Real Options Selected Bibliography:
the Dixit & Pindyck Book
The "Selected Bibliography" list covers many important
aspects of the options thinking. There are comments, for each book or
paper, available by clicking their titles. This selection begins with the
main reference, a born-classic text-book by Dixit & Pindyck:
the first text-book exclusively about the real options approach to
investments. This bibliographical item deserves a more detailed comments,
in view of their importance to the theory.
& R.S. Pindyck (1994): Investment under
Princeton University Press, Princeton, N.J., 1994 . Click here for Marco Dias' comments.
See also in the
Contributions Page, An
Instructor's Manual: for the Dixit & Pindyck's Investment
under Uncertainty textbook by Luiz Eduardo Brandão (in
See also this link with the same material plus some spreadsheets.
See also in the
Contributions Page, Matlab
files to solve some problems from the Dixit & Pindyck's Investment
under Uncertainty textbook by Luke J. Sparvero.
Back to the Bibliographical List
Back to Contents (main page)
A.K. & R.S. Pindyck (1994): Investment under Uncertainty
The book brings an excellent theoretical and practical combination of the
new approach, with their mathematics and economic foundations insights,
complemented with industry typical investments decisions examples,
embedded with representative industry's data.. In many times, when
presenting equations, the book gives both, an intuitive/heuristic
explanation and a more rigorous mathematical development.
In a rare case, a paper was written and published (outside of the books'
review section) about this book (Hubbard, 1994).
The book comprises twelve chapters divided into five parts:
- The Part I is the introduction with two chapters.
- The Chapter 1 presents some introductory concepts (investment,
the ability to wait, and so on) and and a comparison between the
modern "Real Options Approach" and the neoclassical
investment theory with their variants like the
"Tobin's q or marginal q" and the
"Jorgenson's user cost of capital".
Both variants of the neoclassical model rely on the very known "net
present value rule".
This chapter presents also an overview of the book, introducing
some concepts like the "hysteresis in
economics", or path dependence. The chapter concludes
with noneconomic applications, focusing themes like marriage,
suicide, legal reform and Constitutions.
- The Chapter 2 presents, through simple two and three period
examples, the value of waiting to invest, into a value maximizing
investment decision context. In the examples, the static net present
value (NPV), or a NPV in "now or never" investment basis,
is calculated and also a dynamic NPV
(when including the flexibility to wait) calculus is performed, and
the difference between the dynamic and the static is the option to
wait value or the opportunity cost of investing now, rather than
waiting and keeping open the option investment opportunity.
This discrete and lattice approach was used only in this chapter (a
continuous time model, with differential equation building approach,
is presented in all of following chapters of the book), and with the
aim to develop the reader's intuition and concepts. The several
examples show the uncertainty effect in the investment decision and
also others parameters effects. The financial options analogy is
more closely analyzed.
The critical price (or the threshold) is presented with the
numerical examples plotted in many charts. The
"bad news principle" is
illustrated by examples. The chapter also introduces the "technical
uncertainty" concept, when the reader begins to believe
that uncertainty always leads to postpone investments: in
many cases, when investment provides information, uncertainty can
perform the opposite effect, stimulating the investment starting
(ex: petroleum exploration and R&D investments). In this
context, a negative NPV project can be optimally undertaken, because
the project has an additional value that not was considered in the
traditional NPV calculations: the reduction of the variance of the
technical uncertainty with the accumulated investment. This value is
called a shadow value because it is not a directly
measurable cash flow.
- The second part of the book is the "Mathematical Background"
with more two chapters:
- The Chapter 3 presents the stochastic process and the Itô's
Lemma. Concepts such as random walk process, Markov
process, Wiener process (Brownian motion), are
presented in a easy way. The geometric Brownian motion with drift (a
special case of the Itô process) is detailed, with equations
and figures showing sample paths and optimal forecast from this
process (see also the figure "Optimal
Forecast of Geometric Brownian Motion" , file gif with
5988 bytes) .
