real options theory. The Dixit model, which is used as a baseline model in this paper, solves the following problem: a firm can choose between using and mothballing a certain production capacity. The cost of producing is fixed while the product price follows a geometric Brownian motion. When is it optimal to produce, considering that switching between production and lay-up is costly? The optimal policy for an idle firm is not to make entry as soon as the expected net present revenue from continuous production in perpetuity exceeds the Marshallian cost – i.e., the entry cost plus the net present cost of production. One should wait for a fixed higher trigger price due to an option argument. Likewise, the active firm should wait for a fixed trigger price lower than the Marshallian one before leaving.

The basic entry–exit model has been extended in several directions. Dixit (1988) adds (initial) construction costs and (final) scrapping costs, Ekern (1993) restricts the number of switches, Brekke and Øksendal (1994) allow for diminishing production capacity over time, and Bar-Ilan and Strange (1996) include investment lags. The entry–exit model has also spurred empirical work on the relationship between investment and uncertainty.1

The objective of this paper is to present a simplified treatment of entry and exit decisions like those above, based on the discount factor approach to investment proposed by Dixit et al. (1999). The core of the discount factor approach in the standard entry–exit setting is to state the expected net present value of the firm as a function of two investment trigger prices, assuming fixed costs and a general, continuous and autonomous Ito price process. Thus, the approach applies to a broad class of stochastic processes, although the geometric Brownian motion is the only one that will be used for examples here. The optimal policy is found by maximizing the value function with respect to the trigger prices. Two trigger prices leave us with two first-order conditions that must be solved simultaneously. Smooth pasting requires a system of four equations, but the two solutions can be shown to coincide by removing two option coefficients from the smooth pasting solution.2

The discount factor approach is so intuitive that it will not be necessary to discuss the technicalities of all model versions when gradually expanding the approach. Most attention is paid to models with investment lags, for which the approach is especially useful. The remaining part of the paper is structured as follows: Section 2 sets up the entry–exit model from Dixit (1989), introducing the discount factor approach and illustrating with numerical examples. Section 3 includes investment lags as in Bar-Ilan and Strange (1996), and corrects an error in that reference.3 Section 4 expands to construction and scrapping as in the entry–exit–scrapping model by Dixit (1988). Section 5 contains a new model, where investment lags are included in the entry–exit–scrapping model, thereby combining the models of the previous sections. This section also studies investment lags, in practice, based on empirical shipping data. Section 6 restricts the number of switches as in the entry–exit model by Ekern (1993), but also expanding to scrapping. Section 7 allows for diminishing production capacity as in Brekke and Øksendal (1994). Section 8 presents a new model where switching occurs as part of a market game with two players. The intention behind this model, which technically is quite similar to the entry–exit model, is mainly to show that the discount factor approach can also be useful in settings that extend beyond the decisions of a single firm. Section 9 concludes.

Section snippets

The entry–exit model

Consider a firm with the option to switch between using and mothballing a certain production capacity as in Dixit (1989). The potential revenue from being an active producer is stochastic. The expected and discounted revenue from continuous production in perpetuity, starting now and discounting at a constant rate ρ>0, follows an autonomous and continuous Ito processdP=f(P)dt+g(P)dz,where dz is the Wiener increment, dt is the time increment, and f(P) and g(P) are continuous and differentiable

Investment lags

Investment took place instantaneously in the model of the previous section. The real world is different. Bar-Ilan and Strange (1996) include a deterministic lag from the time of the entry decision to the time when revenues from production start to flow. A new investment lag applies if the firm decides to enter once again after a lay-up period. Since no revenue or cost applies during the investment lag, there is no cost of postponing exit decisions until the end of the lag even if the price gets

Construction and scrapping

Dixit (1988) embeds the entry–exit model in a setting where lay-up decisions are preceded by construction and succeeded by scrapping. The model yields four trigger prices (in descending order): H (construction), R (entry), L (exit) and S (scrapping). There is no reason to build before the price is high enough for immediate entry, so the first entry will occur right after construction at trigger price H. The next ones occur at price R. The exit and scrapping policies are more complicated as

Construction and scrapping with lags

Dixit (1988) illustrates the entry–exit–scrapping model of the previous section with the shipping industry, where there are investment lags in connection with newbuilding. This section expands the model to allow for such lags with assumptions as in Section 3, but the investment lag now applies to construction instead of entry. Construction takes a fixed amount of time, τ. The price development during construction decides whether the new ship should go into production, be laid up or perhaps even

Restricted switching

The models discussed so far allow for an infinite number of switches. This may be unrealistic, at least without increasing the switching costs. For example, frequent switching could harm the equipment in use. Ekern (1993) restricts the number of switches in the entry–exit model to a fixed number. This section shows how to model such restrictions by the discount factor approach both in the entry–exit setting and the entry–exit–scrapping setting. The entry and exit trigger prices now will depend

Diminishing production capacity

Brekke and Øksendal (1994) develop an entry–exit model where the production capacity is decreasing in the accumulated time of production. As an example, the resource stock of a mine is limited, and the amount that can be extracted per time unit may decrease as the most attractive parts are exploited first.9 Thus, the revenue flow when production takes place is qpp, where q is the current capacity and p

A market game

The last model to be presented is new and different from all previous ones, as switching now will result from a game between two players. It demonstrates in all its simplicity that the discount factor approach can also be used to investigate some multiple-agent strategic investment problems.10

Suppose an owner of a long-lived asset wants full insurance against an event that could destroy the asset. The asset value in terms of what

Concluding remarks

This paper has focused on how a discount factor approach to investment can simplify analysis of entry and exit in the tradition of Dixit, 1988, Dixit, 1989. The baseline model showed that the loss by missing the exact optimal investment thresholds depends highly on the cost parameters and the level of uncertainty. The more advanced models showed that investment lags tend to increase the value of lay-up options. We also concluded that investment lags are not likely to have strong impact on

Acknowledgements

I am grateful to Geir B. Asheim, Avner Bar-Ilan, Avinash Dixit, John M. Marshall and William Strange for comments on an earlier draft.

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