Under the real options approach to investment under uncertainty, agents formulate optimal policies under the assumption that firms’ growth prospects do not vary over time. This paper proposes and solves a model of investment decisions in which the growth rate and volatility of the decision variable shift between different states at random times. A value-maximizing investment policy is derived such that in each regime the firm's investment policy is optimal and recognizes the possibility of a regime shift. Under this policy, investment is intermittent and increases with marginal q. Moreover, investment typically is very small but, in some states, the capital stock jumps. Implications for marginal q and the user cost of capital are also examined.
The notion that regime shifts are important in explaining the cyclical features of
real macroeconomic variables as proposed by Hamilton [15] is noww idely accepted.
Motivated by anecdotal evidence, a pervasive manifestation of this viewis that
regime shifts, by changing firms growth prospects, affect capital accumulation and
investment decisions. On economic grounds, there are indeed reasons to believe that
regime shifts contain the possibility of significant impact on firms policy choices. For
example, business cycle expansion and contraction ‘‘regimes’’ potentially have
sizable effects on the profitability or riskiness of investment and, hence, on firms’
willingness to invest in physical or human capital. Yet, despite these potential effects,
we still know very little about the relation between regime shifts and investment
decisions.
The idea that shifts in a firm’s environment can have first-order effects on its
investment policy can be related to the burgeoning literature on investment decisions
under uncertainty (see the survey by Dixit and Pindyck [9]). In this literature,
investment opportunities are analyzed as options written on real assets and the
optimal investment policy is derived by maximizing the value of the option to invest.
Because option values depend on the riskiness of the underlying asset, volatility is an
important determinant of the optimal investment policy. Despite this observation,
models of investment decisions typically presume that this very parameter is fixed. It
is not difficult to imagine however that as volatility changes over the business cycle,
so does the value-maximizing investment policy.
This paper develops a framework to study the behavior of investment when the
dynamics of the decision variable are subject to discrete regime shifts at random
times. Following Hamilton, we define shifts in regime for a process as ‘‘episodes
across which the behavior of the series is markedly different’’. To emphasize the
impact of regime shifts on investment decisions and capital accumulation, we
construct a simple model of capacity choice that builds on earlier work by Pindyck
[26] and Abel and Eberly [3]. Specifically, we consider an infinitely lived firm that
produces output with its capital stock and variable factors of production. The price
of the firm’s output fluctuates randomly, yielding a stochastic continuous stream of
cash flows. At any time t; the firm can (irreversibly) increase capacity by purchasing
capital. Investment arises when the marginal valuation of capital equals the purchase
price of capital.
Models of investment decisions under uncertainty generally presume that the
firm’s operating profits are subject to a multiplicative shock that evolves according to
a geometric Brownian motion.1 Implicit in this modeling is the assumption that the firm’s growth prospects do not vary over time. This paper solves for the valuemaximizing
investment policy when the growth rate and volatility of the marginal
revenue product of capital are subject to discrete regime shifts. The analysis
demonstrates that, in contrast with standard models of investment, the optimal
decision rule is not described by a simple threshold for the marginal revenue product
of capital. Instead, the optimal investment policy is characterized by a different
threshold for each regime. Moreover, because of the possibility of a regime shift, the
value-maximizing threshold in each regime reflects the possibility for the firm to
invest in the other regimes. That is, a value-maximizing policy is derived such that in
each regime the firm’s investment policy is optimal, conditional on the optimal
investment policy in the other regimes.
An important question is whether regime shifts actually affect growth and capital
accumulation. To answer this question, we examine the implications of the model for
the optimal rate of investment. These implications are generally consistent with
recent evidence on firms’ investment behavior (see [1] or [7]). In particular, the model
predicts that investment is intermittent and increasing with marginal q: Moreover,
the state space of the dynamic investment problem can be partitioned into various
domains including an inaction region where no investment occurs. Outside
of this region, the optimal rate of investment can be in one of two regimes:
infinitesimal or lumpy. Investment is infinitesimal at the investment threshold.
Investment is lumpy in the transient region and at the initial date if the state of the
system is in the action region. Also, while it is always optimal to invest in the action
region, the optimality of investment is regime dependent in the transient region. That
is, regime shifts generate some time-series variation in the present value of future
cash flows to current cash flows that may induce the firm to invest following a regime
shift.
The analysis in the present paper relates to two different strands of literature.
First, from an economic point of view, it relates to the investment literature that
combines real options features—irreversibility and a continuous stochastic process—
with neoclassical features—no indivisibilities. In these models, investment is
intermittent and, in the absence of fixed adjustment costs, involves marginal
adjustments in the stock of capital (see [26,2,6,13]). When fixed adjustment costs are
introduced, investment is intermittent and lumpy, and the optimal policy involves
impulse control techniques (see [4] or [7]). In the present paper, there are no fixed
adjustment costs. Yet, the optimal investment policy involves both marginal
adjustments and jumps in the stock of capital.
From a technical viewpoint, the present paper relates to a series of recent papers
on option pricing with regime shifts (see [15,16,10]). One of our main contributions is
the extension of techniques in these papers to the case of stochastic control problems
where control policies change the underlying diffusion process. In particular, we use
the solution to the optimal stopping problem derived by Guo [15] to analyze the
recurrent investment decision of a firm with divisible capital. Because the firm’s
problem is homogeneous, the recurrent model displays a structure that is similar to
the stopping problem except that the firm obtains a newinvest ment option whenever
it stops.The paper that is most closely related to the present analysis is Driffill and Sola
[11]. These authors also analyze investment decisions when the dynamics of the state
variable can shift between several regimes. One essential difference between the two
papers is that we examine capacity choice and the valuation of interrelated options
whereas they focus on the valuation of a single investment opportunity (in the spirit
of McDonald and Siegel [24]). Another important point of departure is that we solve
our model analytically whereas they solve their model numerically. Finally, we derive
implications for capital accumulation, marginal q; and the user cost of capital, which
are not examined in their paper.
The remainder of the paper is organized as follows. Section 2 presents the basic
model of investment decisions with regime shifts. Section 3 derives the firm’s
objective function and optimality conditions. Section 4 determines the valuemaximizing
investment policy. Section 5 presents simulation results. Section 6
investigates the implications of the optimal investment policy for capital accumulation
and growth. Section 7 analyses marginal q and the user cost of capital. Section 8
concludes.
This paper has analyzed investment decisions under uncertainty when the
dynamics of the decision variable—growth rate and diffusion coefficient—shift
between different states at random times. The main analytical result of the paper is
that the value-maximizing investment policy is such that in each regime the firm’s
investment policy is optimal, conditional on the optimal investment policy in the
other regimes. This optimal investment policy is characterized by a different
investment curve for each regime. Moreover, because of the possibility of a regime
shift, the investment curve in each regime reflects the possibility for the firm to invest
in the other regime.
To determine the implications of the model for investment decisions and capital
accumulation, we showed that the state space of the dynamic investment problem
can be partitioned into various domains including an inaction region where no
investment occurs. Outside of this region, the optimal rate of investment can be in
one of two regimes: infinitesimal or lumpy. Investment is infinitesimal following an
increase of the firm cash flows in a given regime. Investment is lumpy following a
shift from the regime with the highest investment curve to the regime with the lowest
one. That is, the model predicts that with irreversibility and regime shifts investment
is intermittent and increases with marginal q: Moreover, the optimal rate of
investment typically is very small but occasionally exhibits some spurts of capacity
expansion. These predictions are generally consistent with the available empirical
evidence on firms’ investment behavior (see [7] or [1]). The paper also provided an analysis of the determinants of marginal q and the user cost of capital in such an
environment.