The mean-reverting process is analyzed as a more realistic
economic model for some stochastic variables like the price of raw
commodities such as oil or copper. The Ornstein-Uhlenbeck process
equations illustrate this alternative process, and also figures
showing sample paths and optimal forecast, are presented (see
also the figure "Optimal Forecast of Mean-Reverting Motion"
, file gif with 5924 bytes).
The Itô Lemma is introduced as a tool, permitting the
differentiation and integration of Itô process' functions.
The reflecting barriers concept, which occurs often in economic
applications (because of equilibrating mechanisms in the market), is
showed with equations and research extensions are pointed out.
Closing the chapter, the Poisson (jump) process is introduced as a
way to consider the presence of infrequent but discrete jumps for
the stochastic variable, or in others words, some discontinuous
process in a short time interval inside of a prevailing continuous
- The Chapter 4 presents the dynamic optimization under
uncertainty, developing two important techniques: dynamic
programming and contingent claims analysis. These tools, despite
they make different assumptions about financial markets and discount
rates, lead to identical results when the first one adopts the risk
The dynamic programing is showed for two and many periods, and also
the famous Bellman's equations and the
Bellman's Principle of Optimality. Optimal stopping and smooth
pasting techniques are illustrated in Bellman's equations context.
The contingent claims analysis is presented inside the dynamic
replicating portfolio framework. This portfolio is built for to be
risk-free, and consequently is independent of the investors'
preferences: risk-averse or risk-neutral investors, with any
increasing utility function, agree with the same portfolio value.
The method rely on the market equilibria, where no arbitrage
opportunities survives more than a very short period. The relation
of this approach with the CAPM theory are pointed out, and so the
insights about the quantity of the underlying assets that makes the
portfolio risk-free. The smooth pasting condition and also the
Poisson process are analyzed in this context. The equivalence
between this approach and the dynamic programming (with the
risk-neutral approach for the latter) is established.
- The third part of the book is the "firm's decision" with
three important and more practical chapters:
- The Chapter 5 presents the decisive role of the timing in the
investment opportunities decisions and valuations, where the authors
points "the simple NPV rule is not just wrong; it is often very
wrong". The authors shows also, even for the deterministic
case, the timing flexibility adds value to investment opportunity
and creates a value to waiting. The basic model is solved in both
dynamic programming and contingent claims framework, illustrating
their equivalence. About the validity of the contingent claims, that
needs a perfectly correlated asset (spanning), the authors
point out (see p.152) this market value reference is necessary not
only for contingent claims, but also for CAPM: "Without
spanning, there is no theory for determining the 'correct' value for
the discount rate (unless we make restrictive assumptions about
investors' or managers' utility functions). The CAPM, for example,
would not hold, and so it could not be used to calculate a
risk-adjusted discount rate in the usual way".
The basic result for the perpetual investment opportunity can be
summarized in the following equations (see p.142):
The first equation shows the investment opportunity value (F)
equal to a constant (A) times the value of implanted project (V)
with a exponent which is more than one. This exponent is a
function of the parameters (see p.152): risk-free interest rate
(r), dividend yield (or convenience yield), and the volatility
(standard deviation of the rate of variation of the project
value). The second equation, points out the optimal investment
rule: invest if the market value of the project is equal or more
than the threshold (V*) value. The fraction multiplying the
investment (I) is like a wedge which always values more than 1,
and can reach values such as 2 or 3, for real parameters, and
increases with the economic uncertainty.
The characteristics of the optimal investment rule are examined,
presenting four charts which show the effects of the two more
important parameters (volatility and dividend yield) on both, the
decision rule (V*) and the value of the investment opportunity.
The chapter also examines alternative stochastic processes like
the mean-reverting and the combined Brownian motion and Jump
(Poisson) process. The authors choose the Geometric
Ornstein-Uhlenbeck for the specific mean-reverting process which
results, after some substitutions, in the Kummer's equation, which
has a series representation (that converges) for the perpetual
investment opportunity case. In the combined jump/Brownian
process, also a differential equation is developed, which can be
solved numerically for a more general case or, in the case of the
a simplified assumption (in the event that V falls to zero, where
remains forever), there is an analytical solution similar
to the pure Brownian case, but adding the Poisson parameter to the
- The Chapter 6 examines the value of a project (V), as function
of more basic variables like the output price (P). When V is a
constant multiple of P, both follows the same stochastic process and
with the same parameters (expected growth, volatility, dividend
yield), see p.178, because the rates of variation is the same for
both variables. However in general, when V is not a multiple of P,
just the latter follows a geometric Brownian motion. So this chapter
calculates V as function of P (this is used as boundary condition at
the contact), and the investment opportunity (F) as function of P.
If V follows a stochastic process and their parameters are easy to
calculate, the chapter 5 approach (calculating F as a function of V)
is also possible (see p.182).
Several cases of projects characteristics are analyzed, beginning
with the no operational costs case.
In the section 2 the operational costs are considered, and are
allowed temporary stopping and reactivation, in both cases without
The Excel spreadsheet below, available to download, illustrates the
section 2 of the chapter 6. It calculates the NPV with the option to
shut-down and the value of a perpetual option to invest in this
Download the Excel spreadsheet
with 108 KB
Section 3 presents the variable output case, which the profit is a
convex function of P (but is a concave function of the input
parameter, such as labor, so the optimal choice of input maximizes
the profit). V is nonlinear with P.
The more interesting (for petroleum projects) section 4, deals with
depreciation. The exponential decay case is examined for both (see
p.201) without allows re-investment and allowing to invest back
again after the project dies. The first case is more simple, more
realistic for petroleum fields investments, and V is linear in P,
allowing calculate also F as function of V, with the difference
(comparing with chapter 5) that the Poisson parameter is added to
dividend yield for discounting purposes. The sudden death case and a
general case (exponential decay and sudden death, together) are
The last section examines the price and cost uncertainty case, with
correlated stochastic processes. This more general case needs
numerical solution, two smooth pasting conditions, and instead one
threshold point, there is a line of thresholds named "free
boundary". For the case analyzed (project with perpetual life
and no variable output) the homogeneity of the problem allows to
reduce it to one dimension case with analytical solution.
- The Chapter 7 deals with entry & exit model (or hysteresis
model) with some improvements in relation to early works (such as
Dixit, 1989, and Brennan & Schwartz, 1985). The first case
analyzed is an extreme one: if stop the production the entire
capital rust and the firm must incur the whole investment again to
restart it. The model considers the abandon cost, which can be
positive (legal requirements like restoring the site of mine to its
natural condition, and workers indemnity) or negative (alternative
use for a portion of the capital or scrapping value). The
differential equations are showed for the idle firm (which has the
option to invest) and for the active firm (with the option to
abandon), resulting in four nonlinear equations with four
unknowns (two constants and the two thresholds, one to entry and
other to exit), which are solved with the nice approach using the
function G(P). The authors points out
the spread between the two thresholds with rational expectations is
farther apart than the traditional Marshallian microeconomy says
(with static expectations). An example with the copper industry is
presented (which is an empirical evidence of the option approach to
real decisions), with many illustrating charts.
The second section includes the temporary stopping (plants are "mothballed",
and ship/tankers are laid-up), with both lump sum cost for stopping
and a maintenance cost flow. So besides the investment and
abandonment options, the firm has the option to stop without exit,
and later to reactivate paying another sunk cost. In this case four
thresholds must be calculate (entry, exit, stop, reactivate), three
status are allowed to the firm, and five state changing are possible
(the active firm can exit without stopping stage). Two groups of
four differentials equations are presented, and numerically solved
in two examples, the second one about oil tankers, and again with
lots of charts.
In the "Guide to the Literature", the authors mention a
little mistake in the famous earlier Brennan & Schwartz model,
which they confuse the transition between some states, using the
same lower threshold notation for mothballing and for abandonment,
besides the numerical example considered switches between just two
states (active and mothballing, the firm never exits), because the
maintenance cost was zero.
- The fourth part of the book is the "Industry Equilibrium"
with two chapters, and is more useful for economists:
- The Chapter 8 examines the aggregated industry equilibrium under
a perfect competitive environment. In opposite to the monopolist
case (examined into chapters 5-7), the value of waiting for the
competitive firm is zero. However, the threshold value (for entry)
is the same of the previous chapters! When a price of a commodity
rises and one competitive firm invests (entry), all others identical
competitive firms do the same, so the price falls (because
production surplus), and the threshold is an upper reflecting
barrier on the price process. Firms with rational expectations
will require a sufficiently high price to invest, because they know
the price will fall with the competition. This high (ceiling) price
coincides with the monopolist case, but for different reasons.
In section 3 the model is extended by allowing the abandonment
option, and the authors shows that the value of an idle firm is
zero, for this competitive case. The copper industry example is
presented in a more general situation, where is pointed the
traditional microeconomic theory is out of the reality: prices above
long-run average cost without attracting new entries, or prices
below average variable cost without inducing exit.
The section 4 deals only with firm-specific uncertainty (no
multiplicative aggregated stochastic shock) on the demand curve. The
model is settled into two stages: by paying an entry cost R (like
acquiring a patent), firms earn the right to invest in production
(become potential active firms), and waiting has value. Potential
and active firms can die (with Poisson death probability). The entry
and the investment decisions are examined, and also the distribution
of firms in the industry (including the rate of new entry).
In the last section, a general model is derived with both
firm-specific and industry-wide uncertainty on the demand curve,
reaching the eq.43, which is a generalization of early sections
- The Chapter 9 presents the socioeconomic planner dilemma about
policy intervention to provide additional investment incentives and
also the extension of the previous chapter by relaxing the perfect
competitive assumption (number of competitors is small).
The first section examines the correspondence between social
optimality and equilibrium, using the model aggregated uncertainty
on demand. The value of one industry is given by the sum of the
values in place of all the installed firms, and the sum of the
values of options to install all the futures units. In efficient
markets, dynamics and uncertainty are not reasons for policy
intervention, as many times happens: a high price of goods often
calls antitrust measures, and low prices often justifies trade
sanctions to foreign firms. The policy of price controls is other
example. Of course these governments interventions are often
mistakes that can have perverse effects on market efficiency.
In section 2, the authors examines some commonly used policies: (a)
the price ceiling controls fails to achieve its intended goal of
reducing the prices to consumers, the long-run effect is the
opposite (see table 9.1 and figure 9.1), because the investment
reduction and so the long-run supply; (b) the price floor protection
induces an overproduction in declining demand industries (the
long-run number of firms goes to infinity), like farmers in many
countries, which stay so dependent on price supports; (c) the policy
uncertainty creates by governments, as constant changes in tax
policy and trade tariffs, increases the value of waiting (in
detriment of investment), and/or lowers the scale of its investment,
increasing the prices to consumers.
The last section addresses the oligopolistic industry (competition
is not perfect) using the stochastic game theory,
which is quite recent for such applications, still more in
continuous time. The fear of preemption by a rival induces early
entry. The model studies the duopoly case, and calculates the value
of both firms, the leader and the follower, which
have two different thresholds. Even the leader needs an option
premium to entry (NPV > 0 rule is still wrong). The authors
indicates simultaneous investment is a mistake, because reduce the
value of both firms.
- The Part V, Extensions and Applications, comprises the three last
- The Chapter 10 is very important for firms investment decisions:
deals with sequential investment as analogous to compound
options. The investment is staged, so completing one stage gives
the right (not the obligation) to invest in the next one. The
project takes time to built. The model has two state variables: the
amount of capital installed (or number of stages completed) and the
The first model examines the two-stage project, which the
operational profit will come only after the second stage (is not
like a growth option). If the output price falls during the first
stage, the firm can delays the investment in the next stage. In the
second section, the model is generalized and the investment is
continuous, but takes time to built (there is a maximum rate
at which construction can proceed), and investment can be stopped
and later restart costlessly. The table 10.1 illustrates a numerical
example, with project value and the (decreasing with capital
installed) thresholds. In depth analysis of many parameters from the
solution are presented, including the value of construction time
flexibility (the ability to speed up construction).
The third section examines the learning curve model, which the
production costs are decreasing with cumulative production, until it
reaches a irreducible cost level. This is typical in aircraft
industry, but also happens in others sectors, including drilling
oilwells. See the figure Learning
Curve and Deepwaters Wells Drilling (Offshore Brazil). File
with 5327 bytes.
This learning effect increases the value of the firm and reduces
the threshold (see table 10.3).
Section 4 examines cost uncertainty with two components: economic
and technical uncertainties (together in the same variable). The
technical uncertainty has value if the investment can be staged, so
the learning effects (reduction of the variance of the technical
uncertainty) adds value to firm and decrease the threshold required
to continue the investment (see the table 10.4 for the latter). So
technical uncertainty has the opposite effect of economic
For details on technical uncertainty, see in the Tutorial Page the
topic on Economic
and technical uncertainties, and mainly the webpage on
- The Chapter 11 examines the incremental investment and the
capacity choice in a continuous time framework. The firm has a
capital stock (assets in the place) but also has a project's
portfolio and can grow by increasing the capital stock. The
investment condition is the marketable stochastic variable (like
output price) rises above the investment threshold. Projects with
lower thresholds will be first candidates for investment. The choice
of the capacity is performed "ex-post", not "ex-ante"
(see ex-ante capacity choice)
. The chapter focus is more theoretical and for economists.
In the first section the expansion has diminishing returns of
scale (the marginal revenue product is decreasing with the
investment). The optimal investment policy is derived with both,
dynamic Programming and contingent claims. The figure 11.1 (p.362)
illustrate the incremental investment as a sequence of marginal
projects. The effect of uncertainty is presented with an example
using the very known Cobb-Douglas production function. Using this
example, the authors show that a larger volatility (or uncertainty)
means lower long-run average growth rate (less investment). The
depreciation effect also is analyzed, and it acts with closed
analogy for the case of a single project (the interest rate is
increased by the death Poisson parameter).
In the section two, the expansion has increasing returns of
scale. This happen in situations when the firm to encounter an
initial phase of increasing returns, or by geometric reasons. In
this case the optimal is wait until the threshold is reached, and so
make a big investment, in a large capacity (lump investment),
jumping over the whole range of increasing average product.
The third section close the chapter and presents the
adjustment costs, that means the costs of changing the capital
stock too rapidly. This cost are an invention of the economists, in
order to explain the slowness of the investment to respond the
demand shocks. The authors show the classification of these costs,
including convex adjustment costs. Another look in the Tobin's q
theory is presented, including these costs, and so stochastic theory
insights such as the range of inaction for the investment and the
barrier control that arises from irreversibility.
An excellent complement for this chapter is the Pindyck's paper
(see American Economic Review, vol.79, December 1988, pp.969-985).
- The Chapter 12, Applications and Empirical Research is
one of more interesting and practical of the book, a gold key
The section one deals with petroleum E&P investments and
offshore leases valuations. I detail many aspects (including the
deduction of the equations) in the Real Options in Petroleum page.
The authors recommend the more important model in real options
literature for petroleum investments, the Paddock, Siegel &
Smith model. This model makes an analogy with financial options
model (American call on continuous dividend asset). The developed
reserve (producing reserves) is like an asset, the undeveloped
reserve (before the investment) is like a call option, the
development cost (investment) is like the exercise price, and the
relinquishment is like the time to expiration. The dividend yield
analogy is one of more strong aspects of the model, because is
estimate using the cash flows of the project (so, is project
specific). The authors also extend the model using mean-reverting
stochastic process for the developed reserve value, with a
discussion about the necessity or no, of this mathematically more
The section two examines the electric utility firms dilemma due to
the Clean Air Act. The firm has the following three alternatives:
(a) buy pollution allowances; (b) investment for switch the fuels to
the more expensive low-sulfur coal; (c) investment in a very
The section three analyses the timing of environmental policy.
There are two kinds of irreversibility. First the pollution effect
of the greenhouse gas, like global warming, has high irreversibility
degree (the concentration of the gas has a natural decay rate of
about one-half percent per year). Second, the environmental policy
adoption is also irreversible, and implies the imposition of large
sunk costs to firms and society. The problem is divided into two
regions, the no-adopt policy region, and the adopt region, and two
partial differential equations are derived using optimization
problem (minimize the cost for the society) under uncertainty (cost
has a stochastic component). Remarkably, both partial differential
equations have analytical solutions!
The last section examines the empirical issues of the investment
theory. The authors show that the neoclassical investment theory has
been no success in the explaining the aggregate investment behavior.
Even sophisticated econometric test of Tobin's q theory, allowing
delivery lags, cannot find support to their assumptions. The problem
is the irreversibility was overlooked in the traditional tests.
However, the econometric test of real options theory is not an easy
task. First, the equations describing investment under uncertainty
are non-linear. Second, can be difficult to measure the variables or
parameters that reflect the key components of uncertainty. Third,
the problems of aggregation: different firms have different action
thresholds. However, the few econometric tests with the real options
theory has been positives. The authors also show many empirical
evidence of this theory, including the use by managers of high
hurdles rates, in order to make only deep in the money
In short, this book is a must see, for economists, practitioners,
students, sophisticated managers, and researchers.
Back to Selected Bibliography
Hubbard, R.G. (1994):
Investment under Uncertainty: Keeping Ones Option Open
Journal of Economic Literature, vol.32, December 1994, pp.1816-1831
Back to Dixit & Pindyck book's comments
This neoclassical investment model due to the Nobel Laureate J. Tobin,
1969 ("A General Equilibrium Approach to Monetary Theory",
Journal of Money, Credit and Banking, 1, February 1969, pp.15-29),
compares the capitalized value of the marginal investment to its purchase
(replacement) cost.. The ratio is the Tobin's q. If q > 1, the firm
should invest. The firm stops the investment process when their marginal
investment is at such level that q = 1. If q < 1 the firm should not
invest. The capitalized project value can be the observed directly if is
marketable, but frequently it is computed as the expected present value of
the net cash flows.
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Jorgenson's user cost of capital:
This neoclassical investment model due to the D. Jorgenson, 1963 ("Capital
Theory and Investment Behavior", American Economic Review, 53, May
1963, pp.247-259), treats the capital investment as a purchase of a
durable good. The user cost of capital is defined by the rental cost of
the capital, and is determined using parameters like the purchase price,
opportunity cost of funds, depreciation rates, and applicable taxes. The
theory compares the value of an incremental unit of capital (marginal
product) and the user cost (or rental cost): the firm invests when the
marginal product is greater than the user cost and stops when an equality
In this model and in the Tobin's q model, the dynamics is considered (not
satisfactorily) with adaptations such as "adjustment costs" and "delivery
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Hysteresis in Economics:
The firm's status (if is active or if is idle) in the economy is path
dependent, for a range of the firm's output price. This concept is
important at firm level (entry & exit models) and also at
macroeconomic level, in the studies of the aggregated investment from a
specific sector of the economy, and sectorial policies.
When considering an investment and abandonment (entry & exit)
together, the firm's optimal decision is characterized by two thresholds:
one firm's output high price level, when the firm invests in production,
and one low price level, when the firm abandon the project. Now suppose
the current level of the price is somewhere between these two threshold:
What's the status of the firm? Depend of the recent prices' fluctuations
history. If the price descended from a high level that induced entry to
many competitive firms, then the firm's status remain active. However, if
the actual intermediate level was recently preceded by a low prices' level
(that induced exit) then the firm remain idle. Or, as pointed out by the
book, "the current state of the stochastic variable is not enough to
determine the outcome in the economy; a longer history is needed. The
economy is path dependent".
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Dynamic or Expanded NPV:
This expression has been employed by Trigeorgis in several papers, but
the concept can be employed also to refer the NPV calculus in Dixit &
Pindyck's Chapter 2.
As Trigeorgis points out,
Expanded (dynamic) NPV = static NPV + Option Premium
The "Option Premium" can be from the (most) relevant option, or
a "Combined Option Value" (in a multi-options interactions
context). In large companies, with a large project opportunities
portfolio, the different "synergy effect" in each portfolio
combination, can be added to the above expression, if relevant.
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The "bad news principle":
In a world with uncertainty , a "good" project (with positive
net present value) can be postponed if a firm can wait to invest, because
of the "bad news principle": there is a positive probability of
a downward price movement, so that the waiting avoids the possibly losses
from the project investment. The threshold price (that warrants immediate
investment) depends of the size of the downward movement (not of the
upward movement size) and the probability (besides the investment cost, of
course), as shown in the equation 16, p.41.
This "bad news principle" was first pointed out by Bernanke,
1983 ("Irreversibility, Uncertainty, and Cyclical Investment",
Quarterly Journal of Economics, 98, February 1983, pp.85-106), and some
ideas can also be found in Cukierman, 1980 ("The Effects of
Uncertainty on Investment under Risk Neutrality with Endogenous
A nice and instructive argument is presented by Dixit, 1992 (Investment
and Hysteresis, Journal of Economic Perspectives, vol.6, no 1, Winter
1992, pp.107-132, see p.123) illustrating the "bad news principle":
Why the Japanese firms (despite its larger fixed costs) are more
aggressive investors than the American ones? Because they are protected
from the downside risk through the government supports. Then the value of
waiting to invest is quite small, because the downside movement potential
For disinvestment (abandon) the argument turns around and becomes the
"good news principle": the option value of keeping the
operation alive is governed mainly by the upside potential.
See also, in the tutorial page: "The two sides of the uncertainty"
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Bellman's Principle of Optimality:
"An optimal policy has the property that, whatever the initial
action, the remaining choices constitute an optimal policy with respect to
the subproblem starting at the state that results from the initial actions"
(Dixit & Pindyck, 1994, p.100)
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Function G(P) Method
This function is defined by the difference of value between the active
firm and the idle one (original idea from Dixit, 1989). This function is
only defined between the two thresholds (the thick part of the curve).
This function helps the comparative statics job, too.
The geometric approach permits a graphical solution for the investment
and abandonment thresholds (see footnote at p.220). In a very simple Excel
sheet, I performed this method with success: with average of 4 iterations
(3 minutes), I got the solution with error less than 0.1 %.
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Ex-ante Capacity Choice
In many projects (such as ships, factories, and upstream oil industry)
we have an ex ante choice of scale, but prohibitively ex post adjustment
in scale. In some situations the optimal is the largest capacity (if there
is a increasing returns of scale for the largest scale region), even if is
necessary wait for the output price reaches the threshold, and even if the
other alternatives' threshold is under the market output price level. In
others situations an intermediate scale may be the optimal choice (if
there is decreasing returns of scale AND the elasticity of output with
respect to investment is also decreasing). In others situations, the
optimal is a binary decision between the smallest and the largest scale.
See an excellent exposition of this, with graphical and heuristic
argument in the paper of Dixit (1993, Economic Letters, vol.41,
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